Temporal Complexity Measure of Reaction Time Series: Operational vs. Chronological Time
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Abstract Detrended Fluctuation Analysis (DFA) is a well-known method to evaluate scaling indices of time series, categorizing the dynamics of complex systems. In the literature, DFA has been used to study the fluctuations of reaction time RT(n) time series, where n is the trial number. Herein we propose treating each RT(n) as a duration time that changes the representation from operational (trial) time n to chronological (temporal) time t, or RT(t). To do this, we fill each time interval of RT(n) with fixed noise of magnitude 1 and with a randomly determined sign. The fixed noise represents the rigidity (order) and the random change of sign represents the flexibility (randomness) of the generated time series RT(t). Then the DFA algorithm was applied to RT(t) time series to evaluate scaling indices. We show that this new perspective leads to better results in: 1) differentiating scaling indices between low vs. high time-stress conditions and 2) predicting task performance outcomes. The dataset we analyzed is based on a Go-NoGo shooting task which was performed by 30 participants under low and high time-stress conditions in each of six repeated sessions over a three week period.Exponent
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Abstract We show that a computable function $f:\mathbb R\rightarrow \mathbb R$ has Luzin’s property (N) if and only if it reflects $\Pi ^1_1$ -randomness, if and only if it reflects $\Delta ^1_1({\mathcal {O}})$ -randomness, and if and only if it reflects ${\mathcal {O}}$ -Kurtz randomness, but reflecting Martin–Löf randomness or weak-2-randomness does not suffice. Here a function f is said to reflect a randomness notion R if whenever $f(x)$ is R -random, then x is R -random as well. If additionally f is known to have bounded variation, then we show f has Luzin’s (N) if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset '$ -Kurtz randomness. This links classical real analysis with algorithmic randomness.
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A relevant issue in time series analysis is the estimation of long-range dependence, that is, how much future values of a time series depend on current values. One of the ways to verify this dependence is by estimating the Hurst exponent using methods such as detrended fluctuation analysis. Here, we propose a new methodology to estimate the Hurst exponent, named leave one out detrended fluctuation analysis. Furthermore, based on this new estimator for the Hurst exponent, we propose the noise reduction by the leave one out detrended fluctuation analysis method. We apply this new denoising method to electrocardiogram noise reduction. The results presented in this work show that this new methodology outperforms the SureShrink and universal noise reduction methods.
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In this paper, we have studied electroencephalogram (EEG) activity of schizophrenia patients, in resting eyes closed condition, with detrended fluctuation analysis (DFA). The DFA gives information about scaling and long-range correlations in time series. We computed DFA exponents from 30 scalp locations of 18 male neuroleptic-naïve, recent-onset schizophrenia (NRS) subjects and 15 healthy male control subjects. Our results have shown two scaling regions in all the scalp locations in all the subjects, with different slopes, corresponding to two scaling exponents. No significant differences between the groups were found with first scaling exponent (short-range). However, the second scaling exponent (long-range) were significantly lower in control subjects at all scalp locations (p<;0.05, Kruskal-Wallis test). These findings suggest that the long-range scaling behavior of EEG is sensitive to schizophrenia, and this may provide an additional insight into the brain dysfunction in schizophrenia.
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We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has Luzin's property (N) if and only if it reflects $\Pi^1_1$-randomnes, if and only if it reflects $\Delta^1_1(\mathcal O)$-randomness, and if and only if it reflects $\mathcal O$-Kurtz randomness, but reflecting Martin-L\"of randomness or weak-2-randomness does not suffice. Here a function $f$ is said to reflect a randomness notion $R$ if whenever $f(x)$ is $R$-random, then $x$ is $R$-random as well. If additionally $f$ is known to have bounded variation, then we show $f$ has Luzin's (N) if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset'$-Kurtz randomness. This links classical real analysis with algorithmic randomness.
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Abstract This paper is a concept paper, which discusses the definition of randomness, and the sources of randomness in the physical system (the Universe) as well as in the formal mathematical system. I discuss how randomness, through chaos, the second law, the quantum mechanical character of small scales, and stochasticity is an intrinsic property of nature. I then move to our formal mathematical system and show that even in this formal system we cannot do away with randomness and that the randomness in the physical world is consistent with the origins of randomness suggested from the study of mathematical systems. Rules and randomness are blended together and their interaction is shaping all observed forms and structures.
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