Let X be a random variable with a finite expected value and a finite non-zero variance
3.5 CYCLIC COMPONENT TRACING Many processes in mining have a cyclic character
The cyclic character of processes is generated by the periodic character of the organi-sation of work. The cycle of a process can be connected with the calendar—season, day or production progression per day or shift. By looking more carefully at the opera-tion of many machines, it is easy to notice that their exploitaopera-tion process has a periodic nature, and that it can be understood in a different sense. Periods alone can have a more or less stochastic character. Many years ago, a hypothesis was formulated that proved that this cyclic nature of operation can periodically distort the processes of changes of states (Czaplicki 1974, 1975). The hypothesis proclaimed that in some periods of opera-tion time—during the exploitaopera-tion of a technical object (a single machine or a system of machines), the occurrence of some states are more probable than that of others. If this is so, the probability of the appearance of a given state is not constant in time but is rather a function of time and this function is a cyclic one. In addition, a stream of rock being extracted or hauled by transport means very often also has a periodic character. These two functions are frequently correlated with each other. In some cases, a stronger state-ment can be formulated: if a stream of rock being transported increases, and usually has greater dispersion in value, it means that the probability of the occurrence of a failure in the transporting units increases. Therefore, the output of a hauling unit that is calculated as the product of the probability of the work of the unit and the mean mass of the rock being transported gives an incorrect estimation because the higher the mass being trans-ported, the lower the probability of its displacement. Thus, this estimation gives higher values than it should.
Consider the problem of the existence of a cyclic component from a formal point of view.
There are two cases to consider:
a. the period of cycle is known b. the period of cycle is unidentified.
Consider these cases in a sequence.
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3.5.1 The period of cycle is known
If the period of a cycle is known, we can presume that during the cycle one observes consecutive stochastic copies of the same phenomenon. Denote this cycle by [0, T]. Presume that the object14 can be in any of k mutually excluding states, j = 1, 2, …, k. The object is observed N times—that is, records of what was going on with the object during N cycles are given. If we pay attention to one unit of cycle time (it can be any), we notice one out of the k events. If so, the construction of the probability distribution of a random variable that a given state is observed k times in N independ-ent trials (N ≥ k) is now possible. This distribution is multinomial and is given by the formula:
P N
and bj is the multiplicity of the occurrence of the j-th state.
Consider one state of an object. Its realisation in time consists of the realisations in N periods of time. If all of these realisations are put together, the frequency of the occurrence of this state versus the time cycle will be obtained for N independent trials. The relative frequency, in turn, is the j-th estimator that is unbiased, consistent and most efficient for parameter p of the distribution (formula 3.28).
The following hypothesis can be formulated: the method of the exploitation of an object can generate a significant irregularity in the process of the changes of the states of the object.
In other words, some states can occur more frequently in some periods of time during the cycle and some states will be observed less often.
If we have the diagrams of the relative frequency against time in the cycle for all states, the above hypothesis can be verified. It is obvious that a certain irregularity of the process of the appearance of a given state will be visible due to the stochastic character of the process. How-ever, the problem arises of whether the changes that are observed are connected exclusively with the stochastic nature. By reversing the problem, a question can be formulated: how many times can a given state occur in a moment of cycle time in N trials that such an event can be assessed as very rare—so rare that a certain exploitation factor probably generated this irregularity?
Let us simplify our consideration and study only one state. Denote it by s. If it is the only one of interest, it can be specified in the following way:
P b N
14 In some cases the term ‘technical object’ stands for a system of pieces of equipment. If this system is observed during a longer period of time, it may happen that some pieces are withdrawn or added.
A cardinal feature of mine systems is their changeability because the lengths of the hauling distances change almost continuously. If only one piece of equipment is added or withdrawn from the system, the system is not the same. Its characteristics change.
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thus
P b N
b p
s s
bs s b
s
s N bs
b N b
{ s bbbs}}=⎛ (( pps)) .
⎝⎛⎛⎛⎛
⎝⎝⎛⎛⎛⎛ ⎞
⎠⎞⎞⎞⎞
⎠⎠⎞⎞⎞⎞ (3.29)
The multinomial distribution is reduced to a binomial one. The problem of a significant irregularity in the occurrence of state s is reduced to finding the number bs, which has a prob-ability of appearance lower than that presumed, a small level of probprob-ability, say υ (where υ << 1), that is:
N b p bs s b
s
s N b
⎛ b
⎝⎛⎛⎛⎛
⎝⎝⎛⎛⎛⎛ ⎞
⎠⎞⎞⎞⎞
⎠⎠⎞⎞⎞⎞ ( − pppss)N bN bs <υ (3.30). Due to the well-known properties of the binomial distribution, there will be two values bs that fulfil this inequality. Denote them by bs(l ) and bs(u), where bbs( ) bbs( )l (see Figure 3.10). From this figure it is easy to observe that the critical area is determined by level υ; all events that have a probability below this level should be comprehensively considered; we may suspect that their appearance was non-random.
The probability density function of this distribution has a maximum for:
b = (n + 1)p if (n + 1)p ∉
—set of natural numbers and has two maximum values:
b1= (n + 1)p and b2= (n + 1)p − 1, if (n + 1)p ∈ .
Large reliability investigations comprising continuous mechanised systems operating in both underground and surface mining were carried out in Poland in the mid-seventies of the previous century. The point of interest was, among other things, the problem of whether the cyclic character of the work of these systems had an influence on the course of the operation process of these systems.
A histogram illustrating the frequency of the occurrence of a repair state in a certain series system operating in the underground coal mine in the Silesian District of Poland is shown in Figure 3.11. The system consisted of a coal shearer, two armoured flight conveyors and a certain number of belt conveyors that delivered the coal that was won to the shaft bin. The observation consisted of N elementary observations repeated during every morning shift excluding the first hour. Because of the properties of the series system, any repair of any piece of equipment of the system meant a repair state for the whole system. In the figure two probability levels are visible for which the probability υ was presumed to be 0.05.
P
υ
bs
bs(l) (u)
Figure 3.10. Binomial probability density function.
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Looking at Figure 3.11, four events should be considered—all of them connected with the fact that the frequency of the occurrence of a repair state was above the presumed level.
Further analysis comprised an examination of the records to find out whether the reasons for the appearance of this state were repeated. If the reasons were repeated, they generated such a rare event that it can be assessed as not entirely random. Immediately, a recommenda-tion can be formulated to eliminate them from the further operarecommenda-tion of the system. However, if the reasons were different in each, they can be evaluated as purely random and therefore it does not matter how rare this event was15.
It is very important to understand that as a result of the application of a statistical proce-dure, information is obtained that indicates which events should be taken into further compre-hensive consideration. And that is all. Advanced analyses must proceed outside of the area of mathematics; physical aspects have to be taken into account before any final assessment can be made.
To complete these considerations it is necessary to construct an estimator of the unknown probability ps that is given in formula 3.29. Following the relative frequency approach, the number of favourable events is represented by the area of the histogram, whereas all pos-sible events are represented by an extraordinary event when only one state is observed in all N trials. Thus, an interesting estimator is determined by the function
15 The elimination of every first hour of the operation was connected with the fact that the operation of the system was not a full one. For this reason the frequency cw was below the lower critical level as a rule for obvious reasons.
0 T
bs
t bs
cw
(u)
(l)
Figure 3.11. Histogram of the relative frequency cw of the occurrence of a repair state for a certain series system obtained by observing the system during N exploitation shifts.
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ˆs itsi
p =
∑
NT (3.31)where tsi is the i-th time of state s. The denominator determines the total observation time whereas the numerator defines the total time of state s in the period of observation16.
Consider now the second case.
3.5.2 The period of the cycle is unknown
In the theory of stochastic processes X(t), it has been proved that each stochastic process can be decomposed17 into three components:
a. The trend that is associated with the expected value function (the systematic component), S(t) b. The cyclic component, C(t)
c. The pure random component, Ξ(t).
The first two components are deterministic functions whereas the third one is a random one. This comprises the whole stochastic nature of the process. At the very beginning of the analysis of the process, the problem arises as to whether the composition of these three items should be an additive, multiplication and mixed one. It is suggested that if the process being analysed has an explosive character, the multiplication model should be applied. If the course of the process is rather smooth, the additive model would be better.
In mining engineering practice, the majority of the processes that are analysed have no explo-sive character even if they are non-stationary ones. For this reason, in our further considera-tion we presume that the model of the process is additive, multiplicative and mixed one. i.e.:
X(t) = S(t) + C(t) + Ξ(t) (3.32) If this is so, with data usually in the form of a time series, x(t1), x(t2), …, x(tn) that is the discrete observation of the realisation of a certain stochastic process, the first step in the analysis of the process is the identification of the trend of the process. If this func-tion is identified, S(t), then the data should be transformed in order to obtain a new time series:
y (ti) = x(ti) – S(ti); i = 1, 2, …, n (3.33) The above sequence has no trend and for this reason is stationary but still has both a cyclic and a pure component18. From a theoretical point of view we have the following situ-ation. The sequence is a realisation of a mixture of two stochastic processes that are mutu-ally uncorrelated and stationary with average values that equal zero. The properties of these processes are completely different. One process is strictly cyclic; the second one has no such property. Therefore, if these processes overlap, the final process can have a periodicity that is difficult to trace. The intensity of the obliteration of this periodicity depends on the auto-correlation of the cyclic process and increases when its variance increases (Granger and Hatanaka 1969). Nevertheless, it is necessary to identify these processes beginning with the cyclic component.
16 For more on this subject from a mining engineering point of view, see Czaplicki 2010, p. 34–36.
17 Decomposition generally means to express something in terms of a number of independent simpler components.
18 Obviously, there may be a case in which the realisation has no cyclic component.
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There are some methods in mathematical statistics that allow a periodic component in time series to be identified. These methods are: spectral analysis and harmonic analysis.
To describe the concept of the method of spectral analysis, we must introduce the term:
function of the power spectrum, which is the first derivative of the spectrum distribution F(ω) of the stochastic process. This function can be expressed as:
dF(ω) = f2(ω) + σi
2 f2(ω) = dF2(ω)/d(ω) (3.34) where f2(ω) is the spectral density of the process X2(t) that ‘hides’ the cyclic component and σr2 is the r-th variance of the process X
1(t) of strict periodicity.
The relationship (3.34) can be used as a tool for an analysis of periodicity for two main reasons:
a. if the cyclic component equals zero (X1(t) = 0), then the function dF(ω) = f2(ω), which means it covers the spectral density of the process X2(t), which is an absolutely continuous function
b. if there is a cyclic component (X1(t) ≠ 0), then the function dF(ω) is not absolutely con-tinuous; in points ω = ωr the value of the function jumps up because to the value of the function f2(ω) is added the variance σ2r of the r-th component of the process X1(t).
In practice, when a graphical picture of the function dF(ω) is drawn, it is easy to notice such points (ωr; r = 1, 2, …, s) where the value of the function increases drastically. There-fore, one can say on the periodicity of the process in its points 2π/ωr. A difficult problem arises when in some points the function increases only slightly. Unfortunately, estimation the function dF(ω) is usually done with a certain accuracy only and there is no clear indication whether the observed increment of the value of the function is significant or not. However, there are several methods that can be used to dispel any doubts. Different authors recom-mend different tests; however, many of them are complicated procedures.
Let us first consider the idea of harmonic analysis due to its simplicity.
If the observed time series y(ti); i = 1, 2, …, n has no trend, it can be expanded in a Fourier series. In mathematics, a Fourier series decomposes a periodic function into the sum of simple oscillating functions, namely sines and cosines. Following this line of reasoning, we can write:
The estimators of these coefficients are as follows:
• The expected value estimator
∑
• Further estimators of the Euler–Fourier coefficients
=1
The values of these statistics can be applied to verify the hypothesis that a cyclic compo-nent is significant in a given sample.
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Consider the square amplitude of the process. It is given by the equation:
By replacing the unknown variance σ2 with its unbiased estimator (the variance estimated from the sample), one obtains:
( ) )
where y is the estimate of a , i.e. the arithmetic mean calculated from the sample.ˆ0The probability that an event that Â2jj will be α times greater than AAE2 is determined by the
If the level of significance presumed equals α, one can verify a null hypothesis stating that the i-th jump that is the value of the quotient
( )
ÂAEj 2 is significant. If the inequality holdsthen the verified null hypothesis should be rejected. This means that we can presume that the time series shows important oscillations with the period that equals n/i. This regularity holds with the probability 1 – α.
In some cases, information that some periodic oscillations are significant is enough to take the proper decision in relation to the source of the observed data. But in some other cases it is not enough; for instance, we need to predict what the probable course of the process observed in the near future will be. Let us ignore for the time being what the prognosis means and let us conduct our consideration on a decomposition of a time series a little further. Here a more advanced approach will be presented that makes use of the fundamental monograph written by Box and Jenkins (1976).
If all of the significant fluctuations are identified, we are able to construct the cyclic com-ponent function C(t)19. Then, having specified two deterministic components, we are able to identify the third element, a purely stochastic one. Consider the following sequence:
19 A cyclic component can consist of a few functions.
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u(ti) = x(ti) – S(ti) – C(ti); i = 1, 2, …, n (3.43) The above relationship determines the time series of the residuals generated by the differences. It can be presumed that this sequence is a realisation of the unknown purely stochastic component ξt. Usually, it is also presumed that the sequence u(ti) has a zero expected value and the finite and constant standard deviation σξ. In the majority of cases, a stronger presumption is formulated namely: Nξ(0, σξ). Ignoring how strong the presumptions are, it is necessary to verify—by applying the appropriate statistical tests—
whether all of these assumptions hold when confronted with the properties of data in hand.
■ Example 3.8 (Based on Manowska’s Ph.D. dissertation, 2010, Chapter 14)
In Figure 3.12 is a graph of a mass of hard coal sold in Poland versus time counted in months.
The data comprise the period of the turn of the century: the late nineties—beginning of 21st century. A great restructuring of the mining industry (mainly coal) took place in Poland dur-ing this period.
It is easy to notice that the time series is a non-stationary one and that it decreases with time. Therefore, a linear function20 was applied as the first approximation of the trend in the data, i.e.
β ˆ ( )S β t y tS( )( )= α + β += α + β +t u
The classical method of least squares was applied in order to estimate the unknown param-eters a and b (see Chapter 6.2) obtaining the following equation:
ˆ ( )S )= −25.5625.56 +10947+ ζt y
where ζt is a random component. The linear regression function is visible in this Figure.
20 This function makes sense in the interval observed and perhaps, for only a few months ahead. The more proper one should be a decreasing function that tends to a certain horizontal asymptote.
14000
13000
12000
11000
10000
9000
Mass of hard coal sold (thousand tonnes)
8000
7000
6000
0 20 40 60 80
Months y(t)
t
100 120 140 160
Figure 3.12. Mass of hard coal sold versus time.
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It was suspected that there was a cyclic component in the sequence, thus ζt consisted of the constituent and the purely random one.
As the next step, the differences were calculated: y( )( ))))−yˆˆS( )(( for all of the months of the sample. All of these differences are shown in Figure 3.13.
The sequence visible in this figure was the income data for the Fourier analysis with its transform applied. The Matlab 5.3 program was used and the result of the Fourier study is presented in Figure 3.14.
It is important whether a dominating frequency is observed in terms of the module of the spectrum in such a graph. If so, a cyclic component very likely exists.
It was presumed that the general form of a cyclic function is described by the following pattern:
70000
60000
50000
40000
30000
Module of spectrum
Frequency 20000
10000
0 0 0.0192 0.0769 0.1346 0.1932 0.25 0.3077 0.3654 0.4231 0.4808
Figure 3.14. Result of the application of the Fourier transform (Manowska 2010).
3000
y(t) –yS(t) 2000
1000
0
–1000
–2000
–3000
–4000
ˆ
Figure 3.13. Time series of the differences y( )( )))) yˆˆs( )(( (Manowska 2010).
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y tC( )t = sinsi ⎛ tt A
⎝
⎛⎛
⎝⎝
⎞ 2 ⎠⎠⎠
0
π ϕ
Τ0
(3.44)
where: T0—period, number of months in one cycle, ϕ—the phase displacement,
A—the amplitude.
Next, the harmonic analysis was applied that relies on a description of the residuals as the sum of the sinus functions for whatever period, amplitude and phase were selected using the appropriate algorithm. This algorithm relies on the analysis of the mean square error that is the result of the application of the sinus functions. The minimum of this error was a point of interest. In order to search for this minimum, the period was changed from 1 to 156 (sample size), the amplitude varied from 0 to 1000 tonnes and the phase displacement varied from 0 to 2π.
The minimum of the mean square error was 10,400 tonnes for the period equal to 12 and
The minimum of the mean square error was 10,400 tonnes for the period equal to 12 and