scholarly journals Low Computational Cost for Sample Entropy

Entropy ◽  
2018 ◽  
Vol 20 (1) ◽  
pp. 61 ◽  
Author(s):  
George Manis ◽  
Md Aktaruzzaman ◽  
Roberto Sassi
Keyword(s):  
Time Series ◽  
Dimensional Space ◽  
A Priori ◽  
Computational Cost ◽  
Approximate Entropy ◽  
Sample Entropy ◽  
Series Length ◽  
Approximation Techniques ◽  
Computationally Intensive ◽  
Definition Of

Sample Entropy is the most popular definition of entropy and is widely used as a measure of the regularity/complexity of a time series. On the other hand, it is a computationally expensive method which may require a large amount of time when used in long series or with a large number of signals. The computationally intensive part is the similarity check between points in m dimensional space. In this paper, we propose new algorithms or extend already proposed ones, aiming to compute Sample Entropy quickly. All algorithms return exactly the same value for Sample Entropy, and no approximation techniques are used. We compare and evaluate them using cardiac inter-beat (RR) time series. We investigate three algorithms. The first one is an extension of the k d -trees algorithm, customized for Sample Entropy. The second one is an extension of an algorithm initially proposed for Approximate Entropy, again customized for Sample Entropy, but also improved to present even faster results. The last one is a completely new algorithm, presenting the fastest execution times for specific values of m, r, time series length, and signal characteristics. These algorithms are compared with the straightforward implementation, directly resulting from the definition of Sample Entropy, in order to give a clear image of the speedups achieved. All algorithms assume the classical approach to the metric, in which the maximum norm is used. The key idea of the two last suggested algorithms is to avoid unnecessary comparisons by detecting them early. We use the term unnecessary to refer to those comparisons for which we know a priori that they will fail at the similarity check. The number of avoided comparisons is proved to be very large, resulting in an analogous large reduction of execution time, making them the fastest algorithms available today for the computation of Sample Entropy.

scholarly journals On Quantization Errors in Approximate and Sample Entropy

Entropy ◽  
2021 ◽  
Vol 24 (1) ◽  
pp. 73
Author(s):  
Dragana Bajić ◽  
Nina Japundžić-Žigon
Keyword(s):  
Time Series ◽  
Approximate Entropy ◽  
Large Error ◽  
Sample Entropy ◽  
Error Distribution ◽  
Series Length ◽  
Entropy Error ◽  
Coarse Quantization ◽  
True Values ◽  
Matching Probability

Approximate and sample entropies are acclaimed tools for quantifying the regularity and unpredictability of time series. This paper analyses the causes of their inconsistencies. It is shown that the major problem is a coarse quantization of matching probabilities, causing a large error between their estimated and true values. Error distribution is symmetric, so in sample entropy, where matching probabilities are directly summed, errors cancel each other. In approximate entropy, errors are accumulating, as sums involve logarithms of matching probabilities. Increasing the time series length increases the number of quantization levels, and errors in entropy disappear both in approximate and in sample entropies. The distribution of time series also affects the errors. If it is asymmetric, the matching probabilities are asymmetric as well, so the matching probability errors cease to be mutually canceled and cause a persistent entropy error. Despite the accepted opinion, the influence of self-matching is marginal as it just shifts the error distribution along the error axis by the matching probability quant. Artificial lengthening the time series by interpolation, on the other hand, induces large error as interpolated samples are statistically dependent and destroy the level of unpredictability that is inherent to the original signal.

Measuring body temperature time series regularity using approximate entropy and sample entropy

2009 ◽  
Author(s):  
D. Cuesta-Frau ◽  
P. Miro-Martinez ◽  
S. Oltra-Crespo ◽  
M. Varela-Entrecanales ◽  
M. Aboy ◽  
...  
Keyword(s):  
Time Series ◽  
Body Temperature ◽  
Approximate Entropy ◽  
Sample Entropy ◽  
Temperature Time Series ◽  
Temperature Time

scholarly journals Method for calculating loads combination on a building using information measures

2018 ◽  
Vol 212 ◽  
pp. 01033
Author(s):  
Elena Chernetsova ◽  
Anatoly Shishkin
Keyword(s):  
Time Series ◽  
Error Probability ◽  
A Priori ◽  
Monitoring Network ◽  
Physical Nature ◽  
Data Bank ◽  
Information Measure ◽  
Wireless Monitoring ◽  
Information Measures ◽  
Definition Of

A method for calculating loads combination on a building is considered using information measures of the connectivity of signals received from sensors of various physical nature, united in a wireless monitoring network. The method includes the definition of the most powerful information measure on the ensemble of process realizations with known a priori load data by the criterion of connectedness of time series. Then, based on the selected information measure, the connectivity of the signals for the ensemble of realizations of the random process of loads to the building from the network formed by the wireless monitoring data bank of time series is calculated. The volume of the data bank sufficient to make the correct decision about the combination of loads on the building with a predetermined error probability is calculated on the basis of a consistent criterion for the ratio of Wald probabilities. This method is easily algorithmized and can be used to develop an automated decision support system.

scholarly journals The Sub-Polar Gyre Index – a community data set for application in fisheries and environment research

2016 ◽  
Author(s):  
Barbara Berx ◽  
Mark R. Payne
Keyword(s):  
Time Series ◽  
North Atlantic ◽  
North Atlantic Ocean ◽  
Marine Ecosystem ◽  
Series Length ◽  
Data Set ◽  
The North ◽  
Community Data ◽  
Definition Of ◽  
Index Time Series

Abstract. Scientific interest in the sub-polar gyre of the North Atlantic Ocean has increased in recent years. The sub-polar gyre has contracted and weakened, and changes in circulation pathways have been linked to changes in marine ecosystem productivity. To aid fisheries and environmental scientists, we here present a time series of the Sub-Polar Gyre Index (SPG-I) based on monthly mean maps of sea surface height. The established definition of the SPG-I is applied, and the first EOF and PC are presented. Sensitivity to the spatial domain and time series length are explored, but found not to be important factors. Our time series compares well with indices presented previously. The SPG-I time series is freely available online (doi:10.7489/1806-1) and we invite the community to access, apply and publish studies using this index time series.

Physiological time-series analysis using approximate entropy and sample entropy

2000 ◽  
Vol 278 (6) ◽  
pp. H2039-H2049 ◽  
Author(s):  
Joshua S. Richman ◽  
J. Randall Moorman
Keyword(s):  
Time Series ◽  
Approximate Entropy ◽  
Sample Entropy ◽  
Random Numbers ◽  
Data Sets ◽  
Production Methods ◽  
Biological Time ◽  
Biological Studies ◽  
Probabilistic Character ◽  
Improved Accuracy

Entropy, as it relates to dynamical systems, is the rate of information production. Methods for estimation of the entropy of a system represented by a time series are not, however, well suited to analysis of the short and noisy data sets encountered in cardiovascular and other biological studies. Pincus introduced approximate entropy (ApEn), a set of measures of system complexity closely related to entropy, which is easily applied to clinical cardiovascular and other time series. ApEn statistics, however, lead to inconsistent results. We have developed a new and related complexity measure, sample entropy (SampEn), and have compared ApEn and SampEn by using them to analyze sets of random numbers with known probabilistic character. We have also evaluated cross-ApEn and cross-SampEn, which use cardiovascular data sets to measure the similarity of two distinct time series. SampEn agreed with theory much more closely than ApEn over a broad range of conditions. The improved accuracy of SampEn statistics should make them useful in the study of experimental clinical cardiovascular and other biological time series.

scholarly journals IX. Memoir on the theory of the partitions of numbers. - Part VI. Partitions in two-dimensional space, to which is added an adumbration of the theory of the partitions in three-dimensional space

1912 ◽  
Vol 211 (471-483) ◽  
pp. 345-373 ◽  
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Dimensional Space ◽  
A Priori ◽  
Three Dimensional ◽  
Large Measure ◽  
Lattice Function ◽  
The Subject ◽  
Definition Of

I Resume the subject of Part V. of this Memoir by inquiring further into the generating function of the partitions of a number when the parts are placed at the nodes of an incomplete lattice, viz., of a lattice which is regular but made up of unequal rows. Such a lattice is the graph of the line partition of a number. In Part V. I arrived at the expression of the generating function in respect of a two- row lattice when the past magnitude is unrestricted. This was given in Art. 16 in the form GF ( ∞ ; a, b ) = (1) + x b +1 (a - b) / (1) (2) ... (a+1). (1) (2) ... (b). I remind the reader that the determination of the generating function, when the part magnitude is unrestricted, depends upon the determination of the associated lattice function (see Art. 5, loc . cit .). This function is assumed to be the product of an expression of known form and of another function which I termed the inner lattice function (see Art. 10, loc . cit .), and it is on the form of this function that the interest of the investigation in large measure depends. All that is known about it à priori is its numerical value when x is put equal to unity (Art. 10, loc cit . The lattice function was also exhibited as a sum of sub-lattice functions, and it was shown that the generating function, when the part magnitude is restricted, may be expressed as a linear function of them. These sub-lattice functions are intrinsically interesting, hut it will be shown in what follows that they are not of vital importance to the investigation. In fact, the difficulty of constructing them has been turned by the formation and solution of certain functional equations which lead in the first place to the required generating functions, and in the second place to an exhibition of the forms of the sub-lattice functions. To previous definitions I here add the definition of the inner lattice function when there is a restriction upon the part magnitude, and it will be shown that the generating, lattice, and inner lattice functions satisfy certain functional equations both when there is not and when there is a restriction upon the part magnitude.

scholarly journals The Sub-Polar Gyre Index – a community data set for application in fisheries and environment research

2017 ◽  
Vol 9 (1) ◽  
pp. 259-266 ◽  
Author(s):  
Barbara Berx ◽  
Mark R. Payne
Keyword(s):  
Time Series ◽  
North Atlantic Ocean ◽  
Marine Ecosystem ◽  
Principal Component ◽  
Series Length ◽  
Data Set ◽  
The North ◽  
Community Data ◽  
Definition Of ◽  
Index Time Series

Abstract. Scientific interest in the sub-polar gyre of the North Atlantic Ocean has increased in recent years. The sub-polar gyre has contracted and weakened, and changes in circulation pathways have been linked to changes in marine ecosystem productivity. To aid fisheries and environmental scientists, we present here a time series of the Sub-Polar Gyre Index (SPG-I) based on monthly mean maps of sea surface height. The established definition of the SPG-I is applied, and the first EOF (empirical orthogonal function) and PC (principal component) are presented. Sensitivity to the spatial domain and time series length are explored but found not to be important factors in terms of the SPG-I's interpretation. Our time series compares well with indices presented previously. The SPG-I time series is freely available online (http://dx.doi.org/10.7489/1806-1), and we invite the community to access, apply, and publish studies using this index time series.

Time Series Models

2013 ◽  
Author(s):  
James B. Elsner ◽  
Thomas H. Jagger
Keyword(s):  
Time Series ◽  
Time Series Data ◽  
Series Data ◽  
Time Series Models ◽  
Infinite Length ◽  
Network Graph ◽  
Series Length ◽  
Single Time ◽  
Ergodic Hypothesis ◽  
Definition Of

In this chapter, we consider time series models. A time series is an ordered sequence of numbers with respect to time. In climatology, you encounter time-series data in a format given by . . . {h}Tt=1 = {h1,h2,. . . ,hT} (10.1) . . . where the time t is over a given season, month, week, or day and T is the time series length. The aim is to understand the underlying physical processes that produced the series. A trend is an example. Often by simply looking at a time series plot, you can pick out a trend that tells you that the process generating the data is changing. A single time series gives you a sample from the process. Yet under the ergodic hypothesis, a single time series of infinite length contains the same information (loosely speaking) as the collection of all possible series of finite length. In this case, you can use your series to learn about the nature of the process. This is analogous to spatial interpolation encountered in Chapter 9, where the variogram was computed under the assumption that the rainfall field is stationary. Here we consider a selection of techniques and models for time series data. We begin by showing you how to overlay plots as a tool for exploratory analysis. This is done to compare the variation between two series qualitatively. We demonstrate large variation in hurricane counts arising from a constant rate process. We then show techniques for smoothing. We continue with a change-point model and techniques for decomposing a continuous-valued series. We conclude with a unique way to create a network graph from a time series of counts and suggest a new definition of a climate anomaly. A plot showing your variables on a common time axis is an informative exploratory graph. Values from two different series are scaled to have the same relative range so the covariation in the variables can be compared visually. Here you do this with hurricane counts and sea-surface temperature (SST). Begin by loading annual.RData. These data were assembled in Chapter 6.

scholarly journals New Fast ApEn and SampEn Entropy Algorithms Implementation and Their Application to Supercomputer Power Consumption

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 863 ◽  
Author(s):  
Jiří Tomčala
Keyword(s):  
Time Series ◽  
Power Consumption ◽  
Real World ◽  
Software Package ◽  
World System ◽  
Approximate Entropy ◽  
R Package ◽  
Sample Entropy ◽  
R Software

Approximate Entropy and especially Sample Entropy are recently frequently used algorithms for calculating the measure of complexity of a time series. A lesser known fact is that there are also accelerated modifications of these two algorithms, namely Fast Approximate Entropy and Fast Sample Entropy. All these algorithms are effectively implemented in the R software package TSEntropies. This paper contains not only an explanation of all these algorithms, but also the principle of their acceleration. Furthermore, the paper contains a description of the functions of this software package and their parameters, as well as simple examples of using this software package to calculate these measures of complexity of an artificial time series and the time series of a complex real-world system represented by the course of supercomputer infrastructure power consumption. These time series were also used to test the speed of this package and to compare its speed with another R package pracma. The results show that TSEntropies is up to 100 times faster than pracma and another important result is that the computational times of the new Fast Approximate Entropy and Fast Sample Entropy algorithms are up to 500 times lower than the computational times of their original versions. At the very end of this paper, the possible use of this software package TSEntropies is proposed.

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