Entropy estimation

In various science/engineering applications, such as independent component analysis,^[1] image analysis,^[2] genetic analysis,^[3] speech recognition,^[4] manifold learning,^[5] and time delay estimation^[6] it is useful to estimate the differential entropy of a system or process, given some observations.

The simplest and most common approach uses histogram-based estimation, but other approaches have been developed and used, each with their own benefits and drawbacks.^[7] The main factor in choosing a method is often a trade-off between the bias and the variance of the estimate^[8] although the nature of the (suspected) distribution of the data may also be a factor.^[7]

Histogram estimator

The histogram approach uses the idea that the differential entropy,

${\displaystyle H(X) = -\int_\mathbb{X} f(x)\log f(x)\,dx}$

can be approximated by producing a histogram of the observations, and then finding the discrete entropy

${\displaystyle H(X) = - \sum_{i=1}^nf(x_i)\log \left(\frac{f(x_i)}{w(x_i)} \right) }$

of that histogram (which is itself a maximum-likelihood (ML) estimate of the discretized frequency distribution), where w is the width of the ith bin. Histograms can be quick to calculate, and simple, so this approach has some attractions. However, the estimate produced is biased, and although corrections can be made to the estimate, they may not always be satisfactory.^[9]

A method better suited for multidimensional probability density functions (pdf) is to first make a pdf estimate with some method, and then, from the pdf estimate, compute the entropy. A useful pdf estimate method is e.g. Gaussian mixture modeling (GMM), where the expectation maximization (EM) algorithm is used to find an ML estimate of a weighted sum of Gaussian pdf's approximating the data pdf.

Estimates based on sample-spacings

If the data is one-dimensional, we can imagine taking all the observations and putting them in order of their value. The spacing between one value and the next then gives us a rough idea of (the reciprocal of) the probability density in that region: the closer together the values are, the higher the probability density. This is a very rough estimate with high variance, but can be improved, for example by thinking about the space between a given value and the one m away from it, where m is some fixed number.^[7]

The probability density estimated in this way can then be used to calculate the entropy estimate, in a similar way to that given above for the histogram, but with some slight tweaks.

One of the main drawbacks with this approach is going beyond one dimension: the idea of lining the data points up in order falls apart in more than one dimension. However, using analogous methods, some multidimensional entropy estimators have been developed.^[10][11]

Estimates based on nearest-neighbours

For each point in our dataset, we can find the distance to its nearest neighbour. We can in fact estimate the entropy from the distribution of the nearest-neighbour-distance of our datapoints.^[7] (In a uniform distribution these distances all tend to be fairly similar, whereas in a strongly nonuniform distribution they may vary a lot more.)

Bayesian estimator

When in under-sampled regime, having a prior on the distribution can help the estimation. One such Bayesian estimator was proposed in the neuroscience context known as the NSB (Nemenman–Shafee–Bialek) estimator.^[12][13] The NSB estimator uses a mixture of Dirichlet prior, chosen such that the induced prior over the entropy is approximately uniform.

Estimates based on expected entropy

A new approach to the problem of entropy evaluation is to compare the expected entropy of a sample of random sequence with the calculated entropy of the sample. The method gives very accurate results, but it is limited to calculations of random sequences modeled as Markov chains of the first order with small values of bias and correlations. This is the first known method that takes into account the size of the sample sequence and its impact on the accuracy of the calculation of entropy.^[14][15]

References

1. Dinh-Tuan Pham (2004) Fast algorithms for mutual information based independent component analysis. In Signal Processing. Volume 52, Issue 10, 2690–2700, doi:10.1109/TSP.2004.834398
2. Chang, C.-I.; Du, Y.; Wang, J.; Guo, S.-M.; Thouin, P.D. (2006) Survey and comparative analysis of entropy and relative entropy thresholding techniques. In Vision, Image and Signal Processing, Volume 153, Issue 6, 837–850, doi:10.1049/ip-vis:20050032
3. Martins, D. C. et al. (2008) Intrinsically Multivariate Predictive Genes. In Selected Topics in Signal Processing. Volume 2, Issue 3, 424–439, doi:10.1109/JSTSP.2008.923841
4. Gue Jun Jung; Yung-Hwan Oh (2008) Information Distance-Based Subvector Clustering for ASR Parameter Quantization. In Signal Processing Letters, Volume 15, 209–212, doi:10.1109/LSP.2007.913132
5. Costa, J.A.; Hero, A.O. (2004), Geodesic entropic graphs for dimension and entropy estimation in manifold learning. In Signal Processing, Volume 52, Issue 8, 2210–2221, doi:10.1109/TSP.2004.831130
6. Benesty, J.; Yiteng Huang; Jingdong Chen (2007) Time Delay Estimation via Minimum Entropy. In Signal Processing Letters, Volume 14, Issue 3, March 2007 157–160 doi:10.1109/LSP.2006.884038
7. J. Beirlant, E. J. Dudewicz, L. Gyorfi, and E. C. van der Meulen (1997) Nonparametric entropy estimation: An overview. In International Journal of Mathematical and Statistical Sciences, Volume 6, pp. 17– 39.
8. T. Schürmann, Bias analysis in entropy estimation. In J. Phys. A: Math. Gen, 37 (2004), pp. L295–L301. doi:10.1088/0305-4470/37/27/L02
9. G. Miller (1955) Note on the bias of information estimates. In Information Theory in Psychology: Problems and Methods, pp. 95–100.
10. E. G. Learned-Miller (2003) A new class of entropy estimators for multi-dimensional densities, in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (ICASSP’03), vol. 3, April 2003, pp. 297–300.
11. I. Lee (2010) Sample-spacings based density and entropy estimators for spherically invariant multidimensional data, In Neural Computation, vol. 22, issue 8, April 2010, pp. 2208–2227.
12. Ilya Nemenman, Fariel Shafee, William Bialek (2003) Entropy and Inference, Revisited. Advances in Neural Information Processing
13. Ilya Nemenman, William Bialek, de Ruyter (2004) Entropy and information in neural spike trains: Progress on the sampling problem. Physical Review E
14. Marek Lesniewicz (2014) Expected Entropy as a Measure and Criterion of Randomness of Binary Sequences In Przeglad Elektrotechniczny, Volume 90, 1/2014, pp. 42– 46.
15. Marek Lesniewicz (2016) Analyses and Measurements of Hardware Generated Random Binary Sequences Modeled as Markov Chains In Przeglad Elektrotechniczny, Volume 92, 11/2016, pp. 268-274.