Why sequencing data is modeled as negative binomial

The goal of most sequencing experiments is to identify differences in gene expression between biological conditions such as the influence of a disease-linked genetic mutation or drug treatment. Fitting the correct statistical model to the data is an essential step before making inferences about differentially expressed genes. The negative binomial (NB) distribution has emerged as the model of choice to fit sequencing data. While the NB distribution is bread-and-butter to a statistician, the average experimental biologist may not be very familiar with it.

A first intuition

In a standard sequencing experiment (RNA-Seq), we map the sequencing reads to the reference genome and count how many reads fall within a given gene (or exon). This means that the input for the statistical analysis are discrete non-negative integers (“counts”) for each gene in each sample. The total number of reads for each sample tends to be in the millions, while the counts per gene vary considerably but tend to be in the tens, hundreds or thousands. Therefore, the chance of a given read to be mapped to any specific gene is rather small. Discrete events that are sampled out of a large pool with low probability sounds very much like a Poisson process. And indeed it is. In fact, earlier iterations of RNA-Seq analysis modeled sequencing data as a Poisson distribution. There is one problem, however. The variability of read counts in sequencing experiments tends to be larger than the Poisson distribution allows.

A fundamental property of the Poisson distribution is that its variance is equal to the mean. Here I plotted the gene-wise means versus their variance of the “bottomly” experiment provided by the ReCount project. The code to produce this plot can be found on Github.

nb_mean_var

It is obvious that the variance of counts is generally greater than their mean, especially for genes expressed at a higher level. This phenomenon is called “overdispersion“. The NB distribution is similar to a Poisson distribution but has an extra parameter called the “clumping” or “dispersion” parameter. It is like a Poisson distribution with more variance. Note, how the NB estimates of the mean-variance relationship (blue line) fits the observed values quite well. Thus, a reasonable first intuition of why the NB distribution is a proper way of fitting count data is that the dispersion parameter allows the extra wiggle room to model the “extra” variance that we empirically observe in RNA-Seq experiments.

A more rigorous justification

There are two mathematically equivalent formulations of the NB distribution. In its traditional form, which I will mention only for the sake of completion, the NB distribution estimates the probability of having a number of failures until a specified number of successes occur. An example for an application would be the expected number of games a striker goes without a goal (“failure”) before scoring (“success”). Note that, “success” and “failure” are not value judgements but just the two outcomes of a Bernoulli process and therefore interchangeable. Whenever you see the NB distribution used in this form, pay close attention to what is defined as a “success” and a “failure”. In is a common point of notational confusion. This definition is not terribly useful for understanding how the NB distribution relates to RNA-Seq count data.

The second definition sounds more intimidating but is much more useful. The NB distribution can be defined as a Poisson-Gamma mixture distribution. This means that the NB distribution is a weighted mixture of Poisson distributions where the rate parameter \lambda (i.e. the expected counts) is itself associated with uncertainty following a Gamma distribution. This sounds very similar to our earlier definition as a “Poisson distribution with extra variance”.

While it is convenient to have a distribution that fits our empirical observations it is not quite satisfying without a more theoretical justification. When comparing samples of different conditions we usually have multiple replicates of each condition. Those replicates need to be independent for statistical inference to be valid. Such replicates are called “biological” replicates because they come from independent animals, dishes, or cultures. In contrast, splitting a sample in two and running it through the sequencer twice would be a “technical” replicate. In general, there is more variance associated with biological replicates than technical replicates. If we assume that our samples are biological replicates, it is not surprising that the same transcript is present at slightly different levels in each sample, even under the same conditions. In other words, the Poisson process in each sample has a slightly different expected count parameter. This is the source of the “extra” variance (overdispersion) we observe in sequencing data. In the framework of the NB distribution, it is accounted for by allowing Gamma-distributed uncertainty about the expected counts (the Poisson rate) for each gene. Conversely, if we were to deal with technical replicates, there should be no overdispersion and a simple Poisson model would be adequate.

The variance (dispersion) \sigma^2 of a NB distribution can be expressed as function of the mean \mu and the dispersion parameter \alpha.

\sigma^2 = \mu + \alpha \mu^2

From this formula it is evident that the dispersion is always greater than the mean for \alpha > 0. If \alpha \rightarrow 0, the NB distribution is a Poisson distribution.

Dispersion estimates

Finally, a short note on the practical implications of estimating the dispersion of sequencing data. In a standard sequencing experiment, we have to be content with few biological replicates per condition due to the high costs associated with sequencing experiments and the large amount of time that goes into library preparations. This makes the gene-wise estimates of dispersion rather unreliable. Modern RNA-Seq analysis tools such as DESeq2 and edgeR combine the gene-wise dispersion estimate with an estimate of the expected dispersion rate based on all genes. This Bayesian “shrinkage” of the variance has already been applied successfully in microarray analysis. Although the implementation of this method varies between analysis tools, the concept of using information from the whole data set has emerged as a powerful technique to mitigate the shortcomings of having few replicates.

Why sequencing data is modeled as negative binomial

Analyzing quantitative PCR data the tidy way

Previously in this series on tidy data: Taking up the cudgels for tidy data


One of the most challenging aspects of working with data is how easy it is to get lost. Even if the data sets are small. Multiple levels of hierarchy and grouping quickly confuse our human brains (at least mine). Recording such data in two dimensional spreadsheets naturally leads to blurring of the distinction between observation and variable. Such data requires constant reformatting and its structure may not be intuitive to your fellow researcher.

Here are the two main rules about tidy data as defined in Hadley Wickham’s paper:

  1. Each variable forms a column
  2. Each observation forms a row

A variable is an “attribute” of a given data point that describes the conditions when it was taken. Variables often are categorical (but they don’t need to be). For example, the gene tested or the genotype associated with a given measurement would be a categorical variables.

An observation is a measurement associated with an arbitrary number of variables. There are no measurements that are taken under identical conditions. Each observation is uniquely described by the variables and should form its own row.

Let’s look at an example of a typical recording of quantitative PCR data in Excel.

qpcr_excel_messy

We have measurements for three different genotypes (“control”, “mutant1”, “mutant2”) from three separate experiments (“exp1”, “exp2”, “exp3”) with three replicates each (“rep1”, “rep2”, “rep3”).

Looking at the columns, we see that information on “experiment” and “replicate” are stored in the names of the columns rather than the entries of the columns. This will have to be changed.

There clearly are multiple measurements per row. More precisely, it looks like we have a set of nine measurements for each genotype. But this is not entirely true. Experiments are considered statistically independent as they are typically performed at different times and with different cells. They capture the full biological variability and we call them “biological replicates”. The repeated measurements done in each experiment are not statistically independent because they come from the same sample preparation and thus only capture sources of variance that originate from sample handling or instrumentation. We call them “technical replicates”. Technical replicates cannot be used for statistical inference that requires “statistical independence”, such as a t-test. As you can see, we have an implicit hierarchy in our data that is not expressed in the structure of the data representation shown above.

We will untangle all those complications one by one using R tools developed by Hadley Wickham and others to represent the same data in a tidy format suitable for statistical analysis and visualization. For details about how the code works, please consult the many excellent tutorials on dplyr, tidyr, ggplot2, and broom.

messy <- read.csv("qpcr_messy.csv", row.names = 1)

This is the original data read into R. Let’s get started.

qpcr_R_messy

Row names should form their own column

The “genotype” information is recorded as row names. “Genotype” clearly is a variable, so we should make “genotype” a full column.

tidy <- data.frame(messy) %>%
# make row names a column
mutate(genotype = rownames(messy))

qpcr_R_messy2What are our variables?

Next, we need to think about what are our variables. We have already identified “genotype” but what are the other ones? The way we do this is to ask ourselves what kind of information we would need to uniquely describe each observation. The experiment and replicate number are essential to differentiate each quantitative PCR measurement, so we need to create separate columns for “experiment” and “replicate”. We will do this in two steps. First we use “gather” to convert tabular data from wide to long format (we could have also used the more general “melt” function from the “reshape2” package). The former column names (e.g. “exp1_rep1”) are saved into a temporary column called “sample”. As this column contains information about two variables (“experiment” and “replicate”), we need to separate it into two columns to conform with the “each variable forms a column” rule. To do this, we use “separate” to split “sample” into the two columns “experiment” and “replicate”.

tidy <- tidy %>%
    # make each row a single measurement 
    gather(key = sample, value = measurement, -genotype) %>%
    # make each column a single variable
    separate(col = sample, into = c("experiment", "replicate"), sep = "_")

qpcr_R_messy3Here are the first 10 columns of the “tidy” representation of the initial Excel table. Before we can do statistical tests and visualization, we have to take care of one more thing.

Untangling implicit Domain specific hierarchies

Remember what we said before about the two different kind of replicates. Only data from biological replicates (“experiments”) are considered statistically independent samples, while technical replicates (“replicate”) are not. One common approach is to average the technical replicates (“replicate”) before any statistical test is applied. With tidy data, this is simple.

data <- tidy %>%
    # calculate mean of technical replicates by genotype and experiment
    group_by(genotype, experiment) %>%
    summarise(measurement = mean(measurement)) %>%
    ungroup()

Having each variable as its own column makes the application of the same operation onto different groups straightforward. In our case, we calculate the mean of technical replicates for each genotype and experiment combination.

Now, the data is ready for analysis.

TIdy Statistical analysis of quantitative pcr data

The scientific rational for a quantitative PCR experiment is to find out whether the number of transcripts for a given gene is different between two or more conditions. We have measurements for one transcript in three distinct genotypes (“control”, “mutant1”, “mutant2”). Biological replicates are considered independent and measurements are assumed to be normally distributed around a “true” mean value. A t-test would be an appropriate choice for the comparison of two genotypes. In this case, we have three genotypes, so we will use one-way anova followed by Tukey’s post-hoc test.

mod <- data %>%
    # set "control" as reference
    mutate(genotype = relevel(factor(genotype), ref = "control")) %>%
    # one-way anova and Tukey's post hoc test
    do(tidy(TukeyHSD(aov(measurement ~ genotype, data = .))))

We generally want to compare the effect of a genetic mutation to a “control” condition. We therefore set the reference of “genotype” to “control”.

Using base R statistics functions like “aov” and “TukeyHSD” in a tidy data analysis workflow can pose problems because they were not created with the idea of “dplyr”-style piping (“%>%”) in mind. Piping requires that the input and output of each function is a data frame and that the input is the first argument of the function. The “aov” function neither takes the input data frame as its first argument, nor does it return a data frame but a specialized “aov” object. To add insult to injury, the “TukeyHSD” function only works with such a specialized “aov” object as input.

In situations like this, the “do” function comes in handy. Within the “do” function, the input of the previous line is accessible through the dot character, so we can use an arbitrary function within “do” and just refer to the input data at the appropriate place with “.”. As a final clean-up, the “tidy” function from the “broom” package makes sure that the output of the line is a data frame.

qpcr_tukeyTukey’s post hoc test thinks “mutant1” is different from “control” but “mutant2” is not. Let’s visualize the results to get a better idea of how the data looks like.

tidy Visualization of quantitative PCR data

We are dealing with few replicates, three in our case, so a bar graph is not the most efficient representation of our data. Plotting the individual data points and the confidence intervals gives us more information using less ink. We will use the “ggplot2” package because it is designed to work with data in the tidy format.

# genotype will be on the x-axis, measurements on the y-axis
ggplot(data, aes(x = genotype, y = measurement, col = experiment)) + 
    # plot the mean of each genotype as a cross
    stat_summary(fun.y = "mean", geom = "point", color = "black", shape = 3, size = 5) +
    # plot the 95% confidence interval for each genotype
    stat_summary(fun.data = "mean_cl_normal", geom = "errorbar", color = "black", width = 0.1) + 
    # we we add the averaged measurements for each experiment
    geom_point(shape = 16, size = 5) +
    theme_classic()

qpcr_visualizationWe can see why the first mutant is different from the “control” sample and the second is not. More replicates would be needed to test whether the small difference in means between “control” and “mutant2” is a true difference or not.

What I have shown here is just the tip of the iceberg. There are many more tools and functions to discover. The more data analysis you do, the more you will realize how important it is not to waste time formatting and reformatting the the data for each step of the analysis. Learning about how to tidy up your data is an important step towards that goal.


The R code can be found on Github.

 

Analyzing quantitative PCR data the tidy way

Taking up the cudgels for tidy data

The abundance of data has led to a revolution in marketing and advertisement as well as in biomedical research. A decade ago, the emerging field of “systems biology” (no pun intended) promised to take basic research to the next level through the use of high-throughput screens and “big data”. Institutes were built, huge projects were funded, but surprisingly little of substance has been accomplished since.

There are three main reasons, I think, two of which are under our control and one is not.

During our first forays into understanding biology as a “system” we underestimated the complexity of even an isolated cell, let alone a multi-cellular organism. A somewhat complete description of even the most basic regulatory mechanisms and pathways remains a dream to this day due to myriads of adaptive mechanisms and cross-talks preventing the formulation of a coherent view. Unfortunately, this problem has to be overcome with improved scientific methodology or analysis and will take time.

There are things we can do right now, however.

Overly optimistic or incorrect interpretation of statistical results and “cherry-picking” of hits of high-throughput screens has led to a surprising number of publications that cannot be replicated or even be reproduced. I have written more extensively about this particular problem I refer to as the “Fisherman’s dilemma“.

The lack of standards for structuring data is another reason that prevents the use of existing data and makes merging data sets from different studies or sources unnecessary painful and time consuming. A common saying is that data science is 80% data cleaning, 20% data analysis. The same is true for bioinformatics, where one needs to wrestle with incomplete and messy datasets or, if it’s your lucky day, just different data formats. This problem is especially prevalent in meta-analysis of scientific data. Arguably, the integration of datasets from different sources is where we would predict to find some of the most important and universal results. Why else spend the time to generate expensive datasets if we don’t use them to compare and cross-reference?

If we were to spent less time on the arduous task of “cleaning” data, we could focus our attention on the question itself and the implementation of the analysis. In recent years, Hadley Wickham and others have developed a suite of R tools that help to “tidy up” messy data and establish and enforce a “grammar of data” that allows easy visualization, statistical modeling, and data merging without the need to “translate” the data for each step of the analysis. Hadley deservedly gets a lot of credit for his dplyr, reshape2, tidyr, and ggplot2 packages, but not nearly enough. At this point David Robinson’s excellent broom package for cleaning results from statistical models should also be mentioned.

The idea of tidy data is surprisingly simple. Here are the two most basic rules (see Hadley’s paper for more details).

  1. Each variable forms a column.
  2. Each observation forms a row.

Here is an example.

messy

This is a standard form of recording biological research data such as data from a PCR experiment with three replicates. At first glance, the data looks pretty tidy. Genes in rows, replicates in columns. What’s wrong here?

In fact, this way of recording data violates both basic rules of a tidy dataset. First, the “replicates” are not distinct variables but instances of the same variable, which violates the first rule. Second, each measurement is a distinct observation and should have its own row. Clearly, this is not the case either.

This is how the same data looks like once cleaned-up.

tidy

Each variable is a column, each observation is a row.

Representing data “the tidy way” is not novel. It has been called the “long” format previously, as opposed to the “wide” (“messy”) format. Although “tidy” and “messy” imply a value judgement, it is important to note that while the tidy/long format has distinct advantages for data analysis, the wide format is often seen as the more intuitive and almost always is the more concise.

The most important advantage of tidy data in data analysis is that there is one way of representing the data in a tidy format, while there are many possible ways of having a messy data structure. Take the example of the messy data from above. Storing replicates in rows and genes in columns (the transpose) would have been an equivalent representation to the one shown above. However, cleaning up both representations results in the same tidy data representation shown. This advantage becomes even more important with datasets that contain multiple variables.

A related, but more technical advantage of the tidy format is that it simplifies the use of loops and vectorized programming (implicit loops) because the “one variable, one column – one observation, one row” structure enforces a “linearization” of the data that is more easily dealt with from a programming perspective.

Having data in a consistent format allows feeding data into visualization and modeling tools without spending time on getting the data in the right shape. Similarly, tidy dataset from different sources can be more easily merged and analyzed together.

While data in marketing is sometimes called “cheap”, research data in science often is generally very expensive, both in terms of time and money. Taking the extra step of recording and sharing data in a “tidy” format, would make data analysis in biomedical research and clinical trials more effective and potentially more productive.

In a follow-up post, I will cover the practical application of some of the R tools developed to work with tidy data using an example most experimental biologists are familiar with: statistical analysis and visualization of quantitative PCR.

Taking up the cudgels for tidy data

What is a large enough sample?

In my previous entry, I tried to clear up some of my own confusion about the Central Limit Theorem (CLT) and explained why it is such a valuable theoretical concept in statistics. To recap, the CLT describes how the means of a random sample of an unknown sampling distribution approach a normal distribution as the sample size n approaches \infty. The uncertainty about our estimate of the mean of the original sampling distribution is given by \sigma / \sqrt{n}, where \sigma is the standard deviation of the sampling distribution. We can see that the larger the sample size, the more certain we are about our estimate of the true mean.

The obvious practical question is what is a large enough sample size? The short answer is, it depends. A sample size of 30 is a pretty save bet for most real life applications.

To investigate the influence of sample size on the convergence of the distribution of the means, I will use simulated sampling from three different sampling distributions. All simulations were done using R. The code can be found on Github.

CLT in (simulated) action

Let’s consider a normal sampling distribution to start with. This is useful to illustrate the idea of how the uncertainty of our estimate of the true mean depends on the sample size n. Here is our normal sampling distribution with \mu = 4 and \sigma = 2.

clt_part2_normal

Now we generate a large number m of random samples each with sample size n and calculate their means. If this confuses you, you are not alone. For now, understand that the only variable we are changing is the sample size n. m will just be a “large number”, such as 10000 in our case, so that we can draw a histogram of 10000 simulated means. We will do this four time, each time with a different sample size of n being either 2, 5, 15, or 30.

The histogram shows the distribution of simulated means and the blue curve illustrates the normal distribution predicted by the CLT with a mean of \mu and a standard deviation of \sigma / \sqrt{n}. In the lower panel, I show quantile-quantile plots to investigate the how well the distribution of the means fits a theoretical normal distribution.

clt_part2_normal_sampling Unsurprisingly, the means of random samples drawn from a perfect normal distribution are themselves normally distributed. Even with a sample size as small as 2. It is intuitive that small sample sizes have more uncertainty associated with our estimate of the true mean, which is reflected by the relatively broad normal distribution of the means. As we increase the sample size the distribution of the means becomes more pointy and narrow, indicating that our estimate of the true mean \mu becomes more and more accurate. Note also, that the y-axis changes as we increase the sample size. This is a visual confirmation that the standard deviation of the distribution of the means is given by \sigma / \sqrt{n}.

Let’s turn to an exponential sampling distribution with \lambda = 1/4 next. Recall that both the mean and standard deviation of an exponential distribution is 1 / \lambda. This one is clearly not normal.

clt_part2_exp

I simulated random samples for different sample sizes as described above for the normal distribtion and calculated the means.

clt_part2_exp_sampling

At smaller sample sizes, the deviation of the actual distribution of the means from the theoretical distribution of the means is obvious. It clearly retains some characteristics of an exponential distribution. As we increase the sample size, the fit becomes better and better, until it eventually morphes into a normal distribution.

Does the CLT hold for an arbitrary distribution? Well, let’s consider this crazy sampling distribution I made up using a combination of normal, exponential and uniform distributions.

clt_part2_crazy

Simulation of random samples using different sample sizes as before.

clt_part2_crazy_sampling

As predicted, the CLT holds even for a non-standard sampling distribution. Granted, I did not challenge the assumptions of the CLT too much using for example an extreme tail (skew). I trust this is good enough to convince you that it would just take a few more samples before convergence.

Why is a sample size of 30 large enough?

Back to our original question: what is a large enough sample? We have seen that the major determinant is the shape of the sampling distribution. The more normal it is to begin with, the fewer samples we will need to reach convergence towards a normal distribution of the means.

In practice we do not generate 10000 random samples (10000 experiments!) to get a distribution of the means. We estimate the mean and standard deviation from a single random sample. The larger the random sample, the better will be our estimate of the true mean \mu and the standard deviation \sigma. This follows directly from the Law of large numbers. It is often recommended in statistics textbooks that as a rule of thumb a sample size of 30 can be considered “large”. But why exactly 30? I think there is a practical and a pragmatic argument to be made.

In the simulations we saw that the distribution of the means of a random sample drawn from a (not too crazy) non-normal sampling distribution will be very close to normal. This means that our estimates of the mean and standard deviation of that distribution will be sufficient to describe the distribution of the means and we can use them in hypothesis testing with some confidence (no pun intended).

A more pragmatic argument would make use of the relationship between the sample size and our uncertainty about the true mean of the sampling distribution. Irrespective of the standard deviation \sigma of the sampling distribution, the standard error \sigma / \sqrt{n} decreases proportional to \sqrt{n}. Common sense dictates that increasing the sample size beyond a certain point will result in ever diminishing gains in precision. Here is a graphical representation of the relationship between the standard error and sample size.

clt_part2_rule_of_thumb

As you can see, a sample size of 30 sits right at the point where the curve stops to have an exponential and starts to have a linear decrease. In other words, a sample size of 30 represents the sweet spot in terms of the most “bang for the buck”, no matter the magnitude of the standard deviation of the original sampling distribution.

You might ask, what if the standard deviation is a large value? Well, then our estimate of the true mean will be pretty bad. We will have to increase the sample size and deal with the fact that gains in precision will be ever smaller as n goes beyond 30.

In biomedical research we often face the situation that even a sample size of 30 is unattainable in terms of time or money. Fortunately, there is a solution for that dilemma: Student’s t-distribution. I will investigate how the CLT relates to the t-distribution and hypothesis testing in the next post.


Reproducibility

The full R code is available on Github.

What is a large enough sample?

Tidy unnesting

At least once a week, I have to work with a data set that has “nested” measurements in one of its columns.

data_untidySuch data violates rule #2 of Hadley Wickam’s definition of tidy data. Not all observations are in separate rows. In order to work with this data set, we generally want to “unnest” those measurements to make each a separate row.

data_tidy

The “untidy” way

Ordinarily, I would convert the untidy data to tidy data using a cumbersome sequence of messy commands that involve splitting the entries in “value” into lists, then counting the elements of each list, and replicating each row of the data frame accordingly.

library(stringr)
# split "value" into lists
values <- str_split(data$value, ";")
# count number of elements for each list
n <- sapply(values, length)
# replicate rownames of data based on elements for each list
row_rep <- unlist(mapply(rep, rownames(data), n))
# replicate rows of original data
data_tidy <- data[row_rep, ]
# replace nested measurements with unnested measurements
data_tidy$value <- unlist(values)
# reformat row names
rownames(data_tidy) <- seq(nrow(data_tidy))

The “tidy” way

Is it really necessary to go through all those (untidy!) steps to tidy up that data set? It turns out, it is not. In comes the “unnest” function in the “tidyr” package.

library(tidyr)
# use dplyr/magrittr style piping
data_tidy2 <- data %>% 
  # split "value" into lists
  transform(value = str_split(value, ";")) %>%
  # unnest magic
  unnest(value)

Much tidier! Just split and “unnest”. On top of that, “unnest” nicely fits into a “dplyr”-style data processing workflow using “magrittr” piping (“%>%”)

all.equal(data_tidy, data_tidy2)

The results of both methods are equivalent. Your choice, I have made mine!

Synonymous fission yeast gene names

This is how the workflow would look like using an ID mapping file of synonymous gene names of the fission yeast Schizosaccharomyces pombe. The file can be obtained from the “Pombase” website.

# read data
raw <- read.delim("sysID2product.tsv", header = FALSE, stringsAsFactors = FALSE)
names(raw) <- c("orf", "symbol", "synonyms", "protein")
# unnest
data <- raw %>% 
  transform(synonyms = str_split(synonyms, ",")) %>% 
  unnest(synonyms)

Tidy code is almost as much of a blessing as tidy data.


Reproducibility

The full R code is available on Github.

Tidy unnesting

PCA – Part 5: Eigenpets

In this post scriptum to my series on Principal Component Analysis (PCA) I will show how PCA can be applied to image analysis. Given a number of images of faces, PCA decomposes those images into “eigenfaces” that are the basis of some facial recognition algorithms. Eigenfaces are the eigenvectors of the image data matrix. I have shown in Part 3 of this series that the eigenvectors that capture a given amount of variance of the data can be used to obtain an approximation of the original data using fewer dimensions. Likewise, we can approximate the image of a human face by a weighted combination of eigenfaces.

From images to matrices

A digital image is just a matrix of numbers. Fair game for PCA. You might ask yourself, though, how to coerce many matrices into a single data matrix with samples in the rows and features in the columns? Just stack the columns of the image matrices to get a single long vector and then stack the image vectors to obtain the data matrix. For example, a 64 by 64 image matrix would result in a 4096-element image vector, and 100 such image vectors would be stacked into a 100 by 4096 data matrix.

The more elements a matrix has, the more computationally expensive it becomes to do the matrix factorization that yields the eigenvectors and eigenvalues. As the number of images (samples) is usually much smaller than the number of pixels (features), it is more efficient to compute the eigenvectors of the transpose of the data matrix with the pixels in the rows and the images in the columns.

Fortunately, there is an easy way to get from the eigenvectors of the covariance matrix \boldsymbol A^T \boldsymbol A to those of the covariance matrix \boldsymbol A \boldsymbol A^T. The eigenvector equation of \boldsymbol A^T \boldsymbol A is

\boldsymbol A^T \boldsymbol A \vec{v} = \lambda \vec{v}

Multiplying \boldsymbol A to the left on both sides gives us important clues about the relationship between the eigenvectors and eigenvalues between the two matrices.

\boldsymbol A \boldsymbol A^T (\boldsymbol A \vec{v}) = \lambda (\boldsymbol A \vec{v})

The eigenvalues of the covariance matrices \boldsymbol A^T \boldsymbol A and \boldsymbol A \boldsymbol A^T are identical. If \boldsymbol A is an m by n matrix and m < n, there will be m nonzero eigenvalues and n-m zero eigenvalues.

To get from an eigenvector \vec{v} of \boldsymbol A^T \boldsymbol A to an eigenvector of \boldsymbol A \boldsymbol A^T, we just need to multiply by \boldsymbol A on the left.

What does an “eigenpet” look like?

For this demonstration, I will not be using images of human faces but, in line with the predominant interests of the internet, faces of cats and dogs. I obtained this data set when I took the course “Computational Methods for Data Analysis” on Coursera and converted the original data to text files to make them more accessible to R. The data can be found on Github.

library(RCurl)
# read data from Github
cats <- read.table(text = getURL("https://raw.githubusercontent.com/bioramble/pca/master/cat.csv"), sep = ",")
dogs <- read.table(text = getURL("https://raw.githubusercontent.com/bioramble/pca/master/dog.csv"), sep = ",")
# combine cats and dogs into single data frame
pets <- cbind(cats, dogs)

The data matrix already is in a convenient format. Each 64 by 64 pixel image has been converted into a 4096 pixel vector and each of the 160 image vector is stacked vertically to obtain a data matrix with dimensions 4096 by 160. As you might have guessed, there are 80 cats and 80 dogs in the data set.

The distinction between what is a sample and what is a feature becomes a little blurred in this analysis. Technically, the 160 images are the samples and the 4096 pixels are the features, so we should do PCA on a 160 by 4096 matrix. However, as discussed above it is more convenient to operate with the transpose of this matrix, which is why image data usually is prepared in its transposed form with features in rows and samples in columns.

As we have seen in theoretically in Part 2 of this series, it is necessary to center (and scale) the features before performing PCA. We are dealing with 8-bit greyscale pixel values that are naturally bounded between 0 and 255, so they already are on the same scale. Moreover, all features measure the same quantity (brightness), so it is customary to center the data by subtracting the “average” image from the data.

# compute "average" pet
pet0 <- rowMeans(pets)
cat0 <- rowMeans(pets[, 1:80])
dog0 <- rowMeans(pets[, 81:160])

So what does the average pet, the average cat, average dog look like?

# create grey scale color map
greys <- gray.colors(256, start = 0, end = 1)
# convenience function to plot pets
show_image <- function(v, n = 64, col = greys) {
    # transform image vector back to matrix
    m <- matrix(v, ncol = n, byrow = TRUE)
    # invert columns to obtain right orientation
    # plot using "image"
    image(m[, nrow(m):1], col = col, axes = FALSE)
}
# plot average pets
for (i in list(pet0, cat0, dog0)) {
    show_image(i)
}

pca_part5_fig1As expected, the average pet has features of both cats and dogs, while the average cat and dog are quite recognizable as members of their respective species.

Let’s run PCA on the data set and see what an “eigenpet” looks like.

# subtract average pet
pets0 <- pets - pet0
# run pca
pc <- prcomp(pets0, center = FALSE, scale. = FALSE)

Ordinarily, we would find the eigenvector matrix in the “rotation” object return to us by “prcomp”. Remember, that we did PCA on the transpose of the data matrix, so we have to do some additional work to get to the eigenvectors of the original data. I have shown above that to get from the eigenvectors of \boldsymbol A^T \boldsymbol A to the eigenvectors of \boldsymbol A \boldsymbol A^T, we just need to multiply by \boldsymbol A on the left. \boldsymbol A, in this case, is our centered data matrix “pets0”.

Note that the unscaled eigenvectors of the eigenfaces are equivalent to the projection of the data onto the eigenvectors of the transposed data matrix. Therefore, we do not have to explicitly compute them but can use the “pc$x” object.

# obtain unscaled eigenvectors of eigenfaces
u_unscaled <- as.matrix(pets0) %*% pc$rotation
# this turns out to be the same as the projection of the data 
# stored in "pc$x"
all.equal(u_unscaled, pc$x)

Both “u_unscaled” and “pc$x” are unscaled versions of the eigenvectors, which means that they are not unitary matrices. For plotting, this does not matter because the images will be scaled automatically. If scaled eigenvectors are important, it is more convenient to use singular value decomposition and use the left eigenvalues stored in the matrix “u”.

# singular value decomposition of data
sv <- svd(pets0)
# left eigenvalues are stored in matrix "u"
u <- sv$u

Let’s look at the first couple of “eigenpets” (eigenfaces).

# display first 6 eigenfaces
for (i in seq(6)) {
    show_image(pc$x[, i])
}

pca_part5_fig2Some eigenfaces are definitely more cat-like and others more dog-like. We also see a common issue with eigenfaces. The directions of highest variance are usually associated with the lighting conditions of the original images. Whether they are predominantly light or dark dominates the first couple of components. In this particular case, it appears that the images of cats have stronger contrasts between foreground and background. If we were to do face recognition, we would either preprocess the images to have comparable lighting conditions or exclude the first couple of eigenfaces.

Reconstruction of Pets using “Eigenpets”

Using the eigenfaces we can approximate or completely reconstruct the original images. Let’s see how this looks like with a couple of different variance cut-offs.

# a number of variance cut-offs
vars <- c(0.2, 0.5, 0.8, 0.9, 0.98, 1)
# calculate cumulative explained variance
var_expl <- cumsum(pc$sdev^2) / sum(pc$sdev^2)
# get the number of components that explain a given amount of variance
npc <- sapply(vars, function(v) which(var_expl >= v)[1])
# reconstruct four cats and four dogs
for (i in seq(79, 82)) {
    for (j in npc) {
        # project data using "j" principal components
        r <- pc$x[, 1:j] %*% t(pc$rotation[, 1:j])
        show_image(r[, i])
        text(x = 0.01, y = 0.05, pos = 4, labels = paste("v =", round(var_expl[j], 2)))
        text(x = 0.99, y = 0.05, pos = 2, labels = paste("pc =", j))
    }
}

pca_part5_fig3Every pet starts out looking like a cat due to the dominance of lighting conditions. Somewhere between 80% and 90% of captured variance, every pet is clearly identifiable as either a cat or a dog. Even at 90% of capture variance, we use less than one third of all components for our approximation of the original image. This may be enough for a machine learning algorithm to tell apart a cat from a dog with high confidence. We see, however, that a lot of the detail of the image is contained in the last two thirds of the components.


Reproducibility

The full R code is available on Github.


Further Reading

Scholarpedia – Eigenfaces

Jeff Jauregui – Principal Component Analysis with Linear Algebra


PCA SERIES

Part 1: An Intuition

Part 2: A Look Behind The Curtain

Part 3: In the Trenches

Part 4: Potential Pitfalls

Part 5: Eigenpets

PCA – Part 5: Eigenpets

PCA – Part 3: In the Trenches

Now that we have an intuition of what principal component analysis (PCA) is and understand some of the mathematics behind it, it is time we make PCA work for us.

Practical examples of PCA typically use Ronald Fisher’s famous “Iris” data set, which contains four measurements of leaf lenghts and widths of three subspecies of Iris flowers. To mix things up a little bit, I will use a data set that is closer to what you would encounter in the real world.

The “Human Activity Recognition Using Smartphones Data Set” available from the UCI Machine Learning Repository contains a total of 561 triaxial acceleration and angular velocity measurements of 30 subjects performing different movements such as sitting, standing, and walking. The researchers collected this data set to ask whether those measurements would be sufficient to tell the type of activity of the person. Instead of focusing on this classification problem, we will look at the structure of the data and investigate using PCA whether we can express the information contained in the 561 different measurements in a more compact form.

I will be using a subset of the data containing the measurements of only three subjects. As always, the code used for the pre-processing steps of the raw data can be found on GitHub.

Step 1: Explore the data

Let’s first load the pre-processed subset of the data into our R session.

# read data from Github
measurements <- read.table(text = getURL("https://raw.githubusercontent.com/bioramble/pca/master/pca_part3_measurements.txt"))
description <- read.table(text = getURL("https://raw.githubusercontent.com/bioramble/pca/master/pca_part3_description.txt"))

It’s always a good idea to check for a couple of basic things first. The big three I usually check are:

  • What are dimensions of the data, i.e. how many rows and columns?
  • What type of features are we dealing with, i.e. categorical, ordinal, continuous?
  • Are there any missing values?

The answer to those three questions will determine the amount of additional data munging we have to do before we can use the data for PCA.

# what are the dimensions of the data?
dim(measurements)
# what type of data are the features?
table(sapply(measurements, class))
# are there missing values?
any(is.na(measurements))

The data contains 990 samples (rows) with 561 measurements (columns) each. Clearly too many measurements for visualizing on a scatterplot. The measurements are all of type “numeric”, which means we are dealing with continuous variables. This is great because categorical and ordinal variable are not handled well by PCA. Those need to be “dummy coded“. We also don’t have to worry about missing values. Strategies for handling missing values are a topic on its own.

Before we run PCA on the data, we should look at the correlation structure of the features. If there are features, i.e. measurements in our case, that are highly correlated (or anti-correlated), there is redundancy within the data set and PCA will be able to find a more compact representation of the data.

# feature correlation before PCA
cor_m <- cor(measurements, method = "pearson")
# use only upper triangular matrix to avoid redundancy
upt_m <- cor_m[upper.tri(cor_m)]
# plot correlations as histogram
hist(upt_m, prob = TRUE)
# plot correlations as image
image.plot(cor_m, axes = FALSE)

The code was simplified for clarity. The full version can be found in the script.

pca_part3_fig1We see in the histogram on the left that there is a considerable number of highly correlated features, most of them positively correlated. Those features show up as yellow in the image representation to the right. PCA will likely be able to provide us with a good lower dimensional approximation of this data set.

Step 2: Run PCA

After all the preparation, running PCA is just one line of code. Remember, that we need to at least center the data before using PCA (Why? see Part 2). Scaling is technically only necessary if the magnitude of the features are vastly different. Note, that the data appears to be already centered and scaled from the get-go.

# run PCA
pc <- prcomp(measurements, center = TRUE, scale. = TRUE)

Depending on your system and the number of features of your data this may take a couple of seconds.

The call to “prcomp” has constructed new features by linear combinations of the old features and sorted them by their and weighted by the amount of variance they explain. Because the new features are the eigenvectors of the feature covariance matrix, they should be orthogonal, and hence uncorrelated, by definition. Let’s visualize this directly.

The new representation of the data is stored as a matrix named “x” in the list object we get back from “prcomp”. In our case, the matrix would be stored as “pc$x”.

# feature correlation before PCA
cor_r <- cor(pc$x, method = "pearson")
# use only upper triangular matrix to avoid redundancy
upt_r <- cor_r[upper.tri(cor_r)]
# plot correlations as histogram
hist(upt_r, prob = TRUE)
# plot correlations as image
image.plot(cor_r, axes = FALSE)

pca_part3_fig2

The new features are clearly no longer correlated to each other. As everything seems to be in order, we can now focus on the interpretation of the results.

Step 3: Interpret the results

The first thing you will want to check is how much variance is explained by each component. In PCA speak, this can be visualized with a “scree plot”. R conveniently has a built-in function to draw such a plot.

# draw a scree plot
screeplot(pc, npc = 10, type = "line")

pca_part3_fig3

This is about as good as it gets. A large amount of the variance is captured by the first principal component followed by a sharp decline as the remaining components gradually explain less and less variance approaching zero.

The decision of how many components we should use to get a good approximation of the data has to be made on a case-by-case basis. The cut-offs for the percent explained variance depends on the kind of data you are working with and its inherent covariance structure. The majority of the data sets you will encounter are not nearly as well behaved as this one, meaning that the decline in explained variance is much more shallow. Common cut-offs range from 80% to 95% of explained variance.

Let’s look at how many components we would need to explain a given amount of variance. In the R implementation of PCA, the variances explained by each principle component are stored in a vector called “sdev”. As the name implies, these are standard deviations or the square roots of the variances, which in turn are scaled versions of the eigenvalues. We will need to take the squares “sdev” to get back the variances.

# calculate explained variance as cumulative sum
# sdev are the square roots of the variance
var_expl <- cumsum(pc$sdev^2) / sum(pc$sdev^2)
# plot explained variance
plot(c(0, var_expl), type = "l", lwd = 2, ylim = c(0, 1), 
     xlab = "Principal Components", ylab = "Variance explained")
# plot number of components needed to for common cut-offs of variance explained
vars <- c(0.8, 0.9, 0.95, 0.99)
for (v in vars) {
npc <- which(var_expl > v)[1]
    lines(x = c(0, npc, npc), y = c(v, v, 0), lty = 3)
    text(x = npc, y = v - 0.05, labels = npc, pos = 4)
    points(x = npc, y = v)
}

pca_part3_fig4

The first principle component on its own explains more than 50% of the variance and we need only 20 components to get up to 80% of the explained variance. Fewer than 30% of the components (162 out of 561) are needed to capture 99% of the variance in the data set. This is a dramatic reduction of complexity. Being able to approximate the data set with a much smaller number of features can greatly speed up downstream analysis and can help to visualize the data graphically.

Finally, let’s investigate whether “variance” translates to “information”. In other words, do the prinicipal components associated with the largest eigenvalues discriminate between the different human activities?

If the class labels (“activities” in our case) are known, a good way to do look at the “information content” of the principal components is to look at scatter plots of the first couple of components and color-code the samples by class label. This code gives you a bare bones version of the figure shown below. The complete code can be found on Github.

# plot the first 8 principal components against each other
for(p in seq(1, 8, by = 2)) {
  plot(pc$x[, p:(p+1)], pch = 16, 
       col = as.numeric(description$activity_name))
}

pca_part3_fig5

We have seen previously that the first component alone explains about half of the variance and in this figure we see why. It almost perfectly separates non-moving “activities” (“laying”, “sitting”, “standing”) from moving activities (various types of “walking”). The second component does a reasonable job at telling the difference between walking and walking upstairs. As we move down the list, there remains visible structure but distinctions become somewhat less clear. One conclusion we can draw from this visualization is that it will most likely be most difficult to tell “sitting” apart from “standing” as none of the dimensions seems to be able to distinguish red and green samples. Oddly enough, the fifth component does a pretty good job of separating “laying” from “sitting” and “standing”.

Recap

PCA can be a powerful technique to obtain low dimensional approximations of data with lots of redundant features. The “Human Activity Recognition Using Smartphones Data Set” used in this tutorial is a particularly good example of that. Most real data sets will not be reduced to a few components so easily while retaining most of the information. But even cutting the number of features in half can lead to considerable time savings when using machine learning algorithms.

Here are a couple of useful questions when approaching a new data set to apply PCA to:

  1. Are the features numerical or do I have to convert categorial features?
  2. Are there missing values and if yes, which strategy do I apply to deal with them?
  3. What is the correlation structure of the data? Will PCA be effective in this case?
  4. What is the distribution of variances after PCA? Do I see a steep or shallow decline in explained variance?
  5. How much “explained variance” is a good enough approximation of the data? This is usually a compromise between how much potential information I am willing to sacrifice for cutting down computation time of follow-up analyses.

In the final part of this series, we will discuss some of the limitations of PCA.


Addendum: Understanding “prcomp”

The “prcomp” function is very convenient because it caclulates all the numbers we could possible want from our PCA analysis in one line. However, it is useful to know how those number were generated.

The three most frequently used objects returned by “prcomp” are

  • “rotation”: right eigenvectors (“feature eigenvectors”)
  • “sdev”: square roots of scaled eigenvalues
  • “x”: projection of original data onto the new features

Rotation

In Part 2, I mentioned that software implementations of PCA usually compute the eigenvectors of the data matrix using singular value decomposition (SVD) rather than eigendecomposition of the covariance matrix. In fact, R’s “prcomp” is no exception.

“Rotation” is a matrix whose columns are the right eigenvalues of the original data. We can reconstruct “rotation” using SVD.

# perform singular value decomposition on centered and scaled data
sv <- svd(scale(measurements))
# "prcomp" stores right eigenvectors in "rotation"
w <- pc$rotation
dimnames(w) = NULL
# "svd" stores right eigenvectors in matrix "v"
v <- sv$v
# check if the two matrices are equal
all.equal(w, v)

Sdev

Singular values are the square roots of the eigenvalues as we have seen in Part 2. “sdev” stands for standard deviation and thus stores the square roots of the variances. Thus, the squares of “sdev” and the squares of the singular values are directly proportional to each other and the scaling factor is the number of rows of the original data matrix minus 1.

# relationship between singular values and "sdev"
all.equal(sv$d^2/(nrow(sv$u)-1), pc$sdev^2)

x

The projection of the orignal data (“measurements”) onto its eigenbasis is automatically calculated by “prcomp” through its default argument “retx = TRUE” and stored in “x”. We can manually recreate the projection using matrix-matrix multiplication.

# manual projection of data
all.equal(pc$x, scale(measurements) %*% pc$rotation)

If we wanted to obtain a projection of the data onto a lower dimensional subspace, we just determine the number of components needed and subset the columns of matrix “x”. For example, if we wanted to get an approximation of the original data preserving 90% of the variance, we take the first 52 columns of “x”.

# projection of original data preserving 90% of variance
y90 <- pc$x[, 1:52]
# note that this is equivalent matrix multiplication with 
# the first 52 eigenvectors
all.equal(y90, scale(measurements) %*% pc$rotation[, 1:52])


Reproducibility

The full R code is available on Github.


Further reading

IRIS data set

Sebastian Raschka – Principle Component Analysis in 3 Simple Steps


PCA SERIES

Part 1: An Intuition

Part 2: A Look Behind The Curtain

Part 3: In the Trenches

Part 4: Potential Pitfalls

Part 5: Eigenpets

PCA – Part 3: In the Trenches