Computer Lab 6. Minitab Project Report. Time Series Plot of x. Year

Transcription

1 Compuer Lab Problem. Lengh of Growing Season in England Miniab Projec Repor Time Series Plo of x x Year The ime series plo indicaes a consan rend up o abou 9, hen he lengh of growing season ends o increase. This is no very clear, and he sample ACF and PACF (below) show ha he daa migh acually be a realizaion of an uncorrelaed random variable. I would sugges ha he lengh of growing season in England flucuaes abou some consan mean. Auocorrelaion Auocorrelaion Funcion for Lengh of Growing Season in England (wih % significance limis for he auocorrelaions) Parial Auocorrelaion Funcion for Lengh of Growing Season in England (wih % significance limis for he parial auocorrelaions) Parial Auocorrelaion I migh be reasonable o look a a par of he daa only, for example since 9, o see if here is any increasing rend. The ime series plo below shows he daa since 9 ogeher wih a sraigh line fi.

2 Trend Analysis Plo for x from 9 Linear Trend Model m = * Variable Acual Fis x from 9 8 Accuracy Measures MAPE 9.8 MAD. MSD Index 8 A sligh increase is indicaed by he model fi. A linear model fi suggess ha firs differencing would derend he daa well. To perform ARIMA(p,d,q) modeling of he daa we will firs examine he sample ACF and PACF of he differenced daa o see wha are he possible values of he parameers p and q. Time Series Plo of nablax nablax Year Auocorrelaion Funcion for nabla (wih % significance limis for he auocorrelaions) Parial Auocorrelaion Funcion for nabla (wih % significance limis for he parial auocorrelaions) Auocorrelaion Parial Auocorrelaion

3 Auocorrelaion Funcion: nablax ACF T LBQ Parial Auocorrelaion Funcion: nablax PACF T The sample ACF and PACF sugges MA() for he differenced daa, hence ARIMA(,,) could be a good choice of he model for he original daa since 9. Final Esimaes of Parameers Type Coef SE Coef T P MA.9 8. Modified Box-Pierce (Ljung-Box) Chi-Square saisic Time Series Plo for x from 9 (wih forecass and heir 9% confidence limis) 8 Chi-Square DF 7 P-Value Forecass from period 7 9% Limis Period Forecas Lower Upper x from 9 8 Time Indeed he MA parameer of he model is highly significan. The forecas of he lengh of growing season for nex five years is consan, equal o abou days, wih quie large predicion limis of abou and days. The fied model can be wrien as ARIMA(,,) wih he MA parameer esimaed as θˆ = -.9, x = z.9z, where z is a realizaion of Whie Noise random variable. Had we considered he daa as a realizaion of an uncorrelaed random variable, hen he only indicaion of fuure values would be he mean of he series, ha is, abou 7 days. The residuals indeed show a Whie Noise characerisics, ha is are uncorrelaed wih zero mean and a consan variance, as can be seen a he diagnosic picures below.

4 ACF of s for x from 9 (wih % significance limis for he auocorrelaions) PACF of s for x from 9 (wih % significance limis for he parial auocorrelaions) Auocorrelaion Parial Auocorrelaion Plos for x from 9 99 Normal Probabiliy Plo 8 Versus Fis 9 Percen Fied Value Frequency 7... Hisogram 8 - Versus Order Observaion Order

5 Problem. Paleoclimaic Glacial Varves. Miniab Projec Repor 8 Time Series Plo of Paleoclimaic Glacial Varves x 8 89 Index 78 7 There is no clear increasing or decreasing rend bu somewha wavy paern can be seen, indicaing non-consan mean. There is no seasonaliy in he daa. The ime series plo also shows non-consan variance; a ransformaion is necessary o sabilize i. Box-Cox Plo of x Lower CL Upper CL Lambda (using 9.% confidence) Esimae Lower CL - Upper CL. Rounded Value SDev Limi Lambda The Box-Cox ransformaion indicaes logarihm as he opimal ransformaion for hese daa.

6 Time Series Plo of logx Time Series Plo of Nabla(logx) logx Nabla(logx) - 89 Index Index 78 7 The plo on he lef hand side shows he log-ransformed series. Is variance is indeed sabilized, he wavy paern is however mainained. Differencing he ransformed daa removes he wavy rend as i can be seen in he plo on he righ hand side. This suggess d = in an ARIMA(p,d,q) model. To find possible values of p and q we will examine he sample ACF and PACF of he ransformed and differenced daa (log x ). Auocorrelaion Funcion: nabla(logx) ACF T LBQ Parial Auocorrelaion Funcion: nabla(logx) PACF T Auocorrelaion Funcion for nabla(logx) (wih % significance limis for he auocorrelaions) Parial Auocorrelaion Funcion for nabla(logx) (wih % significance limis for he parial auocorrelaions).. Auocorrelaion Parial Auocorrelaion These wo funcions indicae an MA() model wih a negaive value of. Fiing ARIMA(,,) o he ransformed daa, logx, we obain he following oupu.

7 ARIMA Model: logx Final Esimaes of Parameers Type Coef SE Coef T P MA.777. Modified Box-Pierce (Ljung-Box) Chi-Square saisic 8 Chi-Square DF 7 P-Value 7 The MA parameer is indeed significan, however, he Ljung-Box saisics show ha he ARIMA(,,) does no compleely accoun for he correlaions in he daa (he p-values are very small and we should rejec he hypohesis ha correlaions of he indicaed groups of noise values are non-significan). Indeed, he plos of he sample ACF and PACF below show significan correlaion a lag. ACF of s for x (wih % significance limis for he auocorrelaions) PACF of s for x (wih % significance limis for he parial auocorrelaions) Auocorrelaion Parial Auocorrelaion Hence, i is worh rying anoher ARIMA model. Adding AR par o he model, ha is, fiing ARIMA(,,) improves he residuals and gives boh parameers, and, significan. The new fied model can be wrien as (.8B) x ( 88B) (.8B)( B)x ( 88B) z = z or = or x.8x +.8x = z 88z Now, z mees he requiremens of a whie noise variable. ARIMA Model: logx Type Coef SE Coef T P AR.8.9 MA 88. Modified Box-Pierce (Ljung-Box) Chi-Square saisic 8 Chi-Square DF P-Value Forecass from period 9% Limis Period Forecas Lower Upper Acual

8 ACF of s for log x (wih % significance limis for he auocorrelaions) PACF of s for log x (wih % significance limis for he parial auocorrelaions) Auocorrelaion Parial Auocorrelaion Plos for x Normal Probabiliy Plo Versus Fis Percen Fied Value. Frequency Hisogram - Versus Order Observaion Order The forecased values for nex five observaions slighly increase; he predicion inervals are quie large hough. T im e S e r ie s P lo fo r lo g x (w ih fo re ca s s a n d h e ir 9 % co nfid e nce lim is) log x Tim e