AUTOCORRELATION ANALYSIS

There are some processes in mining engineering that depend on many factors that can be grouped together to create two specific sets: the properties of the object that is the point of the investigation and the main characteristic features of its operational process. They can comprise many elementary components such as: material fatigue, corrosion, friction wear, local weight loss, pitting and so on, all of which concern technical objects. However, there are some other processes running in the rocks surrounding a mine that also depend on many features and it is difficult to take all of them into account. In such cases, the course of an interesting variable can be described by the values of it that were noted in the past. This leads to the application of an autoregression model, which can be used if the values of the variable that were noted in sequent moments of time depend on each other.

There are also some other cases when the dependence of the actual values of the variable being investigated, for which the values were recorded in the past, are significant. This can concern a purely random component in time series.

Let us consider two examples.

Module of spectrum

Frequency

52.00

25000

20000

15000

10000

5000

0

13.00 7.43 5.20 4.00 3.25 2.74 2.36 2.08

Figure 3.18. Result of the application of the Fourier transform for the third residuals of the time series (Manowska 2010).

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■ Example 3.9

The object of consideration was a hoist head rope with a triangular shape of strands working in the main shaft of an underground mine. The point of interest was the course of the wear of the rope. Observations were made every ν = 10³ hoist cycles and the number of breaks in the wires were noted. The empirical data are shown in Figure 3.19. The plot of the theoretical function is also visible in this figure:

N_i a _i N N aaν^b

for which the estimates of the unknown structural parameters were: a = 6.48 × 10⁻⁵ and b = 3.22. These estimates were obtained after the linearisation of this power function and after the application of the least squares method.

The goodness of the estimation using this power function was investigated by analysing the residuals that were determined by the formula:

εi= ni – N_i

The graph of these differences is presented in Figure 3.20.

40 60 80 100

100 200 240

10 N_i

n_i

110 νi × 10³

Figure 3.19. Plot of the total number of cracks in the wires of the hoist head rope vs. the number of winds executed by the hoist; n_i empirical plot, N_i theoretical plot.

0 10 20 30 40

–20 –10 0 10 20 30

εi

i

Figure 3.20. Graph of sequent residuals εi.

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This graph is not a typical realisation of a purely random process with a zero mean and a constant variance, which is what might be expected.

Let us investigate the residuals by searching for internal stochastic relationships.

One of assumptions of the classical method of least squares says that the residuals should not be correlated with each other, i.e.

E( _{iii jj})=0 ii≠j (3.45)

The term correlation was introduced here in relation to the stationary testing of the realisa-tion of the random variable that was observed. Recall that correlarealisa-tion is a certain type of sto-chastic relationship. It relies on such regularity that when the values of one variable increase (or decrease), on average the values of the second variable decrease or increase. Thus, it is a statistical relationship. If the consideration concerns only one variable, we say it is an autocorrelation.

Calculate the correlation coefficients of first, second and third order: r₁^(a), r₂^(a), r₃^(a). This means that we investigate the interdependence between the two sequences that are noted; the original sequence and the sequence derived by the first, second and third beginning elements, respectively. Note, that the number of elements taken under consideration decreases one by one when one calculates the sequent autocorrelation coefficients. As the measure of correla-tion, the classic Pearson’s linear correlation coefficient can be applied if a sample is given in the form (x_i, y_i; i = 1, 2, …, N). The coefficient can be expressed as:

In the case when the autocorrelation coefficient of the first order is being calculated, pat-tern (3.46) takes the form:

r ^{i i i}

Notice, that for an increasing sample size, the difference in the mean values becomes negligible.

One can construct further autocorrelation coefficient formulas in a similar way.

The results of the calculation of the autocorrelation of the residuals in the sample were as follows:

r₁ rr₂^a rr

r r

r^{( )a} rrrrrrr₂^{( )aa} 0 866. rrrrrrrr₃₃₃^{( )( )a} =0 8840.

The values obtained are high, but we have no idea whether they are significant in a statis-tical sense. In order to answer this question, the Durbin-Watson test is usually the one that comes to mind (Durbin 1953, Durbin and Watson 1950, 1951). However, the Durbin-Watson statistic is only valid for stochastic regressors and first order autoregressive schemes (such as AR(1)). Furthermore, it is not relevant in many cases; for example, if the error distribution is not normal, or if it concerns the dependent variable in a lagged form as an independent variable. In these cases, it is not an appropriate test for autocorrelation. The tests that are

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suggested and which do not have these limitations are the Breusch-Godfrey test (Breusch 1979, Godfrey 1978, 1988) as well as the Pawłowski J test (Pawłowski 1973). Because the first of these tests is well-known and much simpler to apply when compared to the latter test, further reasoning will be performed using this kind of examination.

The Breusch-Godfrey statistic is determined by the following formula:

χχχχ²( )) ((( ))

( )

A verified hypothesis is H₀ : ρc= 0, i.e. there is no autocorrelation of the order c in the ran-dom variable being tested. If the following inequality holds:

χχχχ²( )) ((( ))

( )

then there is no ground to reject the null hypothesis. Otherwise, one can presume that the autocorrelation of the order c is significant.

Making all of the necessary calculations and reading the critical values from the table of χ² distribution for a presumed level of significance α = 0.05 (Table 9.4), we have:

37.35 (3.84) 32.97 (5.99) 32.04 (7.82)

where the first number is the empirical value and the corresponding critical one is in the brackets.

The Breusch-Godfrey test can also be supported by the F-Snedecor’s statistic by making use of the well-known relationship between the χ² statistic and the F-Snedecor’s statistic. It has been proven using a simulation technique for small samples that such an approach is better than that one based on the χ² statistic. In the case being analysed, it does not matter which statistic is applied (either χ² or F-Snedecor’s), the result of verification is identical: all empirical values are significant.

By translating this result into engineering language, one can say that there is a significant dependence between the degree of rope wear in a given moment of time and the degree of rope wear a while ago and two whiles before. It also means that the wear process of the rope has a memory and—as investigations showed (Czaplicki 2010, Chapter 5)—this memory can be constant with time in a stochastic sense but it can also be variable depending on the

number of winds that have been executed by the rope. ◀

Some important remarks can be formulated in connection with autocorrelation testing when random variables are the objects of engineering interest.

• If during a statistical investigation, the autocorrelation of the random variable being tested was traced, then one can be almost certain that there is a physical reason generating this statistical regularity.

• Autocorrelation means that there is a ‘memory’ in the process that is observed and the future state depends—as a rule—on the state just before, sometimes on some states that happened earlier.

• If the autocorrelation was traced, it gives ground to formulate the supposition that the adequate model describing the course of the random variable being tested is an autore-gression function and vice versa.

• If autocorrelation was found, the further investigation should be focused on recognising the physical grounds that are the source of this autocorrelation. This can provide important knowledge on the nature of the process that is being observed which can allow for some

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counteraction if this phenomenon is disadvantageous (if possible) or can allow engineers to make use of it if it is useful.