Sampling from a probability distribution

When simulating any system with randomness, sampling from a probability distribution is necessary. Usually, you’ll just need to sample from a normal or uniform distribution and thus can use a built-in random number generator. However, for the time when a built-in function does not exist for your distribution, here’s a simple algorithm.

Let’s say you have some probability density function (PDF) \rho(x) on some domain [x_{\min},x_{\max}] and you want to generate a set of numbers that follows this probability law. This is actually very simple to do although those new to the field may not know how. Generally, any programming language or environment has a built-in random number generator for real numbers (up to machine precision) between 0 and 1. Thus, what you want to do is to find a way to transform these random numbers, which are uniformly distributed on the domain [0,1] to the random numbers you want. The first thing to notice is that the cumulative distribution function (CDF) for your PDF, P(x) = \int_{x_{\min}}^x \rho(x') dx' is a function that ranges over the interval [0,1], since it is a probability. Thus, the algorithm is: 1) generate a random number r between o and 1, and 2) plug it into the inverse of P(x). The numbers you get out satisfy your distribution (i.e. P(x)=r \Rightarrow x = P^{-1}(r) ).

If you’re lucky enough to have a closed form expression of the inverse of the CDF P^{-1}(r) then you can just plug the random numbers directly into your function, otherwise you have to do a little more work. If you don’t have a closed form expression for the CDF then you can just solve the equation P(y)-r=0 for x. Newton’s method will generally converge very quickly.  You just loop over the command x = x – (P(x)-r)/rho(x) until you reach the tolerance you want. If you don’t have a closed form expression of the CDF then the simplest way is construct the CDF numerically by computing the N dimensional vector v[j]=\sum_{i=0}^j \rho(jh-x_{\min}) h, for some discretization parameter h such that x_{\max}= Nh .  If the PDF is defined on the entire real line then you’ll have to decide on what you want the minimum x_{\min} to be. If the distribution has thin tails, then you just need to go out until the probability is smaller than your desired error tolerance. You then find the index j such that v[j] is closest to r (i.e. in Julia you could use findmin(abs(v[j]-r))) and your random number is x = hj.

Here is why the algorithm works. What we are doing is transforming the PDF of r, which we can call u(r), to the desired PDF for x. The transformation we use is r = P(x). Thus, u(r) dr = \rho(x) dx or \rho(x) = u(r) dr/dx. Now, u(r) is just 1 on the interval [0,1] and dr/dx is the PDF of x as expected. Thus, inverting the CDF for uniform random numbers always gives you a new set of random numbers that satisfies the new PDF. Trivial perhaps, but really useful.

R vs Matlab

It has now been almost half a year since I switched from Matlab to open source software and I’ve been amazed at how easy the transition has been. I had planned to replace Matlab with Python, Julia, and R but I have found that R and Julia have been sufficient for my requirements. Maxima is also more than adequate to replace Mathematica. I have become particularly attached to R especially after I started to use R Studio as the interface. I had only used R before as just a statistics tool but it really is a complete programming platform like Matlab. It has very nice graphics capabilities and I find the language very nice to program in. I really like its use of lists where I can pile sets of any type and any size into one object. I also like how R Studio can save your work into Projects, which keeps the whole environment and history in one place. I can then switch between multiple projects and everything comes back. The only thing I miss from Matlab is the command completion history feature, where I could easily find a previous command by just typing the first few letters. Also, I haven’t quite figured out how to output R data into a text file seamlessly yet. I seem to always get extraneous row or column information. I use Julia when I want to write a code that needs to loop fast but for everything else I’ve been reaching for R.

New paper on genomics

James Lee and I have a new paper out: Lee and Chow, Conditions for the validity of SNP-based heritability estimation, Human Genetics, 2014. As I summarized earlier (e.g. see here and here), heritability is a measure of the proportion of the variance of some trait (like height or cholesterol levels) due to genetic factors. The classical way to estimate heritability is to regress standardized (mean zero, standard deviation one) phenotypes of close relatives against each other. In 2010, Jian Yang, Peter Visscher and colleagues developed a way to estimate heritability directly from the data obtained in Genome Wide Association Studies (GWAS), sometimes called GREML.  Shashaank Vattikuti and I quickly adopted this method and computed the heritability of metabolic syndrome traits as well as the genetic correlations between the traits (link here). Unfortunately, our methods section has a lot of typos but the corrected Methods with the Matlab code can be found here. However, I was puzzled by the derivation of the method provided by the Yang et al. paper.  This paper is our resolution.  The technical details are below the fold.

 

Continue reading

Paper on compressed sensing and genomics

New paper on the arXiv. The next step after the completion of the Human Genome Project, was the search for genes associated with diseases such as autism or diabetes. However, after spending hundreds of millions of dollars, we find that there are very few common variants of genes with large effects. This doesn’t mean that there aren’t genes with large effect. The growth hormone gene definitely has a large effect on height. It just means that variations of genes that are common among people have small effects on the phenotype. Given the results of Fisher, Wright, Haldane and colleagues, this was probably expected as the most likely scenario and recent results measuring narrow-sense heritability directly from genetic markers (e.g. see this) confirms this view.

Current GWAS microarrays consider about a million or two markers and this is increasing rapidly. Narrow-sense heritability refers to the additive or linear genetic variance, which means the phenotype is given by the linear model y= Z\beta + \eta, where y is the phenotype vector, Z is the genotype matrix, \beta are all the genetic effects we want to recover, and \eta are all the nonadditive components including environmental effects. This is a classic linear regression problem. The problem comes when the number of coefficients \beta far exceeds the number of people in your sample, which is the case for genomics. Compressed sensing is a field of high dimensional statistics that addresses this specific problem. People such as David Donoho, Emmanuel Candes and Terence Tao have proven under fairly general conditions that if the number of nonzero coefficients are sparse compared to the number samples, then the effects can be completely recovered using L1 penalized optimization algorithms such as the lasso or approximate message passing. In this paper, we show that these ideas can be applied to genomics.

Here is Steve Hsu’s summary of the paper

Application of compressed sensing to genome wide association studies and genomic selection

(Submitted on 8 Oct 2013)

We show that the signal-processing paradigm known as compressed sensing (CS) is applicable to genome-wide association studies (GWAS) and genomic selection (GS). The aim of GWAS is to isolate trait-associated loci, whereas GS attempts to predict the phenotypic values of new individuals on the basis of training data. CS addresses a problem common to both endeavors, namely that the number of genotyped markers often greatly exceeds the sample size. We show using CS methods and theory that all loci of nonzero effect can be identified (selected) using an efficient algorithm, provided that they are sufficiently few in number (sparse) relative to sample size. For heritability h2 = 1, there is a sharp phase transition to complete selection as the sample size is increased. For heritability values less than one, complete selection can still occur although the transition is smoothed. The transition boundary is only weakly dependent on the total number of genotyped markers. The crossing of a transition boundary provides an objective means to determine when true effects are being recovered. For h2 = 0.5, we find that a sample size that is thirty times the number of nonzero loci is sufficient for good recovery.

Comments: Main paper (27 pages, 4 figures) and Supplement (5 figures) combined
Subjects: Genomics (q-bio.GN); Applications (stat.AP)
Cite as: arXiv:1310.2264 [q-bio.GN]
(or arXiv:1310.2264v1 [q-bio.GN] for this version)

Most of neuroscience is wrong

John Ioannidis has a recent paper in Nature Reviews Neuroscience arguing that many results in neuroscience are wrong. The argument follows his previous papers of why most published results are wrong (see here and here) but emphasizes the abundance of studies with small sample sizes in neuroscience. This both reduces the chances of finding true positives and increases the chances of obtaining false positives. Under powered studies are also susceptible to what is called the “winner’s curse” where the effect sizes of true positives are artificially amplified. My take is that any phenomenon with a small effect should be treated with caution even if it is real. If you really wanted to find what causes a given disease then you probably want to find something that is associated with all cases, not just in a small percentage of them.

Bayesian model comparison Part 2

In a previous post, I summarized the Bayesian approach to model comparison, which requires the calculation of the Bayes factor between two models. Here I will show one computational approach that I use called thermodynamic integration borrowed from molecular dynamics. Recall, that we need to compute the model likelihood function

P(D|M)=\int P((D|M,\theta)P(\theta|M) d\theta     (1)

for each model where P(D|M,\theta) is just the parameter dependent likelihood function we used to find the posterior probabilities for the parameters of the model.

The integration over the parameters can be accomplished using the Markov Chain Monte Carlo, which I summarized previously here. We will start by defining the partition function

Z(\beta) = \int P(D|M,\theta)^\beta P(\theta| M) d\theta    (2)

where \beta is an inverse temperature. The derivative of the log of the partition function gives

\frac{d}{d\beta}\ln Z(\beta)=\frac{\int d\theta \ln[P(D |\theta,M)] P(D | \theta, M)^\beta P(\theta|M)}{\int d\theta \ P(D | \theta, M)^\beta P(\theta | M)}    (3)

which is equal to the ensemble average of \ln P(D|\theta,M). However, if we assume that the MCMC has reached stationarity then we can replace the ensemble average with a time average \frac{1}{T}\sum_{i=1}^T \ln P(D|\theta, M).  Integrating (3) over \beta from 0 to 1 gives

\ln Z(1) = \ln Z(0) + \int \langle \ln P(D|M,\theta)\rangle d\beta

From (1) and (2), we see that  Z(1)=P(D|M), which is what we want to compute  and Z(0)=\int P(\theta|M) d\theta=1.

Hence, to perform Bayesian model comparison, we simply run the MCMC for each model at different temperatures (i.e. use P(D|M,\theta)^\beta as the likelihood in the standard MCMC) and then integrate the log likelihoods Z(1) over \beta at the end. For a Gaussian likelihood function, changing temperature is equivalent to changing the data “error”. The higher the temperature the larger the presumed error. In practice, I usually run at seven to ten different values of \beta and use a simple trapezoidal rule to integrate over \beta.  I can even do parameter inference and model comparison in the same MCMC run.

Erratum, 2013-5-2013,  I just fixed an error in the final formula

Prediction requires data

Nate Silver has been hailed in the media as a vindicated genius for correctly predicting the election. He was savagely attacked before the election for predicting that Obama would win handily. Kudos also go to Sam Wang, Pollester.com, electoral-vote.com, and all others who simply took the data obtained from the polls seriously. Hence, the real credit should go to all of the polling organizations for collectively not being statistically biased.  It didn’t matter if single organizations were biased one way or the other as long as they were not correlated in their biases. The true power of prediction in this election was that the errors of the various pollsters were independently distributed. However, even if you didn’t take the data at face value, you could still reasonably predict the election. Obama had an inherent advantage because he had more paths to winning 270 electoral votes. Suppose there were 8 battleground states and Romney needed to win at least 6 of them. Hence, Romney had 28 ways to win while Obama had 228 ways to win. If the win probability was approximately a half in each of these states, which is what a lot of people claimed,  then Romney has slightly more than one in ten chance of winning, which is close to the odds given by Sam. The only way Romney’s odds would increase is if the state results were correlated in his favour. However, it would take a lot of correlated bias to predict that Romney was a favourite.

 

Erratum, Nov 9 2012:  Romney actually has 37 ways and Obama 219 in my example.  The total must add up to 2^8=256.  I forgot to include the fact that Romney could also win 7 of 8 states or all states in his paths to winning.