Forecasting based on creeping trend with harmonic weights Creeping trend can be used if variable changes irregularly in time. We use OLS to estimate parameters of partial trends.

Step I Determine the smoothing constant 1<k<n. The most often used k=3. The quality of smoothing depends on the smoothing constant. How to select the smoothing constant? Let’s have a look at your data. Detect the first turning point.

turning point

Step I cont. If great variation in a short time can be observed, small value of smoothing constant need to be selected. If small variation in a short time can be observed, great value of smoothing constant may be selected. Greater value of smoothing constant causes greater smoothing of data (with great values of smoothing constant, time series data react slowly to any changes that may occur).

Step II Estimation of parameters with OLS for partial trends (smoothing constant, k, is the number of cases for each partial trend).

Step III Determine smoothed values, (fitted values). For a given t from 2 to n-1, there is a set of approximants calculated from the partial trend equation.

Step IV Determine mean smoothed value for t. Mean smoothed value is the mean of smoothed values for time period t.

Step V Determine trend growth for mean smoothed values

Step VI Give weight for trend growth. Weights are in ascending order – this way the newest information are the most important. Weight must sum up to 1. formula for calculating weights:

Step VI cont. (weight can be found in statistical tables of harmonic weights if the number of growths is settled).

Step VII Determine mean trend growth as the weighted average of trend growth with harmonic weights.

Step VIII Forecast for time period T

Step IX Confidence interval for forecast requires calculating u T

Step IX cont. u α depends on normality of residuals. 1.If we didn’t reject the null hypothesis (residuals distribution is roughly normal), and n>30 u can be found in normal distribution tables. For sample size n<30 we should use t-Student distribution table (level of significance alpha and n-2 degrees of freedom)

Step IX cont. u α depends on normality of residuals. 2. If we did reject the null hypothesis (residuals distribution is not normal), or we didn’t check the normality of residuals, u α can be calculated from Tchebyshev inequality:)

Step IX cont. Standard error of the trend growth harmonic weight mean trend growth trend growth for time period t

Step IX cont. Confidence interval for forecast (at the level of confidence 1-alpha) forecast for T standard error of the trend growth

Example – step I The following data present the monthly sales (from January to June). The creeping trends method with harmonic weight will let us to construct the forecast for September (T=9). Smoothing constant k=3 (the most often used, in this case it is hard to say which k would be appropriated).

Example – step I Month JanuaryFebruaryMarchAprilMayJune t y t (in thousands zł)

Example – step II i Time interval from t i to t i+2 Time series fragment y i, y i+1, y i+2 Y Partial trend equation 1t = 1, 2, 3y1y1 y2y2 y3y y 1 (t) = 53,74 + 2,5t 2t = 2, 3, 4y2y2 y3y3 y4y y 2 (t) = 49,32 + 6t 3t = 3, 4, 5y3y3 y4y4 y5y y 3 (t) = 14, t 4t = 4, 5, 6y4y4 y5y5 y6y y 4 (t) = 68,16 + 3t Partial trends for k = 3

Example – step III and IV Smoothed (fitted) values (step III) and mean smoothed values (step IV)

Example – step V Trend growths for mean smoothed values

Example – step VI Harmonic weights

Example VI cont. Harmonic weights – if you don’t want to calculate them, find in harmonic weights tables

Example – step VII Mean trend growth

Example – step VIII Forecast for T=9 Expected sales for September will be zl.