Levels of time series are considered as the function of time
The procedure of smoothing has 3 steps choice of the form of function; determination of parameters of the function; receiving the smoothed values of the levels of series on the basis of the function
Lets consider this method on the example of linear trend equation where a & b – parameters; t – time
The best way to use linear trend in cases, when the preliminary investigation shows, that levels of series change with approximately the same speed, i.e. when chain absolute increases are approximately equal
Parameters a & b are determined by the least square method LSM
The usage of LSM gives the following system of equations for determining the parameters:
The given system of equations can be significantly simplified, if we enumerate the time in the way, that
If the series contains odd number of levels, then the central level of series is enumerated as 0. Levels to the side of decrease of time are enumerated by -1;- 2;-3..., to the side of increase of time – by 1;2;3...
If the series contains even number of levels, then the closest levels to the center are enumerated by -1 and 1, then numeration is the same as with odd number of levels but only with the step 2:...-5,-3,-1,+1,+3,+5...