## Combining Multiple Time Series Using Regression Analysis

Multiple regression is a statistical technique that can be used to analyze the relationship between a single dependent variable and several independent variables. The objective of multiple regression analysis is to use the independent variables whose values are known to predict the value of the single dependent value. Each predictor value is weighed, the weights denoting their relative contribution to the overall prediction.

Here Y is the dependent variable, and X1,^,Xn are the n independent variables. In calculating the weights, a, b1,^,bn, regression analysis ensures maximal prediction of the dependent variable from the set of independent variables. This is usually done by least squares estimation.

This approach can be applied to analyze multivariate time series data when one of the variables is dependent on a set of other variables. We can model the dependent variable Y on the set of independent variables. At any time instant when we are given the values of the independent variables, we can predict the value of Y from Eq. 1.

Handbook of Biosurveillance ISBN 0-12-369378-0 SAT HON yes FRI SUN THE THJ SPIT HON MED FRI SUN

JftN-01-2005 JW-10-2005 JftN-19-2005 JAN-28-2M5 FEB-CE-2005 FEB-15-2005 FEB-24-2005 HNR-05-2005 NPtR-lJ-2005 NfiR-23-2005 M=F-01-200S HPR-1C-2M5

figure 15.1 Synthetic time series for sales of two products.

SAT HON yes FRI SUN THE THJ SPIT HON MED FRI SUN

JftN-01-2005 JW-10-2005 JftN-19-2005 JAN-28-2M5 FEB-CE-2005 FEB-15-2005 FEB-24-2005 HNR-05-2005 NPtR-lJ-2005 NfiR-23-2005 M=F-01-200S HPR-1C-2M5

figure 15.1 Synthetic time series for sales of two products.

In time series analysis, it is possible to do regression analysis against a set of past values of the variables. This is known as autoregression (AR). Let us consider n variables. We have a time series corresponding to each variable. At time t, the vector Zt represents the values of the n variables. The general autoregressive model assumes that Zt can be represented as: