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Two perspectives about how a measured time series is analyzed. (a) The linear autoregressive perspective takes the premise that each measure in time entails the summing of random and independent factors. (b) The multifractal perspective takes the premise that each measure entails interactions among component processes at many nested scales.

Two perspectives about how a measured time series is analyzed. (a) The linear autoregressive perspective takes the premise that each measure in time entails the summing of random and independent factors. (b) The multifractal perspective takes the premise that each measure entails interactions among component processes at many nested scales.

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The creativity and emergence of biological and psychological behavior tend to be nonlinear—biological and psychological measures contain degrees of irregularity. The linear model might fail to reduce these measurements to a sum of independent random factors (yielding a stable mean for the measurement), implying nonlinear changes over time. The pres...

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... linear stricture, a stable autocorrelation is handy for linearly modeling, but trotting out the whole autocorrelation (N-1 coefficients for an N-length measurement) to confirm causal modeling, though, maybe of limited theoretical value (Gilden, 2009). The autocorrelation entails that the contribution of activity at past lags is always independent (Fig. 3A). Here is where the interpretation for our wordreading example may stop feeling recognizable: a stationary autocorrelation function with long-memory implies that all past words matter, but they do so independently. That is, the time that, you, the reader spent reading the word 'independently' at the end of the last sentence depended on ...
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... for anyone doubting that they impact the measurement. Interactions across timescales are not just ineffable truths but may find a quantitative expression that can generalize and support formalism. The scaling relationships we have noticed in the autocorrelations might not be just the coincidental sum of independently estimable contributions (Fig. 3B). Instead, they might be shadows cast by a thoroughly different nonlinear form of cause than articulated by linear modeling-one of causes cascading across scale rather than hopping from one independent point in time to the next. They might be control parameters governing or predicting the sequence of psychological experiences as they ...
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... analysis of linear or nonlinear changes over time evaluates whether or not the measured series can be effectively modeled as a sum of independent random factors (Fig. 3A). Nonlinear changes can mean that the series is not well-modeled as merely a sum. The question of whether it is a sum is a foundational mathematical issue and goes deeper than the relatively superficial question of whether changes over time 'look' linear to the eye. Linear changes over time can include curvilinearity; some nonlinear ...
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... failure of a measured behaviors' nonlinearity to reduce to a sum has taken on a growing urgency to psychological sciences (e.g., Riley & van Orden, 2005). Multifractality can provide us deeper insights into the failure of a series' nonlinearity to reduce to a sum of independent random factors (Fig. 3B). It is quickly becoming clear that the failure of a series' nonlinearity to reduce to a sum is important to psychology. Beyond resemblance to old Gestalt wisdom that wholes differ from sums of parts, estimates of multifractality have predicted outcomes in executive function, perception, or cognition, such as in reaction time , gaze ...
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... in a regression model, no matter whether each value of the t-statistic is itself significantly large for each row of the dataset supporting that regression model. Fig. 9. IAAFT surrogate analysis to identify whether the measured series exhibits multifractal nonlinearity. We reintroduce the schematics in the top panels that had appeared in Fig. 3. Top-left panel schematizes an understanding of time-series variation in which each value depends on independent contributions of past values, with independent past-value contributions reflecting the average effect of events at specific time lags. The top-right panel schematizes the very different understanding of timeseries variation ...
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... be clear, each of the three original series passes the Theiler's (1990) test for multifractality due to nonlinearity because they all have a spectrum wider than all three corresponding sets of IAAFT surrogates' spectra. We have only depicted five surrogates' spectra in Figure S23: Multifractal spectrum for the text-to-speech (TTS) synthesizer: Voice Dream's Samantha, enhanced's absolute-amplitude waveform (red curve) on the same axes as the multifractal spectra for five IAAFT surrogates (dark blue, yellow, purple, green, and light blue) complete with final rank-ordered replacement of values. These surrogates had been built first with the same amplitude spectrum as the original series, then with the values of the original series replaced according to rank. ...
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... analysis of linear or nonlinear changes over time evaluates whether or not the measured time series can be effectively modeled as a sum of independent random factors, or the sum of the measurements (Fig. 1a). Nonlinear changes can mean that the time series is not well-modeled as merely a sum. The question of whether it is a sum is a foundational mathematical issue and goes deeper than the relatively superficial question of whether changes over time 'look' linear to the eye. Linear changes over time can include curvilinearity; some ...
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... can provide us deeper insights into the failure of a time series' nonlinearity to reduce to a sum of independent random factors (Fig. 1b). It is quickly becoming clear that the failure of a time series's nonlinearity to reduce to a sum is important to psychology. Beyond resemblance to old Gestalt wisdom that wholes differ from sums of parts, estimates of multifractality have predicted outcomes in executive function, perception, or cognition, such as in reaction time ...

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The creativity and emergence of biological and psychological behavior tend to be nonlinear—biological and psychological measures contain degrees of irregularity. The linear model might fail to reduce these measurements to a sum of independent random factors (yielding a stable mean for the measurement), implying nonlinear changes over time. The pres...