Permutation Entropy: Enhancing Discriminating Power by Using Relative Frequencies Vector of Ordinal Patterns Instead of Their Shannon Entropy

Entropy Information of Pulse Dynamics in Three Stages of Pregnancy

The aim of the present study is to use entropy to explore the change of pulse generated by normal pregnant women with gestational. Firstly, the subjects were divided into early (E), middle (M), and late (L) three stages according to gestational age. Then, pulse signals of the Chi position of 90 pregnant women at different gestational ages were collected. Secondly, the four entropies, namely fuzzy entropy (FuEn), approximate entropy (ApEn), sample entropy (SamEn), and permutation entropy (PerEn), were applied to the analysis of the long-term pulse changes of the pregnancy. Finally, the related information about pulse in different stages of pregnancy is given by the analysis of four kinds of entropy. Furthermore, the statistical tests are conducted for further comparison, and the descriptive statistics and the results are presented. In addition, boxplots are employed to show the distribution of four entropies of pregnancy. This work has studied the changes in pulse during pregnancy from quantitative and qualitative aspects. Our results show that entropy improves the diagnostic value of pulse analysis during pregnancy and could be applied to facilitate noninvasive diagnosis of pregnant women's physiological signals in the future.

Slope Entropy Characterisation: The Role of the δ Parameter

Many time series entropy calculation methods have been proposed in the last few years. They are mainly used as numerical features for signal classification in any scientific field where data series are involved. We recently proposed a new method, Slope Entropy (SlpEn), based on the relative frequency of differences between consecutive samples of a time series, thresholded using two input parameters, γ and δ. In principle, δ was proposed to account for differences in the vicinity of the 0 region (namely, ties) and, therefore, was usually set at small values such as 0.001. However, there is no study that really quantifies the role of this parameter using this default or other configurations, despite the good SlpEn results so far. The present paper addresses this issue, removing δ from the SlpEn calculation to assess its real influence on classification performance, or optimising its value by means of a grid search in order to find out if other values beyond the 0.001 value provide significant time series classification accuracy gains. Although the inclusion of this parameter does improve classification accuracy according to experimental results, gains of 5% at most probably do not support the additional effort required. Therefore, SlpEn simplification could be seen as a real alternative.

Time-Delay Identification Using Multiscale Ordinal Quantifiers

Time-delayed interactions naturally appear in a multitude of real-world systems due to the finite propagation speed of physical quantities. Often, the time scales of the interactions are unknown to an external observer and need to be inferred from time series of observed data. We explore, in this work, the properties of several ordinal-based quantifiers for the identification of time-delays from time series. To that end, we generate artificial time series of stochastic and deterministic time-delay models. We find that the presence of a nonlinearity in the generating model has consequences for the distribution of ordinal patterns and, consequently, on the delay-identification qualities of the quantifiers. Here, we put forward a novel ordinal-based quantifier that is particularly sensitive to nonlinearities in the generating model and compare it with previously-defined quantifiers. We conclude from our analysis on artificially generated data that the proper identification of the presence of a time-delay and its precise value from time series benefits from the complementary use of ordinal-based quantifiers and the standard autocorrelation function. We further validate these tools with a practical example on real-world data originating from the North Atlantic Oscillation weather phenomenon.

Classification of Actigraphy Records from Bipolar Disorder Patients Using Slope Entropy: A Feasibility Study

Bipolar Disorder (BD) is an illness with high prevalence and a huge social and economic impact. It is recurrent, with a long-term evolution in most cases. Early treatment and continuous monitoring have proven to be very effective in mitigating the causes and consequences of BD. However, no tools are currently available for a massive and semi-automatic BD patient monitoring and control. Taking advantage of recent technological developments in the field of wearables, this paper studies the feasibility of a BD episodes classification analysis while using entropy measures, an approach successfully applied in a myriad of other physiological frameworks. This is a very difficult task, since actigraphy records are highly non-stationary and corrupted with artifacts (no activity). The method devised uses a preprocessing stage to extract epochs of activity, and then applies a quantification measure, Slope Entropy, recently proposed, which outperforms the most common entropy measures used in biomedical time series. The results confirm the feasibility of the approach proposed, since the three states that are involved in BD, depression, mania, and remission, can be significantly distinguished.

Fever Time Series Analysis Using Slope Entropy. Application to Early Unobtrusive Differential Diagnosis

Fever is a readily measurable physiological response that has been used in medicine for centuries. However, the information provided has been greatly limited by a plain thresholding approach, overlooking the additional information provided by temporal variations and temperature values below such threshold that are also representative of the subject status. In this paper, we propose to utilize continuous body temperature time series of patients that developed a fever, in order to apply a method capable of diagnosing the specific underlying fever cause only by means of a pattern relative frequency analysis. This analysis was based on a recently proposed measure, Slope Entropy, applied to a variety of records coming from dengue and malaria patients, among other fever diseases. After an input parameter customization, a classification analysis of malaria and dengue records took place, quantified by the Matthews Correlation Coefficient. This classification yielded a high accuracy, with more than 90% of the records correctly labelled in some cases, demonstrating the feasibility of the approach proposed. This approach, after further studies, or combined with more measures such as Sample Entropy, is certainly very promising in becoming an early diagnosis tool based solely on body temperature temporal patterns, which is of great interest in the current Covid-19 pandemic scenario.

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.

Slope Entropy: A New Time Series Complexity Estimator Based on Both Symbolic Patterns and Amplitude Information

The development of new measures and algorithms to quantify the entropy or related concepts of a data series is a continuous effort that has brought many innovations in this regard in recent years. The ultimate goal is usually to find new methods with a higher discriminating power, more efficient, more robust to noise and artifacts, less dependent on parameters or configurations, or any other possibly desirable feature. Among all these methods, Permutation Entropy (PE) is a complexity estimator for a time series that stands out due to its many strengths, with very few weaknesses. One of these weaknesses is the PE’s disregarding of time series amplitude information. Some PE algorithm modifications have been proposed in order to introduce such information into the calculations. We propose in this paper a new method, Slope Entropy (SlopEn), that also addresses this flaw but in a different way, keeping the symbolic representation of subsequences using a novel encoding method based on the slope generated by two consecutive data samples. By means of a thorough and extensive set of comparative experiments with PE and Sample Entropy (SampEn), we demonstrate that SlopEn is a very promising method with clearly a better time series classification performance than those previous methods.