Time series analysis - lecture 5 Dynamic regression with exponentially decreasing coefficients in the transfer function Consider the dynamic regression model where Yt = the forecast variable (output series); Xt = the explanatory variable (input series); Nt = an ARIMA-process representing the combined effect of all other factors influencing Yt and (B) = ( 1+ B + 2B2 + … + kBk) If k is large, then Time series analysis - lecture 5

Parsimonious parameterization of dynamic regression models where Yt = the forecast variable (output series); Xt = the explanatory variable (input series); Nt = an ARIMA-process representing the combined effect of all other factors influencing Yt Important special cases: s = 0 and r = 0 Time series analysis - lecture 5

Time series analysis - lecture 5 Selecting the order of a general dynamic regression model - values to be determined We need to determine: the values of r, s, and b the values of p, d, and q in the ARIMA(p, d, q) model of the noise Nt the values of P, D, and Q of a seasonal ARIMA model, if such data are analysed Time series analysis - lecture 5

Time series analysis - lecture 5 Selecting the order of a general dynamic regression model - LTF identification Step 1: Fit a multiple regression model with a low order AR model for the noise Step 2: If the errors from the regression appear to be non-stationary, then difference Y and X Step 3: Determine b, r, and s by inspecting the -weights in the regression Step 4: Compute the series of noise terms and fit an ARMA-model to these terms Step 5: Refit the entire model using the new ARMA model for the noise Time series analysis - lecture 5

Proc ARIMA in SAS – stationary inputs and outputs proc arima data=timeseri.gasfurnace; identify var=CO2 crosscorr=gasrate; estimate p=1 input=(3$/(1)gasrate); run; Time series analysis - lecture 5

Proc ARIMA in SAS - nonstationary inputs and outputs proc arima data=timeseri.gasfurnace; identify var=CO2 (1) crosscorr=gasrate(1); estimate p=1 input=(3$/(1)gasrate); run; Time series analysis - lecture 5

Intervention analysis where Yt = the forecast variable (output series); Xt = a step or pulse function; Nt = the combined effect of all other factors influencing Yt (the noise); (B) = ( 0 +  1B +  2B2 + … +  kBk), where k is the order of the transfer function Time series analysis - lecture 5

Intervention analysis - delayed or decayed response Delayed response: Xt is a step function Decayed response: Xt is a pulse function Time series analysis - lecture 5

Statistical methods for trend detection Concordant pair Linear regression Smoothing techniques Non-parametric tests Discordant pair Concordant pair Time series analysis - lecture 5

Time series analysis - lecture 5 The classical Mann-Kendall test for monotone trend in a single time series of data Test statistic: where Under H0, the test statistic T is approximately normal with mean zero and variance one Basic idea: Consider all pairs (Xi, Xj) of observations. Subtract the number of “discordant pairs” from the number of “concordant pairs” Time series analysis - lecture 5

Time series analysis - lecture 5 Mann-Kendall tests for monotone trends in data collected over several seasons Correction for short-term serial dependence Hirsch&Slack (1984): Compute a Mann-Kendall statistic Tj for each season j=1,…,m and form T=T1+…+Tm Derive the variance of T by using the Dietz-Killeen estimator of the covariance matrix of (T1,…,Tm) Year Q1 Q2 Q3 Q4 1996 5.5 4.3 3.5 4.6 1997 6.8 3.7 4.9 1998 3.1 2.6 2.1 1999 3.0 2.4 3.4 2000 2.5 1.9 2.7 2001 2.8 2.3 2002 2003 2004 Test statistic T1 T2 T3 T4 Time series analysis - lecture 5

Time series analysis - lecture 5 Mann-Kendall tests for monotone trends in data collected at several sites Correction for dependence between sites Loftis et al. (1991): Compute a Mann-Kendall statistic Tj for each site j=1,…,m and form T=T1+…+Tm Derive the variance of T by using the Dietz-Killeen estimator of the covariance matrix of (T1,…,Tm) Year Stn1 Stn2 Stn3 Stn4 1996 5.5 4.3 3.5 4.6 1997 6.8 3.7 4.9 1998 3.1 2.6 2.1 1999 3.0 2.4 3.4 2000 2.5 1.9 2.7 2001 2.8 2.3 2002 2003 2004 Test statistic T1 T2 T3 T4 Time series analysis - lecture 5

From multiple time series of data to a smooth trend surface Time series analysis - lecture 5

Time series analysis - lecture 5 A simple model for simultaneous smoothing and adjustment for a single covariate Let be the observed response for the jth coordinate the ith year, and let denote a contemporaneous value of a covariate. Assume that . Response Deterministic trend Impact of covariate Random error Time series analysis - lecture 5

Time series analysis - lecture 5 A semiparametric model for simultaneous smoothing and adjustment for several covariates Let be the observed response for the jth class the ith year, and let denote contemporaneous values of covariates. Assume that . Random error Response Deterministic trend Impact of covariate Impact of covariate Time series analysis - lecture 5

Gradient smoothing Penalty of irregular interannual variation . Penalty of irregular interannual variation Penalty of irregular variation along the gradient Time series analysis - lecture 5

Spatial smoothing along a gradient Temporal smoothing across years Smoothing of the trend function in models of time series data representing several sites along a gradient Spatial smoothing along a gradient Temporal smoothing across years Time series analysis - lecture 5

Sequential smoothing across seasons Temporal smoothing across years Smoothing of the trend function in models of time series data representing several seasons Sequential smoothing across seasons Temporal smoothing across years Time series analysis - lecture 5

Time series analysis - lecture 5 Smoothing of the trend function in models of time series data representing several sectors Circular smoothing across sectors Temporal smoothing across years Time series analysis - lecture 5

Total phosphorus concentrations at Dagskärsgrund in Lake Vänern Time series analysis - lecture 5

Time series analysis - lecture 5 Detection of an unknown level shift - the Standard Normal Homogeneity Test Let X1, …, Xn be a sequence of independent normal random variables with variance one H0: E(Xj) = 0 for 1  j  n H1: E(Xj) = 1 for j  m E(Xj) = 2 for j > m Test statistic where and are the average responses before and after the shift Time series analysis - lecture 5

Detection of a level shift and change-point detection Time series analysis - lecture 5

Time series analysis - lecture 5 Total phosphorus concentrations at Dagskärsgrund in Lake Vänern -simultaneous smoothing and change-point detection Time series analysis - lecture 5

Time series analysis - lecture 5 Ratio between TOC (total organic carbon) and permanganate consumption in Swedish rivers Time series analysis - lecture 5

Time series analysis - lecture 5 Ratio between TOC (total organic carbon) and permanganate consumption in Swedish rivers - simultaneous smoothing and change-point detection Time series analysis - lecture 5

Time series analysis - lecture 5 Uncertainty assessment and hypothesis testing in joint models of smooth trends and change-points Resampling techniques are needed Standard resampling techniques must be modified to accommodate correlated residuals Time series analysis - lecture 5