# Planet R

## August 25, 2016

### CRANberries

#### New package velox with initial version 0.1.0

Package: velox
Type: Package
Title: Fast Raster Manipulation and Extraction
Version: 0.1.0
Date: 2016-08-24
Author: Philipp Hunziker
Maintainer: Philipp Hunziker <hunzikp@gmail.com>
Description: C++ accelerated raster manipulation and extraction.
Imports: methods, Rcpp (>= 0.11.6), raster, rgeos, rgdal, sp
NeedsCompilation: yes
Packaged: 2016-08-24 20:22:13 UTC; hunzikp
LazyData: TRUE
RoxygenNote: 5.0.1
Repository: CRAN
Date/Publication: 2016-08-25 10:01:02

#### New package sfc with initial version 0.1.0

Package: sfc
Type: Package
Title: Substance Flow Computation
Version: 0.1.0
Authors@R: person("Hu", "Sheng", email = "shenghu@nju.edu.cn", role = c("aut", "cre"))
Description: Provides a function sfc() to compute the substance flow with the input files --- "data" and "model". If sample.size is set more than 1, uncertainty analysis will be executed while the distributions and parameters are supplied in the file "data".
Depends: R (>= 3.1.0), dplyr, tidyr
Imports: stats, utils, triangle, zoo, sna
URL: https://github.com/ctfysh/sfc
BugReports: https://github.com/ctfysh/sfc/issues
LazyData: TRUE
RoxygenNote: 5.0.1
NeedsCompilation: no
Packaged: 2016-08-25 00:30:56 UTC; tiger
Author: Hu Sheng [aut, cre]
Maintainer: Hu Sheng <shenghu@nju.edu.cn>
Repository: CRAN
Date/Publication: 2016-08-25 10:01:01

#### New package qfasar with initial version 1.0.0

Package: qfasar
Type: Package
Title: Quantitative Fatty Acid Signature Analysis in R
Version: 1.0.0
Authors@R: person("Jeffrey F.", "Bromaghin", email = "jbromaghin@usgs.gov", role = c("aut", "cre"))
Description: An implementation of Quantitative Fatty Acid Signature Analysis (QFASA) in R. QFASA is a method of estimating the diet composition of predators. The fundamental unit of information in QFASA is a fatty acid signature (signature), which is a vector of proportions describing the composition of fatty acids within lipids. Signature data from at least one predator and from samples of all potential prey types are required. Calibration coefficients, which adjust for the differential metabolism of individual fatty acids by predators, are also required. Given those data inputs, a predator signature is modeled as a mixture of prey signatures and its diet estimate is obtained as the mixture that minimizes a measure of distance between the observed and modeled signatures. A variety of estimation options and simulation capabilities are implemented. Please refer to the vignette for additional details and references.
LazyData: TRUE
Imports: Rsolnp (>= 1.16)
RoxygenNote: 5.0.1
Suggests: knitr, rmarkdown
VignetteBuilder: knitr
Packaged: 2016-08-24 21:42:58 UTC; jbromaghin
Date: 2016-07-26
NeedsCompilation: no
Author: Jeffrey F. Bromaghin [aut, cre]
Maintainer: Jeffrey F. Bromaghin <jbromaghin@usgs.gov>
Repository: CRAN
Date/Publication: 2016-08-25 10:00:58

#### New package memo with initial version 1.0

Package: memo
Type: Package
Title: In-Memory Caching for Repeated Computations
Version: 1.0
Date: 2016-8-22
Author: Peter Meilstrup <peter.meilstrup@gmail.com>
Maintainer: Peter Meilstrup <peter.meilstrup@gmail.com>
Description: A simple in-memory, LRU cache that can be wrapped around any function to memoize it. The cache can be keyed on a hash of the input data (using 'digest') or on pointer equivalence.
Imports: digest
Suggests: testthat (>= 0.2), knitr, rmarkdown
Collate: 'lru.R' 'cache.R' 'getPointer.R' 'memo-description.r'
VignetteBuilder: knitr
RoxygenNote: 5.0.1
NeedsCompilation: yes
Packaged: 2016-08-24 22:42:15 UTC; peter
Repository: CRAN
Date/Publication: 2016-08-25 10:00:55

## August 22, 2016

### Bioconductor Project Working Papers

#### Variable Selection for Estimating the Optimal Treatment Regimes in the Presence of a Large Number of Covariate

Most of existing methods for optimal treatment regimes, with few exceptions, focus on estimation and are not designed for variable selection with the objective of optimizing treatment decisions. In clinical trials and observational studies, often numerous baseline variables are collected and variable selection is essential for deriving reliable optimal treatment regimes. Although many variable selection methods exist, they mostly focus on selecting variables that are important for prediction (predictive variables) instead of variables that have a qualitative interaction with treatment (prescriptive variables) and hence are important for making treatment decisions. We propose a variable selection method within a general classification framework to select prescriptive variables and estimate the optimal treatment regime simultaneously. In this framework, an optimal treatment regime is equivalently defined as the one that minimizes a weighted misclassification error rate and the proposed method forward sequentially select prescriptive variables by minimizing this weighted misclassification error. A main advantage of this method is that it specifically targets selection of prescriptive variables and in the meantime is able to exploit predictive variables to improve performance. The method can be applied to both single- and multiple- decision point setting. The performance of the proposed method is evaluated by simulation studies and application to an clinical trial.

## August 21, 2016

### Dirk Eddelbuettel

#### RcppEigen 0.3.2.9.0

A new upstream release 3.2.9 of Eigen is now reflected in a new RcppEigen release 0.3.2.9.0 which got onto CRAN late yesterday and is now going into Debian. Once again, Yixuan Qiu did the heavy lifting of merging upstream (and two local twists we need to keep around). Another change is by James "coatless" Balamuta who added a row exporter.

The NEWS file entry follows.

#### Changes in RcppEigen version 0.3.2.9.0 (2016-08-20)

• Updated to version 3.2.9 of Eigen (PR #37 by Yixuan closing #36 from Bob Carpenter of the Stan team)

• An exporter for RowVectorX was added (thanks to PR #32 by James Balamuta)

Courtesy of CRANberries, there is also a diffstat report for the most recent release.

This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings.

## August 19, 2016

### Dirk Eddelbuettel

#### RQuantLib 0.4.3: Lots of new Fixed Income functions

A release of RQuantLib is now on CRAN and in Debian. It contains a lot of new code contributed by Terry Leitch over a number of pull requests. See below for full details but the changes focus on Fixed Income and Fixed Income Derivatives, and cover swap, discount curves, swaptions and more.

In the blog post for the previous release 0.4.2, we noted that a volunteer was needed for a new Windows library build of QuantLib for Windows to replace the outdated version 1.6 used there. Josh Ulrich stepped up, and built them. Josh and I tried for several month to get the win-builder to install these, but sadly other things took priority and we were unsuccessful. So this release will not have Windows binaries on CRAN as QuantLib 1.8 is not available there. Instead, you can use the ghrr drat and do

if (!require("drat")) install.packages("drat")
install.packages("RQuantLib")

to fetch prebuilt Windows binaries from the ghrr drat. Everybody else gets sources from CRAN.

The full changes are detailed below.

#### Changes in RQuantLib version 0.4.3 (2016-08-19)

• Changes in RQuantLib code:

• Discount curve creation has been made more general by allowing additional arguments for day counter and fixed and floating frequency (contributed by Terry Leitch in #31, plus some work by Dirk in #32).

• Swap leg parameters are now in combined variable and allow textual description (Terry Leitch in #34 and #35)

• BermudanSwaption has been modfied to take option expiration and swap tenors in order to enable more general swaption structure pricing; a more general search for the swaptions was developed to accomodate this. Also, a DiscountCurve is allowed as an alternative to market quotes to reduce computation time for a portfolio on a given valuation date (Terry Leitch in #42 closing issue #41).

• A new AffineSwaption model was added with similar interface to BermudanSwaption but allowing for valuation of a European exercise swaption utlizing the same affine methods available in BermudanSwaption. AffineSwaption will also value a Bermudan swaption, but does not take rate market quotes to build a term structure and a DiscountCurve object is required (Terry Leitch in #43).

• Swap tenors can now be defined up to 100 years (Terry Leitch in #48 fising issue #46).

• Additional (shorter term) swap tenors are now defined (Guillaume Horel in #49, #54, #55).

• New SABR swaption pricer (Terry Leitch in #60 and #64, small follow-up by Dirk in #65).

• Use of Travis CI has been updated and switch to maintained fork of deprecated mainline.

Courtesy of CRANberries, there is also a diffstat report for the this release. As always, more detailed information is on the RQuantLib page. Questions, comments etc should go to the rquantlib-devel mailing list off the R-Forge page. Issue tickets can be filed at the GitHub repo.

This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings.

## August 16, 2016

### Journal of Statistical Software

#### Mixed Frequency Data Sampling Regression Models: The R Package midasr

When modeling economic relationships it is increasingly common to encounter data sampled at different frequencies. We introduce the R package midasr which enables estimating regression models with variables sampled at different frequencies within a MIDAS regression framework put forward in work by Ghysels, Santa-Clara, and Valkanov (2002). In this article we define a general autoregressive MIDAS regression model with multiple variables of different frequencies and show how it can be specified using the familiar R formula interface and estimated using various optimization methods chosen by the researcher. We discuss how to check the validity of the estimated model both in terms of numerical convergence and statistical adequacy of a chosen regression specification, how to perform model selection based on a information criterion, how to assess forecasting accuracy of the MIDAS regression model and how to obtain a forecast aggregation of different MIDAS regression models. We illustrate the capabilities of the package with a simulated MIDAS regression model and give two empirical examples of application of MIDAS regression.

#### Analyzing State Sequences with Probabilistic Suffix Trees: The PST R Package

This article presents the PST R package for categorical sequence analysis with probabilistic suffix trees (PSTs), i.e., structures that store variable-length Markov chains (VLMCs). VLMCs allow to model high-order dependencies in categorical sequences with parsimonious models based on simple estimation procedures. The package is specifically adapted to the field of social sciences, as it allows for VLMC models to be learned from sets of individual sequences possibly containing missing values; in addition, the package is extended to account for case weights. This article describes how a VLMC model is learned from one or more categorical sequences and stored in a PST. The PST can then be used for sequence prediction, i.e., to assign a probability to whole observed or artificial sequences. This feature supports data mining applications such as the extraction of typical patterns and outliers. This article also introduces original visualization tools for both the model and the outcomes of sequence prediction. Other features such as functions for pattern mining and artificial sequence generation are described as well. The PST package also allows for the computation of probabilistic divergence between two models and the fitting of segmented VLMCs, where sub-models fitted to distinct strata of the learning sample are stored in a single PST.

#### Bayesian Nonparametric Mixture Estimation for Time-Indexed Functional Data in R

We present growfunctions for R that offers Bayesian nonparametric estimation models for analysis of dependent, noisy time series data indexed by a collection of domains. This data structure arises from combining periodically published government survey statistics, such as are reported in the Current Population Study (CPS). The CPS publishes monthly, by-state estimates of employment levels, where each state expresses a noisy time series. Published state-level estimates from the CPS are composed from household survey responses in a model-free manner and express high levels of volatility due to insufficient sample sizes. Existing software solutions borrow information over a modeled time-based dependence to extract a de-noised time series for each domain. These solutions, however, ignore the dependence among the domains that may be additionally leveraged to improve estimation efficiency. The growfunctions package offers two fully nonparametric mixture models that simultaneously estimate both a time and domain-indexed dependence structure for a collection of time series: (1) A Gaussian process (GP) construction, which is parameterized through the covariance matrix, estimates a latent function for each domain. The covariance parameters of the latent functions are indexed by domain under a Dirichlet process prior that permits estimation of the dependence among functions across the domains: (2) An intrinsic Gaussian Markov random field prior construction provides an alternative to the GP that expresses different computation and estimation properties. In addition to performing denoised estimation of latent functions from published domain estimates, growfunctions allows estimation of collections of functions for observation units (e.g., households), rather than aggregated domains, by accounting for an informative sampling design under which the probabilities for inclusion of observation units are related to the response variable. growfunctions includes plot functions that allow visual assessments of the fit performance and dependence structure of the estimated functions. Computational efficiency is achieved by performing the sampling for estimation functions using compiled C++.

#### Mixture Experiments in R Using mixexp

This article discusses the design and analysis of mixture experiments with R and illustrates the use of the recent package mixexp. This package provides functions for creating mixture designs composed of extreme vertices and edge and face centroids in constrained mixture regions where components are subject to upper, lower and linear constraints. These designs cannot be created by other R packages. mixexp also provides functions for graphical display of models fit to data from mixture experiments that cannot be created with other R packages.

## August 14, 2016

### Dirk Eddelbuettel

#### rfoaas 1.0.0

The big 1.0.0 is here! Following the unsurpassed lead of the FOAAS project, we have arrived at a milestone: Release 1.0.0 is now on CRAN.

The rfoaas package provides an interface for R to the most excellent FOAAS service--which itself provides a modern, scalable and RESTful web service for the frequent need to tell someone to f$#@ off. Release 1.0.0 brings fourteen (!!) new access points: back(), bm(), gfy(), think(), keep(), single_(), look(), looking(), no(), give(), zero(), pulp(), sake(), and anyway(). All with documentation and testing. Even more neatly, thanks to a very pull request by Tyler Hunt, we can now issue random FOs via the getRandomFO() function! As usual, CRANberries provides a diff to the previous CRAN release. Questions, comments etc should go to the GitHub issue tracker. More background information is on the project page as well as on the github repo This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings. ## August 07, 2016 ### Dirk Eddelbuettel #### drat 0.1.1: Updates schmupdates! One year ago (tomorrow) drat 0.1.0 was released. It held up rather well, but a number of small fixes and enhancements piled up, along with somewhat-finished to still-raw additions to the examples/ sections. With that, we are happy to announce drat release 0.1.1 which arrived on CRAN earlier today. drat stands for drat R Archive Template, and helps with easy-to-create and easy-to-use repositories for R packages. Since its inception in early 2015 it has found reasonably widespread adoption among R users because repositories is what we use. In other words, friends don't let friends use install_github(). Just kidding. Maybe. Or not. This version 0.1.1 builds on the previous release from one year ago. Several users sent in nicely focused pull request, and I added a bit of spit and polish here and there. The NEWS file (added belatedly in this release) summarises the release as follows: #### Changes in drat version 0.1.1 (2016-08-07) • Changes in drat functionality • Use dir.exists, leading to versioned Depends on R (>= 3.2.0) • Optionally pull remote before insert (Mark in PR #38) • Fix support for dots (Jan G. in PR #40) • Accept dots in package names (Antonio in PR #48) • Switch to htpps URLs at GitHub (Colin in PR #50) • Support additional fields in PACKAGE file (Jan G. in PR #54) • Changes in drat documentation • Further improvements and clarifications to vignettes • Travis script switched to run.sh from our fork • This NEWS file was (belatedly) added Courtesy of CRANberries, there is a comparison to the previous release. More detailed information is on the drat page. This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings. ## July 26, 2016 ### Statistical Modelling #### A class of mixture models for multidimensional ordinal data In rating surveys, people are requested to express preferences on several aspects related to a topic by selecting a category in an ordered scale. For such data, we propose a model defined by a mixture of a uniform distribution and a Sarmanov distribution with CUB (combination of uniform and shifted binomial) marginal distributions (D'Elia and Piccolo, 2005). This mixture generalizes the CUB model to the multivariate case by taking into account the association among answers of the same individual to the items of a questionnaire. It also allows us to distinguish two kinds of uncertainty: specific uncertainty, related to the indecision for single items, and global uncertainty referred to the respondent's hesitancy in completing the whole questionnaire. A simulation and a real case study highlight the usefulness of the new methodology. #### Partitioned conditional generalized linear models for categorical responses Abstract: In categorical data analysis, several regression models have been proposed for hierarchically structured responses, such as the nested logit model, the two-step model or the partitioned conditional model for partially ordered set. The specifications of these models are heterogeneous and they have been formally defined for only two or three levels in the hierarchy. Here, we introduce the class of partitioned conditional generalized linear models (PCGLMs) that encompasses all these models and is defined for any number of levels in the hierarchy. The hierarchical structure of these models is fully specified by a partition tree of categories. Using the genericity of the recently introduced (r,F,Z) specification of generalized linear models (GLMs) for categorical responses, it is possible to use different link functions and explanatory variables for each partitioning step. PCGLMs thus constitute a very flexible framework for modelling hierarchically structured categorical responses including partially ordered responses. #### Sums of smooth exponentials to decompose complex series of counts Representing the conditional mean in Poisson regression directly as a sum of smooth components can provide a realistic model of the data generating process. Here, we present an approach that allows such an additive decomposition of the expected values of counts. The model can be formulated as a penalized composite link model and can, therefore, be estimated by a modified iteratively weighted least-squares algorithm. Further shape constraints on the smooth additive components can be enforced by additional penalties, and the model is extended to two dimensions. We present two applications that motivate the model and demonstrate the versatility of the approach. #### Bayesian semiparametric density ratio modelling with applications to medical malpractice reform This study examines the efficacy of tort reforms instituted throughout the country during the last decade, improving upon existing semiparametric density ratio estimation (DRE) methodologies in the process. DRE is a well-known semiparametric modelling technique that has been used for well over two decades. Although the approach has been demonstrated to be extremely useful in statistical modelling, it has suffered from one main limitation—the methodology has thus far not been capable of modelling individual-level heterogeneity. We address this issue by presenting a novel adaptation of DRE to model individual level heterogeneity. We do so by marginalizing the associated empirical likelihood function involving density ratios to provide an overall distribution of the entire population despite having extremely limited initial information about each individual in the dataset. We apply this approach to medical malpractice loss data from the previous decade to quantify the probability of changes in tort losses. Our results demonstrate the success of a number of recently implemented malpractice reforms. Comparisons to existing DRE methods, as well as standard regression methods, illustrate the efficacy of our approach. ### RCpp Gallery #### RcppHoney Introduction ## Rationale In C++ we often have containers that are not compatible with R or Rcpp with data already in them (std::vector, std::set, etc.). One would like to be able to operate on these containers without having to copy them into Rcpp structures like IntegerVector. RcppHoney aims to address this problem by providing operators and functions with R semantics that can be used on any iterator-based container. ## Introduction RcppHoney allows any iterator-based container to be “hooked” in. Once a container type is hooked to RcppHoney, it is granted operators (+, -, *, /, etc.) and a host of other mathematical functions that can be run on it. It also becomes interoperable with any other hooked data structure. This lets us write expressions that look like std::vector + Rcpp::IntegerVector + log(Rcpp::NumericVector) and get the expected results. ## Implementation RcppHoney has several structures that are hooked in by default. Currently they are std::vector, std::set, and Rcpp::VectorBase. The ability to hook in custom structures is also provided. All operators and functions are implemented as expression templates to minimize memory usage and enhance performance. The goal here is to only copy the data into an R compatible structure when we must (i.e. when we return it to R). This is achieved through the use of the RcppHoney::operand class. RcppHoney::operand provides an iterable interface to the result types of operators and functions. RcppHoney currently provides all the basic mathematical operators (+, -, *, /) as well as some common functions (abs, sin, cos, exp, etc.). Eventually all of the functionality provided by Rcpp::sugar as well as anything else we can think of will be supported. Enough about the abstract though…let’s see it in action. ## Example The following example shows how to hook in a custom data structure (in this case std::list) as well as the types of expressions that can be created once a data structure is hooked in. ## Conclusion RcppHoney is a powerful tool for allowing different container types to interoperate under Rcpp. It can save development time as well as help the user generate faster and more readable code. RcppHoney is available via CRAN though as it is still in an alpha state and changing rapidly, it is recommended that you install it from source. Source code is available at github.com/dcdillon/RcppHoney. ## July 18, 2016 ### R you ready? #### Populating data frame cells with more than one value ### Data frames are lists Most R users will know that data frames are lists. You can easily verify that a data frame is a list by typing d <- data.frame(id=1:2, name=c("Jon", "Mark")) d   id name 1 1 Jon 2 2 Mark  is.list(d)  [1] TRUE  However, data frames are lists with some special properties. For example, all entries in the list must have the same length (here 2), etc. You can find a nice description of the differences between lists and data frames here. To access the first column of d, we find that it contains a vector (and a factor in case of column name). Note, that [[ ]] is an operator to select a list element. As data frames are lists, they will work here as well. is.vector(d[[1]])  [1] TRUE  ### Data frame columns can contain lists A long time, I was unaware of the fact, that data frames may also contain lists as columns instead of vectors. For example, let’s assume Jon’s children are Mary and James, and Mark’s children are called Greta and Sally. Their names are stored in a list with two elements. We can add them to the data frame like this: d$children <-  list(c("Mary", "James"), c("Greta", "Sally"))
d

 id name children
1 1 Jon Mary, James
2 2 Mark Greta, Sally


A single data frame entry in column children now contains more than one value. Given that the column is a list, not a vector, we cannot go as usual when modifying an entry of the column. For example, to change Jon’s children, we cannot do

> d[1 , "children"] <- c("Mary", "James", "Thomas")

Error in [<-.data.frame(*tmp*, 1, "children", value = c("Mary", "James", :
replacement has 3 rows, data has 1


Taking into account the list structure of the column, we can type the following to change the values in a single cell.

d[1 , "children"][[1]] <- list(c("Mary", "James", "Thomas"))

# or also

d$children[1] <- list(c("Mary", "James", "Thomas")) d   id name children 1 1 Jon Mary, James, Thomas 2 2 Mark Greta, Sally  You can also create a data frame having a list as a column using the <tt>data.frame</tt> function, but with a little tweak. The list column has to be wrapped inside the function <tt>I</tt>. This will protect it from several conversions taking place in <tt>data.frame</tt> (see <tt>?I</tt> documentation). d <- data.frame(id = 1:2, name = c("Jon", "Mark"), children = I(list(c("Mary", "James"), c("Greta", "Sally"))) )  This is an interesting feature, which gives me a deeper understanding of what a data frame is. But when exactly would I want to use it? I have not encountered the need to use it very often yet (though of course there may be plenty of situations where it makes sense). But today I had a case where this feature seemed particularly useful. ### Converting lists and data frames to JSON I had two separate types of information. One stored in a data frame and the other one in a list Referring to the example above, I had d <- data.frame(id=1:2, name=c("Jon", "Mark")) d   id name 1 1 Jon 2 2 Mark  and ch <- list(c("Mary", "James"), c("Greta", "Sally")) ch  [[1]] [1] "Mary" "James" [[2]] [1] "Greta" "Sally"  I needed to return an array of JSON objects which look like this. [ { "id": 1, "name": "Jon", "children": ["Mary", "James"] }, { "id": 2, "name": "Mark", "children": ["Greta", "Sally"] } ]  Working with the superb jsonlite package to convert R to JSON, I could do the following to get the result above. library(jsonlite) l <- split(d, seq(nrow(d))) # convert data frame rows to list l <- unname(l) # remove list names for (i in seq_along(l)) # add element from ch to list l[[i]] <- c(l[[i]], children=ch[i]) toJSON(l, pretty=T, auto_unbox = T) # convert to JSON  The results are correct, but getting there involved quite a number of tedious steps. These can be avoided by directly placing the list into a column of the data frame. Then jsonlite::toJSON takes care of the rest. d$children <- list(c("Mary", "James"), c("Greta", "Sally"))
toJSON(d, pretty=T, auto_unbox = T)

[
{
"id": 1,
"name": "Jon",
"children": ["Mary", "James"]
},
{
"id": 2,
"name": "Mark",
"children": ["Greta", "Sally"]
}
]


Nice :) What we do here, is basically creating the same nested list structure as above, only now it is disguised as a data frame. However, this approach is much more convenient.

## July 07, 2016

### Bioconductor Project Working Papers

#### Practical Targeted Learning from Large Data Sets by Survey Sampling

We address the practical construction of asymptotic confidence intervals for smooth (i.e., pathwise differentiable), real-valued statistical
parameters by targeted learning from independent and identically
distributed data in contexts where sample size is so large that it poses
computational challenges. We observe some summary measure of all data and select a sub-sample from the complete data set by Poisson rejective sampling with unequal inclusion probabilities based on the summary measures. Targeted learning is carried out from the easier to handle sub-sample. We derive a central limit theorem for the targeted minimum loss estimator (TMLE) which enables the construction of the confidence intervals. The inclusion probabilities can be optimized to reduce the asymptotic variance of the TMLE. We illustrate the procedure with two examples where the parameters of interest are variable importance measures of an exposure (binary or continuous) on an outcome. We also conduct a simulation study and comment on its results.

#### Estimation of long-term area-average PM2.5 concentrations for area-level health analyses

Introduction: There is increasing evidence of an association between individual long-term PM2.5 exposure and human health. Mortality and morbidity data collected at the area-level are valuable resources for investigating corresponding population-level health effects. However, PM2.5 monitoring data are available for limited periods of time and locations, and are not adequate for estimating area-level concentrations. We developed a general approach to estimate county-average concentrations representative of population exposures for 1980-2010 in the continental U.S.

Methods: We predicted annual average PM2.5 concentrations at about 70,000 census tract centroids, using a point prediction model previously developed for estimating annual average PM2.5 concentrations in the continental U.S. for 1980-2010. We then averaged these predicted PM2.5 concentrations in all counties weighted by census tract population. In sensitivity analyses, we compared the resulting estimates to four alternative county average estimates using MSE-based R2 in order to capture both systematic and random differences in estimates. These estimates included crude aggregates of regulatory monitoring data, averages of predictions at residential addresses in Southern California, and two sets of averages of census tract centroid predictions unweighted by population and interpolated from predictions at 25-km national grid coordinates.

Results: The county-average mean PM2.5 was 14.40 (standard deviation=3.94) µg/m3 in 1980 and decreased to 12.24 (3.24), 10.42 (3.30), and 8.06 (2.06) µg/m3 in 1990, 2000, and 2010, respectively. These estimates were moderately related with crude averages in 2000 and 2010 when monitoring data were available (R2= 0.70-0.82) and almost identical to the unweighted averages in all four decennial years. County averages were also consistent with the county averages derived from residential estimates in Southern California (0.95-0.96). We found grid-based estimates of county-average PM2.5 were more consistent with our estimates when we also included monitoring data (0.95-0.98) than grid-only estimates (0.91-0.96); both had slightly lower concentrations than census tract-based estimates.

Conclusions: Our approach to estimating population representative area-level PM2.5 concentrations is consistent with averages across residences. These exposure estimates will allow us to assess health impacts of ambient PM2.5 concentration in datasets with area-level health data.

## Introduction

A while back, I saw a post on StackOverflow where the user was trying to use Rcpp::sugar::sum() on an RcppParallel::RVector. Obviously, this does not work (as Rcpp Sugar pertains to Rcpp types, but not RcppParallel which cannot rely on SEXP-based representation to allow multi-threaded execution). It raised the question “Why doesn’t something more generic exist to provide functions with R semantics that can be used on arbitrary data structures?” As a result, I set out to create a set of such functions in Rcpp::algorithm which follow the pattern of std::algorithm.

## Rcpp::algorithm

Currently Rcpp::algorithm contains only a few simple functions. If these are found to be useful, more will be added. Examples of using the currently implemented iterator-based functions are below.

### log, exp, and sqrt

Through the coding of these simple “algorithms”, a few needs arose.

First, the ability to deduce the appropriate C numeric type given an Rcpp iterator was necessary. This gave birth to the Rcpp::algorithm::helpers::decays_to_ctype and Rcpp::algorithm::helpers::ctype type traits. Given a type, these allow you to determine whether it can be cast to a C numeric type and which type that would be.

Second, the need arose for more information about R types. This gave birth to the Rcpp::algorithm::helpers::rtype traits. These are defined as follows:

These additional benefits may actually prove more useful than the algorithms themselves. Only time will tell.

## Wrapping Up

There are now some simple iterator-based algorithms that can be used with any structure that supports iterators. They apply the same semantics as the analogous Rcpp::sugar functions, but give us more flexibility in their usage. If you find these to be useful, feel free to request more.

## Introduction

Consider a need to be able to interface with a data type that is not presently supported by Rcpp. The data type might come from a new library, or from within one of our own applications. In either cases, Rcpp is faced with an issue of consciousness as the new data type is not similar to known types so the autocoversion or seamless R to C++ integration cannot be applied correctly. The issue is two-fold as we need to consider both directions:

1. Converting from R to C++ using Rcpp::as<T>(obj)
2. Converting from C++ to R using Rcpp::wrap(obj)

Luckily, there is a wonderful Rcpp vignette called Extending Rcpp that addresses custom objects. However, the details listed are more abstracted than one would like. So, I am going to try to take you through the steps with a bit of commentary. Please note that the approach used is via Templates and partial specialization and will result in some nice automagic at the end.

The overview of the discussion will focus on:

• Stage 1 - Forward Declarations
• Stage 2 - Including the Rcpp Header
• Stage 3 - Implementation of Forward Declarations
• Stage 4 - Testing Functionality
• Stage 5 - All together

## Explanation of Stages

### Stage 1 - Forward Declarations

In the first stage, we must declare our intent to the features we wish to use prior to engaging Rcpp.h. To do so, we will load a different header file and add some definitions to the Rcpp::traits namespace.

Principally, when we start writing the file, the first header that we must load is RcppCommon.h and not the usual Rcpp.h!! If we do not place the forward declaration prior to the Rcpp.h call, we will be unable to appropriately register our extension.

Then, we must add in the different plugin markup for sourceCpp() to set the appropriate flags during the compilation of the code. After the plugins, we will include the actual headers that we want to use. In this document, we will focus on Boost headers for the concrete example. Lastly, we must add two special Rcpp function declaration, Rcpp::as<T>(obj) and Rcpp::wrap(obj), within the Rcpp::traits namespace. To enable multiple types, we must create an Exporter class instead of a more direct call to template <> ClassName as( SEXP ).

### Stage 2 - Include the Rcpp.h header

It might seem frivolous to have a stage just to declare import order, but if Rcpp.h is included before the forward declaration then Rcpp::traits is not updated and we enter the abyss. Template programming can be delicate, respecting this include order is one of many small details one must get right.

Thus:

### Stage 3 - Implementing the Declarations

Now, we must actually implement the forward declarations. In particular, the only implementation that will be slightly problematic is the as<> since the wrap() is straight forward.

#### wrap()

To implement wrap() we must appeal to a built-in type conversion index within Rcpp which is called Rcpp::traits::r_sexptype_traits<T>::rtype. From this, we are able to obtain an int containing the RTYPE and then construct an Rcpp::Vector. For the construction of a matrix, the same ideas hold true.

#### as()

For as<>(), we need to consider the template that will be passed in. Furthermore, we setup a typedef directly underneath the Exporter class definition to easily define an OUT object to be used within the get() method. Outside of that, we use the same trick to move back and forth from a C++ T template type to an R type (implemented as one of several SEXP types).

In order to accomplish the as<>, or the direct port from R to C++, I had to do something dirty: I copied the vector contents. The code that governs this output is given within the get() of the Exporter class. You may wish to spend some time looking into changing the assignment using pointers perhaps. I am not very well versed with ublas so I did not see an easy approach to resolve the pointer pass.

### Stage 4 - Testing

Okay, let’s see if what we worked on paid off (spoiler It did! spoiler). To check, we should look at two different areas:

1. Trace diagnostics within the function and;
2. An automagic test.

Both of which are given below. Note that I’ve opted to shorten the ublas setup to just be:

#### Trace Diagnostics

Test Call:

Results:

Converting from Rcpp::NumericVector to ublas::vector<double>
Running output test with ublas::vector<double>
1
2
3
4
Converting from ublas::vector<double> to Rcpp::NumericVector
Running output test with Rcpp::NumericVector
1
2
3
4


This test performed as expected. Onto the next test!

#### Automagic test

Test Call:

Results:

[1] 1.0 2.0 3.2 1.2


Success!

### Stage 5 - All together

Here is the combination of the above code chunks given by stage. If you copy and paste this into your .cpp file, then everything should work. If not, let me know.

## Closing Remarks

Whew… That was a lot. Hopefully, the above provided enough information as you may want to extend this post’s content past 1D vectors to perhaps a ublas::matrix and so on. In addition, then you now have the autoconvert magic of Rcpp for ublas::vector<double>! Moreover, all one needs to do is specify the either the parameters or return type of the function to be ublas::vector<double> – and Voilà, automagic conversion!

## June 23, 2016

### RCpp Gallery

#### Working with Rcpp::StringVector

Vectors are fundamental containers in R. This makes them equally important in Rcpp. Vectors can be useful for storing multiple elements of a common class (e.g., integer, numeric, character). In Rcpp, vectors come in the form of NumericVector, CharacterVector, LogicalVector, StringVector and more. Look in the header file Rcpp/include/Rcpp/vector/instantiation.h for more types. Here we explore how to work with Rcpp::StringVector as a way to manage non-numeric data.

We typically interface with Rcpp by creating functions. There are several ways to include Rcpp functions in R. The examples here can be copied and pasted into a text file named ‘source.cpp’ and compiled with the command Rcpp::sourceCpp("source.cpp") made from the R command prompt.

## Initialization

Here we create a simple function which initializes an empty vector of three elements in size and returns it.

We can call this function from R as follows.

[1] "" "" ""


The first two lines are pretty much mandatory and you should copy and paste them into all your code until you understand them. The first line tells the program to use Rcpp. The second line exports this function for use, as opposed to keeping it as an internal function that is unavailable to users. Some people like to include using namespace Rcpp; to load the namespace. I prefer to use the scope operator (::) when calling functions. This is a matter of style and is therefore somewhat arbitrary. Whatever your perspective on this, its best to maintain consistency so that your code will be easier for others to understand.

We see that we’ve returned a vector of length three. We can also see that the default value is a string which contains nothing (“”). This is not a vector of NAs (missing data), even though NAs are supported by Rcpp::StringVector.

## Accessing elements

The individual elements of a StringVector can be easily accessed. Here we’ll create an Rcpp function that accepts an Rcpp::StringVector as an argument. We’ll print the elements from within Rcpp. Then we’ll return the vector to R.

After we’ve compiled it we can call it from R.

[1] "apple"  "banana" "orange"

i is: 0, the element value is: apple
i is: 1, the element value is: banana
i is: 2, the element value is: orange

[1] "apple"  "banana" "orange"


We see that the R vector contains the elements “apple”, “banana” and “orange.” Within Rcpp we print each element to standard out with Rcpp::Rcout << statements. And we see that these values are returned to the vector x2.

We’ve also introduced the method .size() which returns the number of elements in an object. This brings up an important difference among C++ and R. Many function names in R may contain periods. For example, the function name write.table() delimits words with a period. However, in C++ the period indicates a method. This means that C++ object names can’t include a period. Camel code or underscores are good alternatives.

There are at least two other important issues to learn from the above example. First, in R we typically access elements with square brackets. While some C++ objects are also accessed with square brackets, the Rcpp::StringVector is accessed with either parentheses or square brackets. In the case of the Rcpp::StringVector these appear interchangeable. However, be very careful, they are different in other containers. A second, but very important, difference between R and C++ is that in R the vectors are 1-based, meaning the first element is accessed with a 1. In C++ the vectors are zero-based, meaning the first element is accessed with a zero. This creates an opportunity for one-off errors. If you notice that the number of elements you’ve passed to C++ and back are off by one element, this would be something good to check.

### Elements of elements

In C++, a std::string can be see as a vector of chars. Individual elements of a Rcpp::StringVector behave similarly. Accessing each element of a StringVector is similar to working with a std::string. Here we access each character of the second element of our StringVector.

And call the code from R.

i is: 0, element is: b
i is: 1, element is: a
i is: 2, element is: n
i is: 3, element is: a
i is: 4, element is: n
i is: 5, element is: a


We see that we’ve accessed and printed the individual characters of the second element of the vector. We accomplish this by using the square brackets to access element one of the vector, and then use a second set of square brackets to access each character of this element.

## Modifying elements

The modification of elements is fairly straight forward. We use the index (begining at zero) to modify the vector elements.

[1] "apple"      "watermelon" "kumquat"


We’ve successfully changed the second element from ‘banana’ to ‘watermelon’ and the third element from ‘orange’ to ‘kumquat.’ This also illustrates that Rcpp::StringVectors are flexible in their use of both square and round brackets. Trying that with standard library containers will usually result in an error.

In the above example we’ve passed an Rcpp::StringVector to a function and returned a new Rcpp::StringVector. By copying the container in this manner it may seem intuitive to work on it. If efficient use of memory is desired it is important to realize that pointers are being passed to the Rcpp function. This means we can create a function which returns void and modifies the elements we’re interested in modifying without the overhead of copying the container.

[1] "apple"      "watermelon" "orange"


## Erasing elements

If we want to remove an element from a StringVector we can use the .erase() method.

And see our changes with R code.

[1] "apple"  "orange"


We see that we’ve erased the second element from the array.

## Growing and shrinking Rcpp::StringVectors

If you have an Rcpp::StringVector and you want to add elements, you can use the method .push_back(). While Rcpp has push functionality, it does not appear to have pop functionality. However, using techniques learned above, we could use object.erase(object.size()) to attain similar functionality. Here I illustrate their use to remove an element and then add two elements.

And implement our example in R.

i is: 0, the element value is: apple
i is: 1, the element value is: banana

[1] "apple"   "banana"  "avocado" "kumquat"


From the Rcpp output we see that we’ve removed the last element from the vector. We also see that we’ve added two elements to the ‘back’ of the vector.

If we want to add to the front of our vector we can accomplish that as well. There does not appear to be ‘push_front’ or ‘pop_front’ methods, but we have the tools necessary to accomplish these tasks. We use the erase and insert methods to push and pop to the front of our Rcpp::StringVector.

And implement our example in R.

[1] "kumquat" "avocado" "orange"


In general, growing and shrinking data structures comes with a performance cost. And if you’re interested in Rcpp, you’re probably interested in performance. You’ll typically be better off setting a container size and sticking with it. But there are times when growing and shrinking your container can be really helpful. My recommendation is to use this functionality sparingly.

## Missing data

In R we handle missing data with ‘NAs.’ In C++ the concept of missing data does not exist. Instead, some sort of placeholder, such as -999, has to be used. The Rcpp containers do support missing data to help make the interface between R and C++ easy. We can see this by continuing our existing example, but use it to set the second element as missing.

[1] "apple"  NA       "orange"


## Finding other methods

The Rcpp header files contain valuable information about objects defined in Rcpp. However, they’re rather technical and may not be very approachable to the novice. (This document is an attempt to help users bridge that gap between a novice and someone who reads headers.) If you don’t know where the header files are, you can use .libPaths() to list the locations of your libraries. In one of these locations you should find a directory called ‘Rcpp.’ Within this directory you should find a directory named ‘include’ which is where the headers are. For example, the header for the String object on my system is at: Rcpp/include/Rcpp/String.h

## Type conversion

Once we have our data in an Rcpp function we may want to make use of the functionality of C++ containers. This will require us to convert our Rcpp container to another C++ form. Once we’ve processed these data we may want convert them back to Rcpp (so we can return them to R). A good example is converting an element of a StringVector to a std::string.

### Implicit type conversion

Conversion from an Rcpp:StringVector to a std::string is a compatible conversion. This means we can accomplish this implicitly by simply setting the value of one container to the other.

[1] "apple"  "banana" "orange"


Note that while we have to load each element of the std::vector individually. However, the loading of the Rcpp::StringVector has been vectorized so that it works similar to R vectors.

### Explicit type conversion

In some instances we may need explicit type conversion. Rcpp provides an ‘as’ method to accomplish this.

[1] "apple"  "banana" "orange"


Type conversion is a lengthy topic and is frequently specific to the types which are being converted to and from. Hopefully this introduction is enough to get you started with the tools provided in Rcpp.

## Attributes

R objects include attributes which help describe the object. This is another concept that is absent in C++. Again, the Rcpp objects implement attributes to help us and to maintain a behavior that is similar to R.

       pome       berry hesperidium
"apple"    "banana"    "orange"

    pome    berry   citrus
"apple" "banana" "orange"


Here we’ve stored the names of the Rcpp:StringVector in a std::vector of strings. We’ve then modified one of the elements and reset the names attribute with this changed vector. This illustrates the use of standard library containers along with those provided by Rcpp. But we need to be a little careful of what we’re doing here. If we store the values of our vector in a vector of std::string we lose our attributes because neither a std::vector or std::string has attributes.

[1] "apple"  "banana" "orange"


Note that while we can assign a vector of strings to a Rcpp::StringVector we can not do the inverse. Instead we need to assign each element to the vector of strings. And we need to remember to keep our square brackets and round brackets associated with the correct data structure.

Below are some links I’ve found useful in writing this document. Hopefully you’ll find them as gateways for your exploration of Rcpp.

Once you’ve crossed from R to C++ there are many of sources of information online. One of my favorites is included below.

## May 13, 2016

### Journal of the Royal Statistical Society: Series A

#### Probability and Statistics by Example 1: Basic Probability and Statistics Y. Suhov M. Kelbert 2014 Cambridge Cambridge University Press 470 pp., $84.99 ISBN 978‐1‐107‐60358‐5 #### A Certain Uncertainty: Nature's Random Ways M. P. Silverman 2014 Cambridge Cambridge University Press xvi + 618 pp., €175.00 ISBN 978‐1‐107‐03281‐1 ## February 28, 2016 ### Journal of the Royal Statistical Society: Series C #### Bayesian two‐stage dose finding for cytostatic agents via model adaptation #### A general angular regression model for the analysis of data on animal movement in ecology #### Two‐stage model for time varying effects of zero‐inflated count longitudinal covariates with applications in health behaviour research #### Dependence modelling with regular vine copula models: a case‐study for car crash simulation data ## January 29, 2016 ### Journal of the Royal Statistical Society: Series B #### Joint estimation of multiple graphical models from high dimensional time series #### Bootstrapping the portmanteau tests in weak auto‐regressive moving average models #### Making a non‐parametric density estimator more attractive, and more accurate, by data perturbation #### Sequential selection procedures and false discovery rate control ## December 27, 2015 ### Alstatr #### R and Python: Gradient Descent One of the problems often dealt in Statistics is minimization of the objective function. And contrary to the linear models, there is no analytical solution for models that are nonlinear on the parameters such as logistic regression, neural networks, and nonlinear regression models (like Michaelis-Menten model). In this situation, we have to use mathematical programming or optimization. And one popular optimization algorithm is the gradient descent, which we're going to illustrate here. To start with, let's consider a simple function with closed-form solution given by $$f(\beta) \triangleq \beta^4 - 3\beta^3 + 2.$$ We want to minimize this function with respect to$\beta. The quick solution to this, as what calculus taught us, is to compute for the first derivative of the function, that is $$\frac{\text{d}f(\beta)}{\text{d}\beta}=4\beta^3-9\beta^2.$$ Setting this to 0 to obtain the stationary point gives us \begin{align} \frac{\text{d}f(\beta)}{\text{d}\beta}&\overset{\text{set}}{=}0\nonumber\\ 4\hat{\beta}^3-9\hat{\beta}^2&=0\nonumber\\ 4\hat{\beta}^3&=9\hat{\beta}^2\nonumber\\ 4\hat{\beta}&=9\nonumber\\ \hat{\beta}&=\frac{9}{4}. \end{align} The following plot shows the minimum of the function at\hat{\beta}=\frac{9}{4}$(red line in the plot below). R ScriptNow let's consider minimizing this problem using gradient descent with the following algorithm: 1. Initialize$\mathbf{x}_{r},r=0$2. while$\lVert \mathbf{x}_{r}-\mathbf{x}_{r+1}\rVert > \nu$3.$\mathbf{x}_{r+1}\leftarrow \mathbf{x}_{r} - \gamma\nabla f(\mathbf{x}_r)$4.$r\leftarrow r + 1$5. end while 6. return$\mathbf{x}_{r}$and$r$where$\nabla f(\mathbf{x}_r)$is the gradient of the cost function,$\gamma$is the learning-rate parameter of the algorithm, and$\nu$is the precision parameter. For the function above, let the initial guess be$\hat{\beta}_0=4$and$\gamma=.001$with$\nu=.00001$. Then$\nabla f(\hat{\beta}_0)=112$, so that $\hat{\beta}_1=\hat{\beta}_0-.001(112)=3.888.$ And$|\hat{\beta}_1 - \hat{\beta}_0| = 0.112> \nu$. Repeat the process until at some$r$,$|\hat{\beta}_{r}-\hat{\beta}_{r+1}| \ngtr \nu$. It will turn out that 350 iterations are needed to satisfy the desired inequality, the plot of which is in the following figure with estimated minimum$\hat{\beta}_{350}=2.250483\approx\frac{9}{4}$. R Script with PlotPython ScriptObviously the convergence is slow, and we can adjust this by tuning the learning-rate parameter, for example if we try to increase it into$\gamma=.01$(change gamma to .01 in the codes above) the algorithm will converge at 42nd iteration. To support that claim, see the steps of its gradient in the plot below. If we try to change the starting value from 4 to .1 (change beta_new to .1) with$\gamma=.01$, the algorithm converges at 173rd iteration with estimate$\hat{\beta}_{173}=2.249962\approx\frac{9}{4}(see the plot below). Now let's consider another function known as Rosenbrock defined as $$f(\mathbf{w})\triangleq(1 - w_1) ^ 2 + 100 (w_2 - w_1^2)^2.$$ The gradient is \begin{align} \nabla f(\mathbf{w})&=[-2(1 - w_1) - 400(w_2 - w_1^2) w_1]\mathbf{i}+200(w_2-w_1^2)\mathbf{j}\nonumber\\ &=\left[\begin{array}{c} -2(1 - w_1) - 400(w_2 - w_1^2) w_1\\ 200(w_2-w_1^2) \end{array}\right]. \end{align} Let the initial guess be\hat{\mathbf{w}}_0=\left[\begin{array}{c}-1.8\\-.8\end{array}\right]$,$\gamma=.0002$, and$\nu=.00001$. Then$\nabla f(\hat{\mathbf{w}}_0)=\left[\begin{array}{c} -2914.4\\-808.0\end{array}\right]$. So that $$\nonumber \hat{\mathbf{w}}_1=\hat{\mathbf{w}}_0-\gamma\nabla f(\hat{\mathbf{w}}_0)=\left[\begin{array}{c} -1.21712 \\-0.63840\end{array}\right].$$ And$\lVert\hat{\mathbf{w}}_0-\hat{\mathbf{w}}_1\rVert=0.6048666>\nu$. Repeat the process until at some$r$,$\lVert\hat{\mathbf{w}}_r-\hat{\mathbf{w}}_{r+1}\rVert\ngtr \nu$. It will turn out that 23,374 iterations are needed for the desired inequality with estimate$\hat{\mathbf{w}}_{23375}=\left[\begin{array}{c} 0.9464841 \\0.8956111\end{array}\right]$, the contour plot is depicted in the figure below. R Script with Contour PlotPython ScriptNotice that I did not use ggplot for the contour plot, this is because the plot needs to be updated 23,374 times just to accommodate for the arrows for the trajectory of the gradient vectors, and ggplot is just slow. Finally, we can also visualize the gradient points on the surface as shown in the following figure. R ScriptIn my future blog post, I hope to apply this algorithm on statistical models like linear/nonlinear regression models for simple illustration. #### R: Principal Component Analysis on Imaging Ever wonder what's the mathematics behind face recognition on most gadgets like digital camera and smartphones? Well for most part it has something to do with statistics. One statistical tool that is capable of doing such feature is the Principal Component Analysis (PCA). In this post, however, we will not do (sorry to disappoint you) face recognition as we reserve this for future post while I'm still doing research on it. Instead, we go through its basic concept and use it for data reduction on spectral bands of the image using R. ### Let's view it mathematically Consider a line$L$in a parametric form described as a set of all vectors$k\cdot\mathbf{u}+\mathbf{v}$parameterized by$k\in \mathbb{R}$, where$\mathbf{v}$is a vector orthogonal to a normalized vector$\mathbf{u}$. Below is the graphical equivalent of the statement: So if given a point$\mathbf{x}=[x_1,x_2]^T$, the orthogonal projection of this point on the line$L$is given by$(\mathbf{u}^T\mathbf{x})\mathbf{u}+\mathbf{v}$. Graphically, we mean$Proj$is the projection of the point$\mathbf{x}$on the line, where the position of it is defined by the scalar$\mathbf{u}^{T}\mathbf{x}$. Therefore, if we consider$\mathbf{X}=[X_1, X_2]^T$be a random vector, then the random variable$Y=\mathbf{u}^T\mathbf{X}$describes the variability of the data on the direction of the normalized vector$\mathbf{u}$. So that$Y$is a linear combination of$X_i, i=1,2$. The principal component analysis identifies a linear combinations of the original variables$\mathbf{X}$that contain most of the information, in the sense of variability, contained in the data. The general assumption is that useful information is proportional to the variability. PCA is used for data dimensionality reduction and for interpretation of data. (Ref 1. Bajorski, 2012) To better understand this, consider two dimensional data set, below is the plot of it along with two lines ($L_1$and$L_2$) that are orthogonal to each other: If we project the points orthogonally to both lines we have, So that if normalized vector$\mathbf{u}_1$defines the direction of$L_1$, then the variability of the points on$L_1$is described by the random variable$Y_1=\mathbf{u}_1^T\mathbf{X}$. Also if$\mathbf{u}_2$is a normalized vector that defines the direction of$L_2$, then the variability of the points on this line is described by the random variable$Y_2=\mathbf{u}_2^T\mathbf{X}$. The first principal component is one with maximum variability. So in this case, we can see that$Y_2$is more variable than$Y_1$, since the points projected on$L_2$are more dispersed than in$L_1$. In practice, however, the linear combinations$Y_i = \mathbf{u}_i^T\mathbf{X}, i=1,2,\cdots,p$is maximized sequentially so that$Y_1$is the linear combination of the first principal component,$Y_2$is the linear combination of the second principal component, and so on. Further, the estimate of the direction vector$\mathbf{u}$is simply the normalized eigenvector$\mathbf{e}$of the variance-covariance matrix$\mathbf{\Sigma}$of the original variable$\mathbf{X}$. And the variability explained by the principal component is the corresponding eigenvalue$\lambda$. For more details on theory of PCA refer to (Bajorski, 2012) at Reference 1 below. As promised we will do dimensionality reduction using PCA. We will use the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) data from (Barjorski, 2012), you can use other locations of AVIRIS data that can be downloaded here. However, since for most cases the AVIRIS data contains thousands of bands so for simplicity we will stick with the data given in (Bajorski, 2012) as it was cleaned reducing to 152 bands only. ### What is spectral bands? In imaging, spectral bands refer to the third dimension of the image usually denoted as$\lambda$. For example, RGB image contains red, green and blue bands as shown below along with the first two dimensions$x$and$y$that define the resolution of the image. These are few of the bands that are visible to our eyes, there are other bands that are not visible to us like infrared, and many other in electromagnetic spectrum. That is why in most cases AVIRIS data contains huge number of bands each captures different characteristics of the image. Below is the proper description of the data. ### Data The Airborne Visible/Infrared Imaging Spectrometer (AVIRIS), is a sensor collecting spectral radiance in the range of wavelengths from 400 to 2500 nm. It has been flown on various aircraft platforms, and many images of the Earth’s surface are available. A 100 by 100 pixel AVIRIS image of an urban area in Rochester, NY, near the Lake Ontario shoreline is shown below. The scene has a wide range of natural and man-made material including a mixture of commercial/warehouse and residential neighborhoods, which adds a wide range of spectral diversity. Prior to processing, invalid bands (due to atmospheric water absorption) were removed, reducing the overall dimensionality to 152 bands. This image has been used in Bajorski et al. (2004) and Bajorski (2011a, 2011b). The first 152 values in the AVIRIS Data represent the spectral radiance values (a spectral curve) for the top left pixel. This is followed by spectral curves of the pixels in the first row, followed by the next row, and so on. (Ref. 1 Bajorski, 2012) To load the data, run the following code: Above code uses EBImage package, and can be installed from my previous post. ### Why do we need to reduce the dimension of the data? Before we jump in to our analysis, in case you may ask why? Well sometimes it's just difficult to do analysis on high dimensional data, especially on interpreting it. This is because there are dimensions that aren't significant (like redundancy) which adds to our problem on the analysis. So in order to deal with this, we remove those nuisance dimension and deal with the significant one. To perform PCA in R, we use the function princomp as seen below: The structure of princomp consist of a list shown above, we will give description to selected outputs. Others can be found in the documentation of the function by executing ?princomp. • sdev - standard deviation, the square root of the eigenvalues$\lambda$of the variance-covariance matrix$\mathbf{\Sigma}$of the data, dat.mat; • loadings - eigenvectors$\mathbf{e}$of the variance-covariance matrix$\mathbf{\Sigma}$of the data, dat.mat; • scores - the principal component scores. Recall that the objective of PCA is to find for a linear combination$Y=\mathbf{u}^T\mathbf{X}$that will maximize the variance$Var(Y)$. So that from the output, the estimate of the components of$\mathbf{u}$is the entries of the loadings which is a matrix of eigenvectors, where the columns corresponds to the eigenvectors of the sequence of principal components, that is if the first principal component is given by$Y_1=\mathbf{u}_1^T\mathbf{X}$, then the estimate of$\mathbf{u}_1$which is$\mathbf{e}_1$(eigenvector) is the set of coefficients obtained from the first column of the loadings. The explained variability of the first principal component is the square of the first standard deviation sdev, the explained variability of the second principal component is the square of the second standard deviation sdev, and so on. Now let's interpret the loadings (coefficients) of the first three principal components. Below is the plot of this, Base above, the coefficients of the first principal component (PC1) are almost all negative. A closer look, the variability in this principal component is mainly explained by the weighted average of radiance of the spectral bands 35 to 100. Analogously, PC2 mainly represents the variability of the weighted average of radiance of spectral bands 1 to 34. And further, the fluctuation of the coefficients of PC3 makes it difficult to tell on which bands greatly contribute on its variability. Aside from examining the loadings, another way to see the impact of the PCs is through the impact plot where the impact curve$\sqrt{\lambda_j}\mathbf{e}_j$are plotted, I want you to explore that. Moving on, let's investigate the percent of variability in$X_i$explained by the$j$th principal component, below is the formula of this, $$\nonumber \frac{\lambda_j\cdot e_{ij}^2}{s_{ii}},$$ where$s_{ii}$is the estimated variance of$X_i$. So that below is the percent of explained variability in$X_i$of the first three principal components including the cumulative percent variability (sum of PC1, PC2, and PC3), For the variability of the first 33 bands, PC2 takes on about 90 percent of the explained variability as seen in the above plot. And still have great contribution further to 102 to 152 bands. On the other hand, from bands 37 to 100, PC1 explains almost all the variability with PC2 and PC3 explain 0 to 1 percent only. The sum of the percentage of explained variability of these principal components is indicated as orange line in the above plot, which is the cumulative percent variability. To wrap up this section, here is the percentage of the explained variability of the first 10 PCs. PC1PC2PC3PC4PC5PC6PC7PC8PC9PC10 Table 1: Variability Explained by the First Ten Principal Components for the AVIRIS data. 82.05717.1760.3200.1820.0940.0650.0370.0290.0140.005 Above variability were obtained by noting that the variability explained by the principal component is simply the eigenvalue (square of the sdev) of the variance-covariance matrix$\mathbf{\Sigma}$of the original variable$\mathbf{X}$, hence the percentage of variability explained by the$j$th PC is equal to its corresponding eigenvalue$\lambda_j$divided by the overall variability which is the sum of the eigenvalues,$\sum_{j=1}^{p}\lambda_j$, as we see in the following code, ### Stopping Rules Given the list of percentage of variability explained by the PCs in Table 1, how many principal components should we take into account that would best represent the variability of the original data? To answer that, we introduce the following stopping rules that will guide us on deciding the number of PCs: 1. Scree plot; 2. Simple fare-share; 3. Broken-stick; and, 4. Relative broken-stick. The scree plot is the plot of the variability of the PCs, that is the plot of the eigenvalues. Where we look for an elbow or sudden drop of the eigenvalues on the plot, hence for our example we have Therefore, we need return the first two principal components based on the elbow shape. However, if the eigenvalues differ by order of magnitude, it is recommended to use the logarithmic scale which is illustrated below, Unfortunately, sometimes it won't work as we can see here, it's just difficult to determine where the elbow is. The succeeding discussions on the last three stopping rules are based on (Bajorski, 2012). The simple fair-share stopping rule identifies the largest$k$such that$\lambda_k$is larger than its fair share, that is larger than$(\lambda_1+\lambda_2+\cdots+\lambda_p)/p$. To illustrate this, consider the following: Thus, we need to stop at second principal component. If one was concerned that the above method produces too many principal components, a broken-stick rule could be used. The rule is that it identifies the principal components with largest$k$such that$\lambda_j/(\lambda_1+\lambda_2+\cdots +\lambda_p)>a_j$, for all$j\leq k$, where $$\nonumber a_j = \frac{1}{p}\sum_{i=j}^{p}\frac{1}{i},\quad j =1,\cdots, p.$$ Let's try it, Above result coincides with the first two stopping rule. The draw back of simple fair-share and broken-stick rules is that it do not work well when the eigenvalues differ by orders of magnitude. In such case, we then use the relative broken-stick rule, where we analyze$\lambda_j$as the first eigenvalue in the set$\lambda_j\geq \lambda_{j+1}\geq\cdots\geq\lambda_{p}$, where$j < p$. The dimensionality$k$is chosen as the largest value such that$\lambda_j/(\lambda_j+\cdots +\lambda_p)>b_j$, for all$j\leq k$, where $$\nonumber b_j = \frac{1}{p-j+1}\sum_{i=1}^{p-j+1}\frac{1}{i}.$$ Applying this to the data we have, According to the numerical output, the first 34 principal components are enough to represent the variability of the original data. ### Principal Component Scores The principal component scores is the resulting new data set obtained from the linear combinations$Y_j=\mathbf{e}_j(\mathbf{x}-\bar{\mathbf{x}}), j = 1,\cdots, p$. So that if we use the first three stopping rules, then below is the scores (in image) of PC1 and PC2, If we base on the relative broken-stick rule then we return the first 34 PCs, and below is the corresponding scores (in image).  Click on the image to zoom in. ### Residual Analysis Of course when doing PCA there are errors to be considered unless one would return all the PCs, but that would not make any sense because why would someone apply PCA when you still take into account all the dimensions? An overview of the errors in PCA without going through the theory is that, the overall error is simply the excluded variability explained by the$k$th to$p$th principal components,$k>j$. ### Reference #### R: k-Means Clustering on Imaging Enough with the theory we recently published, let's take a break and have fun on the application of Statistics used in Data Mining and Machine Learning, the k-Means Clustering. k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. (Wikipedia, Ref 1.) We will apply this method to an image, wherein we group the pixels into k different clusters. Below is the image that we are going to use,  Colorful Bird From Wall321 We will utilize the following packages for input and output: 1. jpeg - Read and write JPEG images; and, 2. ggplot2 - An implementation of the Grammar of Graphics. ### Download and Read the Image Let's get started by downloading the image to our workspace, and tell R that our data is a JPEG file. ### Cleaning the Data Extract the necessary information from the image and organize this for our computation: The image is represented by large array of pixels with dimension rows by columns by channels -- red, green, and blue or RGB. ### Plotting Plot the original image using the following codes: ### Clustering Apply k-Means clustering on the image: Plot the clustered colours: Possible clusters of pixels on different k-Means: Originalk = 6 Table 1: Different k-Means Clustering. k = 5k = 4 k = 3k = 2 I suggest you try it! ### Reference 1. K-means clustering. Wikipedia. Retrieved September 11, 2014. ## December 16, 2015 ### Alstatr #### R and Python: Theory of Linear Least Squares In my previous article, we talked about implementations of linear regression models in R, Python and SAS. On the theoretical sides, however, I briefly mentioned the estimation procedure for the parameter$\boldsymbol{\beta}$. So to help us understand how software does the estimation procedure, we'll look at the mathematics behind it. We will also perform the estimation manually in R and in Python, that means we're not going to use any special packages, this will help us appreciate the theory. ### Linear Least Squares Consider the linear regression model, $y_i=f_i(\mathbf{x}|\boldsymbol{\beta})+\varepsilon_i,\quad\mathbf{x}_i=\left[ \begin{array}{cccc} 1&x_{11}&\cdots&x_{1p} \end{array}\right],\quad\boldsymbol{\beta}=\left[\begin{array}{c}\beta_0\\\beta_1\\\vdots\\\beta_p\end{array}\right],$ where$y_i$is the response or the dependent variable at the$i$th case,$i=1,\cdots, N$. The$f_i(\mathbf{x}|\boldsymbol{\beta})$is the deterministic part of the model that depends on both the parameters$\boldsymbol{\beta}\in\mathbb{R}^{p+1}$and the predictor variable$\mathbf{x}_i$, which in matrix form, say$\mathbf{X}$, is represented as follows $\mathbf{X}=\left[ \begin{array}{cccccc} 1&x_{11}&\cdots&x_{1p}\\ 1&x_{21}&\cdots&x_{2p}\\ \vdots&\vdots&\ddots&\vdots\\ 1&x_{N1}&\cdots&x_{Np}\\ \end{array} \right].$$\varepsilon_i$is the error term at the$i$th case which we assumed to be Gaussian distributed with mean 0 and variance$\sigma^2$. So that $\mathbb{E}y_i=f_i(\mathbf{x}|\boldsymbol{\beta}),$ i.e.$f_i(\mathbf{x}|\boldsymbol{\beta})$is the expectation function. The uncertainty around the response variable is also modelled by Gaussian distribution. Specifically, if$Y=f(\mathbf{x}|\boldsymbol{\beta})+\varepsilon$and$y\in Y$such that$y>0, then \begin{align*} \mathbb{P}[Y\leq y]&=\mathbb{P}[f(x|\beta)+\varepsilon\leq y]\\ &=\mathbb{P}[\varepsilon\leq y-f(\mathbf{x}|\boldsymbol{\beta})]=\mathbb{P}\left[\frac{\varepsilon}{\sigma}\leq \frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]\\ &=\Phi\left[\frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right], \end{align*} where\Phi$denotes the Gaussian distribution with density denoted by$\phi$below. Hence$Y\sim\mathcal{N}(f(\mathbf{x}|\boldsymbol{\beta}),\sigma^2). That is, \begin{align*} \frac{\operatorname{d}}{\operatorname{d}y}\Phi\left[\frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]&=\phi\left[\frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]\frac{1}{\sigma}=\mathbb{P}[y|f(\mathbf{x}|\boldsymbol{\beta}),\sigma^2]\\ &=\frac{1}{\sqrt{2\pi}\sigma}\exp\left\{-\frac{1}{2}\left[\frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]^2\right\}. \end{align*} If the data are independent and identically distributed, then the log-likelihood function ofyis, \begin{align*} \mathcal{L}[\boldsymbol{\beta}|\mathbf{y},\mathbf{X},\sigma]&=\mathbb{P}[\mathbf{y}|\mathbf{X},\boldsymbol{\beta},\sigma]=\prod_{i=1}^N\frac{1}{\sqrt{2\pi}\sigma}\exp\left\{-\frac{1}{2}\left[\frac{y_i-f_i(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]^2\right\}\\ &=\frac{1}{(2\pi)^{\frac{n}{2}}\sigma^n}\exp\left\{-\frac{1}{2}\sum_{i=1}^N\left[\frac{y_i-f_i(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]^2\right\}\\ \log\mathcal{L}[\boldsymbol{\beta}|\mathbf{y},\mathbf{X},\sigma]&=-\frac{n}{2}\log2\pi-n\log\sigma-\frac{1}{2\sigma^2}\sum_{i=1}^N\left[y_i-f_i(\mathbf{x}|\boldsymbol{\beta})\right]^2. \end{align*} And because the likelihood function tells us about the plausibility of the parameter\boldsymbol{\beta}$in explaining the sample data. We therefore want to find the best estimate of$\boldsymbol{\beta}$that likely generated the sample. Thus our goal is to maximize the likelihood function which is equivalent to maximizing the log-likelihood with respect to$\boldsymbol{\beta}$. And that's simply done by taking the partial derivative with respect to the parameter$\boldsymbol{\beta}$. Therefore, the first two terms in the right hand side of the equation above can be disregarded since it does not depend on$\boldsymbol{\beta}$. Also, the location of the maximum log-likelihood with respect to$\boldsymbol{\beta}$is not affected by arbitrary positive scalar multiplication, so the factor$\frac{1}{2\sigma^2}$can be omitted. And we are left with the following equation, $$\label{eq:1} -\sum_{i=1}^N\left[y_i-f_i(\mathbf{x}|\boldsymbol{\beta})\right]^2.$$ One last thing is that, instead of maximizing the log-likelihood function we can do minimization on the negative log-likelihood. Hence we are interested on minimizing the negative of Equation (\ref{eq:1}) which is $$\label{eq:2} \sum_{i=1}^N\left[y_i-f_i(\mathbf{x}|\boldsymbol{\beta})\right]^2,$$ popularly known as the residual sum of squares (RSS). So RSS is a consequence of maximum log-likelihood under the Gaussian assumption of the uncertainty around the response variable$y$. For models with two parameters, say$\beta_0$and$\beta_1$the RSS can be visualized like the one in my previous article, that is Performing differentiation under$(p+1)$-dimensional parameter$\boldsymbol{\beta}is manageable in the context of linear algebra, so Equation (\ref{eq:2}) is equivalent to \begin{align*} \lVert\mathbf{y}-\mathbf{X}\boldsymbol{\beta}\rVert^2&=\langle\mathbf{y}-\mathbf{X}\boldsymbol{\beta},\mathbf{y}-\mathbf{X}\boldsymbol{\beta}\rangle=\mathbf{y}^{\text{T}}\mathbf{y}-\mathbf{y}^{\text{T}}\mathbf{X}\boldsymbol{\beta}-(\mathbf{X}\boldsymbol{\beta})^{\text{T}}\mathbf{y}+(\mathbf{X}\boldsymbol{\beta})^{\text{T}}\mathbf{X}\boldsymbol{\beta}\\ &=\mathbf{y}^{\text{T}}\mathbf{y}-\mathbf{y}^{\text{T}}\mathbf{X}\boldsymbol{\beta}-\boldsymbol{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}+\boldsymbol{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{X}\boldsymbol{\beta} \end{align*} And the derivative with respect to the parameter is \begin{align*} \frac{\operatorname{\partial}}{\operatorname{\partial}\boldsymbol{\beta}}\lVert\mathbf{y}-\mathbf{X}\boldsymbol{\beta}\rVert^2&=-2\mathbf{X}^{\text{T}}\mathbf{y}+2\mathbf{X}^{\text{T}}\mathbf{X}\boldsymbol{\beta} \end{align*} Taking the critical point by setting the above equation to zero vector, we have \begin{align} \frac{\operatorname{\partial}}{\operatorname{\partial}\boldsymbol{\beta}}\lVert\mathbf{y}-\mathbf{X}\hat{\boldsymbol{\beta}}\rVert^2&\overset{\text{set}}{=}\mathbf{0}\nonumber\\ -\mathbf{X}^{\text{T}}\mathbf{y}+\mathbf{X}^{\text{T}}\mathbf{X}\hat{\boldsymbol{\beta}}&=\mathbf{0}\nonumber\\ \mathbf{X}^{\text{T}}\mathbf{X}\hat{\boldsymbol{\beta}}&=\mathbf{X}^{\text{T}}\mathbf{y}\label{eq:norm} \end{align} Equation (\ref{eq:norm}) is called the normal equation. If\mathbf{X}$is full rank, then we can compute the inverse of$\mathbf{X}^{\text{T}}\mathbf{X}, \begin{align} \mathbf{X}^{\text{T}}\mathbf{X}\hat{\boldsymbol{\beta}}&=\mathbf{X}^{\text{T}}\mathbf{y}\nonumber\\ (\mathbf{X}^{\text{T}}\mathbf{X})^{-1}\mathbf{X}^{\text{T}}\mathbf{X}\hat{\boldsymbol{\beta}}&=(\mathbf{X}^{\text{T}}\mathbf{X})^{-1}\mathbf{X}^{\text{T}}\mathbf{y}\nonumber\\ \hat{\boldsymbol{\beta}}&=(\mathbf{X}^{\text{T}}\mathbf{X})^{-1}\mathbf{X}^{\text{T}}\mathbf{y}.\label{eq:betahat} \end{align} That's it, since both\mathbf{X}$and$\mathbf{y}$are known. ### Prediction If$\mathbf{X}$is full rank and spans the subspace$V\subseteq\mathbb{R}^N$, where$\mathbb{E}\mathbf{y}=\mathbf{X}\boldsymbol{\beta}\in V$. Then the predicted values of$\mathbf{y}$is given by, $$\label{eq:pred} \hat{\mathbf{y}}=\mathbb{E}\mathbf{y}=\mathbf{P}_{V}\mathbf{y}=\mathbf{X}(\mathbf{X}^{\text{T}}\mathbf{X})^{-1}\mathbf{X}^{\text{T}}\mathbf{y},$$ where$\mathbf{P}$is the projection matrix onto the space$V$. For proof of the projection matrix in Equation (\ref{eq:pred}) please refer to reference (1) below. Notice that this is equivalent to $$\label{eq:yhbh} \hat{\mathbf{y}}=\mathbb{E}\mathbf{y}=\mathbf{X}\hat{\boldsymbol{\beta}}.$$ ### Computation Let's fire up R and Python and see how we can apply those equations we derived. For purpose of illustration, we're going to simulate data from Gaussian distributed population. To do so, consider the following codes R ScriptPython ScriptHere we have two predictors x1 and x2, and our response variable y is generated by the parameters$\beta_1=3.5$and$\beta_2=2.8$, and it has Gaussian noise with variance 7. While we set the same random seeds for both R and Python, we should not expect the random values generated in both languages to be identical, instead both values are independent and identically distributed (iid). For visualization, I will use Python Plotly, you can also translate it to R Plotly. Now let's estimate the parameter$\boldsymbol{\beta}$which by default we set to$\beta_1=3.5$and$\beta_2=2.8$. We will use Equation (\ref{eq:betahat}) for estimation. So that we have R ScriptPython ScriptThat's a good estimate, and again just a reminder, the estimate in R and in Python are different because we have different random samples, the important thing is that both are iid. To proceed, we'll do prediction using Equations (\ref{eq:pred}). That is, R ScriptPython ScriptThe first column above is the data y and the second column is the prediction due to Equation (\ref{eq:pred}). Thus if we are to expand the prediction into an expectation plane, then we have You have to rotate the plot by the way to see the plane, I still can't figure out how to change it in Plotly. Anyway, at this point we can proceed computing for other statistics like the variance of the error, and so on. But I will leave it for you to explore. Our aim here is just to give us an understanding on what is happening inside the internals of our software when we try to estimate the parameters of the linear regression models. ### Reference 1. Arnold, Steven F. (1981). The Theory of Linear Models and Multivariate Analysis. Wiley. 2. OLS in Matrix Form ## May 12, 2015 ### Chris Lawrence #### That'll leave a mark Here’s a phrase you never want to see in print (in a legal decision, no less) pertaining to your academic research: “The IRB process, however, was improperly engaged by the Dartmouth researcher and ignored completely by the Stanford researchers.” Whole thing here; it’s a doozy. ## April 14, 2015 ### R you ready? #### Beautiful plots while simulating loss in two-part procrustes problem Today I was working on a two-part procrustes problem and wanted to find out why my minimization algorithm sometimes does not converge properly or renders unexpected results. The loss function to be minimized is $\displaystyle L(\mathbf{Q},c) = \| c \mathbf{A_1Q} - \mathbf{B_1} \|^2 + \| \mathbf{A_2Q} - \mathbf{B_2} \|^2 \rightarrow min$ with $\| \cdot \|$ denoting the Frobenius norm, $c$ is an unknown scalar and $\mathbf{Q}$ an unknown rotation matrix, i.e. $\mathbf{Q}^T\mathbf{Q}=\mathbf{I}$. $\;\mathbf{A_1}, \mathbf{A_2}, \mathbf{B_1}$, and $\mathbf{B_1}$ are four real valued matrices. The minimum for $c$ is easily found by setting the partial derivation of $L(\mathbf{Q},c)$ w.r.t $c$ equal to zero. $\displaystyle c = \frac {tr \; \mathbf{Q}^T \mathbf{A_1}^T \mathbf{B_1}} { \| \mathbf{A_1} \|^2 }$ By plugging $c$ into the loss function $L(\mathbf{Q},c)$ we get a new loss function $L(\mathbf{Q})$ that only depends on $\mathbf{Q}$. This is the starting situation. When trying to find out why the algorithm to minimize $L(\mathbf{Q})$ did not work as expected, I got stuck. So I decided to conduct a small simulation and generate random rotation matrices to study the relation between the parameter $c$ and the value of the loss function $L(\mathbf{Q})$. Before looking at the results for the entire two-part procrustes problem from above, let’s visualize the results for the first part of the loss function only, i.e. $\displaystyle L(\mathbf{Q},c) = \| c \mathbf{A_1Q} - \mathbf{B_1} \|^2 \rightarrow min$ Here, $c$ has the same minimum as for the whole formula above. For the simulation I used $\mathbf{A_1}= \begin{pmatrix} 0.0 & 0.4 & -0.5 \\ -0.4 & -0.8 & -0.5 \\ -0.1 & -0.5 & 0.2 \\ \end{pmatrix} \mkern18mu \qquad \text{and} \qquad \mkern36mu \mathbf{B_1}= \begin{pmatrix} -0.1 & -0.8 & -0.1 \\ 0.3 & 0.2 & -0.9 \\ 0.1 & -0.3 & -0.5 \\ \end{pmatrix}$ as input matrices. Generating many random rotation matrices $\mathbf{Q}$ and plotting $c$ against the value of the loss function yields the following plot. This is a well behaved relation, for each scaling parameter $c$ the loss is identical. Now let’s look at the full two-part loss function. As input matrices I used $\displaystyle A1= \begin{pmatrix} 0.0 & 0.4 & -0.5 \\ -0.4 & -0.8 & -0.5 \\ -0.1 & -0.5 & 0.2 \\ \end{pmatrix} \mkern18mu , \mkern36mu B1= \begin{pmatrix} -0.1 & -0.8 & -0.1 \\ 0.3 & 0.2 & -0.9 \\ 0.1 & -0.3 & -0.5 \\ \end{pmatrix}$ $A2= \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{pmatrix} \mkern18mu , \mkern36mu B2= \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{pmatrix}$ and the following R-code. # trace function tr <- function(X) sum(diag(X)) # random matrix type 1 rmat_1 <- function(n=3, p=3, min=-1, max=1){ matrix(runif(n*p, min, max), ncol=p) } # random matrix type 2, sparse rmat_2 <- function(p=3) { diag(p)[, sample(1:p, p)] } # generate random rotation matrix Q. Based on Q find # optimal scaling factor c and calculate loss function value # one_sample <- function(n=2, p=2) { Q <- mixAK::rRotationMatrix(n=1, dim=p) %*% # random rotation matrix det(Q) = 1 diag(sample(c(-1,1), p, rep=T)) # additional reflections, so det(Q) in {-1,1} s <- tr( t(Q) %*% t(A1) %*% B1 ) / norm(A1, "F")^2 # scaling factor c rss <- norm(s*A1 %*% Q - B1, "F")^2 + # get residual sum of squares norm(A2 %*% Q - B2, "F")^2 c(s=s, rss=rss) } # find c and rss or many random rotation matrices # set.seed(10) # nice case for 3 x 3 n <- 3 p <- 3 A1 <- round(rmat_1(n, p), 1) B1 <- round(rmat_1(n, p), 1) A2 <- rmat_2(p) B2 <- rmat_2(p) x <- plyr::rdply(40000, one_sample(3,3)) plot(x$s, x$rss, pch=16, cex=.4, xlab="c", ylab="L(Q)", col="#00000010")  This time the result turns out to be very different and … beautiful :) Here, we do not have a one to one relation between the scaling parameter and the loss function any more. I do not quite know what to make of this yet. But for now I am happy that it has aestethic value. Below you find some more beautiful graphics with different matrices as inputs. Cheers! ## February 24, 2015 ### Douglas Bates # RCall: Running an embedded R in Julia I have used R (and S before it) for a couple of decades. In the last few years most of my coding has been in Julia, a language for technical computing that can provide remarkable performance for a dynamically typed language via Just-In-Time (JIT) compilation of functions and via multiple dispatch. Nonetheless there are facilities in R that I would like to have access to from Julia. I created the RCall package for Julia to do exactly that. This IJulia notebook provides an introduction to RCall. This is not a novel idea by any means. Julia already has PyCall and JavaCall packages that provide access to Python and to Java. These packages are used extensively and are much more sophisticated than RCall, at present. Many other languages have facilities to run an embedded instance of R. In fact, Python has several such interfaces. The things I plan to do using RCall is to access datasets from R and R packages, to fit models that are not currently implemented in Julia and to use R graphics, especially the ggplot2 and lattice packages. Unfortunately I am not currently able to start a graphics device from the embedded R but I expect that to be fixed soon. I can tell you the most remarkable aspect of RCall although it may not mean much if you haven't tried to do this kind of thing. It is written entirely in Julia. There is absolutely no "glue" code written in a compiled language like C or C++. As I said, this may not mean much to you unless you have tried to do something like this, in which case it is astonishing. ## January 16, 2015 ### Modern Toolmaking #### caretEnsemble My package caretEnsemble, for making ensembles of caret models, is now on CRAN. Check it out, and let me know what you think! (Submit bug reports and feature requests to the issue tracker) ## January 15, 2015 ### Gregor Gorjanc #### cpumemlog: Monitor CPU and RAM usage of a process (and its children) Long time no see ... Today I pushed the cpumemlog script to GitHub https://github.com/gregorgorjanc/cpumemlog. Read more about this useful utility at the GitHub site. ## December 15, 2014 ### R you ready? #### QQ-plots in R vs. SPSS – A look at the differences We teach two software packages, R and SPSS, in Quantitative Methods 101 for psychology freshman at Bremen University (Germany). Sometimes confusion arises, when the software packages produce different results. This may be due to specifics in the implemention of a method or, as in most cases, to different default settings. One of these situations occurs when the QQ-plot is introduced. Below we see two QQ-plots, produced by SPSS and R, respectively. The data used in the plots were generated by: set.seed(0) x <- sample(0:9, 100, rep=T)  SPSS R qqnorm(x, datax=T) # uses Blom's method by default qqline(x, datax=T)  There are some obvious differences: 1. The most obvious one is that the R plot seems to contain more data points than the SPSS plot. Actually, this is not the case. Some data points are plotted on top of each in SPSS while they are spread out vertically in the R plot. The reason for this difference is that SPSS uses a different approach assigning probabilities to the values. We will expore the two approaches below. 2. The scaling of the y-axis differs. R uses quantiles from the standard normal distribution. SPSS by default rescales these values using the mean and standard deviation from the original data. This allows to directly compare the original and theoretical values. This is a simple linear transformation and will not be explained any further here. 3. The QQ-lines are not identical. R uses the 1st and 3rd quartile from both distributions to draw the line. This is different in SPSS where of a line is drawn for identical values on both axes. We will expore the differences below. # QQ-plots from scratch To get a better understanding of the difference we will build the R and SPSS-flavored QQ-plot from scratch. ## R type In order to calculate theoretical quantiles corresponding to the observed values, we first need to find a way to assign a probability to each value of the original data. A lot of different approaches exist for this purpose (for an overview see e.g. Castillo-Gutiérrez, Lozano-Aguilera, & Estudillo-Martínez, 2012b). They usually build on the ranks of the observed data points to calculate corresponding p-values, i.e. the plotting positions for each point. The qqnorm function uses two formulae for this purpose, depending on the number of observations $n$ (Blom’s mfethod, see ?qqnorm; Blom, 1958). With $r$ being the rank, for $n > 10$ it will use the formula $p = (r - 1/2) / n$, for $n \leq 10$ the formula $p = (r - 3/8) / (n + 1/4)$ to determine the probability value $p$ for each observation (see the help files for the functions qqnorm and ppoint). For simplicity reasons, we will only implement the $n > 10$ case here. n <- length(x) # number of observations r <- order(order(x)) # order of values, i.e. ranks without averaged ties p <- (r - 1/2) / n # assign to ranks using Blom's method y <- qnorm(p) # theoretical standard normal quantiles for p values plot(x, y) # plot empirical against theoretical values  Before we take at look at the code, note that our plot is identical to the plot generated by qqnorm above, except that the QQ-line is missing. The main point that makes the difference between R and SPSS is found in the command order(order(x)). The command calculates ranks for the observations using ordinal ranking. This means that all observations get different ranks and no average ranks are calculated for ties, i.e. for observations with equal values. Another approach would be to apply fractional ranking and calculate average values for ties. This is what the function rank does. The following codes shows the difference between the two approaches to assign ranks. v <- c(1,1,2,3,3) order(order(v)) # ordinal ranking used by R  ## [1] 1 2 3 4 5  rank(v) # fractional ranking used by SPSS  ## [1] 1.5 1.5 3.0 4.5 4.5  R uses ordinal ranking and SPSS uses fractional ranking by default to assign ranks to values. Thus, the positions do not overlap in R as each ordered observation is assigned a different rank and therefore a different p-value. We will pick up the second approach again later, when we reproduce the SPSS-flavored plot in R.1 The second difference between the plots concerned the scaling of the y-axis and was already clarified above. The last point to understand is how the QQ-line is drawn in R. Looking at the probs argument of qqline reveals that it uses the 1st and 3rd quartile of the original data and theoretical distribution to determine the reference points for the line. We will draw the line between the quartiles in red and overlay it with the line produced by qqline to see if our code is correct. plot(x, y) # plot empirical against theoretical values ps <- c(.25, .75) # reference probabilities a <- quantile(x, ps) # empirical quantiles b <- qnorm(ps) # theoretical quantiles lines(a, b, lwd=4, col="red") # our QQ line in red qqline(x, datax=T) # R QQ line  The reason for different lines in R and SPSS is that several approaches to fitting a straight line exist (for an overview see e.g. Castillo-Gutiérrez, Lozano-Aguilera, & Estudillo-Martínez, 2012a). Each approach has different advantages. The method used by R is more robust when we expect values to diverge from normality in the tails, and we are primarily interested in the normality of the middle range of our data. In other words, the method of fitting an adequate QQ-line depends on the purpose of the plot. An explanation of the rationale of the R approach can e.g. be found here. ## SPSS type The default SPSS approach also uses Blom’s method to assign probabilities to ranks (you may choose other methods is SPSS) and differs from the one above in the following aspects: • a) As already mentioned, SPSS uses ranks with averaged ties (fractional rankings) not the plain order ranks (ordinal ranking) as in R to derive the corresponding probabilities for each data point. The rest of the code is identical to the one above, though I am not sure if SPSS distinguishes between the $n 10$ case. • b) The theoretical quantiles are scaled to match the estimated mean and standard deviation of the original data. • c) The QQ-line goes through all quantiles with identical values on the x and y axis. n <- length(x) # number of observations r <- rank(x) # a) ranks using fractional ranking (averaging ties) p <- (r - 1/2) / n # assign to ranks using Blom's method y <- qnorm(p) # theoretical standard normal quantiles for p values y <- y * sd(x) + mean(x) # b) transform SND quantiles to mean and sd from original data plot(x, y) # plot empirical against theoretical values  Lastly, let us add the line. As the scaling of both axes is the same, the line goes through the origin with a slope of $1$. abline(0,1) # c) slope 0 through origin  The comparison to the SPSS output shows that they are (visually) identical. # Function for SPSS-type QQ-plot The whole point of this demonstration was to pinpoint and explain the differences between a QQ-plot generated in R and SPSS, so it will no longer be a reason for confusion. Note, however, that SPSS offers a whole range of options to generate the plot. For example, you can select the method to assign probabilities to ranks and decide how to treat ties. The plots above used the default setting (Blom’s method and averaging across ties). Personally I like the SPSS version. That is why I implemented the function qqnorm_spss in the ryouready package, that accompanies the course. The formulae for the different methods to assign probabilities to ranks can be found in Castillo-Gutiérrez et al. (2012b). The implentation is a preliminary version that has not yet been thoroughly tested. You can find the code here. Please report any bugs or suggestions for improvements (which are very welcome) in the github issues section. library(devtools) install_github("markheckmann/ryouready") # install from github repo library(ryouready) # load package library(ggplot2) qq <- qqnorm_spss(x, method=1, ties.method="average") # Blom's method with averaged ties plot(qq) # generate QQ-plot ggplot(qq) # use ggplot2 to generate QQ-plot  # Literature 1. Technical sidenote: Internally, qqnorm uses the function ppoints to generate the p-values. Type in stats:::qqnorm.default to the console to have a look at the code. ## October 20, 2014 ### Modern Toolmaking #### For faster R on a mac, use veclib ## Update: The links to all my github gists on blogger are broken, and I can't figure out how to fix them. If you know how to insert gitub gists on a dynamic blogger template, please let me known. In the meantime, here are instructions with links to the code: First of all, use homebrew to compile openblas. It's easy! Second of all, you can also use homebrew to install R! (But maybe stick with the CRAN version unless you really want to compile your own R binary) To use openblas with R, follow these instructions: https://gist.github.com/zachmayer/e591cf868b3a381a01d6#file-openblas-sh To use veclib with R, follow these intructions: https://gist.github.com/zachmayer/e591cf868b3a381a01d6#file-veclib-sh ## OLD POST: Inspired by this post, I decided to try using OpenBLAS for R on my mac. However, it turns out there's a simpler option, using the vecLib BLAS library, which is provided by Apple as part of the accelerate framework. If you are using R 2.15, follow these instructions to change your BLAS from the default to vecLib: However, as noted in r-sig-mac, these instructions do not work for R 3.0. You have to directly link to the accelerate framework's version of vecLib: Finally, test your new blas using this script: On my system (a retina macbook pro), the default BLAS takes 141 seconds and vecLib takes 43 seconds, which is a significant speedup. If you plan to use vecLib, note the following warning from the R development team "Although fast, it is not under our control and may possibly deliver inaccurate results." So far, I have not encountered any issues using vecLib, but it's only been a few hours :-). UPDATE: you can also install OpenBLAS on a mac: If you do this, make sure to change the directories to point to the correct location on your system (e.g. change /users/zach/source to whatever directory you clone the git repo into). On my system, the benchmark script takes ~41 seconds when using openBLAS, which is a small but significant speedup. ## September 19, 2014 ### Chris Lawrence #### What could a federal UK look like? Assuming that the “no” vote prevails in the Scottish independence referendum, the next question for the United Kingdom is to consider constitutional reform to implement a quasi-federal system and resolve the West Lothian question once and for all. In some ways, it may also provide an opportunity to resolve the stalled reform of the upper house as well. Here’s the rough outline of a proposal that might work. • Devolve identical powers to England, Northern Ireland, Scotland, and Wales, with the proviso that local self-rule can be suspended if necessary by the federal legislature (by a supermajority). • The existing House of Commons becomes the House of Commons for England, which (along with the Sovereign) shall comprise the English Parliament. This parliament would function much as the existing devolved legislatures in Scotland and Wales; the consociational structure of the Northern Ireland Assembly (requiring double majorities) would not be replicated. • The House of Lords is abolished, and replaced with a directly-elected Senate of the United Kingdom. The Senate will have authority to legislate on the non-devolved powers (in American parlance, “delegated” powers) such as foreign and European Union affairs, trade and commerce, national defense, and on matters involving Crown dependencies and territories, the authority to legislate on devolved matters in the event self-government is suspended in a constituent country, and dilatory powers including a qualified veto (requiring a supermajority) over the legislation proposed by a constituent country’s parliament. The latter power would effectively replace the review powers of the existing House of Lords; it would function much as the Council of Revision in Madison’s original plan for the U.S. Constitution. As the Senate will have relatively limited powers, it need not be as large as the existing Lords or Commons. To ensure the countries other than England have a meaningful voice, given that nearly 85% of the UK’s population is in England, two-thirds of the seats would be allocated proportionally based on population and one-third allocated equally to the four constituent countries. This would still result in a chamber with a large English majority (around 64.4%) but nonetheless would ensure the other three countries would have meaningful representation as well. ## September 12, 2014 ### R you ready? #### Using colorized PNG pictograms in R base plots Today I stumbled across a figure in an explanation on multiple factor analysis which contained pictograms. Figure 1 from Abdi & Valentin (2007), p. 8. I wanted to reproduce a similar figure in R using pictograms and additionally color them e.g. by group membership . I have almost no knowledge about image processing, so I tried out several methods of how to achieve what I want. The first thing I did was read in an PNG file and look at the data structure. The package png allows to read in PNG files. Note that all of the below may not work on Windows machines, as it does not support semi-transparency (see ?readPNG). library(png) img <- readPNG(system.file("img", "Rlogo.png", package="png")) class(img)  ## [1] "array"  dim(img)  ## [1] 76 100 4  The object is a numerical array with four layers (red, green, blue, alpha; short RGBA). Let’s have a look at the first layer (red) and replace all non-zero entries by a one and the zeros by a dot. This will show us the pattern of non-zero values and we already see the contours. l4 <- img[,,1] l4[l4 > 0] <- 1 l4[l4 == 0] <- "." d <- apply(l4, 1, function(x) { cat(paste0(x, collapse=""), "\n") })  To display the image in R one way is to raster the image (i.e. the RGBA layers are collapsed into a layer of single HEX value) and print it using rasterImage. rimg <- as.raster(img) # raster multilayer object r <- nrow(rimg) / ncol(rimg) # image ratio plot(c(0,1), c(0,r), type = "n", xlab = "", ylab = "", asp=1) rasterImage(rimg, 0, 0, 1, r)  Let’s have a look at a small part the rastered image object. It is a matrix of HEX values. rimg[40:50, 1:6]  ## [1,] "#C4C5C202" "#858981E8" "#838881FF" "#888D86FF" "#8D918AFF" "#8F938CFF" ## [2,] "#00000000" "#848881A0" "#80847CFF" "#858A83FF" "#898E87FF" "#8D918BFF" ## [3,] "#00000000" "#8B8E884C" "#7D817AFF" "#82867EFF" "#868B84FF" "#8A8E88FF" ## [4,] "#00000000" "#9FA29D04" "#7E827BE6" "#7E817AFF" "#838780FF" "#878C85FF" ## [5,] "#00000000" "#00000000" "#81857D7C" "#797E75FF" "#7F827BFF" "#838781FF" ## [6,] "#00000000" "#00000000" "#898C8510" "#787D75EE" "#797E76FF" "#7F837BFF" ## [7,] "#00000000" "#00000000" "#00000000" "#7F837C7B" "#747971FF" "#797E76FF" ## [8,] "#00000000" "#00000000" "#00000000" "#999C9608" "#767C73DB" "#747971FF" ## [9,] "#00000000" "#00000000" "#00000000" "#00000000" "#80847D40" "#71766EFD" ## [10,] "#00000000" "#00000000" "#00000000" "#00000000" "#00000000" "#787D7589" ## [11,] "#00000000" "#00000000" "#00000000" "#00000000" "#00000000" "#999C9604"  And print this small part. plot(c(0,1), c(0,.6), type = "n", xlab = "", ylab = "", asp=1) rasterImage(rimg[40:50, 1:6], 0, 0, 1, .6)  Now we have an idea of how the image object and the rastered object look like from the inside. Let’s start to modify the images to suit our needs. In order to change the color of the pictograms, my first idea was to convert the graphics to greyscale and remap the values to a color ramp of may choice. To convert to greyscale there are tons of methods around (see e.g. here). I just pick one of them I found on SO by chance. With R=Red, G=Green and B=Blue we have brightness = sqrt(0.299 * R^2 + 0.587 * G^2 + 0.114 * B^2)  This approach modifies the PNG files after they have been coerced into a raster object. # function to calculate brightness values brightness <- function(hex) { v <- col2rgb(hex) sqrt(0.299 * v[1]^2 + 0.587 * v[2]^2 + 0.114 * v[3]^2) /255 } # given a color ramp, map brightness to ramp also taking into account # the alpha level. The defaul color ramp is grey # img_to_colorramp <- function(img, ramp=grey) { cv <- as.vector(img) b <- sapply(cv, brightness) g <- ramp(b) a <- substr(cv, 8,9) # get alpha values ga <- paste0(g, a) # add alpha values to new colors img.grey <- matrix(ga, nrow(img), ncol(img), byrow=TRUE) } # read png and modify img <- readPNG(system.file("img", "Rlogo.png", package="png")) img <- as.raster(img) # raster multilayer object r <- nrow(img) / ncol(img) # image ratio s <- 3.5 # size plot(c(0,10), c(0,3.5), type = "n", xlab = "", ylab = "", asp=1) rasterImage(img, 0, 0, 0+s/r, 0+s) # original img2 <- img_to_colorramp(img) # modify using grey scale rasterImage(img2, 5, 0, 5+s/r, 0+s)  Great, it works! Now Let’s go and try out some other color palettes using colorRamp to create a color ramp. plot(c(0,10),c(0,8.5), type = "n", xlab = "", ylab = "", asp=1) img1 <- img_to_colorramp(img) rasterImage(img1, 0, 5, 0+s/r, 5+s) reds <- function(x) rgb(colorRamp(c("darkred", "white"))(x), maxColorValue = 255) img2 <- img_to_colorramp(img, reds) rasterImage(img2, 5, 5, 5+s/r, 5+s) greens <- function(x) rgb(colorRamp(c("darkgreen", "white"))(x), maxColorValue = 255) img3 <- img_to_colorramp(img, greens) rasterImage(img3, 0, 0, 0+s/r, 0+s) single_color <- function(...) "#0000BB" img4 <- img_to_colorramp(img, single_color) rasterImage(img4, 5, 0, 5+s/r, 0+s)  Okay, that basically does the job. Now we will apply it to the wine pictograms. Let’s use this wine glass from Wikimedia Commons. It’s quite big so I uploaded a reduced size version to imgur . We will use it for our purposes. # load file from web f <- tempfile() download.file("http://i.imgur.com/A14ntCt.png", f) img <- readPNG(f) img <- as.raster(img) r <- nrow(img) / ncol(img) s <- 1 # let's create a function that returns a ramp function to save typing ramp <- function(colors) function(x) rgb(colorRamp(colors)(x), maxColorValue = 255) # create dataframe with coordinates and colors set.seed(1) x <- data.frame(x=rnorm(16, c(2,2,4,4)), y=rnorm(16, c(1,3)), colors=c("black", "darkred", "garkgreen", "darkblue")) plot(c(1,6), c(0,5), type="n", xlab="", ylab="", asp=1) for (i in 1L:nrow(x)) { colorramp <- ramp(c(x[i,3], "white")) img2 <- img_to_colorramp(img, colorramp) rasterImage(img2, x[i,1], x[i,2], x[i,1]+s/r, x[i,2]+s) }  Another approach would be to modifying the RGB layers before rastering to HEX values. img <- readPNG(system.file("img", "Rlogo.png", package="png")) img2 <- img img[,,1] <- 0 # remove Red component img[,,2] <- 0 # remove Green component img[,,3] <- 1 # Set Blue to max img <- as.raster(img) r <- nrow(img) / ncol(img) # size ratio s <- 3.5 # size plot(c(0,10), c(0,3.5), type = "n", xlab = "", ylab = "", asp=1) rasterImage(img, 0, 0, 0+s/r, 0+s) img2[,,1] <- 1 # Red to max img2[,,2] <- 0 img2[,,3] <- 0 rasterImage(as.raster(img2), 5, 0, 5+s/r, 0+s)  To just colorize the image, we could weight each layer. # wrap weighting into function weight_layers <- function(img, w) { for (i in seq_along(w)) img[,,i] <- img[,,i] * w[i] img } plot(c(0,10), c(0,3.5), type = "n", xlab = "", ylab = "", asp=1) img <- readPNG(system.file("img", "Rlogo.png", package="png")) img2 <- weight_layers(img, c(.2, 1,.2)) rasterImage(img2, 0, 0, 0+s/r, 0+s) img3 <- weight_layers(img, c(1,0,0)) rasterImage(img3, 5, 0, 5+s/r, 0+s)  After playing around and hard-coding the modifications I started to search and found the EBimage package which has a lot of features for image processing that make ones life (in this case only a bit) easier. library(EBImage) f <- system.file("img", "Rlogo.png", package="png") img <- readImage(f) img2 <- img img[,,2] = 0 # zero out green layer img[,,3] = 0 # zero out blue layer img <- as.raster(img) img2[,,1] = 0 img2[,,3] = 0 img2 <- as.raster(img2) r <- nrow(img) / ncol(img) s <- 3.5 plot(c(0,10), c(0,3.5), type = "n", xlab = "", ylab = "", asp=1) rasterImage(img, 0, 0, 0+s/r, 0+s) rasterImage(img2, 5, 0, 5+s/r, 0+s)  EBImage is a good choice and fairly easy to handle. Now let’s again print the pictograms. f <- tempfile(fileext=".png") download.file("http://i.imgur.com/A14ntCt.png", f) img <- readImage(f) # will replace whole image layers by one value # only makes sense if there is a alpha layer that # gives the contours # mod_color <- function(img, col) { v <- col2rgb(col) / 255 img = channel(img, 'rgb') img[,,1] = v[1] # Red img[,,2] = v[2] # Green img[,,3] = v[3] # Blue as.raster(img) } r <- nrow(img) / ncol(img) # get image ratio s <- 1 # size # create random data set.seed(1) x <- data.frame(x=rnorm(16, c(2,2,4,4)), y=rnorm(16, c(1,3)), colors=1:4) # plot pictograms plot(c(1,6), c(0,5), type="n", xlab="", ylab="", asp=1) for (i in 1L:nrow(x)) { img2 <- mod_color(img, x[i, 3]) rasterImage(img2, x[i,1], x[i,2], x[i,1]+s*r, x[i,2]+s) }  Note, that above I did not bother to center each pictogram to position it correctly. This still needs to be done. Anyway, that’s it! Mission completed. ### Literature Abdi, H., & Valentin, D. (2007). Multiple factor analysis (MFA). In N. Salkind (Ed.), Encyclopedia of Measurement and Statistics (pp. 1–14). Thousand Oaks, CA: Sage Publications. Retrieved from https://www.utdallas.edu/~herve/Abdi-MFA2007-pretty.pdf ## June 18, 2014 ### Chris Lawrence #### Soccer queries answered Kevin Drum asks a bunch of questions about soccer: 1. Outside the penalty area there’s a hemisphere about 20 yards wide. I can’t recall ever seeing it used for anything. What’s it for? 2. On several occasions, I’ve noticed that if the ball goes out of bounds at the end of stoppage time, the referee doesn’t whistle the match over. Instead, he waits for the throw-in, and then immediately whistles the match over. What’s the point of this? 3. Speaking of stoppage time, how has it managed to last through the years? I know, I know: tradition. But seriously. Having a timekeeper who stops the clock for goals, free kicks, etc. has lots of upside and no downside. Right? It wouldn’t change the game in any way, it would just make timekeeping more accurate, more consistent, and more transparent for the fans and players. Why keep up the current pretense? 4. What’s the best way to get a better sense of what’s a foul and what’s a legal tackle? Obviously you can’t tell from the players’ reactions, since they all writhe around like landed fish if they so much as trip over their own shoelaces. Reading the rules provides the basics, but doesn’t really help a newbie very much. Maybe a video that shows a lot of different tackles and explains why each one is legal, not legal, bookable, etc.? The first one’s easy: there’s a general rule that no defensive player can be within 10 yards of the spot of a direct free kick. A penalty kick (which is a type of direct free kick) takes place in the 18-yard box, and no players other than the player taking the kick and the goalkeeper are allowed in the box. However, owing to geometry, the 18 yard box and the 10 yard exclusion zone don’t fully coincide, hence the penalty arc. (That’s also why there are two tiny hash-marks on the goal line and side line 10 yards from the corner flag. And why now referees have a can of shaving cream to mark the 10 yards for other free kicks, one of the few MLS innovations that has been a good idea.) Second one’s also easy: the half and the game cannot end while the ball is out of play. Third one’s harder. First, keeping time inexactly forestalls the silly premature celebrations that are common in most US sports. You’d never see the Stanford-Cal play happen in a soccer game. Second, it allows some slippage for short delays and doesn’t require exact timekeeping; granted, this was more valuable before instant replays and fourth officials, but most US sports require a lot of administrative record-keeping by ancillary officials. A soccer game can be played with one official (and often is, particularly at the amateur level) without having to change timing rules;* in developing countries in particular this lowers the barriers to entry for the sport (along with the low equipment requirements) without changing the nature of the game appreciably. Perhaps most importantly, if the clock was allowed to stop regularly it would create an excuse for commercial timeouts and advertising breaks, which would interrupt the flow of the game and potentially reduce the advantages of better-conditioned and more skilled athletes. (MLS tried this, along with other exciting American ideas like “no tied games,” and it was as appealing to actual soccer fans as ketchup on filet mignon would be to a foodie, and perhaps more importantly didn’t make any non-soccer fans watch.) Fourth, the key distinction is usually whether there was an obvious attempt to play the ball; in addition, in the modern game, even some attempts to play the ball are considered inherently dangerous (tackling from behind, many sliding tackles, etc.) and therefore are fouls even if they are successful in getting more ball than human. * To call offside, you’d also probably need what in my day we called a “linesman.” ## May 07, 2014 ### Chris Lawrence #### The mission and vision thing Probably the worst-kept non-secret is that the next stage of the institutional evolution of my current employer is to some ill-defined concept of “university status,” which mostly involves the establishment of some to-be-determined master’s degree programs. In the context of the University System of Georgia, it means a small jump from the “state college” prestige tier (a motley collection of schools that largely started out as two-year community colleges and transfer institutions) to the “state university” tier (which is where most of the ex-normal schools hang out these days). What is yet to be determined is how that transition will affect the broader institution that will be the University of Middle Georgia.* People on high are said to be working on these things; in any event, here are my assorted random thoughts on what might be reasonable things to pursue: • Marketing and positioning: Unlike the situation facing many of the other USG institutions, the population of the two anchor counties of our core service area (Bibb and Houston) is growing, and Houston County in particular has a statewide reputation for the quality of its public school system. Rather than conceding that the most prepared students from these schools will go to Athens or Atlanta or Valdosta, we should strongly market our institutional advantages over these more “prestigious” institutions, particularly in terms of the student experience in the first two years and the core curriculum: we have no large lecture courses, no teaching assistants, no lengthy bus rides to and from class every day, and the vast majority of the core is taught by full-time faculty with terminal degrees. Not to mention costs to students are much lower, particularly in the case of students who do not qualify for need-based aid. Even if we were to “lose” these students as transfers to the top-tier institutions after 1–4 semesters, we’d still benefit from the tuition and fees they bring in and we would not be penalized in the upcoming state performance funding formula. Dual enrollment in Warner Robins in particular is an opportunity to showcase our institution as a real alternative for better prepared students rather than a safety school. • Comprehensive offerings at the bachelor’s level: As a state university, we will need to offer a comprehensive range of options for bachelor’s students to attract and retain students, both traditional and nontraditional. In particular, B.S. degrees in political science and sociology with emphasis in applied empirical skills would meet public and private employer demand for workers who have research skills and the ability to collect, manage, understand, and use data appropriately. There are other gaps in the liberal arts and sciences as well that need to be addressed to become a truly comprehensive state university. • Create incentives to boost the residential population: The college currently has a heavy debt burden inherited from the overbuilding of dorms at the Cochran campus. We need to identify ways to encourage students to live in Cochran, which may require public-private partnerships to try to build a “college town” atmosphere in the community near campus. We also need to work with wireless providers like Sprint and T-Mobile to ensure that students from the “big city” can fully use their cell phones and tablets in Cochran and Eastman without roaming fees or changing wireless providers. • Tie the institution more closely to the communities we serve: This includes both physical ties and psychological ties. The Macon campus in particular has poor physical links to the city itself for students who might walk or ride bicycles; extending the existing bike/walking trail from Wesleyan to the Macon campus should be a priority, as should pedestrian access and bike facilities along Columbus Road. Access to the Warner Robins campus is somewhat better but still could be improved. More generally, the institution is perceived as an afterthought or alternative of last resort in the community. Improving this situation and perception among community leaders and political figures may require a physical presence in or near downtown Macon, perhaps in partnership with the GCSU Graduate Center. * There is no official name-in-waiting, but given that our former interim president seemed to believe he could will this name into existence by repeating it enough I’ll stick with it. The straw poll of faculty trivia night suggests that it’s the least bad option available, which inevitably means the regents will choose something else instead (if the last name change is anything to go by). ## February 17, 2014 ### Seth Falcon #### Have Your SHA and Bcrypt Too ## Fear I've been putting off sharing this idea because I've heard the rumors about what happens to folks who aren't security experts when they post about security on the internet. If this blog is replaced with cat photos and rainbows, you'll know what happened. ## The Sad Truth It's 2014 and chances are you have accounts on websites that are not properly handling user passwords. I did no research to produce the following list of ways passwords are mishandled in decreasing order of frequency: 1. Site uses a fast hashing algorithm, typically SHA1(salt + plain-password). 2. Site doesn't salt password hashes 3. Site stores raw passwords We know that sites should be generating secure random salts and using an established slow hashing algorithm (bcrypt, scrypt, or PBKDF2). Why are sites not doing this? While security issues deserve a top spot on any site's priority list, new features often trump addressing legacy security concerns. The immediacy of the risk is hard to quantify and it's easy to fall prey to a "nothing bad has happened yet, why should we change now" attitude. It's easy for other bugs, features, or performance issues to win out when measured by immediate impact. Fixing security or other "legacy" issues is the Right Thing To Do and often you will see no measurable benefit from the investment. It's like having insurance. You don't need it until you do. Specific to the improper storage of user password data is the issue of the impact to a site imposed by upgrading. There are two common approaches to upgrading password storage. You can switch cold turkey to the improved algorithms and force password resets on all of your users. Alternatively, you can migrate incrementally such that new users and any user who changes their password gets the increased security. The cold turkey approach is not a great user experience and sites might choose to delay an upgrade to avoid admitting to a weak security implementation and disrupting their site by forcing password resets. The incremental approach is more appealing, but the security benefit is drastically diminished for any site with a substantial set of existing users. Given the above migration choices, perhaps it's (slightly) less surprising that businesses choose to prioritize other work ahead of fixing poorly stored user password data. ## The Idea What if you could upgrade a site so that both new and existing users immediately benefited from the increased security, but without the disruption of password resets? It turns out that you can and it isn't very hard. Consider a user table with columns: userid salt hashed_pass  Where the hashed_pass column is computed using a weak fast algorithm, for example SHA1(salt + plain_pass). The core of the idea is to apply a proper algorithm on top of the data we already have. I'll use bcrypt to make the discussion concrete. Add columns to the user table as follows: userid salt hashed_pass hash_type salt2  Process the existing user table by computing bcrypt(salt2 + hashed_pass) and storing the result in the hashed_pass column (overwriting the less secure value); save the new salt value to salt2 and set hash_type to bycrpt+sha1. To verify a user where hash_type is bcrypt+sha1, compute bcrypt(salt2 + SHA1(salt + plain_pass)) and compare to the hashed_pass value. Note that bcrypt implementations encode the salt as a prefix of the hashed value so you could avoid the salt2 column, but it makes the idea easier to explain to have it there. You can take this approach further and have any user that logs in (as well as new users) upgrade to a "clean" bcrypt only algorithm since you can now support different verification algorithms using hash_type. With the proper application code changes in place, the upgrade can be done live. This scheme will also work for sites storing non-salted password hashes as well as those storing plain text passwords (THE HORROR). ## Less Sadness, Maybe Perhaps this approach makes implementing a password storage security upgrade more palatable and more likely to be prioritized. And if there's a horrible flaw in this approach, maybe you'll let me know without turning this blog into a tangle of cat photos and rainbows. ## December 26, 2013 ### Seth Falcon #### A Rebar Plugin for Locking Deps: Reproducible Erlang Project Builds For Fun and Profit ## What's this lock-deps of which you speak? If you use rebar to generate an OTP release project and want to have reproducible builds, you need the rebar_lock_deps_plugin plugin. The plugin provides a lock-deps command that will generate a rebar.config.lock file containing the complete flattened set of project dependencies each pegged to a git SHA. The lock file acts similarly to Bundler's Gemfile.lock file and allows for reproducible builds (*). Without lock-deps you might rely on the discipline of using a tag for all of your application's deps. This is insufficient if any dep depends on something not specified as a tag. It can also be a problem if a third party dep doesn't provide a tag. Generating a rebar.config.lock file solves these issues. Moreover, using lock-deps can simplify the work of putting together a release consisting of many of your own repos. If you treat the master branch as shippable, then rather than tagging each subproject and updating rebar.config throughout your project's dependency chain, you can run get-deps (without the lock file), compile, and re-lock at the latest versions throughout your project repositories. The reproducibility of builds when using lock-deps depends on the SHAs captured in rebar.config.lock. The plugin works by scanning the cloned repos in your project's deps directory and extracting the current commit SHA. This works great until a repository's history is rewritten with a force push. If you really want reproducible builds, you need to not nuke your SHAs and you'll need to fork all third party repos to ensure that someone else doesn't screw you over in this fashion either. If you make a habit of only depending on third party repos using a tag, assume that upstream maintainers are not completely bat shit crazy, and don't force push your master branch, then you'll probably be fine. ## Getting Started Install the plugin in your project by adding the following to your rebar.config file: %% Plugin dependency {deps, [ {rebar_lock_deps_plugin, ".*", {git, "git://github.com/seth/rebar_lock_deps_plugin.git", {branch, "master"}}} ]}. %% Plugin usage {plugins, [rebar_lock_deps_plugin]}.  To test it out do: rebar get-deps # the plugin has to be compiled so you can use it rebar compile rebar lock-deps  If you'd like to take a look at a project that uses the plugin, take a look at CHEF's erchef project. ## Bonus features If you are building an OTP release project using rebar generate then you can use rebar_lock_deps_plugin to enhance your build experience in three easy steps. 1. Use rebar bump-rel-version version=$BUMP to automate the process of editing rel/reltool.config to update the release version. The argument $BUMP can be major, minor, or patch (default) to increment the specified part of a semver X.Y.Z version. If $BUMP is any other value, it is used as the new version verbatim. Note that this function rewrites rel/reltool.config using ~p. I check-in the reformatted version and maintain the formatting when editing. This way, the general case of a version bump via bump-rel-version results in a minimal diff.

2. Autogenerate a change summary commit message for all project deps. Assuming you've generated a new lock file and bumped the release version, use rebar commit-release to commit the changes to rebar.config.lock and rel/reltool.config with a commit message that summarizes the changes made to each dependency between the previously locked version and the newly locked version. You can get a preview of the commit message via rebar log-changed-deps.

3. Finally, create an annotated tag for your new release with rebar tag-release which will read the current version from rel/reltool.config and create an annotated tag named with the version.

## The dependencies, they are ordered

Up to version 2.0.1 of rebar_lock_deps_plugin, the dependencies in the generated lock file were ordered alphabetically. This was a side-effect of using filelib:wildcard/1 to list the dependencies in the top-level deps directory. In most cases, the order of the full dependency set does not matter. However, if some of the code in your project uses parse transforms, then it will be important for the parse transform to be compiled and on the code path before attempting to compile code that uses the parse transform.

This issue was recently discovered by a colleague who ran into build issues using the lock file for a project that had recently integrated lager for logging. He came up with the idea of maintaining the order of deps as they appear in the various rebar.config files along with a prototype patch proving out the idea. As of rebar_lock_deps_plugin 3.0.0, the lock-deps command will (mostly) maintain the relative order of dependencies as found in the rebar.config files.

The "mostly" is that when a dep is shared across two subprojects, it will appear in the expected order for the first subproject (based on the ordering of the two subprojects). The deps for the second subproject will not be in strict rebar.config order, but the resulting order should address any compile-time dependencies and be relatively stable (only changing when project deps alter their deps with larger impact when shared deps are introduced or removed).

## Digression: fun with dependencies

There are times, as a programmer, when a real-world problem looks like a text book exercise (or an interview whiteboard question). Just the other day at work we had to design some manhole covers, but I digress.

Fixing the order of the dependencies in the generated lock file is (nearly) the same as finding an install order for a set of projects with inter-dependencies. I had some fun coding up the text book solution even though the approach doesn't handle the constraint of respecting the order provided by the rebar.config files. Onward with the digression.

We have a set of "packages" where some packages depend on others and we want to determine an install order such that a package's dependencies are always installed before the package. The set of packages and the relation "depends on" form a directed acyclic graph or DAG. The topological sort of a DAG produces an install order for such a graph. The ordering is not unique. For example, with a single package C depending on A and B, valid install orders are [A, B, C] and [B, A, C].

To setup the problem, we load all of the project dependency information into a proplist mapping each package to a list of its dependencies extracted from the package's rebar.config file.

read_all_deps(Config, Dir) ->
TopDeps = rebar_config:get(Config, deps, []),
Acc = [{top, dep_names(TopDeps)}],
DepDirs = filelib:wildcard(filename:join(Dir, "*")),
Acc ++ [
{filename:basename(D), dep_names(extract_deps(D))}
|| D <- DepDirs ].


Erlang's standard library provides the digraph and digraph_utils modules for constructing and operating on directed graphs. The digraph_utils module includes a topsort/1 function which we can make use of for our "exercise". The docs say:

Returns a topological ordering of the vertices of the digraph Digraph if such an ordering exists, false otherwise. For each vertex in the returned list, there are no out-neighbours that occur earlier in the list.

To figure out which way to point the edges when building our graph, consider two packages A and B with A depending on B. We know we want to end up with an install order of [B, A]. Rereading the topsort/1 docs, we must want an edge B => A. With that, we can build our DAG and obtain an install order with the topological sort:

load_digraph(Config, Dir) ->
G = digraph:new(),
Nodes = all_nodes(AllDeps),
[ digraph:add_vertex(G, N) || N <- Nodes ],
%% If A depends on B, then we add an edge A <= B
[
|| Dep <- DepList ]
|| {Item, DepList} <- AllDeps, Item =/= top ],
digraph_utils:topsort(G).

%% extract a sorted unique list of all deps
all_nodes(AllDeps) ->
lists:usort(lists:foldl(fun({top, L}, Acc) ->
L ++ Acc;
({K, L}, Acc) ->
[K|L] ++ Acc
end, [], AllDeps)).


The digraph module manages graphs using ETS giving it a convenient API, though one that feels un-erlang-y in its reliance on side-effects.

The above gives an install order, but doesn't take into account the relative order of deps as specified in the rebar.config files. The solution implemented in the plugin is a bit less fancy, recursing over the deps and maintaining the desired ordering. The only tricky bit being that shared deps are ignored until the end and the entire linearized list is de-duped which required a . Here's the code:

order_deps(AllDeps) ->
Top = proplists:get_value(top, AllDeps),
order_deps(lists:reverse(Top), AllDeps, []).

order_deps([], _AllDeps, Acc) ->
de_dup(Acc);
order_deps([Item|Rest], AllDeps, Acc) ->
ItemDeps = proplists:get_value(Item, AllDeps),
order_deps(lists:reverse(ItemDeps) ++ Rest, AllDeps, [Item | Acc]).

de_dup(AccIn) ->
WithIndex = lists:zip(AccIn, lists:seq(1, length(AccIn))),
UWithIndex = lists:usort(fun({A, _}, {B, _}) ->
A =< B
end, WithIndex),
Ans0 = lists:sort(fun({_, I1}, {_, I2}) ->
I1 =< I2
end, UWithIndex),
[ V || {V, _} <- Ans0 ].


## Conclusion and the end of this post

The great thing about posting to your blog is, you don't have to have a proper conclusion if you don't want to.

# Probabilistic bug hunting

Have you ever run into a bug that, no matter how careful you are trying to reproduce it, it only happens sometimes? And then, you think you've got it, and finally solved it - and tested a couple of times without any manifestation. How do you know that you have tested enough? Are you sure you were not "lucky" in your tests?

In this article we will see how to answer those questions and the math behind it without going into too much detail. This is a pragmatic guide.

## The Bug

The following program is supposed to generate two random 8-bit integer and print them on stdout:


#include <stdio.h>
#include <fcntl.h>
#include <unistd.h>

/* Returns -1 if error, other number if ok. */
int get_random_chars(char *r1, char*r2)
{
int f = open("/dev/urandom", O_RDONLY);

if (f < 0)
return -1;
if (read(f, r1, sizeof(*r1)) < 0)
return -1;
if (read(f, r2, sizeof(*r2)) < 0)
return -1;
close(f);

return *r1 & *r2;
}

int main(void)
{
char r1;
char r2;
int ret;

ret = get_random_chars(&r1, &r2);

if (ret < 0)
fprintf(stderr, "error");
else
printf("%d %d\n", r1, r2);

return ret < 0;
}



On my architecture (Linux on IA-32) it has a bug that makes it print "error" instead of the numbers sometimes.

## The Model

Every time we run the program, the bug can either show up or not. It has a non-deterministic behaviour that requires statistical analysis.

We will model a single program run as a Bernoulli trial, with success defined as "seeing the bug", as that is the event we are interested in. We have the following parameters when using this model:

• $$n$$: the number of tests made;
• $$k$$: the number of times the bug was observed in the $$n$$ tests;
• $$p$$: the unknown (and, most of the time, unknowable) probability of seeing the bug.

As a Bernoulli trial, the number of errors $$k$$ of running the program $$n$$ times follows a binomial distribution $$k \sim B(n,p)$$. We will use this model to estimate $$p$$ and to confirm the hypotheses that the bug no longer exists, after fixing the bug in whichever way we can.

By using this model we are implicitly assuming that all our tests are performed independently and identically. In order words: if the bug happens more ofter in one environment, we either test always in that environment or never; if the bug gets more and more frequent the longer the computer is running, we reset the computer after each trial. If we don't do that, we are effectively estimating the value of $$p$$ with trials from different experiments, while in truth each experiment has its own $$p$$. We will find a single value anyway, but it has no meaning and can lead us to wrong conclusions.

### Physical analogy

Another way of thinking about the model and the strategy is by creating a physical analogy with a box that has an unknown number of green and red balls:

• Bernoulli trial: taking a single ball out of the box and looking at its color - if it is red, we have observed the bug, otherwise we haven't. We then put the ball back in the box.
• $$n$$: the total number of trials we have performed.
• $$k$$: the total number of red balls seen.
• $$p$$: the total number of red balls in the box divided by the total number of green balls in the box.

• If we open the box and count the balls, we can know $$p$$, in contrast with our original problem.
• Without opening the box, we can estimate $$p$$ by repeating the trial. As $$n$$ increases, our estimate for $$p$$ improves. Mathematically: $p = \lim_{n\to\infty}\frac{k}{n}$
• Performing the trials in different conditions is like taking balls out of several different boxes. The results tell us nothing about any single box.

## Estimating $$p$$

Before we try fixing anything, we have to know more about the bug, starting by the probability $$p$$ of reproducing it. We can estimate this probability by dividing the number of times we see the bug $$k$$ by the number of times we tested for it $$n$$. Let's try that with our sample bug:

  $./hasbug 67 -68$ ./hasbug
79 -101
$./hasbug error  We know from the source code that $$p=25%$$, but let's pretend that we don't, as will be the case with practically every non-deterministic bug. We tested 3 times, so $$k=1, n=3 \Rightarrow p \sim 33%$$, right? It would be better if we tested more, but how much more, and exactly what would be better? ### $$p$$ precision Let's go back to our box analogy: imagine that there are 4 balls in the box, one red and three green. That means that $$p = 1/4$$. What are the possible results when we test three times? Red balls Green balls $$p$$ estimate 0 3 0% 1 2 33% 2 1 66% 3 0 100% The less we test, the smaller our precision is. Roughly, $$p$$ precision will be at most $$1/n$$ - in this case, 33%. That's the step of values we can find for $$p$$, and the minimal value for it. Testing more improves the precision of our estimate. ### $$p$$ likelihood Let's now approach the problem from another angle: if $$p = 1/4$$, what are the odds of seeing one error in four tests? Let's name the 4 balls as 0-red, 1-green, 2-green and 3-green: The table above has all the possible results for getting 4 balls out of the box. That's $$4^4=256$$ rows, generated by this python script. The same script counts the number of red balls in each row, and outputs the following table: k rows % 0 81 31.64% 1 108 42.19% 2 54 21.09% 3 12 4.69% 4 1 0.39% That means that, for $$p=1/4$$, we see 1 red ball and 3 green balls only 42% of the time when getting out 4 balls. What if $$p = 1/3$$ - one red ball and two green balls? We would get the following table: k rows % 0 16 19.75% 1 32 39.51% 2 24 29.63% 3 8 9.88% 4 1 1.23% What about $$p = 1/2$$? k rows % 0 1 6.25% 1 4 25.00% 2 6 37.50% 3 4 25.00% 4 1 6.25% So, let's assume that you've seen the bug once in 4 trials. What is the value of $$p$$? You know that can happen 42% of the time if $$p=1/4$$, but you also know it can happen 39% of the time if $$p=1/3$$, and 25% of the time if $$p=1/2$$. Which one is it? The graph bellow shows the discrete likelihood for all $$p$$ percentual values for getting 1 red and 3 green balls: The fact is that, given the data, the estimate for $$p$$ follows a beta distribution $$Beta(k+1, n-k+1) = Beta(2, 4)$$ (1) The graph below shows the probability distribution density of $$p$$: The R script used to generate the first plot is here, the one used for the second plot is here. ### Increasing $$n$$, narrowing down the interval What happens when we test more? We obviously increase our precision, as it is at most $$1/n$$, as we said before - there is no way to estimate that $$p=1/3$$ when we only test twice. But there is also another effect: the distribution for $$p$$ gets taller and narrower around the observed ratio $$k/n$$: ### Investigation framework So, which value will we use for $$p$$? • The smaller the value of $$p$$, the more we have to test to reach a given confidence in the bug solution. • We must, then, choose the probability of error that we want to tolerate, and take the smallest value of $$p$$ that we can. A usual value for the probability of error is 5% (2.5% on each side). • That means that we take the value of $$p$$ that leaves 2.5% of the area of the density curve out on the left side. Let's call this value $$p_{min}$$. • That way, if the observed $$k/n$$ remains somewhat constant, $$p_{min}$$ will raise, converging to the "real" $$p$$ value. • As $$p_{min}$$ raises, the amount of testing we have to do after fixing the bug decreases. By using this framework we have direct, visual and tangible incentives to test more. We can objectively measure the potential contribution of each test. In order to calculate $$p_{min}$$ with the mentioned properties, we have to solve the following equation: $\sum_{k=0}^{k}{n\choose{k}}p_{min} ^k(1-p_{min})^{n-k}=\frac{\alpha}{2}$ $$alpha$$ here is twice the error we want to tolerate: 5% for an error of 2.5%. That's not a trivial equation to solve for $$p_{min}$$. Fortunately, that's the formula for the confidence interval of the binomial distribution, and there are a lot of sites that can calculate it: ## Is the bug fixed? So, you have tested a lot and calculated $$p_{min}$$. The next step is fixing the bug. After fixing the bug, you will want to test again, in order to confirm that the bug is fixed. How much testing is enough testing? Let's say that $$t$$ is the number of times we test the bug after it is fixed. Then, if our fix is not effective and the bug still presents itself with a probability greater than the $$p_{min}$$ that we calculated, the probability of not seeing the bug after $$t$$ tests is: $\alpha = (1-p_{min})^t$ Here, $$\alpha$$ is also the probability of making a type I error, while $$1 - \alpha$$ is the statistical significance of our tests. We now have two options: • arbitrarily determining a standard statistical significance and testing enough times to assert it. • test as much as we can and report the achieved statistical significance. Both options are valid. The first one is not always feasible, as the cost of each trial can be high in time and/or other kind of resources. The standard statistical significance in the industry is 5%, we recommend either that or less. Formally, this is very similar to a statistical hypothesis testing. ## Back to the Bug ### Testing 20 times This file has the results found after running our program 5000 times. We must never throw out data, but let's pretend that we have tested our program only 20 times. The observed $$k/n$$ ration and the calculated $$p_{min}$$ evolved as shown in the following graph: After those 20 tests, our $$p_{min}$$ is about 12%. Suppose that we fix the bug and test it again. The following graph shows the statistical significance corresponding to the number of tests we do: In words: we have to test 24 times after fixing the bug to reach 95% statistical significance, and 35 to reach 99%. Now, what happens if we test more before fixing the bug? ### Testing 5000 times Let's now use all the results and assume that we tested 5000 times before fixing the bug. The graph bellow shows $$k/n$$ and $$p_{min}$$: After those 5000 tests, our $$p_{min}$$ is about 23% - much closer to the real $$p$$. The following graph shows the statistical significance corresponding to the number of tests we do after fixing the bug: We can see in that graph that after about 11 tests we reach 95%, and after about 16 we get to 99%. As we have tested more before fixing the bug, we found a higher $$p_{min}$$, and that allowed us to test less after fixing the bug. ## Optimal testing We have seen that we decrease $$t$$ as we increase $$n$$, as that can potentially increases our lower estimate for $$p$$. Of course, that value can decrease as we test, but that means that we "got lucky" in the first trials and we are getting to know the bug better - the estimate is approaching the real value in a non-deterministic way, after all. But, how much should we test before fixing the bug? Which value is an ideal value for $$n$$? To define an optimal value for $$n$$, we will minimize the sum $$n+t$$. This objective gives us the benefit of minimizing the total amount of testing without compromising our guarantees. Minimizing the testing can be fundamental if each test costs significant time and/or resources. The graph bellow shows us the evolution of the value of $$t$$ and $$t+n$$ using the data we generated for our bug: We can see clearly that there are some low values of $$n$$ and $$t$$ that give us the guarantees we need. Those values are $$n = 15$$ and $$t = 24$$, which gives us $$t+n = 39$$. While you can use this technique to minimize the total number of tests performed (even more so when testing is expensive), testing more is always a good thing, as it always improves our guarantee, be it in $$n$$ by providing us with a better $$p$$ or in $$t$$ by increasing the statistical significance of the conclusion that the bug is fixed. So, before fixing the bug, test until you see the bug at least once, and then at least the amount specified by this technique - but also test more if you can, there is no upper bound, specially after fixing the bug. You can then report a higher confidence in the solution. ## Conclusions When a programmer finds a bug that behaves in a non-deterministic way, he knows he should test enough to know more about the bug, and then even more after fixing it. In this article we have presented a framework that provides criteria to define numerically how much testing is "enough" and "even more." The same technique also provides a method to objectively measure the guarantee that the amount of testing performed provides, when it is not possible to test "enough." We have also provided a real example (even though the bug itself is artificial) where the framework is applied. As usual, the source code of this page (R scripts, etc) can be found and downloaded in https://github.com/lpenz/lpenz.github.io ## December 01, 2013 ### Gregor Gorjanc #### Read line by line of a file in R Are you using R for data manipulation for later use with other programs, i.e., a workflow something like this: 1. read data sets from a disk, 2. modify the data, and 3. write it back to a disk. All fine, but of data set is really big, then you will soon stumble on memory issues. If data processing is simple and you can read only chunks, say only line by line, then the following might be useful: ## Filefile <- "myfile.txt" ## Create connectioncon <- file(description=file, open="r") ## Hopefully you know the number of lines from some other source orcom <- paste("wc -l ", file, " | awk '{ print$1 }'", sep="")n <- system(command=com, intern=TRUE) ## Loop over a file connectionfor(i in 1:n) {  tmp <- scan(file=con, nlines=1, quiet=TRUE)  ## do something on a line of data }
Created by Pretty R at inside-R.org

## August 13, 2013

### Gregor Gorjanc

#### Setup up the inverse of additive relationship matrix in R

Additive genetic covariance between individuals is one of the key concepts in (quantitative) genetics. When doing the prediction of additive genetic values for pedigree members, we need the inverse of the so called numerator relationship matrix (NRM) or simply A. Matrix A has off-diagonal entries equal to numerator of Wright's relationship coefficient and diagonal elements equal to 1 + inbreeding coefficient. I have blogged before about setting up such inverse in R using routine from the ASReml-R program or importing the inverse from the CFC program. However, this is not the only way to "skin this cat" in R. I am aware of the following attempts to provide this feature in R for various things (the list is probably incomplete and I would grateful if you point me to other implementations):
• pedigree R package has function makeA() and makeAinv() with obvious meanings; there is also calcG() if you have a lot of marker data instead of pedigree information; there are also some other very handy functions calcInbreeding(), orderPed(), trimPed(), etc.
• pedigreemm R package does not have direct implementation to get A inverse, but has all the needed ingredients, which makes the package even more interesting
• MCMCglmm R package has function inverseA() which works with pedigree or phlyo objects; there are also handy functions such as prunePed(), rbv()sm2asreml(), etc.
• kinship and kinship2 R packages have function kinship() to setup kinship matrix, which is equal to the half of A; there are also nice functions for plotting pedigrees etc. (see also here)
As I described before, the interesting thing is that setting up inverse of A is easier and cheaper than setting up A and inverting it. This is very important for large applications. This is an old result using the following matrix theory. We can decompose symmetric positive definite matrix as A = LU = LL' (Cholesky decomposition) or as A = LDU = LDL' (Generalized Cholesky decomposition), where L (U) is lower (upper) triangular, and D is diagonal matrix. Note that L and U in previous two equations are not the same thing (L from Cholesky is not equal to L from Generalized Cholesky decomposition)! Sorry for sloppy notation. In order to confuse you even more note that Henderson usually wrote A = TDT'. We can even do A = LSSU, where S diagonal is equal to the square root of D diagonal. This can get us back to A = LU = LL' as LSSU = LSSL' = LSS'L' = LS(LS)' = L'L (be ware of sloppy notation)! The inverse rule says that inv(A) = inv(LDU) = inv(U) inv(D) inv(L) = inv(L)' inv(D) inv(L) = inv(L)' inv(S) inv(S) inv(L). I thank to Martin Maechler for pointing out to the last (obviously) bit to me. In Henderson's notation this would be inv(A) = inv(T)' inv(D) inv(T) = inv(T)' inv(S) inv(S) inv(T) Uf ... The important bit is that with NRM (aka A) inv(L) has nice simple structure - it shows the directed graph of additive genetic values in pedigree, while inv(D) tells us about the precision (inverse variance) of additive genetic values given the additive genetic values of parents and therefore depends on knowledge of parents and their inbreeding (the more they are inbred less variation can we expect in their progeny). Both inv(L) and inv(D) are easier to setup.

Packages MCMCglmm and pedigree give us inv(A) directly (we can also get inv(D) in MCMCglmm), but pedigreemm enables us to play around with the above matrix algebra and graph theory. First we need a small example pedigree. Bellow is an example with 10 members and there is also some inbreeding and some individuals have both, one, or no parents known. It is hard to see inbreeding directly from the table, but we will improve that later (see also here).

ped <- data.frame( id=c(  1,   2,   3,   4,   5,   6,   7,   8,   9,  10),                  fid=c( NA,  NA,   2,   2,   4,   2,   5,   5,  NA,   8),                  mid=c( NA,  NA,   1,  NA,   3,   3,   6,   6,  NA,   9))

Now we will create an object of a pedigree class and show the A = U'U stuff:

## install.packages(pkgs="pedigreemm")library(package="pedigreemm") ped2 <- with(ped, pedigree(sire=fid, dam=mid, label=id)) U <- relfactor(ped2)A &lt;- crossprod(U) round(U, digits=2)## 10 x 10 sparse Matrix of class "dtCMatrix"                                     ##  [1,] 1 . 0.50 .    0.25 0.25 0.25 0.25 . 0.12##  [2,] . 1 0.50 0.50 0.50 0.75 0.62 0.62 . 0.31##  [3,] . . 0.71 .    0.35 0.35 0.35 0.35 . 0.18##  [4,] . . .    0.87 0.43 .    0.22 0.22 . 0.11##  [5,] . . .    .    0.71 .    0.35 0.35 . 0.18##  [6,] . . .    .    .    0.71 0.35 0.35 . 0.18##  [7,] . . .    .    .    .    0.64 .    . .   ##  [8,] . . .    .    .    .    .    0.64 . 0.32##  [9,] . . .    .    .    .    .    .    1 0.50## [10,] . . .    .    .    .    .    .    . 0.66

## To check
U - chol(A)
round(A, digits=2)## 10 x 10 sparse Matrix of class "dsCMatrix"                                     ##  [1,] 1.00 .    0.50 .    0.25 0.25 0.25 0.25 .   0.12##  [2,] .    1.00 0.50 0.50 0.50 0.75 0.62 0.62 .   0.31##  [3,] 0.50 0.50 1.00 0.25 0.62 0.75 0.69 0.69 .   0.34##  [4,] .    0.50 0.25 1.00 0.62 0.38 0.50 0.50 .   0.25##  [5,] 0.25 0.50 0.62 0.62 1.12 0.56 0.84 0.84 .   0.42##  [6,] 0.25 0.75 0.75 0.38 0.56 1.25 0.91 0.91 .   0.45##  [7,] 0.25 0.62 0.69 0.50 0.84 0.91 1.28 0.88 .   0.44##  [8,] 0.25 0.62 0.69 0.50 0.84 0.91 0.88 1.28 .   0.64##  [9,] .    .    .    .    .    .    .    .    1.0 0.50## [10,] 0.12 0.31 0.34 0.25 0.42 0.45 0.44 0.64 0.5
1.
0
0

N
ote tha
t
pedigreem
m package uses Matrix classes in order to store only what we need to store, e.g., matrix U is triangular (t in "dtCMatrix") and matrix A is symmetric (s in "dsCMatrix"). To show the generalized Cholesky A = LDU (or using Henderson notation A = TDT') we use gchol() from the bdsmatrix R package. Matrix T shows the "flow" of genes in pedigree.

## install.packages(pkgs="bdsmatrix")library(package="bdsmatrix")tmp &lt;- gchol(as.matrix(A))D &lt;- diag(tmp)(T <- as(as.matrix(tmp), "dtCMatrix"))## 10 x 10 sparse Matrix of class "dtCMatrix"                                     ##  [1,] 1.000 .      .    .     .    .    . .   .   .##  [2,] .     1.0000 .    .     .    .    . .   .   .##  [3,] 0.500 0.5000 1.00 .     .    .    . .   .   .##  [4,] .     0.5000 .    1.000 .    .    . .   .   .##  [5,] 0.250 0.5000 0.50 0.500 1.00 .    . .   .   .##  [6,] 0.250 0.7500 0.50 .     .    1.00 . .   .   .##  [7,] 0.250 0.6250 0.50 0.250 0.50 0.50 1 .   .   .##  [8,] 0.250 0.6250 0.50 0.250 0.50 0.50 . 1.0 .   .##  [9,] .     .      .    .     .    .    . .   1.0 .## [10,] 0.125 0.3125 0.25 0.125 0.25 0.25 . 0.5 0.5 1

## To chec
k
L
&
lt;
- T %*% diag(sqrt(D))
L - t(U)
Now the A inverse part (inv(A) = inv(T)' inv(D) inv(T) = inv(T)' inv(S) inv(S) inv(T) using Henderson's notation, note that ). The nice thing is that pedigreemm authors provided functions to get inv(T) and D.

(TInv <- as(ped2, "sparseMatrix"))## 10 x 10 sparse Matrix of class "dtCMatrix" (unitriangular)            ## 1   1.0  .    .    .    .    .   .  .    .   .## 2   .    1.0  .    .    .    .   .  .    .   .## 3  -0.5 -0.5  1.0  .    .    .   .  .    .   .## 4   .   -0.5  .    1.0  .    .   .  .    .   .## 5   .    .   -0.5 -0.5  1.0  .   .  .    .   .## 6   .   -0.5 -0.5  .    .    1.0 .  .    .   .## 7   .    .    .    .   -0.5 -0.5 1  .    .   .## 8   .    .    .    .   -0.5 -0.5 .  1.0  .   .## 9   .    .    .    .    .    .   .  .    1.0 .## 10  .    .    .    .    .    .   . -0.5 -0.5 1round(DInv <- Diagonal(x=1/Dmat(ped2)), digits=2)## 10 x 10 diagonal matrix of class "ddiMatrix"##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]##  [1,]    1    .    .    .    .    .    .    .    .     .##  [2,]    .    1    .    .    .    .    .    .    .     .##  [3,]    .    .    2    .    .    .    .    .    .     .##  [4,]    .    .    . 1.33    .    .    .    .    .     .##  [5,]    .    .    .    .    2    .    .    .    .     .##  [6,]    .    .    .    .    .    2    .    .    .     .##  [7,]    .    .    .    .    .    . 2.46    .    .     .##  [8,]    .    .    .    .    .    .    . 2.46    .     .##  [9,]    .    .    .    .    .    .    .    .    1     .## [10,]    .    .    .    .    .    .    .    .    .  2.33 round(t(TInv) %*% DInv %*% TInv, digits=2)## 10 x 10 sparse Matrix of class "dgCMatrix"## ...round(crossprod(sqrt(DInv) %*% TInv), digits=2)
## 10 x 10 sparse Matrix of class "dsCMatrix"
##  [1,]  1.5  0.50 -1.0  .     .     .     .     .     .     .   ##  [2,]  0.5  2.33 -0.5 -0.67  .    -1.00  .     .     .     .   ##  [3,] -1.0 -0.50  3.0  0.50 -1.00 -1.00  .     .     .     .   ##  [4,]  .   -0.67  0.5  1.83 -1.00  .     .     .     .     .   ##  [5,]  .    .    -1.0 -1.00  3.23  1.23 -1.23 -1.23  .     .   ##  [6,]  .   -1.00 -1.0  .     1.23  3.23 -1.23 -1.23  .     .   ##  [7,]  .    .     .    .    -1.23 -1.23  2.46  .     .     .   ##  [8,]  .    .     .    .    -1.23 -1.23  .     3.04  0.58 -1.16##  [9,]  .    .     .    .     .     .     .     0.58  1.58 -1.16## [10,]  .    .     .    .     .     .     .    -1.16 -1.16
  2
.3
3

#
# T
o c
heck
so
l
ve
(A
) - crossprod(sqrt(DInv) %*% TInv)

The second method (using crossprod) is preferred as it leads directly to symmetric matrix (dsCMatrix), which stores only upper or lower triangle. And make sure you do not do cprod(TInv %*% sqrt(DInv)) as it is the wrong order of matrices.

As promised we will display (plot) pedigree by use of conversion functions of matrix objects to graph objects using the following code. Two examples are provided using the graph and igraph packages. The former does a very good job on this example, but otherwise igraph seems to have much nicer support for editing etc.

## source("http://www.bioconductor.org/biocLite.R")## biocLite(pkgs=c("graph", "Rgraphviz"))library(package="graph")library(package="Rgraphviz")g <- as(t(TInv), "graph")plot(g)

## install.packages(pkgs="igraph")library(package="igraph")i &lt;- igraph.from.graphNEL(graphNEL=g)V(i)$label <- 1:10plot(i, layout=layout.kamada.kawai)## tkplot(i) ## July 02, 2013 ### Gregor Gorjanc #### Parse arguments of an R script R can be used also as a scripting tool. We just need to add shebang in the first line of a file (script): #!/usr/bin/Rscript and then the R code should follow. Often we want to pass arguments to such a script, which can be collected in the script by the commandArgs() function. Then we need to parse the arguments and conditional on them do something. I came with a rather general way of parsing these arguments using simply these few lines: ## Collect argumentsargs <- commandArgs(TRUE) ## Default setting when no arguments passedif(length(args) < 1) { args <- c("--help")} ## Help sectionif("--help" %in% args) { cat(" The R Script Arguments: --arg1=someValue - numeric, blah blah --arg2=someValue - character, blah blah --arg3=someValue - logical, blah blah --help - print this text Example: ./test.R --arg1=1 --arg2="output.txt" --arg3=TRUE \n\n") q(save="no")} ## Parse arguments (we expect the form --arg=value)parseArgs <- function(x) strsplit(sub("^--", "", x), "=")argsDF <- as.data.frame(do.call("rbind", parseArgs(args)))argsL <- as.list(as.character(argsDF$V2))names(argsL) <- argsDF$V1 ## Arg1 defaultif(is.null(args$arg1)) {  ## do something} ## Arg2 defaultif(is.null(args$arg2)) { ## do something} ## Arg3 defaultif(is.null(args$arg3)) {  ## do something}

## ... your code here ...
Created by Pretty R at inside-R.org

It is some work, but I find it pretty neat and use it for quite a while now. I do wonder what others have come up for this task. I hope I did not miss some very general solution.

## March 24, 2013

### Romain Francois

#### Moving

This blog is moving to blog.r-enthusiasts.com. The new one is powered by wordpress and gets a subdomain of r-enthusiasts.com.

See you there

## March 17, 2013

### Modern Toolmaking

#### caretEnsemble Classification example

Here's a quick demo of how to fit a binary classification model with caretEnsemble.  Please note that I haven't spent as much time debugging caretEnsemble for classification models, so there's probably more bugs than my last post.  Also note that multi class models are not yet supported.

Right now, this code fails for me if I try a model like a nnet or an SVM for stacking, so there's clearly bugs to fix.

The greedy model relies 100% on the gbm, which makes sense as the gbm has an AUC of 1 on the training set.  The linear model uses all of the models, and achieves an AUC of .5.  This is a little weird, as the gbm, rf, SVN, and knn all achieve an AUC of close to 1.0 on the training set, and I would have expected the linear model to focus on these predictions. I'm not sure if this is a bug, or a failure of my stacking model.