Decomposition of time series on the basis of modified grouping method of Ward

The trend of time series can change its direction. It is assumed that the time interval is divided into subintervals where the trend is given as particular linear function. The problem is how to divide the observation of time series into disjoint and coherent groups where they have linear trend. That is why the problem of the scatter of multivariable observation was first considered. The degree of data spread is measured by means of a coefficient called a discriminant of multivariable observation. It is equal to the sum of volumes of the parallelotops spanned on multidimensional observations. On the basis of it the modifications of the well known generalized variance were introduced. Geometrical properties of those parameters were investigated. The obtained results are used to generalize well-known clustering methods of Ward. One of the advantages of the method is that it finds clusters of high linear dependent multivariate observations. Finally, the results are used to partition a time series into homogeneous groups where observations are close to linear trend. There is considered an example.