summer university on information technology in agriculture and rural development 19-22 august 2006...

Summer University on Information Technology in Agriculture and Rural Development Summer University on Information Technology in Agriculture and Rural Development

19-22 August 2006 Debrecen19-22 August 2006 Debrecen

Special Time Series Special Time Series MethodsMethods

Maria FazekasMaria Fazekas

Debrecen UniversityDebrecen University

HungaryHungary

Summer University on Information Technology in Agriculture and Rural Development Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen19-22 August 2006 Debrecen

Autoregressive moving Autoregressive moving average modelsaverage models Let Let … z… zt-1t-1, z, ztt, z, zt+1t+1, … denote the , … denote the

observations aobservations att equally spaced equally spaced times … t-1, t, t+1, … . times … t-1, t, t+1, … .

For simplicity we assume that the For simplicity we assume that the mean value of zmean value of ztt is zero is zero..

Denote aDenote att, a, at-1t-1, a, at-2t-2, … a sequence , … a sequence of identically distributed of identically distributed uncorrelated random variables uncorrelated random variables with mean 0 and variance with mean 0 and variance aa

22. . The aThe att are called white noise. are called white noise.


ARMA(p,q)ARMA(p,q)

The autoregressive moving The autoregressive moving average model of order p,q average model of order p,q [[ARMA(p,q)ARMA(p,q)]] can be represent with can be represent with the following expressionthe following expression::

zztt==11zzt-1t-1+ … ++ … +ppzzt-pt-p++

++aatt++11aat-1t-1+…++…+qqaat-qt-q..


ARMA(p,q) model with ARMA(p,q) model with backward shift backward shift operatoroperator Let B the backward shift operatorLet B the backward shift operator

BzBztt=z=zt-1t-1, B, Bkkzztt=z=zt-kt-k.. The ARMA(p,q) model with the The ARMA(p,q) model with the

backward shift operator:backward shift operator: (B)z(B)ztt==(B)a(B)att, ,

where where (B)=1-(B)=1-11BB11-…--…-ppBBpp and and

(B)=1+(B)=1+11+…++…+qqBBqq..


SSpecial cases of thepecial cases of the ARMA(p,q) modelsARMA(p,q) models The AR(p)The AR(p) zztt==11zzt-1t-1+…++…+ppzzt-pt-p+a+att.. The MA(q) The MA(q) zztt=a=att++11aat-1t-1+…++…+qqaat-qt-q.. The special case of AR(p); zThe special case of AR(p); ztt==11zzt-1t-1+a+att.. zztt is linearly dependent on the previous is linearly dependent on the previous

observation zobservation zt-1t-1 and the random shock a and the random shock att. . The special case of MA(q); when q=1; The special case of MA(q); when q=1;

zztt=a=att++11aat-1t-1. . zztt is linear expression of the present and is linear expression of the present and

previous random shock. previous random shock.


StationaryStationary

TheThe stationary series has a constant stationary series has a constant mean and variance and covariance mean and variance and covariance structure, which depends only on the structure, which depends only on the difference between two time points.difference between two time points.

The symbol The symbol =1-B is called the =1-B is called the differencing operator.differencing operator.


StationaryStationary

If a series has to be differenced If a series has to be differenced once to obtain stationary series once to obtain stationary series then the original series is called then the original series is called an integrated an integrated ARMA model of ARMA model of order p,1,q or an ARIMA(p,1,q) order p,1,q or an ARIMA(p,1,q) model. model.

If differencing has to be If differencing has to be performed d times to obtain performed d times to obtain stationary stationary series series the model is the model is called an ARIMA(p,d,q) modelcalled an ARIMA(p,d,q) model..


The autocorrelationThe autocorrelation functionfunction

The dependence structure of a The dependence structure of a stationary time series is characterized stationary time series is characterized by the autocorrelation function. by the autocorrelation function.

The autocorrelation function is The autocorrelation function is defined as the correlation between zdefined as the correlation between ztt and zand zt+kt+k; ; kk=correlation(z=correlation(ztt,z,zt+kt+k), k is ), k is called the time lag. called the time lag.

The autocorrelation function is The autocorrelation function is estimated by the empirical estimated by the empirical autocorrelation function: rautocorrelation function: rkk=c=ckk/c/c00..


CCross-correlationross-correlation functionfunction

The relation between two time The relation between two time series zseries ztt and y and ytt is determined by the is determined by the

cross-correlation function: cross-correlation function: zyzy(k)=correlation(z(k)=correlation(ztt,y,yt+kt+k); ); k=0, k=0, 1, 1, 2, … . 2, … .

The cross-correlation function The cross-correlation function determines the correlation between determines the correlation between the two series as a function of the the two series as a function of the time lag k. It may be shown that time lag k. It may be shown that zyzy(k)=(k)=yzyz(-k).(-k).


Model identification andModel identification and checkingchecking

To obtain an adequate ARIMA To obtain an adequate ARIMA model Box and Jenkins have model Box and Jenkins have suggested the following suggested the following procedure:procedure:

Make the series stationary.Make the series stationary. Choose a provisional model, in Choose a provisional model, in

particular by looking at the particular by looking at the empirical autocorrelation function.empirical autocorrelation function.

Estimate the model parameters.Estimate the model parameters.


Model identification Model identification andand checkingchecking Check the adequacy of the model. Check the adequacy of the model.

In particular check the In particular check the autocorrelation function of the autocorrelation function of the residuals.residuals.

If the model does not fit the data If the model does not fit the data adequately one goes back to step adequately one goes back to step 2. And chooses an improved model. 2. And chooses an improved model. Among different models, which Among different models, which represent the data equally well, represent the data equally well, one chooses the simplest one, the one chooses the simplest one, the model with the fewest parameters.model with the fewest parameters.


Estimation of Estimation of confidence intervalconfidence interval

For estimation of the parameter of For estimation of the parameter of first order autoregressive model two first order autoregressive model two methods are well known:methods are well known:

apply apply the standard normal the standard normal distribution as estimationdistribution as estimation, and , and

White methodWhite method.. These methods above cannot be These methods above cannot be applied in applied in nnon-stationary case. on-stationary case.


Estimation of Estimation of confidence intervalconfidence interval Little known for estimation of the Little known for estimation of the

parameter of AR(1) is the parameter of AR(1) is the application of application of estimation for estimation for continuous time case processescontinuous time case processes..

This method can be applied in This method can be applied in each case properly.each case properly.


Mortality rates for cancer of Mortality rates for cancer of cervix cervix for age class 0-64 and over for age class 0-64 and over 6565

year

1997199419911988198519821979197619731970

death

s p

er

100000 p

opula

tion

40

30

20

10

0

0-64 age

group

over 65


Autocorrelation f. mortality Autocorrelation f. mortality rates for cancer of cervix rates for cancer of cervix

o.65o.65

lag

54321

au

toco

rre

latio

n

1,0

,8

,6

,4

,2

,0

-,2

-,4

-,6

-,8

-1,0

over 65

confidence

limits

coefficient


Partial autocorrelation Partial autocorrelation function function

for mortality rates over 65for mortality rates over 65

lag

54321

pa

rtia

l a

uto

co

rre

latio

n

1,0

,8

,6

,4

,2

,0

-,2

-,4

-,6

-,8

-1,0

over 65

confidence limits

coefficient


The stochastic equation for The stochastic equation for age age

class 0-64 and over 65class 0-64 and over 65 ZZtt=0.576z=0.576zt-1t-1++tt

ZZtt=0.703z=0.703zt-1t-1++tt

2 distributions with (K-p-q) degree of freedom2

0-64=1.956

2

over 65=1.651

2

0.05=11.07

Zt=z

t-1+

t


Cross-correlation f. mortality Cross-correlation f. mortality rates rates for cancer of cervix b. for cancer of cervix b. examined examined groups before fitting groups before fitting modelmodel

lag

543210-1-2-3-4-5

cro

ss c

orr

ela

tio

n

1,0

,8

,6

,4

,2

,0

-,2

-,4

-,6

-,8

-1,0

0-64 and over 65

confidence

limits

coefficients


Cross-correlation f. of Cross-correlation f. of residuals for residuals for mortality rates for mortality rates for cancer of cervix cancer of cervix b. examined b. examined groupsgroups

lag

543210-1-2-3-4-5

cro

ss

co

rre

lation

1,0

,8

,6

,4

,2

,0

-,2

-,4

-,6

-,8

-1,0

0-64-over65

confidence

limits

coefficient


Mortality rates f. Mortality rates f. cerebrovascular cerebrovascular diseases b. men-diseases b. men-womenwomen

year

1997199419911988198519821979197619731970

death

s p

er

100000 p

opula

tion

2000

1800

1600

1400

1200

1000

800

over 65

men

women


Stochastic equation of Stochastic equation of mortality mortality rates for women and rates for women and menmen

ZZtt=0.809z=0.809zt-1t-1++tt

ZZtt=0.792z=0.792zt-1t-1++tt

2 distributions with (K-p-q) degree of freedom

2

women=3.886

2

men=1.746

2

0.05=11.07

Zt=z

t-1+

t


Cross-correlation f. of Cross-correlation f. of residuals for residuals for mortality rates for mortality rates for cerebrovascular cerebrovascular diseases b. diseases b. examined groupsexamined groups

lag

543210-1-2-3-4-5

cro

ss c

orr

ela

tio

n

1,0

,8

,6

,4

,2

,0

-,2

-,4

-,6

-,8

-1,0

men-wom;over65

confidence

limits

coefficients


Summary f. mortality Summary f. mortality rates of cerebrovascular rates of cerebrovascular diseasesdiseases It reveals synchronised behaviour It reveals synchronised behaviour

of cerebrovascular diseases of cerebrovascular diseases between the sexes.between the sexes.

Probably is due to exogenous time-Probably is due to exogenous time-varying factors which varying factors which simultaneously influence vascular simultaneously influence vascular mortality rates in both age classes mortality rates in both age classes and in both sexes.and in both sexes.

The cross-correlation function has The cross-correlation function has significance value at timelag k=0 significance value at timelag k=0 but not at other lags on 95% but not at other lags on 95% confidence level.confidence level.


Summary for mortality Summary for mortality rates of cancer of rates of cancer of cervixcervix No such synchronisation for time No such synchronisation for time

series of mortality rates for cancer of series of mortality rates for cancer of cervix for age class 0-64 and over cervix for age class 0-64 and over 65.65.

The cross correlation function has The cross correlation function has not significance value at timelag k=0 not significance value at timelag k=0 on 95% confidence level.on 95% confidence level.

Probably no influence exogenous Probably no influence exogenous time-varying factors for mortality time-varying factors for mortality rates cancer of cervix.rates cancer of cervix.


Standard normal distribution Standard normal distribution as as estimation, White method, the estimation, White method, the

estimation continuous time estimation continuous time seriesseries

0,7920,792

(men)(men)p=0,1p=0,1 p=0,05p=0,05 p=0,01p=0,01

Standard Standard

n. distr.n. distr.(0,5959;0,988(0,5959;0,988

4)4)(0,5523;1,031(0,5523;1,031

7)7)(0,4736;1,110(0,4736;1,110

4)4)

White White methodmethod

(0,5993;0,984(0,5993;0,9847)7)

(0,5596;1,024(0,5596;1,0244)4)

(0,4791;1,110(0,4791;1,1105)5)

Cont. time Cont. time seriesseries

(0,6949;0,942(0,6949;0,9424)4)

(0,6649;0,972(0,6649;0,9723)3)

(0,6116;0,997(0,6116;0,9978)8)


Standard normal distribution Standard normal distribution as as estimation, White method and estimation, White method and the the estimation continuous time estimation continuous time seriesseries

0,8090,809

(women)(women)p=0,1p=0,1 p=0,05p=0,05 p=0,01p=0,01

Standard Standard

n. distr.n. distr.(0,6201;0,997(0,6201;0,997

9)9)(0,5783;1,039(0,5783;1,039

7)7)(0,5025;1,115(0,5025;1,115

5)5)

White White methodmethod

(0,6235;0,994(0,6235;0,9945)5)

(0,5852;1,032(0,5852;1,0327)7)

(0,5078;1,110(0,5078;1,1103)3)

Cont. time Cont. time seriesseries

(0,7149;0,952(0,7149;0,9527)7)

(0,6848;0,978(0,6848;0,9788)8)

(0,6329;0,998(0,6329;0,9985)5)


Conclusion f. the analysis of Conclusion f. the analysis of first first order autoregressive order autoregressive parametersparameters

Applying Applying the normal distributionthe normal distribution as as estimationestimation and and

White methodWhite method - the confidence intervals are - the confidence intervals are near equal.near equal.

The confidence limits can be larger than one The confidence limits can be larger than one applying these methods – it is not acceptable applying these methods – it is not acceptable for stacionary time series.for stacionary time series.

Applying Applying the continuous time processthe continuous time process - the - the confidence intervals are much smaller.confidence intervals are much smaller.

The upper confidence limit is smaller than The upper confidence limit is smaller than one. one. This method can be used in each case.This method can be used in each case.


ReferencesReferences

Andrade, I. C., Clare, A. D., O’Brien, R. J. Andrade, I. C., Clare, A. D., O’Brien, R. J. Thomas, S. H. 1999. Tests for stochastic Thomas, S. H. 1999. Tests for stochastic seasonality applied to daily financial time seasonality applied to daily financial time series. series. Manchester School.Manchester School. 67(1): 39-59. 67(1): 39-59.

Boyles, R. P., Raman, S. 2003. Analysis of Boyles, R. P., Raman, S. 2003. Analysis of Climate Trends in North Carolina (1949-1998). Climate Trends in North Carolina (1949-1998). Environment International.Environment International. 29: 263-275. 29: 263-275.

Box, G. E. P., Tiao, G. C. 1975. Intervention Box, G. E. P., Tiao, G. C. 1975. Intervention analysis with applications to economic and analysis with applications to economic and environmental problems. environmental problems. Journal of the Journal of the American Statistical AssociationAmerican Statistical Association. 70: 70-79.. 70: 70-79.



Frances, P. H., Ooms, M. 1997. A periodic long-Frances, P. H., Ooms, M. 1997. A periodic long-memory model for quarterly UK inflation. memory model for quarterly UK inflation. Int. J. Int. J. ForecastingForecasting. 13: 117-126.. 13: 117-126.

Frances, P. H., Neele, J., Dijk, D. 2001. Modelling Frances, P. H., Neele, J., Dijk, D. 2001. Modelling asymmetric volatility in weekly Dutch asymmetric volatility in weekly Dutch temperature data. temperature data. Environmental Environmental Modelling&Software. Modelling&Software. 16: 16: 131-137.131-137.

Goh, C., Law, R. 2002. Modelling and Forecasting Goh, C., Law, R. 2002. Modelling and Forecasting Tourism Demand for Arrivals with Stochastic Tourism Demand for Arrivals with Stochastic Nonstationary Seasonality and Intervention. Nonstationary Seasonality and Intervention. Tourism Management.Tourism Management. 23: 499-510. 23: 499-510.



Helfenstein, U. 1990. Detecting hidden Helfenstein, U. 1990. Detecting hidden relations between time series of mortality relations between time series of mortality rates. rates. Methods Inf MedMethods Inf Med. 29: 57-60. . 29: 57-60.

Nelson, C. R., Plosser, C. I. 1982. Trends Nelson, C. R., Plosser, C. I. 1982. Trends and Random Walks in Macroeconomic Time and Random Walks in Macroeconomic Time Series: Some Evidence and Implications. Series: Some Evidence and Implications. Journal of Monetary Economics.Journal of Monetary Economics. 139-162. 139-162.

Zhang, X. B., Hogg, W. D., Mekis, E. 2001. Zhang, X. B., Hogg, W. D., Mekis, E. 2001. Spatial and temporal characteristics of Spatial and temporal characteristics of heavy precipitation events over Canada. heavy precipitation events over Canada. Journal of ClimateJournal of Climate. 14: 1923-1936.. 14: 1923-1936.


PublicationsPublications

Fazekas, M.: Time series models for Fazekas, M.: Time series models for analysing mortality data. analysing mortality data. Lecture Notes in Lecture Notes in Computer Science.Computer Science. 2199 (2001), 81-87. 2199 (2001), 81-87.

Fazekas, M.: Applications of seasonal time Fazekas, M.: Applications of seasonal time series for analysing the occurence of series for analysing the occurence of childhoold leukaemia in Hungary. childhoold leukaemia in Hungary. Controlled Controlled Clinical Trials,Clinical Trials, 24 (2003), 101. 24 (2003), 101.

Kis, M.: Analysis of the time series for some Kis, M.: Analysis of the time series for some causes of death. Surján, G. et al. Eds. In causes of death. Surján, G. et al. Eds. In Health Data in the Information Society. Health Data in the Information Society. Studies in Health Technology and Studies in Health Technology and Informatics,Informatics, 90 (2002), 439-443. 90 (2002), 439-443.



Fazekas, M.: Analysing the occurence of Fazekas, M.: Analysing the occurence of childhood leukaemia using seasonal time childhood leukaemia using seasonal time series. Damini, E. et al. Eds. In Knowledge-series. Damini, E. et al. Eds. In Knowledge-Based Intelligent Information Engegineering Based Intelligent Information Engegineering Systems& Allied Technilogies. Systems& Allied Technilogies. Frontiers in Frontiers in Artificial Intelligence and ApplicationsArtificial Intelligence and Applications, 82 , 82 (2002), (2002), 950-954.950-954.

Fazekas, M.: Applications of seasonal time Fazekas, M.: Applications of seasonal time series for analysing the occurence of series for analysing the occurence of childhood leukaemia. In childhood leukaemia. In The New The New Navigators: from Professionals to Patiens.Navigators: from Professionals to Patiens. IOS Press. ME 2003 France, Saint Malo, IOS Press. ME 2003 France, Saint Malo, (2003).(2003).



Fazekas, M.: Application of ARIMA models. Fazekas, M.: Application of ARIMA models. 33rdrd International Conference on International Conference on Telecommunications for Training.Telecommunications for Training. Prague, Prague, Czeh Republic, (2001), 50-54.Czeh Republic, (2001), 50-54.

Fazekas, M.: Application of special time Fazekas, M.: Application of special time series model. series model. EFITA 2001EFITA 2001 Montpellier, Montpellier, France, (2001), 93-94.France, (2001), 93-94.

Fazekas, M.: Application time series models Fazekas, M.: Application time series models on medical research. on medical research. 66th th International International Conference on Applied Informatics.Conference on Applied Informatics. Eger, Eger, (2004). (2004).


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