Mutual information represents a general information theoretic approach to determine the statistical dependence between variables. The concept was initially developed for discrete data. For a system, A, with a finite set of M possible states {a₁, a₂, ... , }, the Shannon entropy H(A) is defined as 17

where p(a_i) denotes the probability of the state a_i. The Shannon entropy is a measure for how evenly the states of A are distributed. The entropy of system A becomes zero if the outcome of a measurement of A is completely determined to be a_j, thus if p(a_j) = 1 and p(a_i) = 0 for all i ≠ j, whereas the entropy becomes maximal if all probabilities are equal. The joint entropy H(A, B) of two systems A and B is defined analogously

This leads to the relation

H(A, B) ≤ H(A) + H(B)     (3)

which fulfils equality only in the case of statistical independence of A and B. Mutual information MI(A, B) can be defined as 17

MI(A, B) = H(A) + H(B) - H(A, B) ≥ 0     (4)

It is zero if A and B are statistically independent and increases the less statistically independent A and B are.

If mutual information is indeed to be used for the analysis of gene-expression data, the continuous experimental data need to be partitioned into discrete intervals, or bins. In the following section, we briefly review the established procedures; a description of how we have extended the basic approach will be provided in the subsequent section.

In the case of discrete data the estimation of the probabilities p(a_i) is straightforward. Many practical applications, however, supply continuous data for which the probability distributions are unknown and have to be estimated. In a widely used approach 7, the calculation of mutual information is based on the binning of data into M discrete intervals a_i, i = 1... M_A. For experimental data consisting of N measurements of a variable x_u, u = 1... N, an indicator function Θ_i counts the number of data points within each bin. The probabilities are then estimated based on the relative frequencies of occurrence

with

For two variables the joint probabilities are calculated analogously from a multivariate histogram. Additionally it has been suggested 14 to adaptively choose the sizes of the bins, so that each bin constructed nearly has a uniform distribution of points. In a different approach, kernel methods are used for the estimation of the probability density of Eq. (5) 181920. Entropies are then calculated by integration of the estimated densities. Recently, an entropy estimator was suggested 21 and showed in an extensive comparison to other commonly used estimators to be superior.

In the classical binning approach, described above, each data point is assigned to one, and only one, bin. For data points near to the border of a bin, small fluctuations due to biological or measurement noise might shift these points to neighbouring bins. Especially for datasets of moderate size, the positions of the borders of the bins can thereby strongly affect the resulting mutual information 18. In a manner analogous to kernel density estimators (KDE), we now present a generalisation to the classical binning in which we aim to overcome some of the drawbacks associated with the simple approach. Within our algorithm, we allow the data points to be assigned to several bins simultaneously. For this, we extended the indicator function Θ(x) to the set of polynomial B-spline functions. Here, we do not provide the mathematical details for these functions since they have been discussed extensively in the literature 222324, but rather focus on the practical applicability. Within the B-spline approach, each measurement is assigned to more than one bin, i, with weights given by the B-spline functions B_i,k. The spline order k determines the shape of the weight functions and thereby the number of bins each of the data points is assigned to. A spline order k = 1 corresponds to the simple binning, as described in the previous section: Each data point is assigned to exactly one bin (Figure 2, left). For k = 3, each data point is assigned to three bins, with the respective weights given by the values of the B-spline functions at the data point (Figure 2, right).

The first step in the definition of the B-spline functions is the definition of a knot vector t_i for a number of bins i = 1... M and one given spline order k = 1... M - 1 22

where the spline order determines the degree of the polynomial functions. The domain of the B-spline functions lies in the interval z ∈ [0, M - k + 1]. To cover the range of the variables, the new indicator function based on the B-spline functions needs to be linearly transformed to map their range. The recursive definition of the B-spline functions are as follows 22

An important property of B-spline functions is the implicit standardisation of coefficients: All weights belonging to one data point sum up to unity.

We show the estimation with the standard binning and our approach ex-emplarily on two artificial variables x = 0.0,0.2,0.4,0.6,0.8,1.0 and y = 0.8,1.0,0.6,0.4,0.0,0.2 for M = 3 bins, spline order k = 2, and the logarithm to basis two.

It has been discussed elsewhere 2526272820 that the estimated mutual information is systematically overestimated for a finite size of N data points. For the simple binning approach, the mean observed mutual information can be calculated explicitly as the deviation from the true mutual information

As can be seen for an example of artificially generated equidistributed random numbers (Figure 3, left), mutual information calculated from the simple binning scales linearly with 1/N, with the slope depending on the number of bins M in accordance with Eq. (12). Figure 3 shows that this scaling is preserved for the extension to B-spline functions, while the slope is significantly decreased for k = 3, compared to the estimation with the simple binning (k = 1). Mutual information calculated from KDE does not show a linear behaviour but rather an asymptotic one with a linear tail for large datasets. The values are slightly increased compared to the ones from the B-spline approach. The entropy estimator gives values comparable to the ones obtained from the B-spline approach.

More importantly, a similar result also holds for the standard deviation of mutual information. As shown in Figure 3 (right), the standard deviation of the mutual information estimated with the simple binning (k = 1) scales with 1/N for statistically independent events 2629. For the B-spline approach (k = 3), this scaling still holds, but the average values are decreased significantly. For the KDE approach, an asymptotic run above the values from the B-spline approach is observed, again with linear tail for large datasets. shows a linear scaling slightly below the simple binning.

The interpretation of any results obtained from the application of mutual information to experimental data is based on testing to see if the calculated results are consistent with a previously chosen null hypothesis. By following the intuitive approach that the null hypothesis assumes the statistical independence of variables, mutual information is tested against a surrogate dataset, which is consistent with this null hypothesis. As discussed previously in more detail 20, one way of generating such a surrogate dataset is by random permutations of the original data. From the mutual information of the original dataset MI(X,Y)^data, the average value obtained from surrogate data <MI(X^surr, Y^surr) >, and its standard deviation σ^surr, the significance S can be formulated as

For each S the null hypothesis can be rejected to a certain level α depending on the underlying distribution. With increasing significance the probability of false positive associations drops.

In the following, we address the influence of the spline order and the number of bins on the estimation of mutual information. Based on 300 data points of an artificially-generated dataset drawn from the distribution shown in Figure 1, we calculate the mutual information for M = 6 bins and different spline orders k = 1... 5 (Figure 4, left).

From 300 shuffled realisations of this dataset, the mean and maximum mutual information are shown with the standard deviation as error-bars. For all spline orders the null hypothesis can be rejected, in accordance with the dataset shown in Figure 1. To estimate the strength of the rejection, we calculate the significance according to Eq. (13) (Figure 4, right). It can be observed that the largest change in the significance of the mutual information occurs in the transition from k = 1 (simple boxes) to k = 2 with an increase by roughly two-fold. Using more sophisticated functions (k ≥ 3) does not further improve the significance. Similar findings have been reported in the context of kernel density estimators 19. The major contribution leading to this increase of the significance is given by the distribution of surrogate data which becomes more narrow for k > 1 leading to smaller standard deviations σ^surr.

The same dataset is used to show the dependency of mutual information on the number of bins for two spline orders k = 1 and k = 3 (Figure 5). Mutual information estimated from data as well as from surrogate data shows a robust run without strong fluctuations within the range of bins shown. From this we can conclude that the choice of the number of bins does not affect the resulting mutual information notably as long as it is chosen to be within a reasonable range.

Again, the significance is calculated (Figure 6) and compared to the significances obtained from the KDE approach and the estimator. It can be observed that the significance of the mutual information calculated with B-spline functions increased roughly by two-fold compared to the simple binning. The significance obtained from KDE is not depending on M and was determined to be similar to the significance estimated from the B-spline approach. The numerically expensive integration of KDE, however, limits the size of utilisable datasets. The KDE run time requirements were (10⁴) times higher than the ones from the B-spline approach. Strategies to simplify the integration step were proposed 20 but have to be used with caution since they assume particular properties of the distribution of experimental data that are in general not fulfilled. The recently introduced entropy estimator produces intermediate significances between the ones from the binning and the B-spline approach for higher bin numbers. For low bin numbers, the significances are relatively poor.

We now turn to the analysis of experimentally measured gene expression data. As shown previously, the application of mutual information to large-scale expression data reveals biologically-relevant clusters of genes 730. In this section, we will not repeat these analyses, but determine if the correlations detected using mutual information are missed using the established linear measures.

Among the most frequently used measures of similarity for clustering co-expressed genes are the Euclidean distance and the Pearson correlation coefficient R 3. If correlations are well described by the Pearson correlation and the distribution of data is approximately Gaussian like, the relationship between the mutual information and the Pearson correlation given by 32

is expected to be fulfilled. Therefore, we calculated both, the mutual information and the Pearson correlation, for two large-scale gene expression datasets (Figure 7). For each pair of genes X and Y we plot the tuple (MI(X,Y), R(X,Y)). In order to address significance, we additionally calculate all tuples from shuffled data.

The first dataset contains cDNA measurements for S. cerevisiae for up to E₁ = 300 experiments 31. To avoid numerical effects arising from different numbers of defined expression values (missing data points) for each gene, we exclusively utilised genes that are fully defined for all experimental conditions resulting in G₁ = 5345 genes. Analysis on this dataset using mutual information has been done before 2032 on rank-ordered data. The rank-ordering lead to homogeneously distributed data and thereby enabled the application of a simplified algorithm for the numerical estimation from kernel density estimators. The utilisation of our B-spline approach allows us to extend this analysis to non rank-ordered data thereby keeping the original distribution of experimental data. In contrast to the previous studies we find for non rank-ordered data that the theoretical prediction of Eq. 14 is no longer a lower bound for the comparison. Many tuples with high Pearson correlation but low mutual information can be detected arising from outlying expression values (Figure 8A). However, pairs of genes with high mutual information and low Pearson correlation, thus indicating a non-linear correlation, are not observed. The only remarkable tuple (marked with an arrow in Figure 7 and shown in Figure 8B) also arises from outlying values.

The second dataset contains cDNA measurements for E₂ = 102 experiments on G₂ = 22608 genes derived from 20 different human tissues 33. In contrast to the first dataset, tuples with low Pearson correlation but high mutual information are indeed detected. For two exemplary chosen tuples (Figure 8C and 8D), clusters of experimental conditions can be clearly detected by eye. Such type of correlations are missed by analyses based exclusively on linear measures, such as the the analysis done in the original publication of this dataset.

For both datasets, tuples calculated from shuffled data (Figure 7, blue data points) result in small values for both similarity measures. Thereby, they indicate a high significance of the original associations. Peaks with high Pearson correlation in the first dataset arise from gene-gene associations with outlying values. Significance values for the exemplarily chosen pairs of genes of the second dataset (Figure 8C, and 8D) were explicitly calculated (Figure 9). They show high significance values for the two examples of observed non-linear correlations on the basis of the mutual information. Compared to this, the significances calculated from the Pearson correlation are poor. In summary, our analysis confirms for the first dataset that the Pearson correlation does not miss any non-linear correlations. As a side effect we are able to detect gene-gene pairs containing outlying values. For the second dataset, however, a substantial amount of non-linear correlations was detected. Gene-gene pairs exemplarily chosen from this fraction show a clustering of data points (experiments) with a high significance. Even though such patterns can be easily found by eye, computational methods need to be applied for the inspection of several hundred million comparisons.