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#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.

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)

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

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.

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)

#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)

#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

#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

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.

#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.”

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)

#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”

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.

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)

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

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

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.

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

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

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.

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.

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).

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.

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].

The topics are the nodes (words) that tend to co-occur together in the same context (next to each other).

We use a combination of clustering and graph community detection algorithm (Blondel et al based on Louvain) to identify the groups of nodes are more densely connected together than with the rest of the network. They are aligned closer to each other on the graph and are given a distinct color.

We use the Jenks elbow cutoff algorithm to select the top prominent nodes that have significantly higher influence than the rest.

Click the Reveal Non-obvious button to remove the most influential words (or the ones you select) from the graph, to see what terms are hiding behind them.

The most influential nodes are either the ones with the highest betweenness centrality — appearing most often on the shortest path between any two randomly chosen nodes (i.e. linking the different distinct communities) — or the ones with the highest degree.

A structural gap shows the two distinct communities (clusters of words) in this graph that are important, but not yet connected. That's where the new potential and innovative ideas may reside.

This measure is based on a combination of the graph's connectivity and community structure, selecting the groups of nodes that would either make the graph more connected if it's too dispersed or that would help maintain diversity if it's too connected.

These are the latent brokers between the topics: the nodes that have an unusually high rate of influence (betweenness centrality) to their freqency — meaning they may appear not as often as the most influential nodes but they are important narrative shifting points.

These are usually brokers between different clusters / communities of nodes, playing not easily noticed and yet important role in this network, like the "grey cardinals" of sorts.

The most prominent relations between the nodes that exist in this graph are shown above. We treat the graph as undirected by default as it allows us to better detect general patterns.

As an option, you can also downloaded directed bigrams above, in case the direction of the relations is important (for any application other than language).

Main Topics

(according to Latent Dirichlet Allocation):

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Most Influential Words

(main topics and words according to LDA):

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