#detrended_fluctuation_analysis or #dfa is a method for determining the statistical #self_affinity of a #signal. It is useful for analysing #time_series that appear to be long-memory processes (diverging correlation time, e.g. #power_law decaying autocorrelation function) or #1f_noise.
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The obtained #exponent is similar to the #hurst_exponent, except that #dfa may also be applied to signals whose underlying statistics (such as #mean and #variance) or dynamics are #non_stationary (changing with time)
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In #dfa the scaling exponent #alpha is calculated as the #slope of a straight line fit to the log #log graph of F(n)}F(n) using leas #squares. an exponent of 0.5 would correspond to #uncorrelated #white_noise, an exponent of 1 is #pink_noise
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Another way to detect #pink_noise is to build a graph where the x axis are the #events while the y axis records a #time_series estimation relative to the #standard_deviation from the #average (#mean) time interval.
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At its essence #pink_noise is based on #self_affinity and #self_similarity, so that no matter what scale you look at, the pattern is #similar (#scale_free)
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#power_spectral_analysis describes distribution of #power across #frequency components composing the #signal - for #pink_noise we have a 1/f relationship — few powerful signals with low frequency, a long tail of less powerful ones (of which there are many) (hence #1f_noise)
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#envelope is a smooth #curve outlining the extremes of a #signal and it is also calculated in #hilbert_transform, which, in turn is used in calculating #dfa or #detrended_fluctuation_analysis
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#detrended_fluctuation_analysis (#dfa) has proven particularly useful, revealing that genetic #variation, normal development, or #disease can lead to differences in the #scale_free #amplitude #modulation of oscillations https://www.frontiersin.org/articles/10.3389/fphys.2012.00450/full
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The reason why #chaotic #variation (#pink_noise) is indicative of a #healthy state is because it reflects #winnerless_competition behind the process. If there's a deviation in this dynamics (eg some #patterns), it could mean that one process is #dominating the rest.
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#self_affinity is a property of #fractal #time_series where the small parts of the whole are #similar to the whole
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#self_affinity processes and #self_similar structures have in common that the statistical #distribution of the measured quantity follows a #power_law function, which is the only mathematical function without a characteristic scale. Self-affine and #self_similar phenomena are therefore called "#scale_free.”
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In #power_law #distribution the #mean would not necessarily be the same as the #median (which is are closer to each other in #normal #distribution)
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A #power_law #distribution means that there is big number of #small #variation and a small number of #big #variation (hence the line with a negative #slope when expressed as a #log)
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In a #1f #signal the lower #frequency objects have larger #amplitude than the higher #frequency objects (#1f_noise) https://www.frontiersin.org/files/Articles/23105/fphys-03-00450-HTML/image_m/fphys-03-00450-g001.jpg
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the #frequency of a certain #size of flower being inversely #proportional to its #size.
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#time_series in which all #frequency are represented with the same #amplitude will lack the rich variability of the #scale_free #time_series and is referred to as "#white_noise”
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To estimate the #scale_free property we calculate the #standard_deviation (#signal in relation to #mean) over the differently sized #time_windows. If as the #time_windows size increases the #standard_deviation also increases, we're dealing with a #scale_free process. If the #scaling_effect is not there, then it's not a scale free process.
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a stationary #random #fluctuating process has a #signal profile, which is #self_affine with a #scaling_exponent α = 0.5
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when we add #memory in the sense that the #probability of an action depends on the previous actions that the walker has made — we will get a process that will exhibit #self_affinity across scales (#scale_free)
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Different classes of processes with #memory exist: #positive_correlation and those with #anti_correlation - anti-correlations can be seen as a #stabilizing mechanism - a future action is more likely to be opposite than the ones made before. In this case on longer windows (time scales) we will have lower #fluctuating so the coefficient will be lower (α 0 to 0.5) - has #memory, #anti_correlation. 0.5 - #random, 0.5 to 1 - has #memory and #positive_correlation (previous actions increase the likelyhood of that action taken again) https://www.frontiersin.org/files/Articles/23105/fphys-03-00450-HTML/image_m/fphys-03-00450-g003.jpg
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for #dfa the signal is transformed into the #cumulative_signal, then it is split into several #windows equal in size on the #log scale. then for each the data is #detrended and #standard_deviation is calculated for each #window. then #fluctuating function is calculated as the mean #standard_deviation for all the #windows. Then we plot that as a graph on #log scales. The #dfa exponent α is the #slope of the trend. If it follows a straight line 45° then it means that with every #window increase we do not have a #proportional increase in the mean of fluctuation (so it is #linear). If it is more, then it is #non_linear and shows that it is in fact #scale_free
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The lower end of the fitting range is at least four samples, because #linear #detrending will perform poorly with less points (Peng et al., 1994). For the high end of the fitting range, #dfa estimates for window sizes >10% of the #signal length are more noisy due to a low number of windows available for averaging (i.e., less than 10 windows). Finally, the 50% overlap between windows is commonly used to increase the number of windows, which can provide a more accurate estimate of the fluctuation function especially for the long-time-scale windows.
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A #brown_noise process can be obtained by successively summing data points in the #white_noise process. https://www.researchgate.net/publication/232236967_A_tutorial_introduction_to_adaptive_fractal_analysis/figures?lo=1
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Using the classical #dfa method, the #cumulative_sum of data are divided into segments, and the #variance of these sums is studied as a function of segment length after linearly detrending them in each segment. https://www.nature.com/articles/s41598-019-42732-7
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In #dfa, data are divided into segments of length L and are #linearly detrended. The #square_root of the #variance (called #fluctuation) of the detrended data is studied as a function of L. It can be shown that a #linear relationship between the #logarithm of the #fluctuation and the #logarithm of L is indicative of a #power_law behavior of the spectrum. https://www.nature.com/articles/s41598-019-42732-7
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If a #linear relationship between the length of a #segment or #time_windows and the strength of the #fluctuation (or the #square_root of the #variance of the #cumulative_signal) exists, the slope of the corresponding line is also referred to as #hurst_exponent.
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For #white_noise the #hurst_exponent or the relation between the #time_windows and the #fluctuation (square root of #variance) will be #linear: when we double the #time_windows the #fluctuation (or #variance of the #cumulative_sum) will also double.
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For #pink_noise #1f_noise the #hurst_exponent will equal #1 and will mean that for #time_windows twice longer the #fluctuation will increase about 4 times. In other words, the the longer is the #time_windows the more #fluctuation occurs (#positive_correlation).
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#hurst_exponent in this context is #alpha_exponent, because we use #alpha_exponent for #non_stationary processes
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if #alpha_exponent is more than 1, it means that for every increase of scale (#time_windows) the cumulative_sum of #fluctuation increases a lot. That means, the longer we look at the process, the more likely it is to have big #fluctuation — there is a tendency in the #short_term to be #small and in the #long_term there's a tendency to be #big.
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the #cumulative_sum of the difference from the #average of a #time_series will be #brown_noise (#random_walk) for the #white_noise
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In contrast, #0.5 < #hurst_exponent < #1 indicates a #correlated process for #f_gn or what is termed a #persistent process for #f_bm. In this case, #increases in the signal (for #f_gn) or increments of the signal (for #f_bm) are likely to followed by further #increase, and #decrease are likely to be followed by #decreases (i.e., a #positive #long_term #correlation). Anti-#persistent and #persistent processes contain #structure that distinguishes them from truly #random sequences of data. (2) (PDF) A tutorial introduction to adaptive fractal analysis. Available from: https://www.researchgate.net/publication/232236967_A_tutorial_introduction_to_adaptive_fractal_analysis [accessed Apr 21 2021].
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The difference between the #exponent or #exponential_decay and the #power_law #decay is that #power_law #decay is slower: there are more values with a low #amplitude in the case of the #power_law https://math.stackexchange.com/questions/164436/difference-between-power-law-distribution-and-exponential-decay
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