# Variance on Extreme Seasonal Time Series

I'm trying to come up with a decent method for forecasting a unique seasonal time series that is involving multiple periods of seasonality: Weekly, Monthly, Quarterly and I am stuck because I have daily data that is so seasonal intra-week that the variance I get is way too amplified.

(i.e. Mondays Value > 3x Sunday Value, weekly seasonality resembles a Sawtooth Sine Wave)

I know I need to seasonally adjust this data, and I am in the midst of teaching myself more about time series analysis but I am trying to find a good way to estimate the variance here.

I've hypothesized calculating the variance on a MA series but I'm not quite sure what the implications of that are, or how reliable that transformed number will be. If you have any tips or directions you think I should go on please let me know!

• Are you including daily seasonality ? It appears from your question that you are not and you probably should be along with holiday effects/long weekend effects/week of the month effects etc, and of course possible level shifts and local time trends. Your usage of the word "variance" is slightly confusing to me. When you say "estimate the variance here" do you mean the expected value OR do you mean that the variation around the expected daily value ? – IrishStat Jan 6 '16 at 16:47
• @IrishStat I am trying to forecast daily seasonality and its effect on a larger period. The series varies so much between days that when I calculate the variance around the expected value the number I get is borderline unusable and barely assists in and predictions. I am asking if there are any serious statistical ramifications to using calculating the Variance of a Moving Averages, or if there are any special characteristics I should consider. I've been using Single smoothed and double smoothed MA (not exponential as single day values cause dramatic shifts.) Any insight is appreciated! – Charles Rider Jan 6 '16 at 17:16
• "calculate the variance around the expected value"..... please explain this. Is this the observed values for a monday ( for example) vis-a-vis the average monday ? – IrishStat Jan 6 '16 at 17:51
• slide 44 .... from autobox.com/pdfs/capable.pdf might help explain predicting daily values . Note the uncertainty in the forecasts tacitly assumes constant variability (second moment) across days ...NOT constant expected value across days. I believe the restriction of constant variability across days has been changed in the most recent version of their software ( which I helped develop) – IrishStat Jan 6 '16 at 17:59
• @IrishStat By Calculate Variance Around Expected Value I mean a variance calculation of Var(MA) = Sum( (E(MA) - MA_i)^2 )/(n-1) – Charles Rider Jan 8 '16 at 16:19