Monday, 21 October 2013

Benoît Mandelbrot, the modern Leonardo Da Vinci

MATHEMATICS is a curious subject. Though often classed as one, it is not really a science. That scientists use it to describe their interpretation of reality is not quite the same thing. Nor, though, is it an art—not, at any rate, in the modern meaning of that word. The aesthetics of the subject, which any mathematician will tell you are the driving force behind his passion, are not obvious to the senses in the way that those of a painting, a symphony or a play are. Yet Benoît Mandelbrot’s celebrity beyond the academy is largely due to art in its modern, sensuous, sense. For the “set” to which he gave his name, when computed, drawn on a complex plane and suitably tinted, appealed greatly to the senses—as a million posters, greetings cards and T-shirts, bought by people who had not the faintest idea what it was, attest.
The Mandelbrot set is a collection of points in the complex-number plane. The formula for calculating these numbers is zn+1 = zn2 + c, where c is a complex number and n (representing the digits 1 to infinity) counts the number of times the calculation has been performed. Z starts as any number you like, and changes with each calculation, the value of zn+1 being used as zn the next time round. Sometimes the value of z remains finite, no matter how large n gets. In that case, c is part of the Mandelbrot set. Sometimes z shoots off to infinity. In that case, c is not part of the set. The boundary between the two is the swirling fractal line that so appeals to the eye, and the colours of the points outside the set indicate how long the calculation takes to start shooting off to infinity.

Benoît Mandelbrot


Probability and Randomness: Other mathematicians of probability like Kolmogorov may be more academic or progressive but Mandelbrot was unique he proved that mathematicians actually understand randomness.In fact Nassim Nicholas Taleb's "Black Swan Theory" is inspired by work of Mandelbrot as Mandelbrot was much concerned about high-risk rare events(Black Swans).Nassim and Mandelbrot collaborated in  research tasks related to risk management and Mandelbrot was later called by Nassim as "poet of randomness".Black Swans were dealt by him in a philosophical and aesthetic way.Mandelbrot was initially a probability guy but later went into other fields of maths and made his name in other fields.In 1960s Mandelbrot presented his ideas on prices of commodity and stock prices and made a contribution on mathematics of randomness in economic theory.Mandelbrot also knew the pitfalls in Bachelier's model.Mandelbrot linked randomness to geometry and made randomness a more natural science.If stock markets were Gaussian then stock market crashed would have happen once in a Billion years. Mandelbrot's randomness methods make the statistics methods look useless. 

Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness. Mandelbrot designed the mathematical object called "Mandelbrot set" and later worked on shapes and fractals of maths and also worked on Chaos Theory. These objects play an important role on aesthetics , music , architecture , poetry , gestures and tones are derived from fractals . Mandelbrot's book "Fractal Geometry of Nature" it made a fame in arts , visual arts and every artistic circle. He was later offered a position in Medicine , all artists used to call Mandelbrot "The Rock Star of Mathematics". Mandelbrot became famous because of the number of applications of mathematics in our society. Mandelbrot was initially a probability guy but later went into other fields of maths and made his name in other fields.

In 1960s Mandelbrot presented his ideas on prices of commodity and stock prices and made a contribution on mathematics of randomness in economic theory. Just is it as easy to reject innocence  than accept it, it is easier to reject a Bell curve than accept it .It's difficult to reject Fractal than accept it because a single event can cause a disaster.


Mandelbrot's Contribution in Finance.


The first formal model for security price changes was put forward by Bachelier (1900). His price difference process in essence sets out the mathematics of Brownian Motion before Einstein and Wiener rediscovered his results in 1905 and 1923 in the context of physical particles, and in particular generates a Normal (i.e. Gaussian) distribution where variance increases proportionally with time. A crucial assumption of Bachelier’s approach is that successive price changes are independent. His dissertation, which was awarded only a “mention honorable” rather than the “mention très honorable” that was essential for recognition in the academic world, remained unknown to the financial world until Osborne (1959), who made no reference to Bachelier’s work, rediscovered Brownian Motion as a plausible model for security price changes.

But in 1963 the famous mathematician Mandelbrot produced a paper pointing out that the tails of security price distributions are far fatter than those of normal distributions (what he called the “Noah effect” in reference to the deluge in biblical times) and recommending instead a class of independent and identically distributed “alpha-stable” Paretian distributions with infinite variance. Towards the end of the paper Mandelbrot observes that the independence assumption in his suggested model does not fully reflect reality in that “on closer inspection … large changes tend to be followed by large changes – of either sign – and small changes tend to be followed by small changes.” Mandelbrot later called this the “Joseph effect” inreference to the biblical account of seven years of plentiful harvests in Egypt followed by seven years of famine. Such a sequence of events would have had an exceptionally low probability of taking place if harvest yields in successive years were independent. While considering how best to model this dependence effect, Mandelbrot came across the work of Hurst (1951, 1955) which dealt with a very strong dependence in natural events such as river flows (particularly in the case of the Nile) from one year to another and developed the Hurst exponent H as a robust statistical measure of dependence. Mandelbrot’s new model ofFractional Brownian Motion, which is described in detail in Mandelbrot & van Ness (1968), is defined by an equation which incorporates the Hurst exponent H. Many financial economists, particularly Cootner (1964), were highly critical of Mandelbrot’s work, mainly because – if he was correct about normal distributions being seriously inconsistent with reality – most of their earlier statistical work, particularly in tests of the Capital Asset Pricing Model and the Efficient Market Hypothesis, would be invalid. Indeed, in his seminal review work on stockmarket efficiency, Fama (1970) describes how non-normal stable distributions of precisely the type advocated by Mandelbrot are more realistic than standard distributions .

“Economists have, however, been reluctant to accept these results, primarily because of the wealth of statistical techniques available for dealing with normal variables and the relative paucity of such techniques for non-normal stable variables.”

Partly because of estimation problems with alpha-stable Paretian distributions and the mathematical complexity of Fractional Brownian Motion, and partly because of the conclusion in Lo (1991) that standard distributions might give an adequate representation of reality, Mandelbrot’s two suggested new models failed to make a major impact on finance theory, and he essentially left the financial scene to pursue other interests such as fractal geometry. However, in his “Fractal Geometry of Nature”, Mandelbrot (1982) commented on what he regarded as the “suicidal” statistical methodologies that were standard in financetheory: “Faced with a statistical test that rejects the Brownian hypothesis that price changes are Gaussian, the economist can try one modification after another until the test is fooled. A popular fix is censorship, hypocritically called ‘rejection of outliers’. One distinguishes the ordinary ‘small’ price changes from the large changes that defeat Alexander’s filters. The former are viewed as random and Gaussian, and treasures of ingenuity are devoted to them … The latter are handled separately, as ‘nonstochastic’.”

Shortly after the “Noah effect” manifested itself with extreme severity in the collapse of Long-Term Capital Management, Mandelbrot (1999) produced a brief article, the cover story of the February 1999 issue of “Scientific American”, in which he used nautical analogies to highlight the foolhardy nature of standard risk models that assumed independent normal distributions. He also pointed out that a more realistic depiction of market fluctuations, namely Fractional Brownian Motion in multifractal trading time, already existed.


In fact he was one of the pioneers in studying the variation of financial prices even before Bchelier's Brownian model became widely accepted in academia and Mandelbrot also knew the pitfalls in Bachelier's model.For this reason many call him as the "father of Quantitative Finance" 



Mandelbrot's contribution in finance fall into three main stages:

He was the first to stress the essential importance, even in a first approximation, of large variations that may occur as sudden price discontinuities. The Brownian model is unjustified in neglecting them. They are not “outliers” one can safely disregard or study separately. To the contrary, their distribution is much more important than that of the "background noise" constituted by the small changes of Brownian motion. He followed this critique in by showing in 1963 that the big discontinuities and the small "noise" fall on a single power-law distribution and represented them by a scenario based on Levy stable distributions. He and Taylor introduced in 1967 the new notion of intrinsic "trading time." In recent years, fractal trading time and his 1963 model have gained wide acceptance.

Secondly, Mandelbrot tackled the fact that the “background noise” of small price changes is of variable “volatility.” This feature was ordinarily viewed as a symptom of non-stationality that must be studied separately. To the contrary, Mandelbrot interpreted this variability as indicating that price changes are far from being statistically independent. In fact, for all practical purposes, their interdependence should be viewed as continuing to an infinitely long term. In particular, it is not limited to the short term that is studied by Markov processes and more recently ARCH and its variants. In fact, it too follows a power-law side of dependence. He followed this critique and illustrated long-dependence by introducing in 1965  a process called fractional Brownian motion which has become very widely used.

Thirdly, he introduced the new notion of multifractality that combines long power-law tails and long power-law dependence. Early on, his work was motivated by the context of turbulence, but he immediately observed and pointed out that in 1972 the same ideas also apply to finance. After a long hiatus while he was developing other aspects of fractal geometry, he returned to finance in the mid-1990s and developed the multifractal scenario theory in detail in his 1997 book "Fractals and Scaling in Finance". The concept of scaling invariance used by Mandelbrot started by being perceived as suspect, because at that time other fields did not use it. However the period after 1972 also saw the growth of a new subfield of statistical physics concerned with “criticality.” The concepts used in that field are similar to those Mandelbrot had been using in finance.In "The Misbehavior of Markets", another popular book by Mandelbrot,he argues that the Gaussian models for financial risk used by economists like William Sharpe and Harry Markowitz should be discarded, since these models do not reflect reality. Mandelbrot argues that fractal techniques may provide a more powerful way to analyze risk.


Fractal Geometry. Complex-number plane. Lewis Carroll, no mean mathematician himself, asked Alice to believe as many as six impossible things before breakfast. These could easily have been two of them.First, then, the complex plane. This is the space on which all numbers, real, imaginary and combinations of the two, can be plotted. A real number is the familiar sort from normal arithmetic. An imaginary one is a multiple of the square root of -1.

Mathematicians struggled for centuries with the question of what, multiplied by itself, gives the answer -1 before one of them, Leonhard Euler, suggested that the best way to deal with the problem was to invent a new symbol (he chose i) and live with the consequences. It works. And, as Euler’s successor, Carl Friedrich Gauss, was to discover, if you plot real numbers on one axis of a graph and imaginary ones on the other, you create a plane that represents both sorts of numbers. Complex numbers, which have a real and an imaginary part added together, are the points on this plane that do not lie on either axis.The invention of complex numbers was a watershed in mathematics. It also marked the moment when maths began to slip away from being part of the armamentarium of any educated person and towards the dizzyingly abstruse field it has become today. But a fractal is something a ten-year-old child might hit on.

In 1975 he invented the word fractal to describe his discoveries. But the breakthrough that made them famous was the ability of computers to plot them in a way that is easy on the eye. Thus were launched the posters, the cards  and the T-shirts.While the ideas behind fractals, iteration and self-similarity are ancient, it took the coining of the term "fractal geometry" in 1975 and the publication of The Fractal Geometry of Nature in French in the same year to give the quest an identity. As Mandelbrot put it, "to have a name is to be" — and the field exploded.Extending fractals into the plane of complex numbers followed in 1979.


Fractal geometry is now being used in work with marine organisms, vegetative ecosystems, earthquake data, the behaviour of density-dependent populations, percolation and aggregation in oil research, and in the formation of lightning. Lightning resembles the diffusion patterns left by water as it permeates soft rock such as sandstone: computer simulations of this effect look exactly like the real thing.
Fractals hold a promise for building better roads, for video compression and even for designing ships that are less likely to capsize. The geometry is already being successfully applied in medical imaging, and the forms generated by the discipline are a source of pleasure in their own right, adding to our aesthetic awareness as we observe fractals everywhere in nature.

This fractal, Cells Alive, is a winner at Benoît Mandelbrot Fractal Art Contest 2009.

What is the length of a country’s coastline? Any encyclopedia will give you a figure. Yet stand by the sea and watch the irregularity of its edge, and you begin to doubt. It is not just a matter of tide and waves. Even measuring the boundary of a static body of water is no mean feat. The closer you look, the more irregular the line. That, at bottom, is what describes a fractal. When you magnify it, it rushes away from you and becomes a simulacrum of its larger self, eventually infinitely long.

Benoît Mandelbrot



Computers and IBM.His interest in computers was immediate, and his use of the new resource grew rapidly. He returned to France, married Aliette Kagan and became a professor at the University of Lille and then at the Centre National de la Recherche Scientifique in Paris. His academic future looked assured. But he felt uncomfortable in that environment, and in 1958 he spent the summer at IBM as a faculty visitor.
The company asked him to work on eliminating the apparently random noise in signal transmissions between computer terminals. The errors were not in fact completely random – they tended to come in bunches. Mandelbrot observed that the degree of bunching remained constant whether he plotted them by the month, the week or by the day. This was another step towards his fractal revelation.
During the 1960s Mandelbrot's quest led him to study galaxy clusters, applying his ideas on scaling to the structure of the universe itself. He scoured through forgotten and obscure journals. He found the clue he was looking for in the work of the mathematician and meteorologist Lewis Fry Richardson: he took a photocopy, and when he returned to consult the volume further, found it had gone to be pulped. Nonetheless, he knew he had struck a rich seam.Mandelbrot also tested the Fractal Theory in financial markets and certain distributions and methods for High Impact and Rare events(Black Swans) using computers and Monte Carlo Methods.
Before all this Dr Mandelbrot worked in the obscurity that modern mathematicians have resigned themselves to. He had followed, albeit belatedly, a path familiar to Jewish intellectuals driven from eastern Europe by the rise of the Nazis. His family fled Poland for France before the second world war and, though they stayed there for the duration, the young Benoît afterwards oscillated between France and the United States before settling for America in 1958. Once there, he worked for IBM. Among other things, he modelled electrical noise. Which, it turns out, is fractal. That it was the transmogrification of his formula by computers which brought him fame is thus appropriate.

Game, set and match?For a time, fractals seemed the answer to everything: the shape of clouds, the growth of organisms, even why the night sky is dark. Then the world lost interest.
Perhaps it should not have. For among Dr Mandelbrot’s beliefs was a conviction that financial-market movements, too, have fractal forms, rather than the familiar bell shapes of “normal” distribution that Gauss also described. If Dr Mandelbrot’s belief was correct, trading models based on Gauss’s distribution are wrong.
That markets are not Gaussian has now been accepted. Dr Mandelbrot’s interpretation, however, has not. Even if it had been, the bankers might not have noticed. They preferred algorithms to geometry.

Artists have been fascinated by geometry for as long as mathematicians have. The studies of Euclid are reflected in the regularities of classical and Renaissance architecture, from the Pantheon in Rome to the Duomo in Florence. But artists and architects were also thinking centuries ago about non-regular, curving geometries. You could argue that fractals give us the mathematics of the Baroque – they were anticipated by Borromini and Bach. I have a facsimile, given away by an Italian newspaper, of part of Leonardo da Vinci's Atlantic Codex, which contains page after page of his attempts to analyse the geometry of twisted, curving shapes.

Mandelbrot was a modern Leonardo, a man who showed the beauty in nature and a man who worked in many fields of mathematics. He was a prophet of the curving universe and gave us, in the endlessly playful geometry of fractals, a visual lexicon for our complex world.

 Fractal Geometry


 Fractal Geometry

 Fractal Geometry

Fractal Geometry

No comments:

Post a Comment