**4. The NHGPPP analysis**

Within self-organized critical systems there are a great number of small events; however, the main changes of the system are associated with extreme events. The theory of extreme values is an area of statistics that is devoted to developing statistical models and techniques for estimating the performance of the unusual. These rare events are those which are far from the bulk of the distribution. However, there is no formal definition of extreme events in many cases, being defined as those events that exceed some threshold of magnitude, though they can also be defined as the maximum or minimum of one variable over a certain period. From a statistical standpoint, the problem of extreme value theory is a problem of extrapolation. The basic idea that leads to such extrapolation is that of finding a good parametric model for the tail of the distribution of the data generated by the process that can then be adjusted for extreme observations.

Overall, there are two approaches to the topic of Extreme Value Theory (EVT), a group of older models, known as Block top models and a new group of models known as "Peaks Over Threshold" (POT). The latter group corresponds to a pre-fixed high threshold models (Coles, 2001; Beguería, 2005). EVT focused on peak values above a value *u*, with these values being distributed as a Generalized Pareto Distribution.

The method characterizes the exceeding of a threshold based on the assumption that the occurrence of excesses on a strict threshold of a series characterized by an independent identically distributed random variable has a Poisson behaviour, and that the excesses have an exponential distribution or - more generally - a Generalized Pareto (GP) (Davison and Smith, 1990; Coles, 2001).

The distribution of excess *Fu* represents the probability of exceeding the threshold "*u*", in at most an amount of "*y*", which is conditioned by the information that has already exceeded the threshold (Cebrian, 1999, Lang et al. 1999; McNeal, 1999).

Definition: a distribution function with two parameters is known as the Generalized Pareto Distribution (GPD).

$$G\_{k,a}\left(y\right) = \begin{cases} 1 - \left(1 - \frac{ky}{a}\right)^{\frac{1}{k}}, & k \neq 0 \\\\ 1 - e^a, & k = 0 \end{cases} \tag{4}$$

Where *a* 0 and *k* is arbitrary, the range of y is: 0 *y a k* if *k y* 0, 0 , if *k* 0 .

The *k* 0 case is just a re-parameterization of one or more forms of the Pareto distribution, but the extension *k* 0 was proposed by Pickands (1975). The case *k* 0 is interpreted as the limit when *k* 0 , (i.e. the exponential distribution).

#### **4.1 Properties for stability threshold**

*Property 1*. If *Y* is a GPD *u* 0 a threshold, then the conditional distribution for excesses over a threshold - the conditional distribution *Y u* given *Y u* - is also distributed as a Generalized Pareto Distribution.

*Property 2*. If *N* has a Poisson distribution with conditional on *N*, where *N* is the number of the exceedances of a threshold and 1 2 *YY Y* , , ..., *<sup>N</sup>* are independent random variables identically distributed as a GP, then max , , ..., *YY Y* 1 2 *<sup>N</sup>* for each *N* follows a Generalized Extreme Value Distribution. Thus, the exceedances satisfy a Poisson process, with excess distributed as a GPD that implies the Classical Distribution of Extreme Values.

Both properties characterize the GPD in the sense that does not exist another family that has these properties.

The excesses of a variable with GPD also follow a GPD - by Property 1 - allowing it to obtain the value(s) of (the) threshold(s) that rise to the extreme values, which also represents a distribution whose parameter values are constant.

Davison and Smith (1990) apply this idea to the expected value of the excess over a threshold *u* in the case where the GPD is a linear function of the threshold (Diaz, 2003; Beguería, 2005; Lin, 2003).

If *k* 1 , *u* 0 and *a uk* 0 then

$$E\left(\mathbf{x} - \boldsymbol{\mu}, \mathbf{x} > \boldsymbol{\mu}\right) = \frac{a - \boldsymbol{\mu}k}{1 + k} \tag{5}$$

On the values above the threshold at which the GPD is adequate, the mean of excess of the sample is,

$$\mathbf{x}\_{\mu} = \frac{\sum\_{i, \mathbf{x}\_i > \mu} (\mathbf{x}\_i - \mu)}{N\_{\mu}} \tag{6}$$

This should be approximately a linear function of *u*, where *Nu* is the number of the exceedance above a predetermined threshold (McNeil and Saladin, 1997; Martínez, 2003, Lin 2003).

Definition: a distribution function with two parameters is known as the Generalized Pareto

1

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*ky <sup>k</sup>*

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The *k* 0 case is just a re-parameterization of one or more forms of the Pareto distribution, but the extension *k* 0 was proposed by Pickands (1975). The case *k* 0 is interpreted as

*Property 1*. If *Y* is a GPD *u* 0 a threshold, then the conditional distribution for excesses over a threshold - the conditional distribution *Y u* given *Y u* - is also distributed as a

*Property 2*. If *N* has a Poisson distribution with conditional on *N*, where *N* is the number of the exceedances of a threshold and 1 2 *YY Y* , , ..., *<sup>N</sup>* are independent random variables identically distributed as a GP, then max , , ..., *YY Y* 1 2 *<sup>N</sup>* for each *N* follows a Generalized Extreme Value Distribution. Thus, the exceedances satisfy a Poisson process, with excess

Both properties characterize the GPD in the sense that does not exist another family that has

The excesses of a variable with GPD also follow a GPD - by Property 1 - allowing it to obtain the value(s) of (the) threshold(s) that rise to the extreme values, which also represents a

Davison and Smith (1990) apply this idea to the expected value of the excess over a threshold *u* in the case where the GPD is a linear function of the threshold (Diaz, 2003;

> , <sup>1</sup> *a uk Ex ux u*

> > , *i*

*u*

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*ix u*

On the values above the threshold at which the GPD is adequate, the mean of excess of the

*x u*

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*N* 

This should be approximately a linear function of *u*, where *Nu* is the number of the exceedance above a predetermined threshold (McNeil and Saladin, 1997; Martínez, 2003,

*k*

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distributed as a GPD that implies the Classical Distribution of Extreme Values.

Where *a* 0 and *k* is arbitrary, the range of y is: 0 *y a k* if *k y* 0, 0 , if *k* 0 .

*G y a*

,

*k a*

the limit when *k* 0 , (i.e. the exponential distribution).

distribution whose parameter values are constant.

**4.1 Properties for stability threshold** 

Generalized Pareto Distribution.

these properties.

sample is,

Lin 2003).

Beguería, 2005; Lin, 2003).

If *k* 1 , *u* 0 and *a uk* 0 then

Distribution (GPD).

Consider the graph of the mean excesses *uz* (the sum of positive differences in the magnitude of the fixed threshold and the EQ magnitude that exceeds that threshold, by the number of excesses) against the threshold *u*:

$$\sum\_{i,x\_i>u} x\_i$$

$$x\_u = \frac{i\_\* x\_i > u}{N\_u} \tag{7}$$

If the assumption that it behaves as a GPD is correct, then the plot should follow a straight line with the intercept 1 *a <sup>k</sup>* and slope 1 *k <sup>k</sup>* . Therefore, it is enough to fit a straight line so as to obtain both parameters (*a*, *k*). This is a relatively simple method for corroborating the linear relationship between the mean excess and the threshold *u* (Davison and Smith, 1990; Coles, 2001; Beguería, 2005, Lin 2003).

The parameters of the GPD for each year are given in Table 2, and the average parameters are calculated and compared in terms of how well they fit with the data annually and how well they make a global settlement for all years. Once obtained, the parameters can be estimated as a GPD function that adjusts the excesses.

Finally the probability of the excesses is obtained by using a non-homogeneous Poisson Pareto process, and for this we need the rate of occurrence and the GPD as a function of the intensity of the NHGPPP, namely:

$$\theta\left(e\right) = \frac{N\_u}{t} \left[1 - \frac{k\left(e - u\right)}{a}\right]^{\frac{1}{k}}\tag{8}$$

The general methodology - as described earlier - is to obtain the variable rate of occurrence of a Non-Homogeneous Poisson Process that we will use for the analysis of the EQs that occurred in Guerrero.
