- freely available
Entropy 2017, 19(9), 437; https://doi.org/10.3390/e19090437
Of those three [i.e., economics, engineering, mathematics], nothing beats my impact on finance and mathematics. Physics—which I fear was least affected—rewarded my work most handsomely.
2. Hurst, and a Brief History of Hydrology Models
An “ideal dam” for a given time period is such that] (a) the outflow is uniform, (b) the reservoir ends the period as full as it began, (c) the dam never overflows, and (d) the capacity is the smallest compatible with (a), (b) and (c).
2.1. Hurst’s Paper
Although in random events groups of high or low values do occur, their tendency to occur in natural events is greater. ... There is no obvious periodicity, but there are long stretches when the floods are generally high, and others when they are generally low. These stretches occur without any regularity either in their time of occurrence or duration.(, §6)
2.2. Reactions to the Hurst Phenomenon
Is it meaningful to talk of a time-invariant mean over thousands of years? If long enough realizations of such time series were available would they in fact be stationary?(, §3.2)
The question of whether natural processes are stationary or not is likely a philosophical one. … there is probably not a single historic time series of which mathematics can tell with certainty whether it is stationary or not … Traditionally, it has been assumed that, in general, the geophysical, biological, economical, and other natural processes are nonstationary but within relatively short time spans can be well approximated by stationary models.
It is conceivable that the [Hurst] phenomenon can be explained probabilistically, starting from the assumption that the variables are not independent … Mathematically, this would require treating the variables as a Markov process.
It has been suggested that serial correlation or dependence [could cause the Hurst phenomenon]. This, however, cannot be true unless the serial dependence is of a very peculiar kind, for with all plausible models of serial dependence the series of values is always approximated by a [Brownian motion] when the time-scale is sufficiently large. A more plausible theory is that the experimental series used by Hurst are, as a result of serial correlation, not long enough for the asymptotic formula to become valid.
So far, such a convergence [to the law] has never been observed in hydrology. Thus, those who consider Hurst’s effect to be transient implicitly attach an undeserved importance to the value of [the sample size] … These scholars condemn themselves to never witness the full asymptotic development of the models they postulate.
Ever since Hurst published his famous plots for some geophysical time series … the by now classical Hurst phenomenon has continued to haunt statisticians and hydrologists. To some, it has become a puzzle to be explained, to others a feature to be reproduced by their models, and to others still, a ghost to be conjured away.
3. Mandelbrot’s Fractional Models
3.1. Initial Studies of Mandelbrot’s Model
We believe that FBMs do provide useful models for a host of natural time series and wish therefore to present their curious properties to scientists, engineers and statisticians.
If the requirement of continuity is abandoned, many other interesting self-similar processes suggest themselves. One may for example replace [the Brownian motion] by a non-Gaussian process whose increments are [-] stable … Such increments necessarily have an infinite variance. “Fractional Lévy-stable random functions” have moreover an infinite span of interdependence.
... current models of statistical hydrology cannot account for either [Noah or Joseph] effect and must therefore be superseded. As a replacement, the ‘self-similar’ models that we propose appear very promising.
3.2. Reactions to Mandelbrot’s Model
The theory of fractional noise is complicated by the motivating assumptions being in continuous time and the realizable version being needed in discrete time.(, §6.2)
It remains for the hydrologist to decide which type of behaviour [low- or high-frequency] is the more important to reproduce for any particular problem. No doubt derivations of FGN’s preserving both high and low frequency effects will eventually emerge and such a choice will not be necessary.(, §2.3)
The past influences the future only through its effect on the present, and thus once a certain state of the process has been reached, it matters little for the future development how it was arrived at.
An ability to simulate, and even successfully predict, a specific phenomenon does not necessarily imply an ability to explain it correctly. A highly successful operational model may turn out to be totally unacceptable from the physical point of view.
though he conceded that there were in fact possible mechanisms in the man-made world, although not, in his view, in the physical one.By what sort of physical mechanism can the influence of, say, the mean temperature of this year at a particular geographic location be transmitted over decades and centuries? What kind of a mechanism is it that has carried the impact of the economic crisis of the 1930s through World War II and the boom of the 1950s all the way into our times and will carry it far beyond?
The consequences of this fundamental idea are hard to accept ... [a]nd many people in many contexts have been arguing strongly against it, ... If infinite dependence is necessary it does not mean that IBM’s details of ten years ago influence IBM today, because there’s no mechanism within IBM for this dependence. However, IBM is not alone. The River Nile is [not] alone. They’re just one-dimensional corners of immensely big systems. The behaviour of IBM stock ten years ago does not influence its stock today through IBM, but IBM the enormous corporation has changed the environment very strongly. The way its price varied, went up, or went up and fluctuated, had discontinuities, had effects upon all kinds of other quantities, and they in turn affect us. And so my argument has always [sic] been that each of these causal chains is totally incomprehensible in detail, [and] probably exponentially decaying. There are so many of them that a very strong dependence may be perfectly compatible. Now I would like to mention that this is precisely the reason why infinite dependence exists, for example, in physics, in a magnet-because [although] two parts far away have very minor dependence along any path of actual dependence, there are so many different paths that they all combine to create a global structure.
Using self-similarity (with ) to extrapolate the correlated behaviour from a finite time span to an asymptotically infinite one is physically completely unjustified. Furthermore, using self-similarity to intrapolate [sic] to a very short time span … is physically absurd.
[The] self-similar model is the only model that predicts for the rescaled range statistic … precisely the same behaviour as Harold Edwin Hurst has observed empirically. To achieve the same agreement with other models, large numbers of ad hoc parameters are required. Thus the model’s justification is empirical, as is ultimately the case for any model of nature.
The preservation within synthetic sequences … [of h] is of prime importance to engineers since it characterizes long term storage behaviour. The use of synthetic sequences which fail to preserve this parameter usually leads to underestimation of long term storage requirements.
4. Fractionally Differenced Models
[Granger] prefers a discrete-time version of FBM that differs a bit from the Type I and Type II algorithm in . Discretization is usually motivated by unquestionable convenience, but I view it as more than a detail. I favor very heavily the models that possess properties of time-invariance or scaling. In these models, no time interval is privileged by being intrinsic. In discrete-time models, to the contrary, a privileged time interval is imposed nonintrinsically.
Conflicts of Interest
|ARFIMA||AutoRegressive Fractionally Integrated Moving Average|
|BDoA||Brownian Domain of Attraction|
|FBM||Fractional Brownian Motion|
|FGN||Fractional Gaussian Noise|
|LRD||Long Range Dependence|
|SRD||Short Range Dependence|
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