Fractal analysis is assessing

fractal In mathematics, a fractal is a Shape, 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 scale ...

characteristics of

data Data ( , ) are a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted for ...

. It consists of several methods to assign a

fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...

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 biosignal, bio signals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in ...

signals, digital images, molecular motion, and

data science Data science is an interdisciplinary academic field that uses statistics, scientific computing, scientific methods, processing, scientific visualization, algorithms and systems to extract or extrapolate knowledge from potentially noisy, stru ...

. Fractal analysis is now widely used in all areas of

science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...

. 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 was formulated which is a generalization of ordinary calculus.

Underlying principles

Fractals have fractional dimensions, which are a measure of

complexity Complexity characterizes the behavior of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to c ...

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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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

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

properties at different scales. This characteristic is termed '' scale invariance'', and can be further categorized as '' self-similarity'' 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''

Types of fractal analysis

There are various types of fractal analysis, including

box counting Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the pr ...

, lacunarity analysis, mass methods, and multifractal analysis. 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.

Applications

Ecology and evolution

Unlike theoretical fractal curves which can be easily measured and the underlying mathematical properties calculated; natural systems are sources of heterogeneity and generate complex space-time structures that may only demonstrate partial self-similarity. Using fractal analysis, it is possible to analyze and recognize when features of complex ecological 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) is a system formed by Organism, organisms in interaction with their Biophysical environment, environment. The Biotic material, biotic and abiotic components are linked together through nutrient cycles and en ...

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 (biology), tissue in vascular plants, the other being phloem; both of these are part of the vascular bundle. The basic function of the xylem is to transport water upward from the roots to parts o ...

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 or positive 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 ...

Architecture, urban design and landscape design

In his publication ''The Fractal Geometry of Nature'', Benoit Mandelbrot suggested fractal theory could be applied to architecture. In this context, Mandelbrot was talking about the self-similar feature of fractal objects, rather than fractal analysis. In 1996, Carl Bovill applied the

method of fractal analysis to Architecture. Bovill’s work, using a manual version of box counting, has since been refined by others and computational approaches have been developed. Fractal analysis is one of the few quantitative analysis methods available to architects and designers to understand the visual complexity of buildings, urban areas and landscapes. Typical uses of fractal analysis of the built environment have been to understand the visual complexity of cities and skylines, the fractal dimensions of works of different architects and the landscape. Combining the fractal analysis of ecology (see above) with fractal analysis of architecture, fractal dimensions have been used to explore the possible relationship between nature and architecture. Promising results suggest further research is needed in this area.

Animal behaviour

Patterns in animal

behaviour Behavior (American English) or behaviour (British English) is the range of actions of Individual, individuals, organisms, systems or Artificial intelligence, artificial entities in some environment. These systems can include other systems or or ...

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, 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, 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 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 * Human anatomy * Diagnostic imaging * Cancer research * Fractal analysis of complex networks *

Classification Classification is the activity of assigning objects to some pre-existing classes or categories. This is distinct from the task of establishing the classes themselves (for example through cluster analysis). Examples include diagnostic tests, identif ...

histopathology Histopathology (compound of three Greek words: 'tissue', 'suffering', and '' -logia'' 'study of') is the microscopic examination of tissue in order to study the manifestations of disease. Specifically, in clinical medicine, histopatholog ...

slides in

medicine Medicine is the science and Praxis (process), practice of caring for patients, managing the Medical diagnosis, diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment, Palliative care, palliation of their injury or disease, ...

* Fractal landscape or

Coast A coast (coastline, shoreline, seashore) is the land next to the sea or the line that forms the boundary between the land and the ocean or a lake. Coasts are influenced by the topography of the surrounding landscape and by aquatic erosion, su ...

line complexity *

Electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that 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 *

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 A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology. In ...

and image compression * Urban growth *

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

* Diagnostic imaging *

Pathology Pathology is the study of disease. 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 the context of modern medical treatme ...

Geology Geology (). is a branch of natural science concerned with the Earth and other astronomical objects, the rocks of which they are composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth ...

Geography Geography (from Ancient Greek ; combining 'Earth' and 'write', literally 'Earth writing') is the study of the lands, features, inhabitants, and phenomena of Earth. Geography is an all-encompassing discipline that seeks an understanding o ...

Archaeology Archaeology or archeology is the study of human activity through the recovery and analysis of material culture. The archaeological record consists of Artifact (archaeology), artifacts, architecture, biofact (archaeology), biofacts or ecofacts, ...

Seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes (or generally, quakes) and the generation and propagation of elastic ...

* Soil studies * Computer and video game design, especially

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

for organic 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 content and algorithms coupled with computer-generated randomness and processing power. I ...

* Fractography and fracture mechanics * Fractal antennas — Small size antennas using fractal shapes * Small angle scattering theory of fractally rough systems * Generation of patterns for camouflage, such as MARPAT * 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. As a type of active management, it stands in contradiction to ...

of price series (see Elliott wave principle) * Fractal calculus