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## Special Time Series Methods

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**Special Time Series Methods**Maria Fazekas Debrecen University Hungary Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Autoregressive moving average models**• Let … zt-1, zt, zt+1, … denote the observations at equally spaced times … t-1, t, t+1, … . • For simplicity we assume that the mean value of zt is zero. • Denote at, at-1, at-2, … a sequence of identically distributed uncorrelated random variables with mean 0 and variance a2. • The at are called white noise. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**ARMA(p,q)**• The autoregressive moving average model of order p,q [ARMA(p,q)] can be represent with the following expression: • zt=1zt-1+ … +pzt-p+ +at+1at-1+…+qat-q. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**ARMA(p,q) model with backward shift operator**• Let B the backward shift operator Bzt=zt-1, Bkzt=zt-k. • The ARMA(p,q) model with the backward shift operator:(B)zt=(B)at, where • (B)=1-1B1-…-pBpand • (B)=1+1+…+qBq. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Special cases of theARMA(p,q) models**• The AR(p) zt=1zt-1+…+pzt-p+at. • The MA(q) zt=at+1at-1+…+qat-q. • The special case of AR(p); zt=1zt-1+at. • zt is linearly dependent on the previous observation zt-1 and the random shock at. • The special case of MA(q); when q=1; zt=at+1at-1. • zt is linear expression of the present and previous random shock. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Stationary**• The stationary series has a constant mean and variance and covariance structure, which depends only on the difference between two time points. • The symbol =1-B is called the differencing operator. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Stationary**• If a series has to be differenced once to obtain stationary series then the original series is called an integrated ARMA model of order p,1,q or an ARIMA(p,1,q) model. • If differencing has to be performed d times to obtain stationary series the model is called an ARIMA(p,d,q) model. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**The autocorrelationfunction**• The dependence structure of a stationary time series is characterized by the autocorrelation function. • The autocorrelation function is defined as the correlation between zt and zt+k; k=correlation(zt,zt+k), k is called the time lag. • The autocorrelation function is estimated by the empirical autocorrelation function: rk=ck/c0. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Cross-correlationfunction**• The relation between two time series zt and yt is determined by the cross-correlation function: zy(k)=correlation(zt,yt+k); k=0, 1, 2, … . • The cross-correlation function determines the correlation between the two series as a function of the time lag k. It may be shown that zy(k)=yz(-k). Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Model identification andchecking**To obtain an adequate ARIMA model Box and Jenkins have suggested the following procedure: • Make the series stationary. • Choose a provisional model, in particular by looking at the empirical autocorrelation function. • Estimate the model parameters. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Model identification andchecking**• Check the adequacy of the model. In particular check the autocorrelation function of the residuals. • If the model does not fit the data adequately one goes back to step 2. And chooses an improved model. Among different models, which represent the data equally well, one chooses the simplest one, the model with the fewest parameters. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Estimation of confidence interval**For estimation of the parameter of first order autoregressive model two methods are well known: • apply the standard normal distribution as estimation, and • White method. These methods above cannot be applied in non-stationary case. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Estimation of confidence interval**• Little known for estimation of the parameter of AR(1) is the application of estimation for continuous time case processes. This method can be applied in each case properly. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Mortality rates for cancer of cervix for age class 0-64**and over 65 Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Autocorrelation f. mortality rates for cancer of cervix**o.65 Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Partial autocorrelation function for mortality rates over**65 Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**The stochastic equation for age class 0-64 and over 65**• Zt=0.576zt-1+t • Zt=0.703zt-1+t Zt=zt-1+t 2 distributions with (K-p-q) degree of freedom 20-64=1.956 20.05=11.07 2over 65=1.651 Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Cross-correlation f. mortality rates for cancer of cervix**b. examined groups before fitting model Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Cross-correlation f. of residuals for mortality rates for**cancer of cervix b. examined groups Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Mortality rates f. cerebrovascular diseases b. men-women**Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Stochastic equation of mortality rates for women and men**• Zt=0.809zt-1+t • Zt=0.792zt-1+t Zt=zt-1+t 2 distributions with (K-p-q) degree of freedom 2women=3.886 20.05=11.07 2men=1.746 Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Cross-correlation f. of residuals for mortality rates for**cerebrovascular diseases b. examined groups Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Summary f. mortality rates of cerebrovascular diseases**• It reveals synchronised behaviour of cerebrovascular diseases between the sexes. • Probably is due to exogenous time-varying factors which simultaneously influence vascular mortality rates in both age classes and in both sexes. • The cross-correlation function has significance value at timelag k=0 but not at other lags on 95% confidence level. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Summary for mortality rates of cancer of cervix**• No such synchronisation for time series of mortality rates for cancer of cervix for age class 0-64 and over 65. • The cross correlation function has not significance value at timelag k=0 on 95% confidence level. • Probably no influence exogenous time-varying factors for mortality rates cancer of cervix. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Standard normal distribution as estimation, White method,**the estimation continuous time series Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Standard normal distribution as estimation, White method**and the estimation continuous time series Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Conclusion f. the analysis of first order autoregressive**parameters • Applying the normal distributionas estimation and • White method - the confidence intervals are near equal. • The confidence limits can be larger than one applying these methods – it is not acceptable for stacionary time series. • Applying the continuous time process - the confidence intervals are much smaller. • The upper confidence limit is smaller than one. This method can be used in each case. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**References**• Andrade, I. C., Clare, A. D., O’Brien, R. J. Thomas, S. H. 1999. Tests for stochastic seasonality applied to daily financial time series. Manchester School. 67(1): 39-59. • Boyles, R. P., Raman, S. 2003. Analysis of Climate Trends in North Carolina (1949-1998). Environment International. 29: 263-275. • Box, G. E. P., Tiao, G. C. 1975. Intervention analysis with applications to economic and environmental problems. Journal of the American Statistical Association. 70: 70-79. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**References**• Frances, P. H., Ooms, M. 1997. A periodic long-memory model for quarterly UK inflation. Int. J. Forecasting. 13: 117-126. • Frances, P. H., Neele, J., Dijk, D. 2001. Modelling asymmetric volatility in weekly Dutch temperature data. Environmental Modelling&Software. 16: 131-137. • Goh, C., Law, R. 2002. Modelling and Forecasting Tourism Demand for Arrivals with Stochastic Nonstationary Seasonality and Intervention. Tourism Management. 23: 499-510. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**References**• Helfenstein, U. 1990. Detecting hidden relations between time series of mortality rates. Methods Inf Med. 29: 57-60. • Nelson, C. R., Plosser, C. I. 1982. Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications. Journal of Monetary Economics. 139-162. • Zhang, X. B., Hogg, W. D., Mekis, E. 2001. Spatial and temporal characteristics of heavy precipitation events over Canada. Journal of Climate. 14: 1923-1936. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Publications**• Fazekas, M.: Time series models for analysing mortality data. Lecture Notes in Computer Science. 2199 (2001), 81-87. • Fazekas, M.: Applications of seasonal time series for analysing the occurence of childhoold leukaemia in Hungary. Controlled Clinical Trials, 24 (2003), 101. • Kis, M.: Analysis of the time series for some causes of death. Surján, G. et al. Eds. In Health Data in the Information Society. Studies in Health Technology and Informatics, 90 (2002), 439-443. Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Publications**• Fazekas, M.: Analysing the occurence of childhood leukaemia using seasonal time series. Damini, E. et al. Eds. In Knowledge-Based Intelligent Information Engegineering Systems& Allied Technilogies. Frontiers in Artificial Intelligence and Applications, 82 (2002), 950-954. • Fazekas, M.: Applications of seasonal time series for analysing the occurence of childhood leukaemia. In The New Navigators: from Professionals to Patiens. IOS Press. ME 2003 France, Saint Malo, (2003). Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Publications**• Fazekas, M.: Application of ARIMA models. 3rd International Conference on Telecommunications for Training. Prague, Czeh Republic, (2001), 50-54. • Fazekas, M.: Application of special time series model. EFITA 2001 Montpellier, France, (2001), 93-94. • Fazekas, M.: Application time series models on medical research. 6th International Conference on Applied Informatics. Eger, (2004). Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen**Thank You for your attention!**Summer University on Information Technology in Agriculture and Rural Development 19-22 August 2006 Debrecen