# Why is NGS read alignment so slow (fast)?

Modern sequencing machines produce a wealth of information. Unfortunately, it comes in the form of a jumble of millions of short sequence fragments (“reads”), for which we don’t know where on the genome they originated from. The job of read alignment tools like Bowtie2, BWA, or STAR is to bring order to chaos and tell us the likely location of a read on a reference genome.

Even if we have access to a high performance computer cluster, the alignment step is computationally expensive and thus rate-limiting in many next-generation sequencing pipelines. In perfect wet-lab biologist jargon, I have heard it referred to as the “overnight step”. Why does it take so long to determine the location of a short fragment to a reference sequence?

The bioinformatician’s question of how to map a read to a genome translates to the computer scientist’s question of how to match a pattern to a string. Luckily, the latter is a well-studied problem.

### A naive approach

The simplest solution would be to take a read (the “pattern”) and slide it along the genomic sequence (the “string”) and compare at each position whether the read matches the genome. This “brute force” approach is perfectly valid and works just fine. Let’s take a look at some numbers to see if it is feasible in our context.

The human genome has approximately 3 billion bases, the typical length of a read is around 100 bases, and a typical experiment has about 10 million reads per sample. This means that a single read has roughly 3 billion possible position and for each position we would have to make 100 comparisons. Those 300 billion comparisons must be made for each of the 10 million reads, so we would end up with the fantastically large number of 3 quintillion comparisons per sample. Usually we have more than one sample. If we allow for even a single mismatch, things get completely out of hand. The brute force approach is clearly not an option.

Note that this is the worst case scenario and that there are better ways of searching a short pattern in a string. Even so, the major problem why the the brute force approach is slow remains: It makes many unnecessary comparisons. In other words, it searches for matches in areas where there is no chance of finding anything useful.

### The power of indexing

When phones still had cords, people used to either memorize numbers or look them up in a phone book. If your friend’s last name started with an “S”, you wouldn’t look for his or her number in the “T” section. The implicit assumption was that all last names were ordered alphabetically and there was no chance that Mrs. Smith was to be found next to Mr. Taylor. By listing the names of people in alphabetical order, a phone book effectively limits the search space and allows you to find any number relatively quickly. A structure that facilitates lookup of large volumes of data using keys is called an “index“.

Like phone books, suffix arrays are structures that are designed for efficient searching of large bodies of text. To construct a suffix array, you sort all substrings of the original string that contain the last character (“suffixes”) in lexicographical order and record the positions relative to the unsorted suffixes. The effect is similar to a phone book. Suffixes starting with the identical character end up next to each other in the array. With a suffix array to guide the search looking up the location of a pattern in a string is extremely fast because every occurrence of the pattern is equivalent to locating the suffixes that begin with the pattern. Two binary searches for the start and end positions of the pattern within the suffix array result in the location of the pattern within the string. There are two problems, however.

The first issue is that we need to invest time to construct the suffix array from the original sequence of characters. Usually that’s not a serious problem because there exist algorithms to build suffix arrays that scale well to extremely long strings such as the human genome. On top, re-use of the index for multiple sequencing experiments will amortize the cost in time that went into building the index.

The second problem is more serious. It takes multiple times the space of the raw string to store a suffix array. This remains true even if we only implicitly store the order of the suffixes rather than the suffixes themselves. The memory footprint of the raw sequence of the human genome is on the order of several gigabytes. Working with a suffix array multiple times that size becomes troublesome if we want to keep it in RAM. This is a good examples that besides accuracy and speed, memory consumption is also an important consideration when determining how useful an algorithm or data structure is.

What we are looking for is a data structure that combines the fast lookup times of a suffix array with the low memory footprint of the brute-force approach.

### The Burrows-Wheeler transform

The Burrows-Wheeler transform (BWT) is a reversible permutation of a sequence of characters that is more “compressible” than the original sequence. The BWT involves lexicographical sorting of all permutations of a string so that identical characters end up next to each other. This facilitates compact encoding as a sequence of ten A’s can be stored as “AAAAAAAAAA” or “10A” without loss of information.

There are important similarities between the suffix array and the BWT. Recall that a suffix array involves lexicographical sorting of all suffixes of a string. The BWT is the last column of a matrix of all lexicographical sorting of all permutations of a string. As the suffixes are necessarily contained in the string permutations and both are lexicographically sorted, the order of the elements in a suffix array and the Burrows-Wheeler permutation matrix must be identical. In fact, the BWT can be efficiently calculated from a suffix array and the BWT implicitly encodes a suffix array.

The BWT allows for a compact representation of the original string but it is by itself not very well suited for fast lookup of the location of a pattern. By augmenting the BWT with a table of ranks of each character in the BWT and a partial suffix arrays, we obtain a data structure that gets very close to the fast pattern matching found in suffix arrays while maintaining a memory footprint close to the raw string. Such a data structure is called an FM-index and is the basis of alignment tools like Bowtie2 and BWA (Burrows-Wheeler Aligner). The commands “bowtie2-build” and “bwa index” build the respective software-specific FM-indices that are used for running the alignment.

### How to handle mismatches?

The FM-index based mapping of reads to the genome is only fast for exact matches. In practice, read mapping must be tolerant to mismatches. Mismatches can occur due to technical reasons such as PCR artifacts or incorrect base calling during the sequencing process. Conversely, true variations between sequences (SNPs, CNV, Indels) are among the most interesting biological results we have obtained from having sequenced thousands of human genomes. We definitely do not want an alignment tool to discard all reads with mismatches by default just because they aren’t perfect matches to the reference genome.

One strategy that is used in modern sequence alignment tools like Bowtie2 is to split the reads into “seeds”. Exact matches of seeds to the genome are found using the FM-index and then extended using variants of more sensitive sequence alignment algorithms like Needleman-Wunsch or Smith-Waterman. In this way, Bowtie2 balances the speed of finding the exact location of reads on the genome with a certain error tolerance that allows the identification of possibly interesting sequence variants.

What constitutes a valid alignment and how it scores can be tuned by the user with the help of command line arguments such as the number of allowed mismatches and gap penalties. This is where the biggest differences between alignment tools is observed. Which alignment tool to use is ultimately a matter of personal preference. In general, BWA is thought to have higher precision and is thus favored in variant calling, while Bowtie2 appears to be faster and more sensitive but may lack some of BWA’s precision.

The development of fast and accurate read alignment tools was an essential contribution to the current boom in genomics research. Without decades of research and algorithm development in computer science, we would be waiting for days or weeks for our read alignments to finish. So what are a few hours?

Other posts on next-generation sequence analysis:

Why we use the negative binomial distribution to model sequencing reads?

# 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.

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.

# 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.

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.

#### 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))


#### What 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 = "_")


Here 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.

Tukey’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()


We 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.

# 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.

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.

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.

# Tidy unnesting

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

Such 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.

### 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
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.

# PCA – Part 4: Potential Pitfalls

In the first three parts of this series on principal component analysis (PCA), we have talked about what PCA can do for us, what it is mathematically, and how to apply it in practice. Today, I will briefly discuss some of the potential caveats of PCA.

### INformation and Noise

PCA looks for the dimensions with highest variance within the data and assumes that high variance is a proxy for “information”. This assumption is usually warranted otherwise PCA would not be useful.

In cases of unsupervised learning, that is if we have no class labels of the data available, looking for structure within the data based on the data itself is our only choice. In a sense, we cannot tell what parts of the data are information and what parts are noise.

If we have class labels available (supervised learning), we could in principle look for dimensions of variance that optimally separate the classes from each other. PCA does not do that. It is “class agnostic” and thus treats “information”-variance and “noise”-variance the same way.

It is possible that principle components associated with small eigenvalues nevertheless carry the most information. In other words, the size of the eigenvalue and the information content are not necessarily correlated. When choosing the number of components to project our data, we could thus lose important information. Luckily, such situations rarely happen in practice. Or we just never realize …

There are other techniques related to PCA that attempt to find dimensions of the data that optimally separate the data based on class labels. The most famous is Fisher’s “Linear Discriminant Analysis” (LDA) and its non-linear cousins “Quadratic Discriminant Analysis” (QDA).

### Interpretability

In Part 3 of this series, we have looked at a data set containing a multitude of motion detection measurements of humans doing various activities. We used PCA to find a lower dimensional representation of those measurements that approximate the data well.

Each of the original measurements were quite tangible (despite their sometimes cryptic names) and therefore interpretable. After PCA, we are left with linear combinations of those original features, which may or may not be interpretable. It is far from guaranteed that the eigenvectors correspond to “real” entities, they may just be convenient summaries of the data.

We will rarely be able to say the first principle component means “X” and the second principle component means “Y”, however tempting it may be based on our preconceived notions of the data. A good example of that is mentioned in Cosma Shalizi’s excellent notes on PCA. Cavalli-Sforza et al. analyzed the distribution of human genes using PCA and interpreted the principal components as patterns of human migration and population expansion. Later, November and Stephens showed that similar patterns could be obtained using simulated data with spatial correlation. As humans are genetically more similar to humans they close to (at least historically), genetic data is necessarily spatially correlated and thus PCA will uncover such structures, even if they do not represent “real” events or are liable to misinterpretation.

### Independence

Linear algebra tells us that eigenvectors are orthogonal to each other. A set of $n$ orthogonal vectors form a basis of an $n$-dimensional subspace. The principle components are eigenvectors of the covariance matrix and the set of principle components form a basis for our data. We also say that the principle components are “uncorrelated”. This becomes obvious when we remember that matrix decomposition is sometimes called “diagonalization”. In the eigendecomposition, the matrix containing the eigenvalues has zeros everywhere but on its diagonal, which contains the eigenvalues.

Variance and covariance are measures in the L2 norm, which means that they involve the second moment or square. Being uncorrelated in the L2 norm does not mean that there is no “correlation” in higher norms, in other words the absence of correlation does not imply independence. In statistics, higher order norms are skew (“tailedness” or third moment) and kurtosis (“peakedness” or fourth moment). Techniques related to PCA such as Independent Component Analysis (IDA) can be used to extract two separate, but convolved signals (“independent components”) from each other based on higher order norms.

The distinction between correlation and independence is a technical point when it comes to the practical application of PCA but certainly worth being aware of.

Cosma Shalizi – Principal Components: Mathematics, Example, Interpretation

Cavalli-Sforza et al. – The History and Geography of Human Genes (1994)

Novembre & Stephens – Interpreting principal component analyses of spatial genetic variation (2008)

### 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

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


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.

We 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)


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")


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)
}


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))
}


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.

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.

##### 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