Long-Memory Processes: Probabilistic Properties and Statistical Methods

When the statistical MMN meets the physical MMN
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Volume 35 , Issue 4. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.

If the address matches an existing account you will receive an email with instructions to retrieve your username. Book Review. Read the full text. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access. Climatic, i. One may say that these values are not climatic in strict sense.

But they are strongly linked to the variability of the climate of a large area, from Mediterranean to the tropics. And they are instrumental. For example, around AD we have a group of low values producing a low climatic value, and around and we have groups of large values. Such grouping would not appear in a climate that would be the synthesis of independent random events. The latter would be more flat as illustrated by the synthetic example of Figure 3.

Another way of viewing the long-term variability of the Nile in Figure 2 is by using the notion of trends, irregularly changing from positive to negative and from mild to steep. Nile River annual minimum water level at Roda Nilometer from Ref. A few missing values at years , , , , , and are filled in using a simple method from Ref.

A synthetic time series from an independent white noise process with same statistics as those of the Nilometer series shown in the caption of Figure 2. Variability over different time scales, trends, clustering and persistence are all closely linked to each other. The former is a more rigorous concept and is mathematically described by the variance or the standard deviation of the averaged process as a function of the averaging time scale, aka climacogram.

In white noise, i. No variability is added at any finite time scale.

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A seemingly realistic stochastic process, which has been widely used for climate, is the Markov process, whose discrete time version is more widely known as the AR 1 process. The characteristic properties of this process are two:.

As a result, when the time scale of interest is fairly larger than this characteristic scale, the process behaves like white noise. It is difficult to explain why this model has become dominant in climatology. Even these two theoretical properties should have hampered its popularity. How could the future be affected just by the latest value and not by the entire past? Could any geophysical process, including climate, be determined by just one mechanism acting on a single time scale?

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The flow in a river not necessarily the Nile may help us understand better the multiplicity of mechanisms producing change and the multiplicity of the relevant time scales see also Ref. Of course none of these changes will be a surprise; rather, it would be a surprise if things remained static. Despite being anticipated, all these changes are not predictable. Does a plurality of mechanisms acting on different scales require a complex stochastic model? Not necessarily. A decade before Hurst detected LTP in natural processes, Andrey Kolmogorov , [15] devised a mathematical model which describes this behaviour using one parameter only, i.

In this model, change is produced at all scales and thus it never becomes similar to white noise, whatever the time scale of averaging is. Specifically, the variance will never become inversely proportional to time scale; it will decrease at a lower rate, inversely proportional to the power 2 — 2 H of the time scale nb.

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A characteristic property of the HK process is that its autocorrelation function is independent of time scale. In other words if there is some correlation in the average temperature between a year and the next one and in fact there is , the same correlation will be between a decade and the next one, a century and the next one, and so on to infinity.

Because there will always be another natural mechanism acting on a bigger scale, which will create change, and thus positive correlation at all lower scales the relationship of change with autocorrelation is better explained in Ref. The HK behaviour seems to be consistent with the principle of extremal entropy production. Are there other records of geophysical processes consistent with the HK behaviour?

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A recent overview paper [17] cites numerous studies where this behaviour has been verified. It also examines several instrumental and proxy climate data series related to temperature and, by superimposing the climacograms of the different series, it obtains an overview of the variability for time scales spanning almost nine orders of magnitude—from 1 month to 50 million years. The overall climacogram supports the presence of HK dynamics in climate with H at least 0. The orbital forcing Milankovitch cycles is also evident in the combined climacogram at time scales between 10 and thousand years.

We may see, for example, that what, according to the classical statistical perception, would require the entire age of the Earth to occur once i. This dramatic difference can help us understand why the choice of a proper stochastic model is relevant for the detection of changes in climate. It may also help us realize how easy it is to fool ourselves, given that our perception of probability may heavily rely on classical statistics.

Both versions result in about the same answer: the probability of having 11 warmest years in 12, or 12 warmest years in 15, is 0.

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Figure 4. If we used the IPCC AR4 terminology [19] we would say that either of these events is exceptionally unlikely to have a natural cause. Interestingly, the present results do not contradict those of a recent study of Zorita, Stocker and von Storch, [20] who examined a similar question and concluded that:. I note, though, that there are differences in the methodology followed here and that in Zorita et al. One may note that the above results, as well as those by Zorita et al.

The data are altered all the time as a result of continuous adaptations and adjustments.

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The generated synthetic datasets are analyzed using the temporal and DM models. Thus these affected costs are definitely not statistically independent. Biometrics , 69 , — The weaker performance of the DM model is expected since it does not explicitly model sampling zeros. Risk identification is one reason early activation of the IPT is essential to project success. While risks may arise from specific causes, they may also be the result of general environmental conditions that are not limited to specific times and places but are pervasive throughout the project. It is unknown a priori how relative abundances of bacterial taxa vary over time and how treatments and abrupt changes in the environment might alter ecological dynamics.

Even the ranks of the different years are changing: for example in the CRU data examined by Koutsoyiannis and Montanari [21] , was rank 1 the warmest of all and was rank 2, while now the ranking of these two years was inverted. But most importantly, the analysis is affected by the Mexican Hat Fallacy MHF , if I am allowed to use this name to describe a neat example of statistical misuse offered by von Storch, [22] in which the conclusion is drawn that:.

The Mexican Hat is not of natural origin but man-made. The fundamental error is that the null hypothesis is not independent of the data which are used to conduct the test. We know a-priori that the Mexican Hat is a rare event, therefore the impossibility of finding such a combination of stones cannot be used as evidence against its natural origin. That is why I prefer other statistical methods of detecting changes [23] , such as the tests proposed by Hamed [24] and by Cohn and Lins [10].

The former relies on a test statistic based on the ranks of all data, rather than a few of them, while the second considers also the magnitude of the actual change, not that of the change in the ranks. Another test statistic was proposed by Rybski et al.

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Note that, to make the test simple, the uncertainty in the estimation of H was not considered even in the latter version thus it could rather be called a pseudo-test. Here I updated the application of this test and I present the results in Figure 5. Figure 5 Updated Fig. The method has the advantages that it uses the entire series not a few values , it considers the actual climatic values not their ranks and it avoids specifying a mathematical form of trend e.

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Furthermore, it is simple: First we calculate the climatic value of each year as the average of the present and the previous 29 years. This is plotted as a pink continuous line in Figure 5, where we can see, among other things, that the latest climatic value is 0. Thus, during the last years the climate has warmed by 0. Note that no subjective smoothing is made here in contrast to the graphs by CRU , and thus the climatic series has length years but with only 5 non-overlapping values , while the annual series has length Our pseudo test relies on climatic differences for different time lags not just that of the latest and earliest values.

For example, assuming a lag of 30 years equal to the period for which we defined a climatic value , the climate difference between and is 0. The value 0. Likewise, we find climatic differences for years , , …, , all for lag Plotting all these we get the series of green triangles shown in Figure 5. We repeat the same procedure for time lags that are multiples of 30 years, namely 60 years red points , 90 years blue points and years purple points. Finally, we calculate, in a way described in Ref. The critical values are different for each lag and are plotted as flat lines with the same colour as the corresponding points.

Practically, as long as the points lie below the corresponding flat lines, nothing significant has happened. This is observed for the entire length of the lag and lag differences. A few of the last points of the lag series exceed the critical value; this corresponds to the combination of high temperatures in the s and low temperatures in the s. But then all points of the lag series lie again below the critical value, indicating no significant change.

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Assuming that the data set we used is representative and does not contain substantial errors, the only result that we can present as fact is that in the last years the climate has warmed by 0. Whether this change is statistically significant or not depends on assumptions. Irrespective of statistical significance, paleoclimate and instrumental data provide evidence that the natural climatic variability, the natural potential for change, is high and concerns all time scales. The mechanisms producing change are many and, in practice, it is more important to quantify their combined effects, rather than try to describe and model each one separately.

From a practical point of view, it could be dangerous to pretend that we are able to provide credible quantitative description of the mechanisms, their causes and effects, and their combined consequences: We know that the mechanisms and their interactions are nonlinear, as well as that the climate model hind casts are poor. He teaches undergraduate and postgraduate courses in hydrometeorology, hydrology, hydraulics, hydraulic works, water resource systems, water resource management, and stochastic modelling. He is an experienced researcher in the areas of hydrological modelling, hydrological and climatic stochastics, analysis of hydrosystems, water resources engineering and management, hydro-informatics, and ancient hydraulic technologies.

His record includes about scientific and technological contributions, spanning from research articles to engineering studies, among which 96 publications in peer reviewed journals.