Fractal analysis is assessing

fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...

characteristics of

data In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpret ...

. It consists of several methods to assign a

fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...

and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from phenomena including topography, natural geometric objects, ecology and aquatic sciences, sound, market fluctuations, heart rates, frequency domain in

electroencephalography Electroencephalography (EEG) is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in the neocorte ...

signals, digital images, molecular motion, and

data science Data science is an interdisciplinary field that uses scientific methods, processes, algorithms and systems to extract or extrapolate knowledge and insights from noisy, structured and unstructured data, and apply knowledge from data across a br ...

. Fractal analysis is now widely used in all areas of

science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...

. An important limitation of fractal analysis is that arriving at an empirically determined fractal dimension does not necessarily prove that a pattern is fractal; rather, other essential characteristics have to be considered. Fractal analysis is valuable in expanding our knowledge of the structure and function of various systems, and as a potential tool to mathematically assess novel areas of study.

Fractal calculus In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...

was formulated which is a generalization of ordinary calculus

Underlying principles

Fractals have fractional dimensions, which are a measure of

complexity Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to ch ...

that indicates the degree to which the objects fill the available space. The fractal dimension measures the change in "size" of a fractal set with the changing observational scale, and is not limited by

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

values. This is possible given that a smaller section of the fractal resembles the entirety, showing the same

statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...

properties at different scales. This characteristic is termed ''

scale invariance In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...

'', and can be further categorized as ''

self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...

'' or '' self-affinity'', the latter scaled anisotropically (depending on the direction). Whether the view of the fractal is expanding or contracting, the structure remains the same and appears equivalently complex. Fractal analysis uses these underlying properties to help in the understanding and characterization of complex systems. It is also possible to expand the use of fractals to the lack of a single characteristic time scale, or pattern. ''Further information on the Origins:

Fractal Geometry In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...

Types of fractal analysis

There are various types of fractal analysis, including box counting, lacunarity analysis, mass methods, and

multifractal analysis A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed ...

. A common feature of all types of fractal analysis is the need for benchmark patterns against which to assess outputs. These can be acquired with various types of fractal generating software capable of generating benchmark patterns suitable for this purpose, which generally differ from software designed to render fractal art. Other types include detrended fluctuation analysis and the Hurst absolute value method, which estimate the

hurst exponent The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst expone ...

. It is suggested to use more than one approach in order to compare results and increase the robustness of one's findings.

Applications

Ecology and evolution

Unlike theoretical fractal curves which can be easily measured and the underlying mathematical properties calculated;

natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...

systems are sources of heterogeneity and generate complex space-time structures that may only demonstrate partial

. Using fractal analysis, it is possible to analyze and recognize when features of complex

ecological Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overlaps wi ...

systems are altered since fractals are able to characterize the natural complexity in such systems. Thus, fractal analysis can help to quantify patterns in nature and to identify deviations from these natural sequences. It helps to improve our overall understanding of

ecosystem An ecosystem (or ecological system) consists of all the organisms and the physical environment with which they interact. These biotic and abiotic components are linked together through nutrient cycles and energy flows. Energy enters the syst ...

s and to reveal some of the underlying structural mechanisms of nature. For example, it was found that the structure of an individual tree’s

xylem Xylem is one of the two types of transport tissue in vascular plants, the other being phloem. The basic function of xylem is to transport water from roots to stems and leaves, but it also transports nutrients. The word ''xylem'' is derived from ...

follows the same architecture as the spatial distribution of the trees in the forest, and that the distribution of the trees in the forest shared the same underlying fractal structure as the branches, scaling identically to the point of being able to use the pattern of the trees’ branches mathematically to determine the structure of the forest stand. The use of fractal analysis for understanding structures, and spatial and temporal complexity in biological systems has already been well studied and its use continues to increase in ecological research. Despite its extensive use, it still receives some

criticism Criticism is the construction of a judgement about the negative qualities of someone or something. Criticism can range from impromptu comments to a written detailed response. , ''"the act of giving your opinion or judgment about the good or bad q ...

Animal behaviour

Patterns in animal

behaviour Behavior (American English) or behaviour (British English) is the range of actions and mannerisms made by individuals, organisms, systems or artificial entities in some environment. These systems can include other systems or organisms as wel ...

exhibit fractal properties on spatial and temporal scales. Fractal analysis helps in understanding the behaviour of animals and how they interact with their environments on multiple scales in space and time. Various animal movement signatures in their respective environments have been found to demonstrate spatially non-linear fractal patterns. This has generated ecological interpretations such as the

Lévy Flight Foraging hypothesis The Lévy flight foraging hypothesis is a hypothesis in the field of biology that may be stated as follows: ''Since Lévy flights and walks can optimize search efficiencies, therefore natural selection should have led to adaptations for Lévy fligh ...

, which has proven to be a more accurate description of animal movement for some species. Spatial patterns and animal behaviour sequences in fractal time have an optimal complexity range, which can be thought of as the homeostatic state on the spectrum where the complexity sequence should regularly fall. An increase or a loss in complexity, either becoming more stereotypical or conversely more random in their behaviour patterns, indicates that there has been an alteration in the functionality of the individual. Using fractal analysis, it is possible to examine the movement sequential complexity of animal behaviour and to determine whether individuals are experiencing deviations from their optimal range, suggesting a change in condition. For example, it has been used to assess welfare of domestic hens, stress in bottlenose dolphins in response to human disturbance, and parasitic infection in Japanese macaques and sheep. The research is furthering the field of behavioural ecology by simplifying and quantifying very complex relationships. When it comes to animal welfare and

conservation Conservation is the preservation or efficient use of resources, or the conservation of various quantities under physical laws. Conservation may also refer to: Environment and natural resources * Nature conservation, the protection and manageme ...

, fractal analysis makes it possible to identify potential sources of stress on animal behaviour, stressors that may not always be discernible through classical behaviour research. This approach is more objective than classical behaviour measurements, such as frequency-based observations that are limited by the counts of behaviours, but is able to delve into the underlying reason for the behaviour. Another important advantage of fractal analysis is the ability to monitor the health of

wild Wild, wild, wilds or wild may refer to: Common meanings * Wild animal * Wilderness, a wild natural environment * Wildness, the quality of being wild or untamed Art, media and entertainment Film and television * ''Wild'' (2014 film), a 2014 A ...

and free-ranging animal populations in their natural habitats without invasive measurements.

Applications include

Applications of fractal analysis include: * Heart rate analysis * Human gait, balance, and activity * Diagnostic imaging * Cancer research * Fractal analysis of complex networks *

Classification Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood. Classification is the grouping of related facts into classes. It may also refer to: Business, organizat ...

histopathology Histopathology (compound of three Greek words: ''histos'' "tissue", πάθος ''pathos'' "suffering", and -λογία '' -logia'' "study of") refers to the microscopic examination of tissue in order to study the manifestations of disease. Sp ...

slides in

medicine Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...

Fractal landscape A fractal landscape is a surface that is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, ...

Coast The coast, also known as the coastline or seashore, is defined as the area where land meets the ocean, or as a line that forms the boundary between the land and the coastline. The Earth has around of coastline. Coasts are important zones in n ...

line complexity *

Electrical Engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

* Enzyme/enzymology ( Michaelis-Menten kinetics) * Generation of new music * Generation of various art forms
Detecting ‘life as we don't know it’ by fractal analysis
*

Search and rescue Search and rescue (SAR) is the search for and provision of aid to people who are in distress or imminent danger. The general field of search and rescue includes many specialty sub-fields, typically determined by the type of terrain the search ...

Signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...

and

image compression Image compression is a type of data compression applied to digital images, to reduce their cost for storage or transmission. Algorithms may take advantage of visual perception and the statistical properties of image data to provide superior re ...

* Urban growth *

Neuroscience Neuroscience is the science, scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, an ...

Diagnostic imaging Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues ( physiology). Medical imaging seeks to r ...

Pathology Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in ...

Geology Geology () is a branch of natural science concerned with Earth and other Astronomical object, astronomical objects, the features or rock (geology), rocks of which it is composed, and the processes by which they change over time. Modern geology ...

Geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...

Archaeology Archaeology or archeology is the scientific study of human activity through the recovery and analysis of material culture. The archaeological record consists of artifacts, architecture, biofacts or ecofacts, sites, and cultural landsc ...

Seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...

Wave propagation in self-similar (fractal) media
* Soil studies * Computer and video game design, especially

computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...

for

organic Organic may refer to: * Organic, of or relating to an organism, a living entity * Organic, of or relating to an anatomical organ Chemistry * Organic matter, matter that has come from a once-living organism, is capable of decay or is the product ...

environments and as part of

procedural generation In computing, procedural generation is a method of creating data algorithmically as opposed to manually, typically through a combination of human-generated assets and algorithms coupled with computer-generated randomness and processing power. In ...

* Fractography and

fracture mechanics Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics ...

Fractal antenna A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the effective length, or increase the perimeter (on inside sections or the outer structure), of material that can receive or transmit electromagnetic radiatio ...

s — Small size antennas using fractal shapes * Small angle scattering theory of fractally rough systems *

T-shirt A T-shirt (also spelled tee shirt), or tee, is a style of fabric shirt named after the T shape of its body and sleeves. Traditionally, it has short sleeves and a round neckline, known as a '' crew neck'', which lacks a collar. T-shirts are genera ...

s and other

fashion Fashion is a form of self-expression and autonomy at a particular period and place and in a specific context, of clothing, footwear, lifestyle, accessories, makeup, hairstyle, and body posture. The term implies a look defined by the fash ...

* Generation of patterns for camouflage, such as

MARPAT MARPAT (short for Marine pattern) is a multi-scale camouflage pattern in use with the United States Marine Corps, designed in 2001 and introduced from late 2002 to early 2005 with the Marine Corps Combat Utility Uniform (MCCUU), which replace ...

* Digital sundial *

Technical analysis In finance, technical analysis is an analysis methodology for analysing and forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and quantitative analysis use many of the sam ...

of price series (see

Elliott wave principle The Elliott Wave Principle, or Elliott wave theory, is a form of technical analysis that finance traders use to analyze financial market cycles and forecast market trends by identifying extremes in investor psychology and price levels, such as hi ...

)
Fractal analysis in music
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Underlying principles

Types of fractal analysis

Applications

Ecology and evolution

Animal behaviour

Applications include

See also

References

Further reading