{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:16:37Z","timestamp":1760242597241,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2017,11,15]],"date-time":"2017-11-15T00:00:00Z","timestamp":1510704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Permutation entropy has become a standard tool for time series analysis that exploits the temporal and ordinal relationships within data. Motivated by a Kullback\u2013Leibler divergence interpretation of permutation entropy as divergence from white noise, we extend pattern-based methods to the setting of random walk data. We analyze random walk null models for correlated time series and describe a method for determining the corresponding ordinal pattern distributions. These null models more accurately reflect the observed pattern distributions in some economic data. This leads us to define a measure of complexity using the deviation of a time series from an associated random walk null model. We demonstrate the applicability of our methods using empirical data drawn from a variety of fields, including to a variety of stock market closing prices.<\/jats:p>","DOI":"10.3390\/e19110615","type":"journal-article","created":{"date-parts":[[2017,11,15]],"date-time":"2017-11-15T11:13:35Z","timestamp":1510744415000},"page":"615","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Random Walk Null Models for Time Series Data"],"prefix":"10.3390","volume":"19","author":[{"given":"Daryl","family":"DeFord","sequence":"first","affiliation":[{"name":"Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Katherine","family":"Moore","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"60005","DOI":"10.1209\/0295-5075\/83\/60005","article-title":"Combinatorial detection of determinism in noisy time series","volume":"83","author":"Zambrano","year":"2008","journal-title":"Europhys. 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