Misplaced Pages

#338661

128-487: In fractal geometry , the generalized Hurst exponent has been denoted by H or H q in honor of both Harold Edwin Hurst and Ludwig Otto Hölder (1859–1937) by Benoît Mandelbrot (1924–2010). H is directly related to fractal dimension , D , and is a measure of a data series' "mild" or "wild" randomness. The Hurst exponent is referred to as the "index of dependence" or "index of long-range dependence". It quantifies

256-402: A power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent : one quantity varies as a power of another. The change is independent of the initial size of those quantities. For instance, the area of a square has a power law relationship with

384-913: A 93% success rate in distinguishing real from imitation Pollocks. Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals. Decalcomania , a technique used by artists such as Max Ernst , can produce fractal-like patterns. It involves pressing paint between two surfaces and pulling them apart. Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art , games, divination , trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles. Hokky Situngkir also suggested

512-726: A category of fractal that has come to be called "self-inverse" fractals. One of the next milestones came in 1904, when Helge von Koch , extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the Koch snowflake . Another milestone came a decade later in 1915, when Wacław Sierpiński constructed his famous triangle then, one year later, his carpet . By 1918, two French mathematicians, Pierre Fatou and Gaston Julia , though working independently, arrived essentially simultaneously at results describing what

640-561: A circular village made of circular houses. According to Pickover , the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them. Indeed, according to various historical accounts, after that point few mathematicians tackled

768-406: A common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. The Tweedie distributions are a family of statistical models characterized by closure under additive and reproductive convolution as well as under scale transformation. Consequently, these models all express a power-law relationship between the variance and the mean. These models have

896-455: A concrete sense, this means fractals cannot be measured in traditional ways. To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as

1024-498: A constant factor c {\displaystyle c} causes only a proportionate scaling of the function itself. That is, where ∝ {\displaystyle \propto } denotes direct proportionality . That is, scaling by a constant c {\displaystyle c} simply multiplies the original power-law relation by the constant c − k {\displaystyle c^{-k}} . Thus, it follows that all power laws with

1152-440: A fractal scales compared to how geometric shapes are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is rep-tiled into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are

1280-505: A full characterization of the tail behavior of many well-known probability distributions, including power-law distributions, distributions with other types of heavy tails, and even non-heavy-tailed distributions. Bundle plots do not have the disadvantages of Pareto Q–Q plots, mean residual life plots and log–log plots mentioned above (they are robust to outliers, allow visually identifying power laws with small values of α {\displaystyle \alpha } , and do not demand

1408-463: A function of log ⁡ n {\displaystyle \log n} , and fitting a straight line; the slope of the line gives H {\displaystyle H} . A more principled approach would be to fit the power law in a maximum-likelihood fashion. Such a graph is called a box plot. However, this approach is known to produce biased estimates of the power-law exponent. For small n {\displaystyle n} there

SECTION 10

#1732771900339

1536-423: A fundamental role as foci of mathematical convergence similar to the role that the normal distribution has as a focus in the central limit theorem . This convergence effect explains why the variance-to-mean power law manifests so widely in natural processes, as with Taylor's law in ecology and with fluctuation scaling in physics. It can also be shown that this variance-to-mean power law, when demonstrated by

1664-442: A hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income). Among them are: A broken power law is a piecewise function , consisting of two or more power laws, combined with a threshold. For example, with two power laws: The pieces of a broken power law can be smoothly spliced together to construct

1792-427: A lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in

1920-770: A lopsided Sierpinsky Gasket". Some works by the Dutch artist M. C. Escher , such as Circle Limit III , contain shapes repeated to infinity that become smaller and smaller as they get near to the edges, in a pattern that would always look the same if zoomed in. Aesthetics and Psychological Effects of Fractal Based Design: Highly prevalent in nature, fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on

2048-419: A more convergent estimate than the maximum likelihood method. It has been applied to study probability distributions of fracture apertures. In some contexts the probability distribution is described, not by the cumulative distribution function , by the cumulative frequency of a property X , defined as the number of elements per meter (or area unit, second etc.) for which X  >  x applies, where x

2176-404: A particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f ( x ) {\displaystyle f(x)} and x {\displaystyle x} , and the straight-line on the log–log plot is often called the signature of

2304-402: A power law relationship, as many non power-law distributions will appear as straight lines on a log–log plot. This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in

2432-505: A power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws. Thus, accurately fitting and validating power-law models is an active area of research in statistics; see below. A power-law x − k {\displaystyle x^{-k}} has

2560-450: A power-law distribution of the form to the data x ≥ x min {\displaystyle x\geq x_{\min }} , where the coefficient α − 1 x min {\displaystyle {\frac {\alpha -1}{x_{\min }}}} is included to ensure that the distribution is normalized . Given a choice for x min {\displaystyle x_{\min }} ,

2688-473: A power-law known as the Pareto distribution (for example, the net worth of Americans is distributed according to a power law with an exponent of 2). On the one hand, this makes it incorrect to apply traditional statistics that are based on variance and standard deviation (such as regression analysis ). On the other hand, this also allows for cost-efficient interventions. For example, given that car exhaust

SECTION 20

#1732771900339

2816-420: A pure power law would allow for arbitrarily large or small values. Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature. The power-law model does not obey the treasured paradigm of statistical completeness. Especially probability bounds,

2944-438: A significant bias in α ^ {\displaystyle {\hat {\alpha }}} , while choosing it too large increases the uncertainty in α ^ {\displaystyle {\hat {\alpha }}} , and reduces the statistical power of our model. In general, the best choice of x min {\displaystyle x_{\min }} depends strongly on

3072-486: A small finite sample-size bias of order O ( n − 1 ) {\displaystyle O(n^{-1})} , which is small when n  > 100. Further, the standard error of the estimate is σ = α ^ − 1 n + O ( n − 1 ) {\displaystyle \sigma ={\frac {{\hat {\alpha }}-1}{\sqrt {n}}}+O(n^{-1})} . This estimator

3200-450: A small set of universality classes. Similar observations have been made, though not as comprehensively, for various self-organized critical systems, where the critical point of the system is an attractor . Formally, this sharing of dynamics is referred to as universality , and systems with precisely the same critical exponents are said to belong to the same universality class . Scientific interest in power-law relations stems partly from

3328-977: A smoothly broken power law. There are different possible ways to splice together power laws. One example is the following: ln ⁡ ( y y 0 + a ) = c 0 ln ⁡ ( x x 0 ) + ∑ i = 1 n c i − c i − 1 f i ln ⁡ ( 1 + ( x x i ) f i ) {\displaystyle \ln \left({\frac {y}{y_{0}}}+a\right)=c_{0}\ln \left({\frac {x}{x_{0}}}\right)+\sum _{i=1}^{n}{\frac {c_{i}-c_{i-1}}{f_{i}}}\ln \left(1+\left({\frac {x}{x_{i}}}\right)^{f_{i}}\right)} where 0 < x 0 < x 1 < ⋯ < x n {\displaystyle 0<x_{0}<x_{1}<\cdots <x_{n}} . When

3456-469: A total of 3 = 9 pieces. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/ r , there are a total of r pieces. Now, consider the Koch curve . It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being

3584-401: A well-defined mean over x ∈ [ 1 , ∞ ) {\displaystyle x\in [1,\infty )} only if k > 2 {\displaystyle k>2} , and it has a finite variance only if k > 3 {\displaystyle k>3} ; most identified power laws in nature have exponents such that the mean is well-defined but

3712-429: A wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of theory of large deviations (also called extreme value theory ), which considers the frequency of extremely rare events like stock market crashes and large natural disasters . It is primarily in the study of statistical distributions that

3840-587: A ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay

3968-417: Is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; this is generally helpful but limited. Authors disagree on the exact definition of fractal , but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in. One point agreed on

Hurst exponent - Misplaced Pages Continue

4096-436: Is a slowly varying function , which is any function that satisfies lim x → ∞ L ( r x ) / L ( x ) = 1 {\displaystyle \lim _{x\rightarrow \infty }L(r\,x)/L(x)=1} for any positive factor r {\displaystyle r} . This property of L ( x ) {\displaystyle L(x)} follows directly from

4224-1225: Is a significant deviation from the 0.5 slope. Anis and Lloyd estimated the theoretical (i.e., for white noise) values of the R/S statistic to be: E [ R ( n ) / S ( n ) ] = { Γ ( n − 1 2 ) π Γ ( n 2 ) ∑ i = 1 n − 1 n − i i , for  n ≤ 340 1 n π 2 ∑ i = 1 n − 1 n − i i , for  n > 340 {\displaystyle \mathbb {E} [R(n)/S(n)]={\begin{cases}{\frac {\Gamma ({\frac {n-1}{2}})}{{\sqrt {\pi }}\Gamma ({\frac {n}{2}})}}\sum \limits _{i=1}^{n-1}{\sqrt {\frac {n-i}{i}}},&{\text{for }}n\leq 340\\{\frac {1}{\sqrt {n{\frac {\pi }{2}}}}}\sum \limits _{i=1}^{n-1}{\sqrt {\frac {n-i}{i}}},&{\text{for }}n>340\end{cases}}} where Γ {\displaystyle \Gamma }

4352-542: Is a variable real number. As an example, the cumulative distribution of the fracture aperture, X , for a sample of N elements is defined as 'the number of fractures per meter having aperture greater than x . Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope). Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow

4480-486: Is accumulated, they continue to grow. These power-law probability distributions are also called Pareto-type distributions, distributions with Pareto tails, or distributions with regularly varying tails. A modification, which does not satisfy the general form above, with an exponential cutoff, is In this distribution, the exponential decay term e − λ x {\displaystyle \mathrm {e} ^{-\lambda x}} eventually overwhelms

4608-588: Is also a power-law function, but with a smaller scaling exponent. For data, an equivalent form of the cdf is the rank-frequency approach, in which we first sort the n {\displaystyle n} observed values in ascending order, and plot them against the vector [ 1 , n − 1 n , n − 2 n , … , 1 n ] {\displaystyle \left[1,{\frac {n-1}{n}},{\frac {n-2}{n}},\dots ,{\frac {1}{n}}\right]} . Although it can be convenient to log-bin

4736-431: Is assumed here that a random sample is obtained from a probability distribution, and that we want to know if the tail of the distribution follows a power law (in other words, we want to know if the distribution has a "Pareto tail"). Here, the random sample is called "the data". Pareto Q–Q plots compare the quantiles of the log-transformed data to the corresponding quantiles of an exponential distribution with mean 1 (or to

4864-467: Is close to 0, because Pareto Q–Q plots are not designed to identify distributions with slowly varying tails. On the other hand, in its version for identifying power-law probability distributions, the mean residual life plot consists of first log-transforming the data, and then plotting the average of those log-transformed data that are higher than the i -th order statistic versus the i -th order statistic, for i  = 1, ...,  n , where n

4992-469: Is convenient to assume a lower bound x m i n {\displaystyle x_{\mathrm {min} }} from which the law holds. Combining these two cases, and where x {\displaystyle x} is a continuous variable, the power law has the form of the Pareto distribution where the pre-factor to α − 1 x min {\displaystyle {\frac {\alpha -1}{x_{\min }}}}

5120-462: Is defined in terms of the asymptotic behaviour of the rescaled range as a function of the time span of a time series as follows; E [ R ( n ) S ( n ) ] = C n H  as  n → ∞ , {\displaystyle \mathbb {E} \left[{\frac {R(n)}{S(n)}}\right]=Cn^{H}{\text{ as }}n\to \infty \,,} where For self-similar time series, H

5248-772: Is dimension dependent, and for 1D and 2D it is H q 1 D = 1 2 , H q 2 D = − 1. {\displaystyle H_{q}^{1D}={\frac {1}{2}},\quad H_{q}^{2D}=-1.} For the popular Lévy stable processes and truncated Lévy processes with parameter α it has been found that H q = q / α , {\displaystyle H_{q}=q/\alpha ,} for q < α {\displaystyle q<\alpha } , and H q = 1 {\displaystyle H_{q}=1} for q ≥ α {\displaystyle q\geq \alpha } . Multifractal detrended fluctuation analysis

Hurst exponent - Misplaced Pages Continue

5376-603: Is directly related to fractal dimension , D , where 1 < D < 2, such that D = 2 - H . The values of the Hurst exponent vary between 0 and 1, with higher values indicating a smoother trend, less volatility, and less roughness. For more general time series or multi-dimensional process, the Hurst exponent and fractal dimension can be chosen independently, as the Hurst exponent represents structure over asymptotically longer periods, while fractal dimension represents structure over asymptotically shorter periods. A number of estimators of long-range dependence have been proposed in

5504-409: Is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially. The median does exist, however: for a power law x , with exponent ⁠ k > 1 {\displaystyle k>1} ⁠ , it takes the value 2 x min , where x min

5632-715: Is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant well-being. Humans appear to be especially well-adapted to processing fractal patterns with fractal dimension between 1.3 and 1.5. When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress. Power law In statistics ,

5760-406: Is equivalent to the popular Hill estimator from quantitative finance and extreme value theory . For a set of n integer-valued data points { x i } {\displaystyle \{x_{i}\}} , again where each x i ≥ x min {\displaystyle x_{i}\geq x_{\min }} , the maximum likelihood exponent is the solution to

5888-567: Is not met by space-filling curves such as the Hilbert curve . Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to Falconer , fractals should be only generally characterized by a gestalt of the following features; As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance,

6016-410: Is now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals. Very shortly after that work was submitted, by March 1918, Felix Hausdorff expanded the definition of "dimension", significantly for

6144-425: Is one method to estimate H ( q ) {\displaystyle H(q)} from non-stationary time series. When H ( q ) {\displaystyle H(q)} is a non-linear function of q the time series is a multifractal system . In the above definition two separate requirements are mixed together as if they would be one. Here are the two independent requirements: (i) stationarity of

6272-620: Is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals. Modeled fractals may be sounds, digital images, electrochemical patterns, circadian rhythms , etc. Fractal patterns have been reconstructed in physical 3-dimensional space and virtually, often called " in silico " modeling. Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above. As one illustration, trees, ferns, cells of

6400-488: Is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion. Images of fractals can be created by fractal generating programs . Because of the butterfly effect , a small change in a single variable can have an unpredictable outcome. Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to

6528-454: Is that fractal patterns are characterized by fractal dimensions , but whereas these numbers quantify complexity (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns. In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension . However, this requirement

SECTION 50

#1732771900339

6656-490: Is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms. Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees. Phenomena known to have fractal features include: Fractals often appear in

6784-574: Is the Euler gamma function . The Anis-Lloyd corrected R/S Hurst exponent is calculated as 0.5 plus the slope of R ( n ) / S ( n ) − E [ R ( n ) / S ( n ) ] {\displaystyle R(n)/S(n)-\mathbb {E} [R(n)/S(n)]} . No asymptotic distribution theory has been derived for most of the Hurst exponent estimators so far. However, Weron used bootstrapping to obtain approximate functional forms for confidence intervals of

6912-488: Is the normalizing constant . We can now consider several properties of this distribution. For instance, its moments are given by which is only well defined for m < α − 1 {\displaystyle m<\alpha -1} . That is, all moments m ≥ α − 1 {\displaystyle m\geq \alpha -1} diverge: when α ≤ 2 {\displaystyle \alpha \leq 2} ,

7040-426: Is the minimum value for which the power law holds. The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the critical exponents of

7168-474: Is the size of the random sample. If the resultant scatterplot suggests that the plotted points tend to stabilize about a horizontal straight line, then a power-law distribution should be suspected. Since the mean residual life plot is very sensitive to outliers (it is not robust), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots. Log–log plots are an alternative way of graphically examining

7296-461: Is the time lag and averaging is over the time window t ≫ τ , {\displaystyle t\gg \tau ,} usually the largest time scale of the system. Practically, in nature, there is no limit to time, and thus H is non-deterministic as it may only be estimated based on the observed data; e.g., the most dramatic daily move upwards ever seen in a stock market index can always be exceeded during some subsequent day. In

7424-428: Is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from

7552-519: The actin filaments in human cells assemble into fractal patterns. Similarly Matthias Weiss showed that the endoplasmic reticulum displays fractal features. The current understanding is that fractals are ubiquitous in cell biology, from proteins , to organelles , to whole cells. Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by Jackson Pollock by pouring paint directly onto horizontal canvasses. Recently, fractal analysis has been used to achieve

7680-430: The infinite regress in parallel mirrors or the homunculus , the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed. This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how

7808-626: The method of expanding bins , implies the presence of 1/ f noise and that 1/ f noise can arise as a consequence of this Tweedie convergence effect. Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or Pareto Q–Q plots ), mean residual life plots and log–log plots . Another, more robust graphical method uses bundles of residual quantile functions. (Please keep in mind that power-law distributions are also called Pareto-type distributions.) It

SECTION 60

#1732771900339

7936-568: The topological dimension . Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set . This exhibition of similar patterns at increasingly smaller scales is called self-similarity , also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge , the shape is called affine self-similar. Fractal geometry lies within

8064-440: The 17th century with notions of recursion , fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano , Bernhard Riemann , and Karl Weierstrass , and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in

8192-554: The 20th century. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension ." Later, seeing this as too restrictive, he simplified and expanded

8320-523: The Africans might have been using a form of mathematics that they hadn't even discovered yet." In a 1996 interview with Michael Silverblatt , David Foster Wallace explained that the structure of the first draft of Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (a.k.a. Sierpinski gasket), but that the edited novel is "more like

8448-462: The Hurst exponent; subseries of smaller length lead to a high variance of the R/S estimates. The basic Hurst exponent can be related to the expected size of changes, as a function of the lag between observations, as measured by E(| X t + τ − X t |). For the generalized form of the coefficient, the exponent here is replaced by a more general term, denoted by q . There are a variety of techniques that exist for estimating H , however assessing

8576-419: The Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e.

8704-527: The Royal Prussian Academy of Sciences. In addition, the quotient difference becomes arbitrarily large as the summation index increases. Not long after that, in 1883, Georg Cantor , who attended lectures by Weierstrass, published examples of subsets of the real line known as Cantor sets , which had unusual properties and are now recognized as fractals. Also in the last part of that century, Felix Klein and Henri Poincaré introduced

8832-410: The above mathematical estimation technique, the function H ( q ) contains information about averaged generalized volatilities at scale τ {\displaystyle \tau } (only q = 1, 2 are used to define the volatility). In particular, the H 1 exponent indicates persistent ( H 1 > 1 ⁄ 2 ) or antipersistent ( H 1 < 1 ⁄ 2 ) behavior of

8960-1051: The accuracy of the estimation can be a complicated issue. Mathematically, in one technique, the Hurst exponent can be estimated such that: H q = H ( q ) , {\displaystyle H_{q}=H(q),} for a time series g ( t ) , t = 1 , 2 , … {\displaystyle g(t),t=1,2,\dots } may be defined by the scaling properties of its structure functions S q {\displaystyle S_{q}} ( τ {\displaystyle \tau } ): S q = ⟨ | g ( t + τ ) − g ( t ) | q ⟩ t ∼ τ q H ( q ) , {\displaystyle S_{q}=\left\langle \left|g(t+\tau )-g(t)\right|^{q}\right\rangle _{t}\sim \tau ^{qH(q)},} where q > 0 {\displaystyle q>0} , τ {\displaystyle \tau }

9088-437: The average and all higher-order moments are infinite; when 2 < α < 3 {\displaystyle 2<\alpha <3} , the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the central moment estimators (like the mean and the variance) for diverging moments will never converge – as more data

9216-431: The cdfs of the data and the power law with exponent α {\displaystyle \alpha } , respectively. As this method does not assume iid data, it provides an alternative way to determine the power-law exponent for data sets in which the temporal correlation can not be ignored. This criterion can be applied for the estimation of power-law exponent in the case of scale-free distributions and provides

9344-503: The collection of much data). In addition, other types of tail behavior can be identified using bundle plots. In general, power-law distributions are plotted on doubly logarithmic axes , which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) cumulative distribution (ccdf) that is, the survival function , P ( x ) = P r ( X > x ) {\displaystyle P(x)=\mathrm {Pr} (X>x)} , The cdf

9472-429: The constant C is a scaling factor to ensure that the total area is 1, as required by a probability distribution. More often one uses an asymptotic power law – one that is only true in the limit; see power-law probability distributions below for details. Typically the exponent falls in the range 2 < α < 3 {\displaystyle 2<\alpha <3} , though not always. More than

9600-410: The continuous version should not be applied to discrete data, nor vice versa. Further, both of these estimators require the choice of x min {\displaystyle x_{\min }} . For functions with a non-trivial L ( x ) {\displaystyle L(x)} function, choosing x min {\displaystyle x_{\min }} too small produces

9728-422: The conventional dimension (which is formally called the topological dimension ). Analytically, many fractals are nowhere differentiable . An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional , its fractal dimension indicates that it locally fills space more efficiently than an ordinary line. Starting in

9856-416: The data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided. The survival function, on the other hand, is more robust to (but not without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a survival function representation is favored over that of

9984-890: The definition to this: "A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples have been formulated and studied. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media and found in nature , technology , art , and architecture . Fractals are of particular relevance in

10112-600: The dependence of the rescaled range on the time span n of observation. A time series of full length N is divided into a number of nonoverlapping shorter time series of length n , where n takes values N , N /2, N /4, ... (in the convenient case that N is a power of 2). The average rescaled range is then calculated for each value of n . For each such time series of length n {\displaystyle n} , X = X 1 , X 2 , … , X n {\displaystyle X=X_{1},X_{2},\dots ,X_{n}\,} ,

10240-753: The ease with which certain general classes of mechanisms generate them. The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems; see also universality above. The ubiquity of power-law relations in physics is partly due to dimensional constraints , while in complex systems , power laws are often thought to be signatures of hierarchy or of specific stochastic processes . A few notable examples of power laws are Pareto's law of income distribution, structural self-similarity of fractals , scaling laws in biological systems , and scaling laws in cities . Research on

10368-591: The estimation of the power-law exponent, which does not assume independent and identically distributed (iid) data, uses the minimization of the Kolmogorov–Smirnov statistic , D {\displaystyle D} , between the cumulative distribution functions of the data and the power law: with where P e m p ( x ) {\displaystyle P_{\mathrm {emp} }(x)} and P α ( x ) {\displaystyle P_{\alpha }(x)} denote

10496-599: The evolution of the definition of fractals, to allow for sets to have non-integer dimensions. The idea of self-similar curves was taken further by Paul Lévy , who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole , described a new fractal curve, the Lévy C curve . Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked

10624-422: The exponent α {\displaystyle \alpha } (Greek letter alpha , not to be confused with scaling factor a {\displaystyle a} used above) is greater than 1 (otherwise the tail has infinite area), the minimum value x min {\displaystyle x_{\text{min}}} is needed otherwise the distribution has infinite area as x approaches 0, and

10752-569: The field of chaos theory because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction). The term "fractal" was coined by the mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus , meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature . The word "fractal" often has different connotations for

10880-432: The foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names , the species richness in clades of organisms, the sizes of power outages , volcanic eruptions, human judgments of stimulus intensity and many other quantities. Empirical distributions can only fit a power law for a limited range of values, because

11008-800: The function is plotted as a log-log plot with horizontal axis being ln ⁡ x {\displaystyle \ln x} and vertical axis being ln ⁡ ( y / y 0 + a ) {\displaystyle \ln(y/y_{0}+a)} , the plot is composed of n + 1 {\displaystyle n+1} linear segments with slopes c 0 , c 1 , . . . , c n {\displaystyle c_{0},c_{1},...,c_{n}} , separated at x = x 1 , . . . , x n {\displaystyle x=x_{1},...,x_{n}} , smoothly spliced together. The size of f i {\displaystyle f_{i}} determines

11136-439: The identification of power-law probability distributions using random samples has been proposed. This methodology consists of plotting a bundle for the log-transformed sample . Originally proposed as a tool to explore the existence of moments and the moment generation function using random samples, the bundle methodology is based on residual quantile functions (RQFs), also called residual percentile functions, which provide

11264-436: The impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant well-being. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create

11392-432: The increments , x ( t + T ) − x ( t ) = x ( T ) − x (0) in distribution. This is the condition that yields longtime autocorrelations. (ii) Self-similarity of the stochastic process then yields variance scaling, but is not needed for longtime memory. E.g., both Markov processes (i.e., memory-free processes) and fractional Brownian motion scale at the level of 1-point densities (simple averages), but neither scales at

11520-484: The issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters". Thus, it was not until two centuries had passed that on July 18, 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered a fractal, having the non- intuitive property of being everywhere continuous but nowhere differentiable at

11648-399: The lay public as opposed to mathematicians, where the public is more likely to be familiar with fractal art than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background. The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with

11776-399: The length of its side, since if the length is doubled, the area is multiplied by 2 , while if the length is tripled, the area is multiplied by 3 , and so on. The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares , cloud sizes,

11904-460: The level of pair correlations or, correspondingly, the 2-point probability density. An efficient market requires a martingale condition, and unless the variance is linear in the time this produces nonstationary increments, x ( t + T ) − x ( t ) ≠ x ( T ) − x (0) . Martingales are Markovian at the level of pair correlations, meaning that pair correlations cannot be used to beat a martingale market. Stationary increments with nonlinear variance, on

12032-409: The literature. The oldest and best-known is the so-called rescaled range (R/S) analysis popularized by Mandelbrot and Wallis and based on previous hydrological findings of Hurst. Alternatives include DFA , Periodogram regression, aggregated variances, local Whittle's estimator, wavelet analysis, both in the time domain and frequency domain . To estimate the Hurst exponent, one must first estimate

12160-562: The log likelihood function becomes: The maximum of this likelihood is found by differentiating with respect to parameter α {\displaystyle \alpha } , setting the result equal to zero. Upon rearrangement, this yields the estimator equation: where { x i } {\displaystyle \{x_{i}\}} are the n {\displaystyle n} data points x i ≥ x min {\displaystyle x_{i}\geq x_{\min }} . This estimator exhibits

12288-443: The mathematical branch of measure theory . One way that fractals are different from finite geometric figures is how they scale . Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which

12416-701: The means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings). That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , which built on earlier work by Lewis Fry Richardson . In 1975, Mandelbrot solidified hundreds of years of thought and mathematical development in coining

12544-440: The most reliable techniques are often based on the method of maximum likelihood . Alternative methods are often based on making a linear regression on either the log–log probability, the log–log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent. For real-valued, independent and identically distributed data, we fit

12672-418: The name "power law" is used. In empirical contexts, an approximation to a power-law o ( x k ) {\displaystyle o(x^{k})} often includes a deviation term ε {\displaystyle \varepsilon } , which can represent uncertainty in the observed values (perhaps measurement or sampling errors) or provide a simple way for observations to deviate from

12800-545: The nervous system, blood and lung vasculature, and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques. The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects, such as coastlines and mountains. A limitation of modeling fractals

12928-416: The origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics , computer science , linguistics , geophysics , neuroscience , systematics , sociology , economics and more. However, much of the recent interest in power laws comes from the study of probability distributions : The distributions of

13056-402: The other hand, induce the longtime pair memory of fractional Brownian motion that would make the market beatable at the level of pair correlations. Such a market would necessarily be far from "efficient". An analysis of economic time series by means of the Hurst exponent using rescaled range and Detrended fluctuation analysis is conducted by econophysicist A.F. Bariviera. This paper studies

13184-400: The particular form of the lower tail, represented by L ( x ) {\displaystyle L(x)} above. More about these methods, and the conditions under which they can be used, can be found in . Further, this comprehensive review article provides usable code (Matlab, Python, R and C++) for estimation and testing routines for power-law distributions. Another method for

13312-399: The pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended. There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield unbiased and consistent answers . Some of

13440-447: The plot tend to converge to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published. A disadvantage of these plots is that, in order for them to provide reliable results, they require huge amounts of data. In addition, they are appropriate only for discrete (or grouped) data. Another graphical method for

13568-399: The power-law behavior at very large values of x {\displaystyle x} . This distribution does not scale and is thus not asymptotically as a power law; however, it does approximately scale over a finite region before the cutoff. The pure form above is a subset of this family, with λ = 0 {\displaystyle \lambda =0} . This distribution is

13696-429: The power-law function (perhaps for stochastic reasons): Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated power function is possible: p ( x ) = C x − α {\displaystyle p(x)=Cx^{-\alpha }} for x > x min {\displaystyle x>x_{\text{min}}} where

13824-507: The practical limits of physical time and space. Models may simulate theoretical fractals or natural phenomena with fractal features . The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis . Some specific applications of fractals to technology are listed elsewhere . Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it

13952-417: The quantiles of a standard Pareto distribution) by plotting the former versus the latter. If the resultant scatterplot suggests that the plotted points asymptotically converge to a straight line, then a power-law distribution should be suspected. A limitation of Pareto Q–Q plots is that they behave poorly when the tail index α {\displaystyle \alpha } (also called Pareto index)

14080-506: The range 0 – 0.5 indicates a time series with long-term switching between high and low values in adjacent pairs, meaning that a single high value will probably be followed by a low value and that the value after that will tend to be high, with this tendency to switch between high and low values lasting a long time into the future, also following a power law. A value of H =0.5 indicates short-memory , with (absolute) autocorrelations decaying exponentially quickly to zero. The Hurst exponent, H ,

14208-652: The realm of living organisms where they arise through branching processes and other complex pattern formation. Ian Wong and co-workers have shown that migrating cells can form fractals by clustering and branching . Nerve cells function through processes at the cell surface, with phenomena that are enhanced by largely increasing the surface to volume ratio. As a consequence nerve cells often are found to form into fractal patterns. These processes are crucial in cell physiology and different pathologies . Multiple subcellular structures also are found to assemble into fractals. Diego Krapf has shown that through branching processes

14336-451: The relative tendency of a time series either to regress strongly to the mean or to cluster in a direction. A value H in the range 0.5–1 indicates a time series with long-term positive autocorrelation, meaning that the decay in autocorrelation is slower than exponential, following a power law ; for the series it means that a high value tends to be followed by another high value and that future excursions to more high values do occur. A value in

14464-488: The requirement that p ( x ) {\displaystyle p(x)} be asymptotically scale invariant; thus, the form of L ( x ) {\displaystyle L(x)} only controls the shape and finite extent of the lower tail. For instance, if L ( x ) {\displaystyle L(x)} is the constant function, then we have a power law that holds for all values of x {\displaystyle x} . In many cases, it

14592-450: The rescaled range is calculated as follows: The Hurst exponent is estimated by fitting the power law E [ R ( n ) / S ( n ) ] = C n H {\displaystyle \mathbb {E} [R(n)/S(n)]=Cn^{H}} to the data. This can be done by plotting log ⁡ [ R ( n ) / S ( n ) ] {\displaystyle \log[R(n)/S(n)]} as

14720-407: The same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference

14848-594: The sharpness of splicing between segments i − 1 , i {\displaystyle i-1,i} . A power law with an exponential cutoff is simply a power law multiplied by an exponential function: In a looser sense, a power-law probability distribution is a distribution whose density function (or mass function in the discrete case) has the form, for large values of x {\displaystyle x} , where α > 1 {\displaystyle \alpha >1} , and L ( x ) {\displaystyle L(x)}

14976-514: The similar properties in Indonesian traditional art, batik , and ornaments found in traditional houses. Ethnomathematician Ron Eglash has discussed the planned layout of Benin city using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that

15104-414: The snowflake has an infinite perimeter. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics , with several notable people contributing canonical fractal forms along the way. A common theme in traditional African architecture is the use of fractal scaling, whereby small parts of the structure tend to look similar to larger parts, such as

15232-427: The suspected cause of typical bending and/or flattening phenomena in the high- and low-frequency graphical segments, are parametrically absent in the standard model. One attribute of power laws is their scale invariance . Given a relation f ( x ) = a x − k {\displaystyle f(x)=ax^{-k}} , scaling the argument x {\displaystyle x} by

15360-454: The system. Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach criticality —can be shown, via renormalization group theory, to share the same fundamental dynamics. For instance, the behavior of water and CO 2 at their boiling points fall in the same universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by

15488-413: The tail of a distribution using a random sample. Taking the logarithm of a power law of the form f ( x ) = a x k {\displaystyle f(x)=ax^{k}} results in: which forms a straight line with slope k {\displaystyle k} on a log-log scale. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for

15616-470: The time varying character of Long-range dependency and, thus of informational efficiency. Hurst exponent has also been applied to the investigation of long-range dependency in DNA , and photonic band gap materials. Fractal geometry In mathematics , a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding

15744-456: The transcendental equation where ζ ( α , x m i n ) {\displaystyle \zeta (\alpha ,x_{\mathrm {min} })} is the incomplete zeta function . The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for α ^ {\displaystyle {\hat {\alpha }}} are not equivalent, and

15872-483: The trend. For the BRW ( brown noise , 1 / f 2 {\displaystyle 1/f^{2}} ) one gets H q = 1 2 , {\displaystyle H_{q}={\frac {1}{2}},} and for pink noise ( 1 / f {\displaystyle 1/f} ) H q = 0. {\displaystyle H_{q}=0.} The Hurst exponent for white noise

16000-469: The two most popular methods, i.e., for the Anis-Lloyd corrected R/S analysis: and for DFA : Here M = log 2 ⁡ N {\displaystyle M=\log _{2}N} and N {\displaystyle N} is the series length. In both cases only subseries of length n > 50 {\displaystyle n>50} were considered for estimating

16128-471: The unique real number D that satisfies 3 = 4. This number is called the fractal dimension of the Koch curve; it is not the conventionally perceived dimension of a curve. In general, a key property of fractals is that the fractal dimension differs from the conventionally understood dimension (formally called the topological dimension). This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable ". In

16256-405: The variance is not, implying they are capable of black swan behavior. This can be seen in the following thought experiment: imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest person entering the room, with a monthly income of about 1 billion US$ . What happens to the average income in the room? Income is distributed according to

16384-607: The word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set , captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". In 1980, Loren Carpenter gave a presentation at the SIGGRAPH where he introduced his software for generating and rendering fractally generated landscapes. One often cited description that Mandelbrot published to describe geometric fractals

#338661