Permutation entropy with vector embedding delays

Shared increased entropy of brain signals across patients with different mental illnesses: A coordinate-based activation likelihood estimation meta-analysis

Entropy is a measurement of brain signal complexity. Studies have found increased/decreased entropy of brain signals in psychiatric patients. There is no consistent conclusion regarding the relationship between the entropy of brain signals and mental illness. Therefore, this meta-analysis aimed to identify consistent abnormalities in the brain signal entropy in patients with different mental illnesses. We conducted a systematic search to collect resting-state functional magnetic resonance imaging (rs-fMRI) studies in patients with psychiatric disorders. This work identified 9 eligible rs-fMRI studies, which included a total of 14 experiments, 67 activation foci, and 1383 subjects. We tested the convergence across their findings by using the activation likelihood estimation method. P-value maps were corrected by using cluster-level family-wise error p < 0.05 and permuting 2000 times. Results showed that patients with different psychiatric disorders shared commonly increased entropy of brain signals in the left inferior and middle frontal gyri, and the right fusiform gyrus, cuneus, precuneus. No shared alterations were found in the subcortical regions and cerebellum in the patient group. Our findings suggested that the increased entropy of brain signals in the cortex, not subcortical regions and cerebellum, might have associations with the pathophysiology across mental illnesses. This meta-analysis study provided the first comprehensive understanding of the abnormality in brain signal complexity across patients with different psychiatric disorders.

Using the Information Provided by Forbidden Ordinal Patterns in Permutation Entropy to Reinforce Time Series Discrimination Capabilities

Despite its widely tested and proven usefulness, there is still room for improvement in the basic permutation entropy (PE) algorithm, as several subsequent studies have demonstrated in recent years. Some of these new methods try to address the well-known PE weaknesses, such as its focus only on ordinal and not on amplitude information, and the possible detrimental impact of equal values found in subsequences. Other new methods address less specific weaknesses, such as the PE results' dependence on input parameter values, a common problem found in many entropy calculation methods. The lack of discriminating power among classes in some cases is also a generic problem when entropy measures are used for data series classification. This last problem is the one specifically addressed in the present study. Toward that purpose, the classification performance of the standard PE method was first assessed by conducting several time series classification tests over a varied and diverse set of data. Then, this performance was reassessed using a new Shannon Entropy normalisation scheme proposed in this paper: divide the relative frequencies in PE by the number of different ordinal patterns actually found in the time series, instead of by the theoretically expected number. According to the classification accuracy obtained, this last approach exhibited a higher class discriminating power. It was capable of finding significant differences in six out of seven experimental datasets-whereas the standard PE method only did in four-and it also had better classification accuracy. It can be concluded that using the additional information provided by the number of forbidden/found patterns, it is possible to achieve a higher discriminating power than using the classical PE normalisation method. The resulting algorithm is also very similar to that of PE and very easy to implement.

Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications

Permutation Entropy (PE) is a time series complexity measure commonly used in a variety of contexts, with medicine being the prime example. In its general form, it requires three input parameters for its calculation: time series length N, embedded dimension m, and embedded delay τ. Inappropriate choices of these parameters may potentially lead to incorrect interpretations. However, there are no specific guidelines for an optimal selection of N, m, or τ, only general recommendations such as N>>m!, τ=1, or m=3,…,7. This paper deals specifically with the study of the practical implications of N>>m!, since long time series are often not available, or non-stationary, and other preliminary results suggest that low N values do not necessarily invalidate PE usefulness. Our study analyses the PE variation as a function of the series length N and embedded dimension m in the context of a diverse experimental set, both synthetic (random, spikes, or logistic model time series) and real–world (climatology, seismic, financial, or biomedical time series), and the classification performance achieved with varying N and m. The results seem to indicate that shorter lengths than those suggested by N>>m! are sufficient for a stable PE calculation, and even very short time series can be robustly classified based on PE measurements before the stability point is reached. This may be due to the fact that there are forbidden patterns in chaotic time series, not all the patterns are equally informative, and differences among classes are already apparent at very short lengths.

Permutation Entropy Based on Non-Uniform Embedding

A novel visualization scheme for permutation entropy is presented in this paper. The proposed scheme is based on non-uniform attractor embedding of the investigated time series. A single digital image of permutation entropy is produced by averaging all possible plain projections of the permutation entropy measure in the multi-dimensional delay coordinate space. Computational experiments with artificially-generated and real-world time series are used to demonstrate the advantages of the proposed visualization scheme.