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Journal of Statistical and Econometric Methods, vol.4, no.1, 2015, 63-71 ISSN: 2241-0384 (print), 2241-0376 (online) Scienpress Ltd, 2015

Some basic properties of cross-correlation functions of n-dimensional vector Anthony E. Usoro1

Abstract

In this work, cross-correlation of multivariate time series was the interest. The design of cross-correlation function at different lags was presented.

is the of the cross- functions, and are the variances훾푋푖푡+푘푋푗푡 of+푙

and vectors respectively. Vector cross-correlation훾푋푖푡 훾function푋푗푡 was derived as

푖푡 푗푡 푋 , 푋 = . A statistical package was used to verify the vector cross 훾푋푖푡+푘푋푗푡+푙 푋푖푡+푘 푋푗푡+푙 휌 훾푋푖푡훾푋푗푡 correlation functions,� with trivariate analysis as a special case. From the results, some properties of vector cross-correlation functions were established.

Keywords: Vector time series; cross- and cross-correlation function

1 Department of Mathematics and , Akwa Ibom State University, Mkpat Enin, Akwa Ibom State. Email: [email protected]

Article Info: Received : January 17, 2015. Revised : February 28, 2015. Published online : March 31, 2015.

64 Cross-correlation functions of n-dimensional vector time series

1 Introduction

In statistics, the term cross-covariance is sometimes used to refer to the covariance corr(X,Y) between two random vectors and , (where =

, , … , = , , … , ). In signal processing,푋 the cross푌 -covariance푋 is often푋1 푋2 called푋푛 푎푛푑cross푌-correlation푌1 푌2 and푌푛 is a of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time between the signals, and it is sometimes called the sliding dot product. In univariate time series, the of a random process describes the correlation between values of the process at different points in time, as a function of the two times or of the time difference. Let be some repeatable process, and be some point in time after the start of

that 푋process. ( may be an integer for a discrete푖 -time process or a real number for a continuous-time푖 process.) Then is the value (or realization) produced by a given run of the process at time . Suppose푋푖 that the process is further known to have defined values for 푖and for all times . Then the definition of 2 the autocorrelation between휇푖 times and 휎is푖 푖 푠 {(푡 )( )] ( , ) = , 퐸 푋푡−휇푡 푋푠−휇푠 휌 푠 푡 휎푡휎푠 where “E” is the operator. It is required to note that the above expression is not well-defined for all time series or processes, because the variance may be zero. If the function is well-defined, its value must lie in the

[-1,1], with 1 indicating perfect 휌correlation and -1 indicating perfect anti- correlation. If is a second-order then the mean and the variance are푋 time-independent, and further the autocorrelation depends휇 only on 2 the difference휎 between and : the correlation depends only on the time-distance between the pair of values푡 but푠 not on their position in time. This further implies that the autocorrelation can be expressed as a function of the time-lag, and that Anthony E. Usoro 65

this would be an even function of the lag = , which implies = + .

This gives the more familiar form, 푘 푠 − 푡 푠 푡 푘 [( )( )] [( )( )] = = [( ) ] [( ) ] 퐸 푋푡−휇 푋푡+푘−휇 퐸 푋푡−휇 푋푡+푘−휇 2 2 2 휌푘 √퐸 푋푡−휇 퐸 푋푡+푘−휇 휎 where and are time series process at lag k time difference. Hence,

autocovariance푋푡 푋 coefficient푡+푘 at lag k, measures the covariance between two values and , a distance훾푘 k apart. The autocorrelation coefficient is defined as the 푍autocovariance푡 푍푡+푘 at lag k divided by variance ( ). The휌 푘plot of against lag k is called훾 푘the function ( )훾,0 while푘=0 the plot of 훾푘 against lag k is called the autocorrelation function (Box 훾and푘 Jenkins 1976). 휌푘 In multivariate time series, cross-correlation or covariance involves more than one process. For instance, and are two processes of which could be

cross-correlated with at lag k.푋 The푡 lag푌푡 k value return by ( , ) estimates푋푡 the ( ) correlation between 푌푡 + ( ), Venables and Ripley푐푐푓 (2002).푋 푌 Storch and Zwiers (2001) described푋 푡 cross푘 푎푛푑-correlation푌 푡 in signal processing and time series. In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a -duration signal for a shorter known feature. It also has application in , signal particle analysis, electron tomographic averaging, cryptanalysis and neurophysiology. In autocorrelation, which is the cross-correlation of a signal with itself, there is always a peak at a lag of zero unless the signal is a trivial zero signal. In theory and Statistics, correlation is always used to include a standardising factor in such a way that correlations have values between -1 and 1. Let ( , ) represent a pair of that are jointly wide sense

stationary.푋푡 푌푡 Then the cross covariance given by Box et al (1984) is ( ) = [( ) ],

훾푥푦 휏 퐸 푋푡 − 휇푥 �푌푡+휏 − 휇푦� 66 Cross-correlation functions of n-dimensional vector time series where and are the of and respectively. The cross-correlation function휇 푥 is휇 the푦 normalized cross-푋covariance푡 푌푡 function. Therefore, 휌푥푦 ( ) ( ) = 훾푥푦 휏 휌푥푦 휏 where and are the 휎of푥 휎processes푦 and respectively. If = 휎for푥 all 휎t,푦 then the cross-correlation function is simply푋푡 the푌푡 autocorrelation 푋function푡 푌푡 for a discrete process of length n defined as { , … , } which known mean and variance, an estimate of the autocorrelation may푋 be1 obtained푋푛 as

1 = 푛−푘( )( ) ( ) ( ) 푅� 푘 2 � 푋푡 − 휇 푋푡+푘 − 휇 for any positive integer k

are known, the estimate is unbiased. If the true mean, this estimate휇 is unbiased. 2 휎If the true mean and variance of the process are not known, there are several : i. if and are replaced by the standard formulas for sample mean and sample 2 variance,휇 then휎 this is a biased estimate. ii. if n-k in the above formula is replaced with n, the estimate is biased. However, it usually has a smaller mean square error, Priestly (1982) and Donald and Walden (1993). iii. if is stationary process, then the following are true

푡 푋 = = , for all t,s and ( , ) = ( ) = ( ),

푡 푠 푥푥 푡 푠 푥푥 푠−푡 푥푥 푇 where T=s-t,휇 is the휇 lag휇 time or the moment퐶 of time 퐶by which the퐶 signal has been shifted. As a result, the autocovariance becomes

( ) = [ ( ) ( ) ] = ( ) ( ) = ( ) , 2 2 푥푥 푡 푡+푇 푡 푡+푇 푥푥 푇 where퐶 푇 represents퐸 �푋 the− 휇autocorrelation��푋 − 휇� in the퐸� 푋signal푋 processi� − 휇ng sense.푅 − 휇

( ) ( )푅=푥푥 , Hoel (1984). 퐶푥푥 푇 2 푅푥푥 푇 휎 Anthony E. Usoro 67

For and , the following properties hold:

푡 푡 1. 푋( ) 1푌

푥푦 ℎ 2. 휌 ( )≤= ( )

푥푦 ℎ 푥푦 −ℎ 3. 휌 ( ) 1휌

푥푦 0 휌 ≠ ( ) 4. ( ) = 훾(푥푦) ℎ ( ) 휌푥푦 ℎ √훾푥 0 훾푦 0 Mardia and Goodall (1993) defined separable cross-correlation function as

( , ) = ( , ) ,

푖푗 1 2 1 2 푖푗 where = [ ] is a ×퐶 푋positive푋 definite휌 푋 푋 matrix푎 and (. , . ) is a valid

correlation퐴 function.푎푖푗 Goulard푝 푝 & Voltz (1992); Wackernage (2003);휌 Ver Hoef and Barry (1998) implied that the cross- covariance function is

( ) = ( ) , 푟 푖푗 1 2 푘=1 푘 1 2 푖푘 푗푘 for an integer 1 퐶 ,푋 where− 푋 =∑[ ] 휌is a 푋 ×− 푋 full푎 rank푎 matrix and (.) are

valid stationary ≤correlation푟 ≤ 푝 functions.퐴 푎Apanasovich푖푗 푝 푟 and Genton (2010) constructed휌푘 valid parametric cross-covariance functions. Apanasovich and Genton proposed a simple methodology based on latent and existing covariance models for univariate covariance, to develop flexible, interpretable and computationally feasible classes of cross-covariance functions in closed forms. They discussed estimation of the models and performed a small simulation study to demonstrate the models. The interest in this work is to extend cross-correlation functions beyond a-two variable case, present the multivariate design of vector cross- covariance and correlation functions and therefore establish some basic properties of vector cross-correlation functions from the analysis of vector cross-correlation functions.

68 Cross-correlation functions of n-dimensional vector time series

2 The design of cross-covariance cross-correlation functions

The matrix of cross-covariance functions is as shown below:

( ), ( ) ( ), ( ) ( ), ( ) . . . ( ), ( )

푋 1푡+푘 푋 1푡+푙 푋 1푡+푘 푋 2푡+푙 푋 1푡+푘 푋 3푡+푙 푋 1푡+푘 푋 푛푡+푙 ( 훾 ), ( ) ( 훾 ), ( ) ( 훾 ), ( ) . . . ( 훾 ), ( )

푋 2푡+푘 푋 1푡+푙 푋 2푡+푘 푋 2푡+푙 푋 2푡+푘 푋 3푡+푙 푋 2푡+푘 푋 푛푡+푙 훾 ( ), ( ) 훾 ( ), ( ) 훾 ( ), ( ) . . . 훾 ( ), ( )

3푡+푘 1푡+푙 3푡+푘 2푡+푙 3푡+푘 3푡+푙 3푡+푘 푛푡+푙 훾 푋 .푋 훾푋 푋 훾푋 푋 훾푋 푋 . . . . .

( ), ( ) ( ), ( ) ( ), ( ) . . . ( ), ( )

푋 푚푡+푘 푋 1푡+푙 푋 푚푡+푘 푋 2푡+푙 푋 푚푡+푘 푋 3푡+푙 푋 푚푡+푘 푋 푛푡+푙 where 훾= 0, … , , = 0훾, … , . 훾 훾

푘The above푎 matrix푙 is a 푏square matrix, and could be reduced to the form,

( ), ( )

푋 푖푡+푘 푋 푗푡+푙 where = 1, … , , = 1, … , , 훾= 0, … , , = 0, … , , ( = ).

From the푖 above cross푚 푗 -covariance푛 푘 matrix, 푎 푙 푏 푛 푚

, = , 훾푋푖푡+푘푋푗푡+푙 푋푖푡+푘 푋푗푡+푙 휌 �훾푋푖푡훾푋푗푡 where, is the matrix of the cross-covariance functions, and

are the 훾variances푋푖푡+푘푋푗푡+푙 of and vectors respectively. Given the above훾푋 matrix,푖푡 훾it푋 is푗푡 required to note that푋 푖푡two vector푋푗푡 processes and can only be cross- correlated at different lags, if either of 푋푖푡+ 푘or 푋 푗푡of+ 푙 has a fixed value zero. That is can be cross-corre푘lated푙푎푔 with푋 푖푡 푙(푙푎푔± 1,2,푋…푗푡 , ), or can be cross-correlated푋푖푡 with ( ± 1,2, … , ). 푋푗푡+푙 푙 푏 푋푗푡 푋푖푡+푘 푘 푎

Anthony E. Usoro 69

3 Analysis of the cross-correlation functions

Given two processes , ( , ) is the cross-correlation

1푡 2푡+푘 between , 푋while,1푡 푎푛푑 푋( 2푡, 휌 푋) is푥 the cross-correlation between 푋1푡 푎푛푑 푋2 푡 ,푎푡 Box푙푎푔 et푘 al (1984).휌 푥2 푡In푥1 푡+this푘 work, three vector processes 푋2푡,푎푛푑 푋1푡 푎푡 푙푎푔 are푘 used to carry out the cross-correlation analysis. For = 푋0,1±푡 1푋,22푡,푎푛푑… ,4,푋 the3푡 following results were obtained with a software: 푘

Lag ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

k 휌 푥1푡 푥2푡+푘 휌 푥2푡 푥1푡+푘 휌 푥1푡 푥3푡+푘 휌 푥3푡 푥1푡+푘 휌 푥2푡 푥3푡+푘 휌 푥3푡 푥2푡+푘 -4 -0.172 0.572 -0.102 0.643 -0.427 -0.350 -3 -0.517 0.405 -0.501 0.410 -0.076 0.042 -2 -0.611 0.098 -0.662 0.067 0.327 0.399 -1 -0.605 -0.290 -0.674 -0.303 0.659 0.697 0 -0.506 -0.506 -0.578 -0.578 0.900 0.900 1 -0.290 -0.605 -0.303 -0.674 0.697 0.659 2 0.098 -0.611 0.067 -0.662 0.399 0.327 3 0.405 -0.517 0.410 -0.501 0.042 -0.076 4 0.572 -0.172 0.643 -0.102 -0.350 -0.427

From the above analysis, the following properties were established:

1. a. , 1, k = 0, = ±1, … , ±b, ,

푋푖푡+푘 푋푗푡+푙 b. 휌 , ≠1, 푓표푟 = 0, 푙 = ±1, … , ±a, 푖 ≠ 푗.

푋푖푡+푘 푋푗푡+푙 2. a. 휌 , ≠ 푓표푟,푙 , for푘 k = 0, = 1, …푖,≠b,푗 ,

푋푖푡+푘 푋푗푡+푙 푋푖푡+푘 푋푗푡−푙 b. 휌 , ≠ 휌 , , for = 0, k푙 = 1, … , a, 푖 ≠ 푗.

푋푖푡+푘 푋푗푡+푙 푋푖푡−푘 푋푗푡+푙 3. a. 휌 , =≠ 휌 , , for 푙k = 0, = 1, … , b, 푖 ≠ 푗 ,

푋푖푡+푘 푋푗푡+푙 푋푗푡+푘 푋푖푡−푙 b. 휌 , = 휌 , , for = 0, 푙 = 1, … , a, 푖 ≠ 푗. 휌푋푖푡+푘 푋푗푡+푙 휌푋푗푡−푘 푋푖푡+푙 푙 푘 푖 ≠ 푗 70 Cross-correlation functions of n-dimensional vector time series

4 Conclusion

The motivation behind this research work was to carry out cross- correlation functions of multivariate time series. Ordinarily, cross-correlation compares two series by shifting one of them relative to the other. In the case of and variables, the variable may be cross-correlated at different lags of , and푋 vice 푌versa. In this work, 푋 and were used as vector time series, 푌using trivariate as a special case of푋 푖푡multivariate푋푗푡 cross-correlation functions. The design of the cross-covariance functions has been displayed in a matrix form. Estimates obtained revealed some basic properties of vector cross-correlation functions.

References

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