Fractal analysis

Fractal analysis studies geometric and statistical patterns through fractal dimensions, scaling behavior, and related measures of complexity. It characterizes structures whose detail changes across scales.

Fractal dimension: A measure describing how a geometric set fills space as observation scale changes. It is the central quantity used to compare pattern complexity across scales.
Self-similarity: A property in which a structure resembles itself across different scales. Repeated resemblance motivates many geometric fractal models and measurements.
Fractal analysis of time series: The study of scale-dependent structure and complexity in sequences observed over time. It applies scaling measures to fluctuations rather than spatial outlines.
Correlation dimension: A dimension estimate based on how pairs of points cluster within increasing distances. It measures point-cloud structure through correlations rather than grid-box counts.
Fractal dimension estimation: The inference of a fractal dimension from sampled, finite, or noisy data. Different estimators can produce different values from the same limited pattern.
Box-counting dimension: A fractal dimension estimated by counting grid boxes that intersect a set at different sizes. Its log-log scaling estimate is a common practical route from image to dimension.
Scale invariance: A property of a system whose relevant form or laws remain unchanged under rescaling. Fractal analysis tests whether observed patterns approach scale-invariant behavior.
Hurst exponent: A statistic describing long-range dependence and scaling behavior in a time series. It summarizes persistence or roughness in signals analyzed across temporal scales.
Information dimension: A dimension derived from the scaling of information entropy across partitions. It weights regions by probability, unlike purely geometric box counting.
Scaling range: The interval of scales over which a measured quantity follows an approximate scaling law. Choosing this interval strongly affects whether a fractal interpretation is credible.

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Fractional dimensionNarrower topic: It uses dimension estimates to compare the complexity of shapes and measured data.