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Modeling Time Series Data. Module 5. A Composite Model. We can fit a composite model of the form: Sales = (Trend) * (Seasonality) * (Cyclicality) * (Error). Trend. - PowerPoint PPT Presentation
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Modeling Time Series Data
Module 5
STy *
A Composite Model
We can fit a composite model of the form:
Sales = (Trend) * (Seasonality) * (Cyclicality) * (Error)
Trend
A linear model captures the general upward (or downward) trend with steady growth.
Trend is the long term level and the pattern of change in the dependent variable. It is estimated as a simple function of the period number (time). Linear regression or method of least squares is used to estimate the trend.
Seasonality
Seasonality captures regular, predictable deviations from the trend. Typical seasons are quarters, weeks, or days.
Seasonality is a cycle with a period of exactly one year. We estimate it as a proportion of trend for each season. Data must
be available on seasonal basis.
Time series decomposition is a method to estimate seasonal component.
Cyclicality
Cyclicality captures the effects of long-term macroeconomic boom-bust cycles. It is often difficult to get enough data to measure accurately.
Composite Model
Any residual deviations are attributed to random error.
Time Series Decomposition
• Start with raw data (y)• Estimate Seasonal Indices
– Compute base trend using centered moving averages (t’)– Estimate seasonal ratios (y/t’)– Average seasonal ratios to get raw seasonal indices– Normalize seasonal indices (s)
• De-seasonalize the raw data (y/s)• Estimate the trend equation using de-seasonalized data
(t)• Forecast y’ = t * s• Calculate error = y – (t*s)
Example: Modeling Trend and Seasonality
Toys R Us Revenue (millions $)
Per Year Qtr Revenue
1 1992 1 1026.00
2 1992 2 1056.00
3 1992 3 1182.00
4 1992 4 2861.00
5 1993 1 1172.00
6 1993 2 1249.00
7 1993 3 1346.00
8 1993 4 3402.00
9 1994 1 1286.00
10 1994 2 1317.00
11 1994 3 1449.00
12 1994 4 3893.00
13 1995 1 1462.00
14 1995 2 1452.00
15 1995 3 1631.00
16 1995 4 4200.00
17 1996 1 1776.25
18 1996 2 1808.25
19 1996 3 1941.75
20 1996 4 4128.75
Example: Computing Moving Averages
Per Year Qtr Revenue Moving Avg
1 1992 1 1026.00
2 1992 2 1056.00
3 1992 3 1182.00 1531.3
4 1992 4 2861.00 1567.8
5 1993 1 1172.00 1616.0
6 1993 2 1249.00 1657.0
7 1993 3 1346.00 1792.3
8 1993 4 3402.00 1820.8
9 1994 1 1286.00 1837.8
10 1994 2 1317.00 1863.5
11 1994 3 1449.00 1986.3
12 1994 4 3893.00 2030.3
13 1995 1 1462.00 2064.0
14 1995 2 1452.00 2109.5
15 1995 3 1631.00 2186.3
16 1995 4 4200.00 2264.8
17 1996 1 1776.25 2353.9
18 1996 2 1808.25 2431.6
19 1996 3 1941.75 2413.8
20 1996 4 4128.75
Calculate Moving Average with span of 4
(1026 + 1056 + 1182 + 2861) 4 = 1531.3
Center Moving Average if using even number of data points
(1531.3 + 1567.8) 2 = 1549.5
Per Year Qtr Revenue Moving Avg Centered MA
1 1992 1 1026.00
2 1992 2 1056.00
3 1992 3 1182.00 1531.3 1549.5
4 1992 4 2861.00 1567.8 1591.9
5 1993 1 1172.00 1616.0 1636.5
6 1993 2 1249.00 1657.0 1724.6
7 1993 3 1346.00 1792.3 1806.5
8 1993 4 3402.00 1820.8 1829.3
9 1994 1 1286.00 1837.8 1850.6
10 1994 2 1317.00 1863.5 1924.9
11 1994 3 1449.00 1986.3 2008.3
12 1994 4 3893.00 2030.3 2047.1
13 1995 1 1462.00 2064.0 2086.8
14 1995 2 1452.00 2109.5 2147.9
15 1995 3 1631.00 2186.3 2225.5
16 1995 4 4200.00 2264.8 2309.3
17 1996 1 1776.25 2353.9 2392.7
18 1996 2 1808.25 2431.6 2422.7
19 1996 3 1941.75 2413.8
20 1996 4 4128.75
Example: Using centered moving averages to estimate base demand
Example: Computing Seasonal Ratios
Calculate the ratio of the revenue to the centered moving average
1182 1549.5 = .7628
Per Year Qtr RevenueMoving
AvgCentered
MA Ratio
1 1992 1 1026.00
2 1992 2 1056.00
3 1992 3 1182.00 1531.3 1549.5 0.7628
4 1992 4 2861.00 1567.8 1591.9 1.7973
5 1993 1 1172.00 1616.0 1636.5 0.7162
6 1993 2 1249.00 1657.0 1724.6 0.7242
7 1993 3 1346.00 1792.3 1806.5 0.7451
8 1993 4 3402.00 1820.8 1829.3 1.8598
9 1994 1 1286.00 1837.8 1850.6 0.6949
10 1994 2 1317.00 1863.5 1924.9 0.6842
11 1994 3 1449.00 1986.3 2008.3 0.7215
12 1994 4 3893.00 2030.3 2047.1 1.9017
13 1995 1 1462.00 2064.0 2086.8 0.7006
14 1995 2 1452.00 2109.5 2147.9 0.6760
15 1995 3 1631.00 2186.3 2225.5 0.7329
16 1995 4 4200.00 2264.8 2309.3 1.8187
17 1996 1 1776.25 2353.9 2392.7 0.7424
18 1996 2 1808.25 2431.6 2422.7 0.7464
19 1996 3 1941.75 2413.8
20 1996 4 4128.75
Example: Calculating raw Seasonal Indices
Calculate the average ratio for each season (quarter).
.7162 + .6949 + .7006 + .7424 4
Raw Seasonal Index = .7135
Per Year Qtr RevenueMoving
AvgCentered
MA RatioAvg
Ratio
1 1992 1 1026.00
2 1992 2 1056.00
3 1992 3 1182.00 1531.3 1549.5 0.7628
4 1992 4 2861.00 1567.8 1591.9 1.7973
5 1993 1 1172.00 1616.0 1636.5 0.7162 0.7135
6 1993 2 1249.00 1657.0 1724.6 0.7242 0.7077
7 1993 3 1346.00 1792.3 1806.5 0.7451 0.7406
8 1993 4 3402.00 1820.8 1829.3 1.8598 1.8444
9 1994 1 1286.00 1837.8 1850.6 0.6949
10 1994 2 1317.00 1863.5 1924.9 0.6842
11 1994 3 1449.00 1986.3 2008.3 0.7215
12 1994 4 3893.00 2030.3 2047.1 1.9017
13 1995 1 1462.00 2064.0 2086.8 0.7006
14 1995 2 1452.00 2109.5 2147.9 0.6760
15 1995 3 1631.00 2186.3 2225.5 0.7329
16 1995 4 4200.00 2264.8 2309.3 1.8187
17 1996 1 1776.25 2353.9 2392.7 0.7424
18 1996 2 1808.25 2431.6 2422.7 0.7464
19 1996 3 1941.75 2413.8
20 1996 4 4128.75
Example: Normalizing Seasonal Indices
Normalize to make sure Seasonal Indices average to 1.0 (or add up to 4 in this case)
.7135 ..7135+.7077+.7406+1.844
= .7124
Per Year Qtr RevenueMoving
AvgCentered
MA RatioAvg
Ratio SI
1 1992 1 1026.00 0.7124
2 1992 2 1056.00 0.7066
3 1992 3 1182.00 1531.3 1549.5 0.7628 0.7394
4 1992 4 2861.00 1567.8 1591.9 1.7973 1.8415
5 1993 1 1172.00 1616.0 1636.5 0.7162 0.7135 0.7124
6 1993 2 1249.00 1657.0 1724.6 0.7242 0.7077 0.7066
7 1993 3 1346.00 1792.3 1806.5 0.7451 0.7406 0.7394
8 1993 4 3402.00 1820.8 1829.3 1.8598 1.8444 1.8415
9 1994 1 1286.00 1837.8 1850.6 0.6949 0.7124
10 1994 2 1317.00 1863.5 1924.9 0.6842 0.7066
11 1994 3 1449.00 1986.3 2008.3 0.7215 0.7394
12 1994 4 3893.00 2030.3 2047.1 1.9017 1.8415
13 1995 1 1462.00 2064.0 2086.8 0.7006 0.7124
14 1995 2 1452.00 2109.5 2147.9 0.6760 0.7066
15 1995 3 1631.00 2186.3 2225.5 0.7329 0.7394
16 1995 4 4200.00 2264.8 2309.3 1.8187 1.8415
17 1996 1 1776.25 2353.9 2392.7 0.7424 0.7124
18 1996 2 1808.25 2431.6 2422.7 0.7464 0.7066
19 1996 3 1941.75 2413.8 0.7394
20 1996 4 4128.75 1.8415
Example: De-Seasonalizing raw data
Deseasonalize observations.
= 1440.2
Per Year
Qtr Revenue
Moving Avg
Centered MA Ratio
Avg Ratio SI DeS
1 1992 1 1026.00 0.7124 1440.2
2 1992 2 1056.00 0.7066 1494.4
3 1992 3 1182.00 1531.3 1549.5 0.7628 0.7394 1598.5
4 1992 4 2861.00 1567.8 1591.9 1.7973 1.8415 1553.6
5 1993 1 1172.00 1616.0 1636.5 0.7162 0.7135 0.7124 1645.1
6 1993 2 1249.00 1657.0 1724.6 0.7242 0.7077 0.7066 1767.6
7 1993 3 1346.00 1792.3 1806.5 0.7451 0.7406 0.7394 1820.3
8 1993 4 3402.00 1820.8 1829.3 1.8598 1.8444 1.8415 1847.4
9 1994 1 1286.00 1837.8 1850.6 0.6949 0.7124 1805.1
10 1994 2 1317.00 1863.5 1924.9 0.6842 0.7066 1863.8
11 1994 3 1449.00 1986.3 2008.3 0.7215 0.7394 1959.6
12 1994 4 3893.00 2030.3 2047.1 1.9017 1.8415 2114.0
13 1995 1 1462.00 2064.0 2086.8 0.7006 0.7124 2052.2
14 1995 2 1452.00 2109.5 2147.9 0.6760 0.7066 2054.9
15 1995 3 1631.00 2186.3 2225.5 0.7329 0.7394 2205.7
16 1995 4 4200.00 2264.8 2309.3 1.8187 1.8415 2280.7
17 1996 1 1776.25 2353.9 2392.7 0.7424 0.7124 2493.3
18 1996 2 1808.25 2431.6 2422.7 0.7464 0.7066 2559.0
19 1996 3 1941.75 2413.8 0.7394 2626.0
20 1996 4 4128.75 1.8415 2242.0
1026 .7124
y’ = y/s
Example: De-Seasonalizing
Fit a regression line to the deseasonalized observations – y’ (using time as the independent variable).
Example: De-Seasonalizing
Use trend to make deseasonalized predictions - T
Per YearQtr Revenue
Moving Avg
Centered MA Ratio Avg Ratio SI DeS Forecast
1 1992 1 1026.00 0.7124 1440.2 1430.3
2 1992 2 1056.00 0.7066 1494.4 1487.3
3 1992 3 1182.00 1531.3 1549.5 0.7628 0.7394 1598.5 1544.2
4 1992 4 2861.00 1567.8 1591.9 1.7973 1.8415 1553.6 1601.1
5 1993 1 1172.00 1616.0 1636.5 0.7162 0.7135 0.7124 1645.1 1658.0
6 1993 2 1249.00 1657.0 1724.6 0.7242 0.7077 0.7066 1767.6 1715.0
7 1993 3 1346.00 1792.3 1806.5 0.7451 0.7406 0.7394 1820.3 1771.9
8 1993 4 3402.00 1820.8 1829.3 1.8598 1.8444 1.8415 1847.4 1828.8
9 1994 1 1286.00 1837.8 1850.6 0.6949 0.7124 1805.1 1885.8
10 1994 2 1317.00 1863.5 1924.9 0.6842 0.7066 1863.8 1942.7
11 1994 3 1449.00 1986.3 2008.3 0.7215 0.7394 1959.6 1999.6
12 1994 4 3893.00 2030.3 2047.1 1.9017 1.8415 2114.0 2056.6
13 1995 1 1462.00 2064.0 2086.8 0.7006 0.7124 2052.2 2113.5
14 1995 2 1452.00 2109.5 2147.9 0.6760 0.7066 2054.9 2170.4
15 1995 3 1631.00 2186.3 2225.5 0.7329 0.7394 2205.7 2227.4
16 1995 4 4200.00 2264.8 2309.3 1.8187 1.8415 2280.7 2284.3
17 1996 1 1776.25 2353.9 2392.7 0.7424 0.7124 2493.3 2341.2
18 1996 2 1808.25 2431.6 2422.7 0.7464 0.7066 2559.0 2398.2
19 1996 3 1941.75 2413.8 0.7394 2626.0 2455.1
20 1996 4 4128.75 1.8415 2242.0 2512.0
56.93 * (1) + 1373.4 =
1430.3
Example: De-Seasonalizing
Per YearQtr Revenue
Moving Avg
Centered MA Ratio
Avg Ratio SI DeS
Forecast ReS
1 1992 1 1026.00 0.7124 1440.2 1430.3 1019.0
2 1992 2 1056.00 0.7066 1494.4 1487.3 1050.9
3 1992 3 1182.00 1531.3 1615.5 0.7317 0.7394 1598.5 1544.2 1141.8
4 1992 4 2861.00 1699.7 1699.7 1.6833 1.8415 1553.6 1601.1 2948.5
- - - - - - - - - - - -
13 1995 1 1462.00 1457.0 1486.0 0.9838 0.7124 2052.2 2113.5 1505.7
14 1995 2 1452.00 1515.0 1850.6 0.7846 0.7066 2054.9 2170.4 1533.7
15 1995 3 1631.00 2186.3 2225.5 0.7329 0.7394 2205.7 2227.4 1647.0
16 1995 4 4200.00 2264.8 2309.3 1.8187 1.8415 2280.7 2284.3 4206.6
17 1996 1 1776.25 2353.9 2392.7 0.7424 0.7124 2493.3 2341.2 1667.9
18 1996 2 1808.25 2431.6 2422.7 0.7464 0.7066 2559.0 2398.2 1694.6
19 1996 3 1941.75 2413.8 0.7394 2626.0 2455.1 1815.4
20 1996 4 4128.75 1.8415 2242.0 2512.0 4625.9
21 0.71241 2568.9 1830.2
22 0.70662 2625.9 1855.5
23 0.73944 2682.8 1983.8
24 1.84153 2739.7 5045.3
Reseasonalize predictions (T*S) to make forecasts into the future.
2568.9 * .71241 = 1830.2
Example: De-Seasonalizing
Plot the forecasts – T*S
Example: De-Seasonalizing
Per Year Qtr Revenue
Reseason-alized forecast Square Error
1 1992 1 1026 1018.983 97.00468
2 1992 2 1056 1050.927 51.54981
3 1992 3 1182 1141.832 2950.947
4 1992 4 2861 2948.503 2257.815
5 1993 1 1172 1181.217 167.3762
6 1993 2 1249 1211.841 2765.401
7 1993 3 1346 1310.219 2341.483
8 1993 4 3402 3367.862 343.6507
9 1994 1 1286 1343.45 6503.07
10 1994 2 1317 1372.755 6225.828
11 1994 3 1449 1478.607 1603.193
12 1994 4 3893 3787.221 3299.437
13 1995 1 1462 1505.684 3759.864
14 1995 2 1452 1533.669 13358.15
15 1995 3 1631 1646.995 467.8932
16 1995 4 4200 4206.581 12.7695
17 1996 1 1776.25 1667.917 23123.7
18 1996 2 1808.25 1694.584 25875.59
19 1996 3 1941.75 1815.382 29205.72
20 1996 4 4128.75 4625.94 72893.41
9865.2
(1026 – 1018.98)2 = 97.0
Average square error
As an alternative goodness of fit measure, calculate Root Mean Square Error.
RMSE = 9865.2 = 99.3
Example: De-Seasonalizing with Statpro
Statpro can be used to calculate seasonal indices. Click on Statpro -> Forecast.
http://www.indiana.edu/~mgtsci/StatPro.html
Example: De-Seasonalizing with Statpro
Select the dependent variable.
Example: De-Seasonalizing with Statpro
Select quarterly data.
Example: De-Seasonalizing with Statpro
Select a span of 4 and a moving average method of deseasonalizing.
Example: De-Seasonalizing with Statpro
Statpro generates the same values that we calculated manually.
(Statpro output)