Mutual Information Between Two Continuous Distributions
PLoS One. 2014; 9(2): e87357.
Mutual Information between Discrete and Continuous Data Sets
Brian C. Ross
Department of Physics, University of Washington, Seattle, Washington, United States of America
Daniele Marinazzo, Editor
Received 2013 Nov 26; Accepted 2013 Dec 19.
- Supplementary Materials
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Script S1: Slow (vector) MI calculator. Estimates MI between two vector or scalar data sets using the nearest-neighbor method.
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GUID: A023BB14-AEDB-4609-AAB3-85587FD6F32B
Script S2: Fast (scalar) MI calculator. Estimates MI between two scalar data sets using the nearest-neighbor method.
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GUID: 6F9570C9-7E93-4D3C-8A01-57F5FBF555DE
Script S3: Binning MI calculator. Estimates MI between two scalar data sets using the binning method.
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GUID: F40B4602-07AB-4288-97B4-CA8B9FB8C21B
Script S4: Testing script. Compares the methods using sampled data drawn from user-defined distributions. This script was used to generate the plots in this paper.
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GUID: FC80087F-3117-41F3-9B26-7AB3E11460B9
Abstract
Mutual information (MI) is a powerful method for detecting relationships between data sets. There are accurate methods for estimating MI that avoid problems with "binning" when both data sets are discrete or when both data sets are continuous. We present an accurate, non-binning MI estimator for the case of one discrete data set and one continuous data set. This case applies when measuring, for example, the relationship between base sequence and gene expression level, or the effect of a cancer drug on patient survival time. We also show how our method can be adapted to calculate the Jensen–Shannon divergence of two or more data sets.
Introduction
Mutual information (MI) [1] is in several ways a perfect statistic for measuring the degree of relatedness between data sets. First, MI will detect any sort of relationship between data sets whatsoever, whether it involves the mean values or the variances or higher moments. Second, MI has a straightforward interpretation as the amount of shared information between data sets (measured in, for example, bits); other statistics such as rank-ordering are harder to interpret. Since MI is grounded in information theory it has an established base of theoretical tools. Finally, MI is insensitive to the size of the data sets. Whereas a 'p-value' test for strict independence can be pushed arbitrarily low by taking a large data set if the variables are even slightly related, MI will simply converge with tight error bounds to a measure of their relatedness.
The MI between two data sets 
and 
can be estimated from the statistics of the 
pairs between the two data sets. (Although MI is straightforward to calculate if the underlying probability distribution is known, that is not usually the case: our knowledge of the distribution generally comes from the sampled data itself, so MI must be estimated from the statistics of our data set.) For example, if we were to compare the day of week (
) with the time of breakfast (
) we might find that when 
is a weekday the corresponding 
is early in the morning, and when 
is Sunday or (especially) Saturday the corresponding 
is somewhat later. MI quantifies the strength of this effect. Importantly, the procedure for estimating MI depends on whether 
and 
take discrete values (e.g. a day of week, a nucleobase, a phenotypic category, etc.), or are real-valued continuous variables (a time of day, a gene expression level, a patient's survival time, etc.). If 
and 
are both discrete, then we can estimate the true frequencies of all combinations of 
pairs by counting the number of times each pair occurs in the data, and straightforwardly use these frequencies to estimate MI. Real-valued data sets are more difficult to deal with, since they are by definition sparsely sampled: most real numbers will not be found in a data set of any size. The common workaround is to lump the continuous variables into discrete 'bins' and then apply a discrete MI estimator, but good sampling requires large bins which destroys resolution. An improved continuous-continuous MI estimator described in Ref. [2] circumvents this tradeoff by using statistics of the spacings between data points and their nearest neighbors. Crucially, their method only works when both variables are real-valued, as the nearest neighbor of a discrete variable is not well-defined.
This paper describes a method for estimating the MI between a discrete data set and a continuous (scalar or vector) data set, using a similar approach to that of Ref. [2]. This is an important statistic simply because so many scientific activities involve a search for significant relationships between discrete and continuous variables. For example, one might use MI to quantify the extent to which nationality (a discrete variable) determines income (continuous); to identify DNA bases (ACGT, discrete) that affect a given gene's expression level (continuous); or to find drugs (given or not: a discrete parameter) that alter cell division rates (continuous data). In the University of Washington Nanopore Physics lab we use this estimator to determine where a given DNA base must sit within the sequencing pore in order to affect the current passing through it, and to quantify the relative influence of different base positions on the current. As we will demonstrate, our nearest-neighbors method estimates MI much more reliably than does the present alternative method of 'binning' the data.
MI between a discrete and a continuous variable is equivalent to a weighted form of the Jensen-Shannon (JS) divergence [3] which is used as a measure of the dissimilarity between two or more continuous probability distributions. We can therefore apply our method to estimate the weighted JS divergence, by storing samples from each distribution to be compared in the continuous data set 
, and using the discrete data set 
to identify which distribution each sample was drawn from. To use our method to estimate the unweighted JS divergence, we would either draw equal numbers of samples from each distribution, or else modify our method somewhat as explained in the Analysis section.
Results
To test our method, we chose two simple distributions 
: a square wave distribution in 
for each value in 
, and a Gaussian distribution in 
for each 
(Figure 2A). Because we knew the exact form of the distributions, we were able to calculate MI exactly using its mathematical definition:
(4)
Next, from each distribution, we constructed test data sets by randomly sampling a certain number 
of 
data pairs. We then independently estimated MI from those data sets using our nearest-neighbor estimator and also using our binning estimator, and compared those estimates to each other and to the exact result. We also compared the MI estimate between our vector and scalar implementations of the nearest-neighbor method. Their results in all cases are in exact agreement with each other. This is a strong check that the scripts were written correctly, since the two estimators were coded quite differently.
Both the nearest-neighbor method and the binning method involve a somewhat arbitrary parameter that must be set by the user. The nearest neighbor method requires that the user specify 
(the 
th neighbor). 
should be some low integer, much less than the number of data points 
, so Figure 2B plots MI estimated by nearest neighbors over the range 
. Likewise, the binning method requires that the user specify the number of data points 
per bin. It is less obvious what the best value of 
should be; Figure 2C plots MI estimated by binning over all possible values 
.
Our first conclusion is that there is a much simpler prescription for setting the 
parameter of the nearest-neighbor estimator than the 
parameter of the binning method. The nearest-neighbor estimator consistently gives good results when 
is set to a low integer. Reference [2] suggests using 
, and that choice works well with our estimator too. By contrast, the binning estimator overestimates MI when 
is low and underestimates MI when 
is high, and although there is guaranteed to be a crossing point where the method is accurate it is hard to guess where that point might be. (In the limit 
the binning method estimates MI to be the entropy of the discrete variable. The actual MI only attains this maximum limit if the sub-distributions 
are all completely separated in 
. In the limit 
the binning method estimates MI to be zero.).
Our second conclusion is that there is no simple way to calculate the optimal binning parameter 
based on simple statistics of the data, such as the total number of data points 
or frequencies with which different discrete symbols occur. For example, the large Gaussian data sets and the large square-wave data sets each have 10000 data points per set, with twice as many red points as blue points on average, and five times more reds than greens. But the best value of 
is ∼100 for the square-wave data set and ∼600 for the Gaussian data sets. This is easiest to see in Figure 3A, which plots the ratio of the median binning error using given 
to the median nearest-neighbors error using 
. We find that there is no choice of 
for which binning is better than nearest-neighbors for both the square wave and Gaussian data sets. Figure 3B shows roughly the same result for the 400-point data sets, which again are statistically similar except in the shape of their distributions in 
.
We conclude that MI estimation by the nearest neighbor method is far more accurate than binning-based MI estimates, barring a lucky guess of the unknowable best value of 
. Furthermore, our nearest-neighbor method is computationally cheap: both computation time and memory usage are proportional to 
for the scalar estimator. Therefore nearest neighbors should be the method of choice for estimating MI in the discrete-continuous case.
Analysis
Here we derive the formula for our nearest-neighbor MI estimator.
Consider a discrete variable 
and the continuous variable 
, drawn from probability density 
. Both 
and 
may be either univariate (composed of scalars) or multivariate (vectors). We will write discrete probability functions as 
and continuous densities using the symbol 
: therefore 
and 
. The mutual information is:
(5)
Here 
denotes an entropy, 
is the probability density for sampling 
irrespective of the value of 
, and 
is the probability density for sampling 
given a particular value of 
. The averages are taken over the full distribution and weighted by 
, and they would be straightforward to calculate if we knew the underlying density functions. Alternatively, each average can be taken over a representative set from 
pairs sampled from the distribution; using this latter interpretation we estimate the MI from the mean of 
and 
at each of our sampled data points. The more points we have, the greater the accuracy.
The remaining task is to estimate the logarithm of two continuous distributions evaluated at given data points. For this we use a nearest-neighbor entropy estimator originally developed by Kozachenko and Leonenko [5] whose proof we will briefly outline. Given a point 
, we define 
as the volume of points centered about 
that are closer to point 
than its 
th neighbor. The estimator uses Bayesian arguments to identify 
with 
(
denotes a probability density that is not to be confused with 
). Approximating the density function 
as being constant throughout the neighborhood of point 
, we find:
(6)
where 
is the beta function [4] and 
is the digamma function. We can now estimate the entropy using the full data set:
(7)
where the average is taken over all sampled data points.
For each sampled data point 
we employ the Kozachenko-Leonenko (KL) entropy estimator twice: once to estimate 
by finding a neighbor from the full set of data points, and once to estimate 
by finding a neighbor in the subset of data points 
for which 
. Notice that we can independently choose the neighbors of the two points: we will pick the 
th neighbor in the reduced distribution and the 
th neighbor from the full distribution. The result is
(8)
There is a systematic averaging error that comes from the fact that the 
th-neighbor KL entropy estimator applied to point 
necessarily computes the average of 
over the volume 
, rather than evaluated exactly at point 
. Following Ref. [2], we attempt to minimize this error by choosing 
and 
so that both uses of the KL entropy estimator use the same neighbor 
. Therefore 
for each data point, and we obtain Eq. 2. The cancellation is only partial; but because the averaging error scales with the number of data pairs as 
whereas the counting error scales as 
, averaging error is generally insignificant except for very small data sets (as we have verified in our tests).
As mentioned before, the mutual information between discrete and continuous data is equivalent to a weighted Jensen-Shannon (JS) divergence between the conditional distributions 
, where the frequencies 
of the discrete symbols 
are the weighting factors. To compute an unweighted JS divergence we need to place all the conditional distributions on equal footing irrespective of their frequencies in the data, by weighting each term in the averages in Eq. 5 by the factor 
where 
is the number of distinct values that 
can take. The result is
(9)
Supporting Information
Script S1
Slow (vector) MI calculator. Estimates MI between two vector or scalar data sets using the nearest-neighbor method.
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Script S2
Fast (scalar) MI calculator. Estimates MI between two scalar data sets using the nearest-neighbor method.
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Script S3
Binning MI calculator. Estimates MI between two scalar data sets using the binning method.
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Script S4
Testing script. Compares the methods using sampled data drawn from user-defined distributions. This script was used to generate the plots in this paper.
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Acknowledgments
The author wishes to acknowledge Vikram Agarwal and Walter Ruzzo for helpful discussions, and Andrew Laszlo, Henry Brinkerhoff, Jens Gunlach and Jenny Mae Samson for valuable comments on this paper.
Funding Statement
Funding for this work was provided by National Institutes of Health/National Human Genome Research Institute grant R01HG005115. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
References
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3. Grosse I, Bernaola-Galván P, Carpena P, Román-Roldán R, Oliver J, et al. (2002) Analysis of symbolic sequences using the jensen-shannon divergence. Physical Review E 65: 041905. [PubMed] [Google Scholar]
4. Abramowitz M, Stegun I (1970) Handbook of mathematical functions. New York: Dover Publishing Inc. [Google Scholar]
5. Kozachenko L, Leonenko NN (1987) Sample estimate of the entropy of a random vector. Problemy Peredachi Informatsii 23: 9–16. [Google Scholar]
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