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 <i>N</i>, embedded dimension <i>m</i>, a...
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doaj-6f935cb9c1684910970f7384c0c63de52020-11-25T00:58:53ZengMDPI AGEntropy1099-43002019-04-0121438510.3390/e21040385e21040385Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its ApplicationsDavid Cuesta-Frau0Juan Pablo Murillo-Escobar1Diana Alexandra Orrego2Edilson Delgado-Trejos3Technological Institute of Informatics, Universitat Politècnica de València, Alcoi Campus, 03801 Alcoi, SpainGrupo de Investigación e Innovación Biomédica (GI2B), Instituto Tecnológico Metropolitano (ITM), Medellín, ColombiaGrupo de Investigación e Innovación Biomédica (GI2B), Instituto Tecnológico Metropolitano (ITM), Medellín, ColombiaCM&P, Instituto Tecnológico Metropolitano (ITM), Medellín, ColombiaPermutation 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 <i>N</i>, embedded dimension <i>m</i>, and embedded delay <inline-formula> <math display="inline"> <semantics> <mi>τ</mi> </semantics> </math> </inline-formula>. Inappropriate choices of these parameters may potentially lead to incorrect interpretations. However, there are no specific guidelines for an optimal selection of <i>N</i>, <i>m</i>, or <inline-formula> <math display="inline"> <semantics> <mi>τ</mi> </semantics> </math> </inline-formula>, only general recommendations such as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>></mo> <mo>></mo> <mi>m</mi> <mo>!</mo> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, or <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>7</mn> </mrow> </semantics> </math> </inline-formula>. This paper deals specifically with the study of the practical implications of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>></mo> <mo>></mo> <mi>m</mi> <mo>!</mo> </mrow> </semantics> </math> </inline-formula>, since long time series are often not available, or non-stationary, and other preliminary results suggest that low <i>N</i> values do not necessarily invalidate PE usefulness. Our study analyses the PE variation as a function of the series length <i>N</i> and embedded dimension <i>m</i> 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 <i>N</i> and <i>m</i>. The results seem to indicate that shorter lengths than those suggested by <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>></mo> <mo>></mo> <mi>m</mi> <mo>!</mo> </mrow> </semantics> </math> </inline-formula> 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.https://www.mdpi.com/1099-4300/21/4/385permutation entropyembedded dimensionshort time recordssignal classificationrelevance analysis |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
David Cuesta-Frau Juan Pablo Murillo-Escobar Diana Alexandra Orrego Edilson Delgado-Trejos |
spellingShingle |
David Cuesta-Frau Juan Pablo Murillo-Escobar Diana Alexandra Orrego Edilson Delgado-Trejos Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications Entropy permutation entropy embedded dimension short time records signal classification relevance analysis |
author_facet |
David Cuesta-Frau Juan Pablo Murillo-Escobar Diana Alexandra Orrego Edilson Delgado-Trejos |
author_sort |
David Cuesta-Frau |
title |
Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications |
title_short |
Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications |
title_full |
Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications |
title_fullStr |
Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications |
title_full_unstemmed |
Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications |
title_sort |
embedded dimension and time series length. practical influence on permutation entropy and its applications |
publisher |
MDPI AG |
series |
Entropy |
issn |
1099-4300 |
publishDate |
2019-04-01 |
description |
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 <i>N</i>, embedded dimension <i>m</i>, and embedded delay <inline-formula> <math display="inline"> <semantics> <mi>τ</mi> </semantics> </math> </inline-formula>. Inappropriate choices of these parameters may potentially lead to incorrect interpretations. However, there are no specific guidelines for an optimal selection of <i>N</i>, <i>m</i>, or <inline-formula> <math display="inline"> <semantics> <mi>τ</mi> </semantics> </math> </inline-formula>, only general recommendations such as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>></mo> <mo>></mo> <mi>m</mi> <mo>!</mo> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, or <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>7</mn> </mrow> </semantics> </math> </inline-formula>. This paper deals specifically with the study of the practical implications of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>></mo> <mo>></mo> <mi>m</mi> <mo>!</mo> </mrow> </semantics> </math> </inline-formula>, since long time series are often not available, or non-stationary, and other preliminary results suggest that low <i>N</i> values do not necessarily invalidate PE usefulness. Our study analyses the PE variation as a function of the series length <i>N</i> and embedded dimension <i>m</i> 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 <i>N</i> and <i>m</i>. The results seem to indicate that shorter lengths than those suggested by <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>></mo> <mo>></mo> <mi>m</mi> <mo>!</mo> </mrow> </semantics> </math> </inline-formula> 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. |
topic |
permutation entropy embedded dimension short time records signal classification relevance analysis |
url |
https://www.mdpi.com/1099-4300/21/4/385 |
work_keys_str_mv |
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