Probability Grid Method for Fisher-Tippett Law

Introduction. Graphical representation of statistical analysis results in the form of probability graphs is widely used in modern software systems, particularly in the analysis of reliability or survival. This allows for the estimation of distribution law parameters and the identification of outliers¹. Estimation of parameters using probability grids is used alongside other well-known methods and in some cases may be preferable. Probabilistic graphs are employed in the processing of resource test results² and the creation of control maps in quality management systems³. The probability grid method enables a visual assessment of the correspondence between a data set and an assumed model for a random variable, as described in the works of M.A. Deryabin [1], S.A. Dobrotin [2], V.L. Shper [3], Ya.I. Bulanov [4], K.S. Ablazova [5], N.P. Velikanova [6], G.Sh. Khazanovich [7] and other modern scientists.

V.E. Kasyanov [8] and A.A. Kotesov [9] propose using one of the forms of generalized distribution of extreme values [10] with a certain type of parameterization, which they propose to call the Fisher–Tippett law, for probabilistic assessment of machine element and structure loading, but differs in that it focuses on maximum values. The Fisher–Tippett law is suitable for estimating reliability indicators in combination with the Weibull law, for example, when applying the load–strength failure model [11].

The graphical representation of the calculation results and the method used to estimate the parameters for the Fisher–Tippett law are not well justified. The scientific literature and regulatory and technical documents do not provide a specific method for estimating these parameters using a probability grid, which limits the practical application of this law. Therefore, the aim of this study is to develop and substantiate a methodology for estimating Fisher–Tippett law using the probability grid method.

Materials and Methods. Estimation of distribution parameters using probability graphs is based on grouping data by intervals and constructing an interval empirical distribution regardless of the assumed theoretical distribution. Therefore, such methods are often called nonparametric or rank-based. A probability grid is constructed for a specific probability distribution law in order to get a linear relationship between variables⁴. Plotting involves linear approximation of an array of empirical points on a probability grid. Therefore, this approach may be considered crude, but it is often used along with others. The probability grid method can be decisive in the case when other methods are untenable. For example, when estimating parameters using the maximum likelihood method, the likelihood function may contain several local maximas. In this case, parameter estimates can be very inaccurate [12].

To justify the probability grid, we need to reduce the probability distribution function to a linear form. The Fisher–Tippett distribution function is defined by the following expression:

(1)

where x — value of a random variable; a, b, c — scale, shape, and shift parameters of the distribution, respectively.

We transform distribution function (1) by taking the logarithm of both the left and right sides. Provided that c > х, we get:

(2)

Obviously, expression (2) is a linear function of the form:

(3)

where х — function variable; q and m — constants.

Comparing (2) and (3), we get:

.

Expression (2) differs from a similar sound to the Weibull distribution with three parameters in that only the right side is different:

Therefore, to construct a probability graph of the Fisher–Tippett law, it is advisable to use the basic provisions of GOST 11.008 and GOST 50779.27. According to these standards, statistical data are plotted on a probability grid during graphical analysis, and then the distribution parameters are estimated. Let us note that the probability grid method is implemented both graphoanalytically and completely analytically. Therefore, to eliminate possible ambiguity, we will call the estimation of parameters using the probability grid method graphical, and the estimation by the maximum likelihood method analytical.

The left side of expression (2) allows us to determine the ordinate of the probability scale for estimating the scale parameter. Let us assume that с – х = а. By substituting this value in (2), we get:

,

,

,

,

,

,

,

. (4)

Result (4) allows us to conclude that the abscissa of the point approximating the line with zero ordinate will be an estimate of the scale parameter.

The decimal logarithm can be used along the abscissa axis of the probability graph. In this case, dependency (2) will take the form:

.

An important aspect of implementing the probability grid method is the initial processing of statistical data, specifically, obtaining an interval series of variations and estimating the empirical distribution function values. To obtain an empirical distribution function, a ranking method is typically used, which involves estimating the position of a distribution based on ordered data, considering the characteristics of the variation series (mean, median, mode, etc.). Therefore, various dependencies are used to determine the ordinates of points, including expressions for approximate estimation [13]. In this case, the choice will be determined by the amount of empirical data, the expected theoretical distribution, and the type of probability graph. This takes into account the need for an adequate description of the extreme members of the variation series [14].

It should be noted that some previous approaches to estimating the empirical distribution function have been criticized, and this may be the subject of a separate discussion [15].

Results. The inverse function method is used to model a set of random data without a specific physical meaning, distributed according to the Fisher–Tippett law.

The inverse distribution function is obtained analytically from expression (1):

,

,

,

,

. (5)

The simulation was conducted using the Matlab 8.6 software package, (Fig. 1), according to the specified parameters a, b, and c. The initial data used for the modeling are presented in Table 1.

The simulation results in the form of a set of random data x_i are presented in Table 2.

An analytical assessment of the scale, shape, and shift parameters has been conducted. The estimates are indicated respectively — a΄, b΄, c΄ (Table 3).

The Fisher–Tippett law, unlike the Weibull law, has a restriction on the right and sets the maximum value for a random variable. Therefore, to obtain a series of variations, it is necessary to order the values in the dataset (sample) from maximum to minimum.

If the sample size is n ≤ 30, then it is not advisable to group the data by intervals. In such cases, each variant should be assigned a rank of j, and an approximation for the median position of these ranks can be used to estimate the values of the empirical distribution function [16]:

, (6)

where x_i — value of the sample variants ordered from maximum to minimum, corresponding to the j-th rank; j — ordinal number of the rank; n — sample size.

Otherwise, for n > 30 it is necessary to group the data by intervals in accordance with the absolute sample size. At the same time, it is recommended to take the number of interval k in the range of 7 ≤ k ≤ 40 depending on sample size n. To group the data, it is necessary to determine the boundaries of the interval by selecting the values X΄ ≤ x_min and X΄΄ ³ x_max, and divide the resulting interval [ X΄; X΄΄] into the intervals of equal length h:

. (7)

Then, an interval variation series should be obtained by determining the number of sample values n_i, that fall within each interval. Each interval is described by abscissa X_i, which defines the position of the ordered data distribution.

For the middle position, the empirical distribution function is estimated using the expression:

, (8)

where X_i — middle of the i-th interval; n_i — number of sample members that fell into the i-th interval; k — number of intervals; n — sample size.

As an example, the values of the empirical distribution function were grouped and calculated (Fig. 2) for the data set from Table 3.

Figure 2 shows the probability value along the ordinate axis; and the values of the dataset (sample) without a specific physical meaning are shown along the abscissa axis.

For data grouping, we assume k = 25, X΄ = 84, X΄΄ = 242 and determined value h = 6.32. One sample value falls into the first three intervals, so they are combined. The total number of intervals — k = 23. Table 4 provides the calculation results.

On the x-axis of the probability graph, we use a scale with decimal logarithms. The calculation results from columns 8 and 9 of Table 4 determine the coordinates of the points for plotting {Lg(X_i); Ln(–Ln(1–(F(Х_i)+F(Х_i+1))))}.

At the next stage, the shift parameter is evaluated. To do this, a smooth curve (rather than a straight line) (Fig. 3, pos. 2) should be drawn through the array of points (Fig. 3, pos. 1).

At the point where the straight line intersects the zero ordinate (Fig. 3, pos. 7), graphical estimation of scale parameter А΄ is made.

Figure 3 shows the probability value along the ordinate axis, and the abscissa axis shows the values of a data set (sample) without a specific physical meaning.

The coordinates of the points of the extreme members of the variation series are denoted by {Х₁; Y₁} and {Х₂; Y₂}, and Y₃ coordinate is estimated:

. (9)

Using ordinate Y₃ on the previously indicated curve, abscissa Х₃ can be determined (Fig. 3, pos. 3). Then shift parameter С΄ is estimated:

. (10)

In the example presented, the extreme terms of the variation series are the midpoints of intervals i = 1 and i = 23 with coordinates {Lg(X₁); Ln(–Ln(1–(F(Х₁)))} and {Lg(X₂₃); Ln(–Ln(1–(F(Х₂₂)+F(Х₂₃)))}. Accordingly, Y₁ = Ln(–Ln(1–(F(Х₁))), Y₂ = Ln(–Ln(1–(F(Х₂₂)+F(Х₂₃))), X₁ = Lg(X₁); X₂ = Lg(X₂₃). As a result, a graphical estimate of shift parameter С΄ = 248.88. We use it to adjust the abscissa of all points, determining values (С΄–Х_i), and plot the points with the corresponding coordinates on the graph (Fig. 3, pos. 4). As you can see, after the correction, the points lined up “more evenly”, which allows us to draw a straight line through them (Fig. 3, pos. 5).

The estimate of the shape parameter corresponds to the angle of inclination of the approximating straight line (Fig. 3, pos. 5) to the abscissa axis. To graphically evaluate the parameter, you can use the coordinates of the points or a special scale (if available). When estimating the shape parameter by coordinates, it is necessary to express the values along the abscissa axis on the scale of the natural logarithm, i.e. use the value Ln(X) instead of Lg(X).

The considered example shows a scale for graphical evaluation of shape parameter B΄ (Fig. 3, pos. 8). To construct the scale, the coordinates of points {Lg(X); Ln(Y)} were calculated based on the set values of the shape parameter (Table 5). The scale is oriented to the reference point with coordinates {0; 0} (Fig. 3, pos. 9). To estimate the shape parameter, it is necessary to draw a straight line parallel to the approximating line through the reference point (Fig. 3, pos. 10).

As a result of data processing, graphical estimates of the parameters of the Fisher–Tippett law were obtained (Table 6).

After evaluating the parameters, we need to perform a check using inverse function (5) with the specified probability values:

. (11)

By calculating the values of inverse distribution function (11) and connecting the resulting points on the graph, you can visually assess the quality of the model. As you can see, the graph of the inverse function (Fig. 4) smoothly describes the array of initial points (Fig. 4, pos. 1 and 2). This suggests that the model is able to accurately describe the data, and the parameter estimation has been performed correctly.

Fig. 4. Checking the model after graphical evaluation of the parameters
1 — starting points with coordinates {Lg(X_i); Ln(–Ln(1–(F(Х_i)+F(Х_i+1)))};
2 — graph of inverse distribution function F^–1(x) with parameters A΄, B΄, С΄

Figure 4 shows the probability value along the ordinate axis, and the abscissa axis shows the values of the data set (sample) without a specific physical meaning.

The results of the verification calculations are presented in Table 7.

As it can be seen, the graphical and analytical estimates of the parameters are close to the parameters set during the modeling of the dataset (a, b, c).

It is not entirely correct to compare the estimates obtained with respect to the specified parameters, however, such a comparison is justified if the specified parameters are taken as the true parameters of the general population, and the set of random data х_i is considered a representative sample. A comparative analysis of graphical and analytical estimates is presented in Table 8.

Comparative analysis has shown that the relative error of graphical estimates does not exceed 2% (δ < 2%). The error of the analytical estimates in this example turned out to be higher, but this does not mean that the analytical method is less accurate.

Discussion and Conclusion. The presented probability grid method for the Fisher–Tippett law is adequate and suitable for practical application. For example, it can be used in software packages or when creating custom applications for graphical representation of statistical analysis results. It opens up the possibility to perform model fitting together with other known methods, even if they are untenable. The proposed method of constructing a scale for graphically estimating the shape parameter can be used to evaluate the shape parameter of the Weibull's law. The obtained analytical dependencies, the provisions of the methodology and the graphic material can be useful in the development of an appropriate national standard.