Forecasting based on creeping Forecasting based on creeping trend with harmonic weightstrend 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.

02468

1012141618

t

Yt

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).

310312314316318320322324326328

January

Ferbuary

March

April

May

June Ju

ly

August

September

October

November

t

Yt

0

5

10

15

20

25

30

January

Ferbuary

March

April

May

June Ju

ly

August

September

October

November

December

January

t

Yt

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.

ty

ty

Step IV

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

ty~

Step V

Determine trend growth for mean smoothed values

1,...,2,1~~11 ntforyyw ttt

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:

1,...,2,11

1

1

11

ntfor

innC

t

i

nt

Step VI cont.

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

tn

10,250 0,7500,111 0,278 0,6110,063 0,146 0,271 0,5210,040 0,090 0,157 0,257 0,4570,028 0,061 0,109 0,158 0,242 0,4080,020 0,044 0,073 0,109 0,156 0,228 0,3700,016 0,034 0,054 0,079 0,111 0,152 0,215 0,3400,012 0,026 0,042 0,061 0,083 0,111 0,148 0,203 0,3140,010 0,021 0,034 0,048 0,065 0,085 0,110 0,143 0,193 0,2930,008 0,017 0,027 0,039 0,052 0,067 0,085 0,108 0,138 0,184 0,2750,007 0,014 0,023 0,032 0,043 0,054 0,068 0,085 0,106 0,134 0,175 0,2590,006 0,012 0,019 0,027 0,036 0,045 0,056 0,069 0,084 0,104 0,129 0,168 0,2450,005 0,011 0,017 0,023 0,030 0,038 0,047 0,057 0,069 0,083 0,101 0,125 0,161 0,2320,004 0,009 0,014 0,020 0,026 0,033 0,040 0,048 0,058 0,069 0,082 0,099 0,121 0,155 0,2210,004 0,008 0,013 0,017 0,023 0,028 0,035 0,041 0,049 0,058 0,069 0,081 0,097 0,118 0,149 0,2110,003 0,007 0,011 0,015 0,020 0,025 0,030 0,036 0,042 0,050 0,058 0,068 0,080 0,094 0,114 0,144 0,2020,003 0,006 0,010 0,012 0,017 0,022 0,026 0,031 0,037 0,043 0,050 0,058 0,067 0,078 0,092 0,111 0,139 0,1940,003 0,006 0,009 0,012 0,016 0,019 0,023 0,028 0,033 0,038 0,044 0,050 0,058 0,067 0,077 0,090 0,108 0,134 0,187

18 1913 14 15 1610 11 12 176 7 8 92 3 4 5

181920

1

14151617

10111213

6789

2345

Step VII

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

1

111

n

tt

nt wCw

Step VIII

Forecast for time period T

wnTyy nT )(~

Step IX

Confidence interval for forecast requires calculating uT

Step IX cont.

uα 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.

uα 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, uα can be calculated from Tchebyshev

inequality:)

Step IX cont.

Standard error of the trend growth

harmonic weightmean 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 January February March April May June

t 1 2 3 4 5 6

yt (in

thousands zł)

53 67 58 79 88 85

Example – step II

iTime

interval from ti to ti+2

Time series fragment yi, yi+1, yi+2

Y Partial trend equation

1 t = 1, 2, 3 y1 y2 y3 53 67 58 y1 (t) =

53,74 + 2,5t

2 t = 2, 3, 4 y2 y3 y4 67 58 79 y2 (t) = 49,32

+ 6t

3 t = 3, 4, 5 y3 y4 y5 58 79 88 y3 (t) = 14,25

+ 15t

4 t = 4, 5, 6 y4 y5 y6 79 88 85 y4 (t) = 68,16

+ 3t

Partial trends for k = 3

Example – step III and IVSmoothed (fitted) values (step III) and mean smoothed values (step IV) ty~

Example – step V

Trend growths for mean smoothed values 1tw

Example – step VIHarmonic weights

457,012

1

3

1

4

1

5

1

5

1

56

1

46

1

36

1

26

1

16

1

16

15

257,02

1

3

1

4

1

5

1

5

1

46

1

36

1

26

1

16

1

16

14

157,03

1

4

1

5

1

5

1

36

1

26

1

16

1

16

13

09,04

1

5

1

5

1

26

1

16

1

16

12

04,05

1

5

1

16

1

16

11

615

614

613

612

611

Ct

Ct

Ct

Ct

Ct

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

tn

10,250 0,7500,111 0,278 0,6110,063 0,146 0,271 0,5210,040 0,090 0,157 0,257 0,4570,028 0,061 0,109 0,158 0,242 0,4080,020 0,044 0,073 0,109 0,156 0,228 0,3700,016 0,034 0,054 0,079 0,111 0,152 0,215 0,3400,012 0,026 0,042 0,061 0,083 0,111 0,148 0,203 0,3140,010 0,021 0,034 0,048 0,065 0,085 0,110 0,143 0,193 0,2930,008 0,017 0,027 0,039 0,052 0,067 0,085 0,108 0,138 0,184 0,2750,007 0,014 0,023 0,032 0,043 0,054 0,068 0,085 0,106 0,134 0,175 0,2590,006 0,012 0,019 0,027 0,036 0,045 0,056 0,069 0,084 0,104 0,129 0,168 0,2450,005 0,011 0,017 0,023 0,030 0,038 0,047 0,057 0,069 0,083 0,101 0,125 0,161 0,2320,004 0,009 0,014 0,020 0,026 0,033 0,040 0,048 0,058 0,069 0,082 0,099 0,121 0,155 0,2210,004 0,008 0,013 0,017 0,023 0,028 0,035 0,041 0,049 0,058 0,069 0,081 0,097 0,118 0,149 0,2110,003 0,007 0,011 0,015 0,020 0,025 0,030 0,036 0,042 0,050 0,058 0,068 0,080 0,094 0,114 0,144 0,2020,003 0,006 0,010 0,012 0,017 0,022 0,026 0,031 0,037 0,043 0,050 0,058 0,067 0,078 0,092 0,111 0,139 0,1940,003 0,006 0,009 0,012 0,016 0,019 0,023 0,028 0,033 0,038 0,044 0,050 0,058 0,067 0,077 0,090 0,108 0,134 0,187

2345

1213

6789

181920

1

14151617

1011

2 3 4 5 6 7 8 9 10 11 12 17 18 1913 14 15 16

Example – step VII

Mean trend growth

097,5

02285,06471,208967,22313,01516,0

)05,0(457,03,10257,031,13157,0

57,209,079,304,01

111

n

tt

nt wCw

Example – step VIII

Forecast for T=9

451,101291,1516,86097,5316,86)69(~)(~

69 wyy

wnTyy nT

Expected sales for September will be 101 451 zl.