SMOOTHING OF TIME SERIES 



Smoothing, which is essentially low-piss filtering, can oe 

 accomplished by a numerical operator caLled the smoothing 

 function.. Such functions consist of a series of fractional 

 values called weights. These weights determine the extent 

 to which each observation of the time series contributes to 

 the smoothed or filtered value. Equation (12) indicates 

 how smoothing is performed by a set of weights. If y t is a 

 smoothed value corresponding to the observation y t in the 

 time series, the computation is given by the equation below: 



N 



V t = ^ *; , : -/ "-, •'■.. ' - :: ■■, ;,, 



\ 



y t + * '' 



= w_ 



n h 



.N + 



k-Tir+J. 



N 













r 

 -1 



y + 



w 



o 



y + 



W 



1 



y + 



,y y (12) 



The weight W Q is called the principal weight and, if the 

 mean of the original series is to be preserved, the sum of 

 the weights composing the weighting function must be unity. 



FREQUENCY RESPONSE OF SOME SMOOTHING FUNCTION) 



1 . "EQUALLY -WEIGHTED RUNNING M£AN"OF 2.V+1 CONSECUTIVE WEIGHTS 



The weighting function is W k = 1/ (2V+l) . Substituting this 

 weighting function in equation (ll), the frequency response 

 becomes 



N 



B{f) = (2tf + l)" 1 1 + 2 \ cos(2ff/'-) 



(13) 



One may use the analytic form of the envelope of the weights 

 V{t ) to obtain a convenient approximation to the frequency 

 response of the equally weighted running mean. If T is the 



