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Spectral analysis of aggregates and products oftime series construed of variable failure ratesN. Singh a & V.S.S. Yadavalli ba Department of Mathematics , Monash University-Clayton , Vic. 3168, Australiab Department of Statistics , University of South Africa , P 0 Box 392, Pretoria, 0003,South AfricaPublished online: 03 Apr 2007.

To cite this article: N. Singh & V.S.S. Yadavalli (1997) Spectral analysis of aggregates and products oftime series construed of variable failure rates, Stochastic Analysis and Applications, 15:4, 629-641, DOI:10.1080/07362999708809498

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STOCHASTIC ANALYSIS AND APPLICATIONS, 15(4), 629-641 (1997)

SPECTRAL ANALYSIS OF AGGREGATES AND PRODUCTS OF TIME SERIES CONSTRUED OF VARIABLE FAILURE RATES

N. Singh

Department of Mathematics Monash University Clayton, Vic. 3168

Australia

and

V.S.S. Yadavalli

Department of Statistics University of South Africa P 0 Box 392, Pretoria 0003

South Africa

ABSTRACT

In this paper, the authors derive spectra of some linear and non-linear combinations of ARMA processes and discusses their application in reliability and surival analysis. Illustrative examples are given.

1. INTRODUCTION

Singh has recently [I] developed an unconventional

approach in time domain to the analysis of observed or

estimated failure rates of complex systems that operate

under changing operational and environmental conditions.

Since such failure rates often exhibit random changes in

level and slope, they can be construed as time series.

At times, the variable failure rates may also exhibit a

periodic (cyclic) pattern depending on the maintenance

and inspection procedures. Then obviously it would be

erroneous to analyse such failure rates using the

Copyright C 1997 by Marcel Dekker, Inc.

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630 SINGH AND YADAVALLI

standard failure distribution approach. Furthermore,

periodcities cannot be taken into account by the

latter approach; and, perhaps, for this reason the

periodicities have been ignored by the reliability

analysts.

Since engineers in general will be more interested

in the analysis of such variable failure rates in the

frequency domain, rather than the time domain, the

author obtains in this paper the spectra of aggregates

and products of time series made up of failure rates.

From a mathematical point of view, the spectrum and

autocovariance function contain equivalent information

concerning the underlying stationary random sequence

{ x } ; however, in practice the spectrum has a more

tangible interpretation in terms of the inherent varia-

tions about the mean. Furthermore, a great strength of

the spectral analysis lies in the independence of peri-

odogram ordinates at different Fourier frequencies.

This leads technically to a very straightforward method

of statistical analysis and testing. In addition, it is

known that the resulting methods are not critically

dependent on the assumption of the underlying random

sequences being normally distributed. However, the

restriction of spectral methods to stationary phenomena

is of ctitical importance. While using spectral methods,

it is presumed that any non-zero mean value, whether

constant or time-dependent, is absent; otherwise it

should be removed by detrending prior to the analysis.

2. BASIC CONCEPTS AND DEFINITIONS

2.1

Let { T , ; n = 1, 2 , . . . } be a sequence of pre-defined equidistant time points on the time

axis and let t, s t, s . . . be a sequence of failure times which constitute a serial time series. D

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SPECTRAL ANALYSIS OF ARMA PROCESSES 63 1

I\

Let 2, denote the failure rate at time t and let 2,

denote the estimate of Z, defined by h 2, = X, if T , . , c t s T , , n = 1 , 2, . . . (2.1.1)

where # of failures in interval ( T , . ~ , T , ]

X, = . (2.1.2) # of survivals at 7 , - I

Then the estimated failure rate sequence {x,} may be

regarded as a time series at equidistant time points.

In general, the series {x,} may be nonstationary and periodic.

Example 2 .1 (Taken from Davis [2] )

A frequency distribution of failures of V 8 0 5 vacuum

tubes used in transmitters is given Figure2.1 and the

estimated failure rates using (2.1.2) are plotted in

Figure2.2 (see appendix I) . The plot of the number of failures in Figure 2.1

clearly indicates that the failure times follow an

exponential distribution and are obviously subjected

to decreasing trend and random fluctuations. This

phenomenon is also supported by the graph of the

failure rates in Figure2.2 which, instead of being

constant, seems to form a stationary time series.

2 . 2 LINEAR COMBINATIONS

Let {x, : t =0, +I, k2, . . . } denote a basic

univariate nonseasonal and stationary time series in the

original time scale t and let B denote the backshipt

operator (B1x, = x,.,) . We define the following:

1. An Overlapping Linear Combination

An overlapping linear combination of the x,'s in the

original time scale is defined by J-1

where the w ' s are constant weights.

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632 SINGH AND YADAVALLI

2. A Non-overlapping Linear Combination

A linear combination of the xt's in the new time scale

defined by J- 1

is called a non-overlapping linear combination, where

t = JT. The weights w,, w,, . . . , w,., in (2.2.1) and

(2.2.2) are assumed to be the known constants, with

w, e 0. Two interesting special cases of (2.2.2) are:

( i) Temporal Aggregation

When w, = w, = . . . =w,-, = w, V, in (2.2.2) is called

a temporal aggregation of the x,., namely,

V, = w(x, + x,-, + . . . +x~.~I;) (2.2.3)

(ii) Systematic Sample

If w, = 1 and w, = 0 for j = 1, 2, . . .J-1, then

V, , T = 0 , kl, . . . in (2.2.2) is called a systematic sample of the xtls in (2.2.1) namely

V, = x , , , T = O , + l , . . . (2.2.4)

3. Contemporaneous Aggregation

Let {x,, , i = 1, 2, . . . ,k} be k independent tirw series, then their linear combination

Zt = a, x,, + a, x,, + . . . + ak xk, (2.2.5)

is called the contemporaneous linear combination of the

xtls.

3.SPECTRA OF LINEAR AND NON-LINEAR COMBINATIONS Assuming that the basic time series {x,) follow an

ARMA model, we first derive in this section the models

for various combinations of x, and then determine their

spectra.

3.1 SPECTRA OF OVERLAPPING LINEAR COMBINATIONS

Let {x,}be a nonseasonal and stationary AR(p)

process defined by Dow

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SPECTRAL ANALYSIS OF ARMA PROCESSES 633

@(B)x, = 6 , (3.1.1)

where

@(B) = 1 - p,B - p2B2 - . . . - p,BP is an AR operator and { E , } is a white-noise sequence with mean 0

variance aC2 . Theorem 3.1

Let {x,} be a nonseasonal and stationary A R ( p )

process defined in (3.1.1) and let V, = x, + x, + . . .+x,.~ be an overlapping linear combination of the

xtls. Then V, is an AR(p) process with the same

coefficients and the error variance equal to (J + l)a,'. The spectrum of V, is then given by

Theorem 3.1 suggests a smoothing technique which

corresponds to forming the overlapping.linear combina-

tion. This means that for any two different values of

t , V, may include some of the same x, . For t ranging

over half years, a simple case is when J = 1, that is,

a moving average of two six monthly values.

Theorem 3.2

Let {x,} be an M A ( q ) process defined by

where O(B) = 1 + 8,B + . . .+&JqBq is an MA operator and {e,} is defined in (3.1). Let V, = x, + x,., +

. . .+x,., be an overlapping linear combination of the

xtls, then V, is an MA(q) process with the same coeffi-

cients and the error variance equal to (J + 1)ue2 and

ae2 = (k + 1) (1 + 8,' + . . .+eq2)oe2. The spectrum of V, is given by

(J + 1) ae2 f,(w) = Is I

2 n

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SINGH AND YADAVALLI

Theorem 3.3

Let {x,} be an ARMA(p, q) process defined by

@(B)x, = O(B)e, (3.1.5)

and let V, = x, + x,., + . . .+x ,-,, then V, is an

ARMA(p,q), with the same coefficients and the error

variance (J + 1) u e 2 . The ue2 can be determined using the

well-known Yule and Walker equations. The spectrum of

V, is given by

3.2 SPECTRA OF NON-OVERLAPPING LINEAR COMBINATIONS

Suppose that the records about the failure times

and/or failure rates are maintained fortnightly and for

some reasons, one is able to analyse only the monthly

data by aggregating the fortnightly figures in the form:

V, = x,+x,-, , T = O , &I, k2, . . . (3.2.1)

where t = 2T. It may be noticed from (3.2.1) that the

linear conbination is non-overlapping.

In general, a non-overlapping linear combination of

the x,'s in the new time scale T is defined by J-1

where t = JT. In (3.2.2) , the weight wj's are assumed to

be known. There are two special cases:

(a) Temporal Aggregation

When all weights in (3.2.2) are equal, we have a

temporal aggregation. Consider

Theorem 3.4

Let {x,} be an AR(1) process defined by

X, = cp x,-, + e, , (3.2.3)

where {e,} is a white-noise sequence with mean 0 and

varaince ae2 and let Dow

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SPECTRAL ANALYSIS OF ARMA PROCESSES 635

where t = JT, then V, is an ARMA (1,l) process defined by

approximately, where

u, = e, + e,-, + . . . + e, .,,.,, (3.2.6)

Hence a,' = J a,' , and

a$ = J 0,' (1 + p2 + 2pJt1) / (1 -* (p2J)

The spectrum of V, is then given by

J a,' (1 + 2 p cos w + p2) f,(w) = -

2 n (I - 2pJ cos w + p2J)

(b) Systematic Sampling

Suppose that failure rates of certain items are

recorded at the end of each semester in a year and

suppose the data are not complete in one of the

semesters; then one can form a yearly series consisting

of data of one semester only, that is,

x2,-, , for the first semester z, = LX2; T = 0, +I, . . . (3.2.8)

Theorem 3.5

Let {x,) be an -(I) process as defined in (3.2.3)

and let Z, be a process as defined in (3.2.8) ; then Z, is

an AR (1) process with error variance equal to ae2 (1 + ( p 2 )

and its spectrum is given by

ue2 (1 + (p2) f,(w) = {I + (p4 - 2p2 COS w}-I (3.2.9)

2 n

3.3 SPECTRA OF CONTEMPORANEOUS AGGREGATION

Theorem 3.6

Let { x ; i = 1, 2, . . . k} be k independent

stationary processes with spectra f,,(w),

i = 1, 2, . . .k; then the spectrum of the linear

combination

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636 SINGH AND YADAVALLI

is given by k

where the ails are constants.

The proof is simple and straightforward, hence

omtted.

Corollary 3 . 1

If a, = l / k ; i = 1 , 2, . . . , k and if f , , ( o ) =

f , , ( u ) = f , ( u ) ; i, j = 1 , 2 , . . . k ; i + j , then

f , ( u ) = f , (w) ( 3 . 3 . 3 )

Exam~le 3 . 1

Let { x i , i = 1 , 2 be two independent A R ( 1 )

processes defined by

Xit = (Pi X i . t - l + uLt ; i = 1 , 2 ( 3 . 3 . 4 )

where p < 1 , i = 1 , 2 ; the {ui,} are two independent

white-noise processes with variences auX2, i = 1 , 2. Then

the spectrum of

z, = Xl , + x2t ( 3 . 3 . 5 )

is given by f , ( w ) = f,, ( w ) + f,, ( w )

where f x i ( u ) = aui2{1 + qi2 - 2 p i cos w}-l/2n

when

(PI = (P2 = (P , U U I 2 = 0 ~ 2 ~ = au2

f,(~) = 2UU2 {l + (P2 - 2 (P COS a}-' / 271 ( 3 . 3 . 6 )

3 . 4 SPECTRA OF THE PRODUCT OF TWO TIME SERIES

Theorem 3 . 7

Let { x } , i = 1, 2 be two independent A R ( 1 )

processes defined at ( 3 . 2 . 3 ) . Then the spectrum of the

produce Z = XI, X,, is given by

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SPECTRAL ANALYSIS OF ARMA PROCESSES

Corollarv 3.2

When cp, = cp, = cp , aUl2 = = aU2, then

f,(w) = au2(1 + q12) / {(I - (p2)(l + p4 - 2p2 COS 0)}2n

(3.4.2)

Theorem 3.8

Let {x,} and {Y,} be two independent MA(1) processes

defined by X, = e, - 6, e,., and Y, = u, - 6, u,~,, (3.4.3)

where {e} and {u,} are two independent white-noise

processes with variances ae2 and aU2 respectively. Then

the spectrum of the product Z, = X,Y, is given by

f,(w) = ae2 aU2 {(I + 012) (1 + 0,') + 2 6, 0, cos w}/ 27~

Corollarv 3.3 (3.4.4)

If a,' = a, = au2 , 0, = 6, = 0, then

f,(~) = a4 {I + O4 + 2 02(1 + cos w)} (3.4.5)

4. EXAMPLES

4.1 Consider the following two m ( 1 ) processes:

X, = 0.8 X,-, + e, and Y, = 0.6 Y,-, + u, (4.4.1)

where e,, u, ,N(O, 1) . The spectra of their various

combinations are given in Appendix 11.

4.2 Spectra of the following two M ( 1 ) processes and

their various combinations are given in Appendix 111:

X, = -0.8 X,., + e, and Y, = 0.6 Y,., - u, (4.1.2)

4.3 Spectra of the following Ar(2) processes and their

various combinations are given in Appendix IV:

X, = 0. ax,., - 0 .4X,., + e, and

Y, = 0.5Y,., + 0 .7Y,., + U, (4.1.3)

REFERENCES

[I] N. Singh, "Stochastic Modelling of aggregates and products of variable failure rate", Statistical Research Report No.235, Department of Mathematics, Monash University, Clayton, Vic.3168, Australia.

[ 2 ] D.J. Davis, 'I An analysis of some failure data", J. Amer. Statist. Soc., Vo1.47, 1952, pp 113-150

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APPENDIX l

SINGH AND YADAVALLI

F I Q U ~ ~ 2 1 D~slribution of failure frequency w~ th operallng 11me - type v805 vacuum tube used n transm~tiei

0 0 0 o ~ I ~ ~ I ~ ~ I ~ I I ~ 1 I l I l 10 20 30 40 50 60 70 80 90 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0

F~qure 2 2 The Fa~lure Rates

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SPECTRAL ANALYSIS OF ARMA PROCESSES

APPENDIX II

- F~gure II 1 F~gure 11.2 Spectrum of 4 = 0.84 . + e, Spectrum ol Y, = 0.W , + u,

0

F~gute ll 3 Spectrum of Ihe sum I = X + Y,

0

Figure 111 4 S~ec t rum of the product Z = X Y,

0

Fgure 1 5 Speclrum of the sum and product z = X I Y, + X Y,

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APPENDIX Ill

I

0 1 2 3 0

F~gure 111 1 Spectrum of X = -0 8 X , + e,

SINGH AND YADAVALLI

0

Fgure 111 2 Spectrum ol Y = 0 BY, + e

1 2

0 8 fh(o) 0 l0M 6 f2(0)iim

0 4

0 2 0 1 2 3 0

0 1 2 3

0 F~gure 111 3 Figure 111.4 Speclrum ol Ihe sum Z, = X, + Y, Spectrum of Ihe product Z, = X Y,

w Figure Ill 5 Spectrum of lhe sum and producl Z - X, + Y, I X, Y.

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SPECTRAL ANALYSIS OF ARMA PROCESSES

APPENDIX lV

F~gure IN1 Figure lV2 Spectrum ol X, = 0 8Y , - 0 4X, + e, Speclrum of Y = 0 5Y, , - 0 7Y, , + e,

0

Figure IV 1 Spectrum of the producl Z, = X, Y,

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