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General topology set theory methodology of mathematics education work with mathematically gifted secondary school students mathematical statistics pedagogical measurement. Climate change also could lead, however, to changes in probability distributions for extreme events. Such changes actually could cause serious problems because the risk and consequences of these transitions are difficult to quantify and identify in advance. The impacts caused by these events have not yet been explored, although they should constitute an essential aspect of any impact and adaptation assessment.
The impacts of extreme event consequences of stochastic climate variability, however, have begun to attract researchers' attention in a related context. As noted in Chapter 18 , changes in mean climate can increase the likelihood that distributed weather will cross thresholds where the consequences and impacts are severe and extreme. This variant of stochastic singularity therefore can change in frequency even if the probability of extreme weather events, measured against the mean, is unaffected by long-term trends.
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There also is a fourth type that generally arises from a combination of all other singularity categories. Such macro-discontinuities may cause damages to natural and human systems that exceed the negative impacts of "ordinary" disasters by several orders of magnitudes.
Only the last three are climate change effects in the proper sense, but the first is important because it translates highly averaged indicators of climate change into the actual trigger acting at the relevant scale. One therefore might assume that singularities tend to be preserved down such a cascade.
Singular events also can arise further down the causal chain. Purely regular geophysical forcing, for example, can cause singular impacts, and singular socioeconomic responses may result from regular impacts. Harmful impacts of climate change generally can be alleviated by adaptation or exacerbated by mismanagement see, e.
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Climate-triggered singular phenomena can generate substantial impacts because their predictability and manageability are low. Such impacts would be considerably reduced if they could be "regularized" by appropriate measures. For example, an ingenious array of seawalls and dikes could transform an extreme storm surge into a mundane inundation that could be controlled by routine contingency procedures.
So too could a long-term policy of retreat from the sea. However, inappropriate flood control structures could wreak havoc, particularly because they foster a false sense of security and actually inspire further coastal development. In summary, singularities tend to produce singularities, as a rule; regularities may turn into singularities under specific conditions, and singularities can be regularized by autonomous ecological processes or judicious societal measures.
Defining the propagation of singular events in the causal cascade or opportunities to convert them into regular events remains a major research challenge. Calendar of Meetings. Meeting Documentation. News and Events. Publications and Data. Technical Papers. Supporting Material. MFS is composed of several points, which highlights the minimum singularity exponent a min , the maximum singularity exponent a max and the singularity exponent a 0 that correspond to the maximal f a.
MFS estimation can be computed by several methods; the well-known Detrended Fluctuation Analysis DFA proposed by Kantelhardten in  is one of the most commonly used in practical applications. DFA consists of five basic steps:. This step converts the signal to a random walk like time series. Divide the profile into non-overlapping N s segments of an equal length of scale s 3. F 2 s,v is computed based on the variance of the series profile Y by subtraction the fitting polynomial fit Y v in the segment v , as in :.
Where k denotes the delayed sample and n the time series samples. Estimation of instantaneous cyclic autocorrelation is the new first step in DFA.
Finally, the steps 2 to 5 of DFA are followed for each instant of time ; for more detailed information see . By analogy with traditional energy, the energy of a TS-MFSD is the sum of the square modules of f t,a on the time axis . SES was proposed in for practical applications in engineering. Given the analysis presented by the authors in  and based on the fractal energy measurement theory, the estimation of the multifractal spectrum distribution energy of the signal can be seen as:.
Where x a m represents a signal that corresponds to a discrete fractal sub-band defined as:. As depicted in Fig. A shows an example of the W a m. Using W a m , we propose to evaluate the energy contained in the This value was selected based on one standard deviation.
Although this is true only for the normal distribution, in this work we use it as a practical rule of thumb . The variability of the SE over time is evidenced in Fig. For each feature, a threshold to discriminate two classes is selected. The performance of the classification can be determined by the confusion matrix shown in Table 2 .
We computed the and the for a set of values of c. The best cut-off point is defined as in . We can see difference between spectrum width and changes in SE distribution in time. Given the dynamic behavior in time of the SE, the visual analysis can be more representative if we focus on the following three lines: the minimum singularity exponent a min t ; the maximum singularity exponent a max t ; and the singularity exponent a 0 t.
The latter corresponds to the maximal f t, a. We can see that these lines for the class 3 signal shows a lower variance in the SE, particularly in a 0 t. This line is analogous to the maximum of the spectrum in multifractal analysis . However, in multifractal spectrum, the maximum is a scalar value, and it cannot capture the changes in time of this feature. Accordingly, we used the standard deviation as features to describe the deviation in time of a 0 t. The same analysis is made for a min t , a max t.
On the other hand, we computed the SES for a more complete analysis of the information contained in the time-singularity spectrum.
We can see the characteristics described in the spectrum, where class 0 and class 1 are wider and skewed to the right. Class 3 is narrower. This behavior is captured by the proposed feature W a m. At last, we computed four features: the We can see the ability of all the features to distinguish Classes. Only standard deviation of a min t does not contain representative information for this task. Taking into account that some authors have suggested that only a high level of fractionation is related with critical sites e. Tabla 3 shows the comparison between our results and a previous study that reported conventional and multifractal features computed in the same database .
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The proposed features in this study outperform those results regarding the discrimination of Class 3 signals with continuous electrical activity.