14.5.6 Minitab Time Series And Forecasting
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14.5.6
Time Series Analysis and Forecasting with Minitab
Unfortunately Minitab does not combine a moving average trend with seasonal decomposition. We shall therefore look at its default method of combining a linear trend with seasonal decomposition and compare its output with s reproduction of the analysis previous worked by hand. In this worksheet we will just analyse seasonal data. Exponential smoothing and curve fitting can be carried out quite easily in Minitab by following the instructions in the Help menu. The quarterly sales, (£0,000s), of a departmental store have been monitored for the past five years with the following information being produced: (Tutorial question 12.1) Total quarterly sales (£0,000s) Year
Quarter 1
Quarter 2
Quarter 3
Quarter 4
2001
48
58
57
65
2002
50
61
59
68
2003
52
62
59
69
2004
52
64
60
73
2005
53
65
60
75
5.6.1 Enter the data: Type all the sales figures in chronological order in one column heading it Sales. 5.6.2 Plot a Sequence graph of Sales to see if an additive model seems appropriate. Graph / Time Series Plot / Simple / Series SALES / Time scale / Calendar / Quarter Year / Start values / Quarter 1 Year 2001 Time Series Plot of Sales 75 70
Sales
65 60
55 50
Quarter Q1 Year 2001
Q3
Q1 2002
Q3
Q1 2003
Q3
Q1 2004
Q3
Q1 2005
Q3
1
5.6.3 Assuming the data to be seasonal, carry out the seasonal decomposition using an additive model: Stat / Time Series / Decomposition / Variable SALES / Seasonal length 4 / Additive Model / Trend plus seasonal / Store Trend, Fits and Residuals. Graphs / Do not display plots / Generate forecasts 4 / Starting from origin 20 Time Series Decomposition for Sales Additive Model Data Sales Length 20 NMissing 0 Fitted Trend Equation Yt = 56.3933 + 0.391118*t Seasonal Indices Period Index 1 -8.82813 2 1.92188 3 -1.14063 4 8.04688 Forecasts Period Forecast 21 55.7786 22 66.9197 23 64.2484 24 73.8270
Save this data as Quarterly sales 5.6.4 Plot a Sequence graph of Sales with Trend to see if an additive model seems appropriate. Graph / Time Series Plot / Multiple SALES and TREND Time Series Plot of Sales, TREN1 Variable Sales TREN1
75 70
Dat a
65 60 55 50 2
4
6
8
10 12 I ndex
14
16
18
20
2
The trend produced is a straight line. With this particular data that looks reasonable. Plot the sales and fitted values on the same graph to check the model fit. Time Series Plot of Sales, FI TS1 Variable Sales FI TS1
75 70
Dat a
65 60 55
50
2
4
6
8
10 12 I ndex
14
16
18
20
5.6.5 Residual analysis: Remember that the residuals should: (a) be small, (b) have a mean of 0, (c) have a standard deviation which is much smaller than that of sales, (d) be normally distributed and (e) be random timewise. The residuals have been saved as RESI1. (a), (b), (c)
Produce descriptive statistics of Sales and RESI1.
Descriptive Statistics: Sales, RESI1 Variable N Sales 20 RESI1 20
(d)
N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum 0 60.50 1.66 7.40 48.00 54.00 60.00 65.00 75.00 0 4.274E-15 0.274 1.227 -2.684 -0.916 -0.0449 0.550 2.737
For RESI1 produce a boxplot and a histogram with a normal plot. Boxplot of RESI 1
Histogram of RESI 1 Normal
3 9
Mean StDev N
8
2
4.263256E-15 1.227 20
7 6
0
Frequency
RESI 1
1
5 4 3
-1
2 1
-2
0
-3
-2
-1
0 RESI1
1
2
3
-3
3
Carry out the K-S hypothesis test for normality: Stat / Basic Statistics / Normality Test / Kolmogorov-Smirnov Probability Plot of RESI 1 Normal 99
Mean StDev N KS P-Value
95 90
4.263256E-15 1.227 20 0.120 > 0.150
80
Percent
70 60 50 40 30 20 10 5
1
-3
(e)
-2
-1
0 RESI 1
1
2
3
Produce a time series plot of the errors.
Stat / Time Series / Decomposition / Seasonal length 4 / Additive Model / Trend plus seasonal Select SALES. and storing Trend, Fits and Residuals. Graphs / Residual plots / Four in one Residual Plots for Sales Residuals Versus t he Fit t ed Values 3.0
90
1.5 Residual
Per cent
Normal Probabilit y Plot of t he Residuals 99
50 10 1
0.0 -1.5 -3.0
-3.0
-1.5
0.0 Residual
1.5
3.0
50
8 6
1.5
4 2 0
60 65 Fitted Value
70
Residuals Versus t he Order of t he Dat a 3.0
Residual
Fr equency
Hist ogram of t he Residuals
55
0.0 -1.5 -3.0
-3
-2
-1
0 1 Residual
2
3
2
4
6
8 10 12 14 Obser vation Or der
16
18
20
Do these look to be a good set of residuals?
4
We are now going to build up a moving trend plus seasonal factor model. 5.6.6 Using Minitab to produce the moving average: Stat / Time Series / Moving Average / Variable SALES / MA length 4 / Centre the moving averages / Store Moving averages. (You may need to plot a separate sequence graph of sales and moving average as the AV line seems displaced on the default production.) Moving Average Length Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
4 Sales 48 58 57 65 50 61 59 68 52 62 59 69 52 64 60 73 53 65 60 75
MA * * 57.250 57.875 58.500 59.125 59.750 60.125 60.250 60.375 60.500 60.750 61.125 61.750 62.375 62.625 62.750 63.000 * *
Predict * * * * * 57.250 57.875 58.500 59.125 59.750 60.125 60.250 60.375 60.500 60.750 61.125 61.750 62.375 62.625 62.750
Error * * * * * 3.750 1.125 9.500 -7.125 2.250 -1.125 8.750 -8.375 3.500 -0.750 11.875 -8.750 2.625 -2.625 12.250
Moving Average Plot for Sales Variable Actual Fits
75
Moving Av erage Length 4
70
Accuracy Measures MAPE 8.9528 MAD 5.6250 MSD 46.9073
Sales
65 60
55 50
2
4
6
8
10 12 I ndex
14
16
18
20
Moving average line appears to be misplaced, so plot lines on a sequence graph.
5
Time Series Plot of Sales, AVER1 Variable Sales AVER1
75
70
Dat a
65
60
55 50
2
4
6
8
10 12 I ndex
14
16
18
20
5.6.7 Head the next column Seasonal and type in the appropriate seasonal factors as produced by the deseasonal composition in Task 3. These were –8.83, +1.92, -1.14 and +8.05 for quarters 1 to 4 respectively. Calculate the FITTED values as AVER1 + SEASONAL Calc / Calculator / Store result in variable Fitted / Expression AVER1 + SEASONAL 5.6.8 Plot a Sequence graph of Sales and Fitted values to see if this model is better. Graph / Time Series Plot / Multiple SALES and FITTED Does this model look a closer fit? It probably does but we need to compare the residuals. Time Series Plot of Sales, Fitted Variable Sales Fitted
75 70
Dat a
65 60 55 50
2
4
6
8
10 12 I ndex
14
16
18
20
6
5.6.9 Calculate the residuals from this model as RESI2 = SALES - FITTED 5.6.10 Compare residuals: (a), (b), (c)
Produce descriptive statistics of Sales and RESI1, RESI2.
Descriptive Statistics: Sales, RESI1, RESI2 Variable N N* Mean SE Mean Sales 20 0 60.50 1.66 RESI1 20 0 4.274E-15 0.274 RESI2 16 4 0.0547 0.209
Maximum 75.00 2.737 2.325
For both sets of errors produce boxplots and histograms with a normal plots. Graphs / Boxplots / Multiple Ys / Simple Boxplot of RESI 1 , RESI 2 3
2
Dat a
1
0
-1
-2
-3 RESI1
RESI2
Graphs / Histogram / With fit and groups Histogram of RESI 1 , RESI 2 Normal 8
Variable RESI 1 RESI 3
7
Mean 4.263256E-15 0.05469
6 Frequency
(d)
StDev Minimum Q1 Median Q3 7.40 48.00 54.00 60.00 65.00 1.227 -2.684 -0.916 -0.0449 0.550 0.835 -1.235 -0.344 0.0175 0.375
StDev N 1.227 20 0.8353 16
5 4 3 2 1 0
-3
-2
-1
0 Dat a
1
2
3
7
Carry out the K-S hypothesis test for normality: Stat / Basic Statistics / NormalityTest / Kolmogorov-Smirnov Probability Plot of RESI 1 Normal 99
Mean StDev N KS P-Value
95 90
4.263256E-15 1.227 20 0.120 > 0.150
80
Percent
70 60 50 40 30 20 10 5
1
-3
-2
-1
0 RESI 1
1
2
3
Probability Plot of RESI 2 Normal 99
Mean StDev N AD P-Value
95 90
0.05469 0.8353 16 0.455 0.234
80
Percent
70 60 50 40 30 20 10 5
1
-2
(e)
-1
0 RESI 2
1
2
Produce a time series plots of both sets of errors. Graph / Time Series Plot RESI1 RESI2
8
Time Series Plot of RESI1, RESI2 2.5
Variable RESI1 RESI2
2.0 1.5
Dat a
1.0 0.5 0.0 -0.5 -1.0 2
4
6
8
10 12 Index
14
16
18
20
Which set look preferable? The set from the moving average model is better. 5.6.11Assuming that the trend from the moving average model, AVER1, is increasing by 0.3 per quarter, what would be your forecasts for the four quarters of 2006? Compare with those produced in Task 5.6.3. Time 18 19 20 21 22 23 24
Trend 63.0 63.3 63.6 63.9 64.2 64.5 64.8
Seasonal +1.92 -1.14 +8.05 -8.83 +1.92 -1.14 +8.05
Forecast
55.07 66.12 63.36 72.85
£551 000 £661 000 £634 000 £729 000
9
Time Series Analysis and Forecasting with Minitab
Unfortunately Minitab does not combine a moving average trend with seasonal decomposition. We shall therefore look at its default method of combining a linear trend with seasonal decomposition and compare its output with s reproduction of the analysis previous worked by hand. In this worksheet we will just analyse seasonal data. Exponential smoothing and curve fitting can be carried out quite easily in Minitab by following the instructions in the Help menu. The quarterly sales, (£0,000s), of a departmental store have been monitored for the past five years with the following information being produced: (Tutorial question 12.1) Total quarterly sales (£0,000s) Year
Quarter 1
Quarter 2
Quarter 3
Quarter 4
2001
48
58
57
65
2002
50
61
59
68
2003
52
62
59
69
2004
52
64
60
73
2005
53
65
60
75
5.6.1 Enter the data: Type all the sales figures in chronological order in one column heading it Sales. 5.6.2 Plot a Sequence graph of Sales to see if an additive model seems appropriate. Graph / Time Series Plot / Simple / Series SALES / Time scale / Calendar / Quarter Year / Start values / Quarter 1 Year 2001 Time Series Plot of Sales 75 70
Sales
65 60
55 50
Quarter Q1 Year 2001
Q3
Q1 2002
Q3
Q1 2003
Q3
Q1 2004
Q3
Q1 2005
Q3
1
5.6.3 Assuming the data to be seasonal, carry out the seasonal decomposition using an additive model: Stat / Time Series / Decomposition / Variable SALES / Seasonal length 4 / Additive Model / Trend plus seasonal / Store Trend, Fits and Residuals. Graphs / Do not display plots / Generate forecasts 4 / Starting from origin 20 Time Series Decomposition for Sales Additive Model Data Sales Length 20 NMissing 0 Fitted Trend Equation Yt = 56.3933 + 0.391118*t Seasonal Indices Period Index 1 -8.82813 2 1.92188 3 -1.14063 4 8.04688 Forecasts Period Forecast 21 55.7786 22 66.9197 23 64.2484 24 73.8270
Save this data as Quarterly sales 5.6.4 Plot a Sequence graph of Sales with Trend to see if an additive model seems appropriate. Graph / Time Series Plot / Multiple SALES and TREND Time Series Plot of Sales, TREN1 Variable Sales TREN1
75 70
Dat a
65 60 55 50 2
4
6
8
10 12 I ndex
14
16
18
20
2
The trend produced is a straight line. With this particular data that looks reasonable. Plot the sales and fitted values on the same graph to check the model fit. Time Series Plot of Sales, FI TS1 Variable Sales FI TS1
75 70
Dat a
65 60 55
50
2
4
6
8
10 12 I ndex
14
16
18
20
5.6.5 Residual analysis: Remember that the residuals should: (a) be small, (b) have a mean of 0, (c) have a standard deviation which is much smaller than that of sales, (d) be normally distributed and (e) be random timewise. The residuals have been saved as RESI1. (a), (b), (c)
Produce descriptive statistics of Sales and RESI1.
Descriptive Statistics: Sales, RESI1 Variable N Sales 20 RESI1 20
(d)
N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum 0 60.50 1.66 7.40 48.00 54.00 60.00 65.00 75.00 0 4.274E-15 0.274 1.227 -2.684 -0.916 -0.0449 0.550 2.737
For RESI1 produce a boxplot and a histogram with a normal plot. Boxplot of RESI 1
Histogram of RESI 1 Normal
3 9
Mean StDev N
8
2
4.263256E-15 1.227 20
7 6
0
Frequency
RESI 1
1
5 4 3
-1
2 1
-2
0
-3
-2
-1
0 RESI1
1
2
3
-3
3
Carry out the K-S hypothesis test for normality: Stat / Basic Statistics / Normality Test / Kolmogorov-Smirnov Probability Plot of RESI 1 Normal 99
Mean StDev N KS P-Value
95 90
4.263256E-15 1.227 20 0.120 > 0.150
80
Percent
70 60 50 40 30 20 10 5
1
-3
(e)
-2
-1
0 RESI 1
1
2
3
Produce a time series plot of the errors.
Stat / Time Series / Decomposition / Seasonal length 4 / Additive Model / Trend plus seasonal Select SALES. and storing Trend, Fits and Residuals. Graphs / Residual plots / Four in one Residual Plots for Sales Residuals Versus t he Fit t ed Values 3.0
90
1.5 Residual
Per cent
Normal Probabilit y Plot of t he Residuals 99
50 10 1
0.0 -1.5 -3.0
-3.0
-1.5
0.0 Residual
1.5
3.0
50
8 6
1.5
4 2 0
60 65 Fitted Value
70
Residuals Versus t he Order of t he Dat a 3.0
Residual
Fr equency
Hist ogram of t he Residuals
55
0.0 -1.5 -3.0
-3
-2
-1
0 1 Residual
2
3
2
4
6
8 10 12 14 Obser vation Or der
16
18
20
Do these look to be a good set of residuals?
4
We are now going to build up a moving trend plus seasonal factor model. 5.6.6 Using Minitab to produce the moving average: Stat / Time Series / Moving Average / Variable SALES / MA length 4 / Centre the moving averages / Store Moving averages. (You may need to plot a separate sequence graph of sales and moving average as the AV line seems displaced on the default production.) Moving Average Length Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
4 Sales 48 58 57 65 50 61 59 68 52 62 59 69 52 64 60 73 53 65 60 75
MA * * 57.250 57.875 58.500 59.125 59.750 60.125 60.250 60.375 60.500 60.750 61.125 61.750 62.375 62.625 62.750 63.000 * *
Predict * * * * * 57.250 57.875 58.500 59.125 59.750 60.125 60.250 60.375 60.500 60.750 61.125 61.750 62.375 62.625 62.750
Error * * * * * 3.750 1.125 9.500 -7.125 2.250 -1.125 8.750 -8.375 3.500 -0.750 11.875 -8.750 2.625 -2.625 12.250
Moving Average Plot for Sales Variable Actual Fits
75
Moving Av erage Length 4
70
Accuracy Measures MAPE 8.9528 MAD 5.6250 MSD 46.9073
Sales
65 60
55 50
2
4
6
8
10 12 I ndex
14
16
18
20
Moving average line appears to be misplaced, so plot lines on a sequence graph.
5
Time Series Plot of Sales, AVER1 Variable Sales AVER1
75
70
Dat a
65
60
55 50
2
4
6
8
10 12 I ndex
14
16
18
20
5.6.7 Head the next column Seasonal and type in the appropriate seasonal factors as produced by the deseasonal composition in Task 3. These were –8.83, +1.92, -1.14 and +8.05 for quarters 1 to 4 respectively. Calculate the FITTED values as AVER1 + SEASONAL Calc / Calculator / Store result in variable Fitted / Expression AVER1 + SEASONAL 5.6.8 Plot a Sequence graph of Sales and Fitted values to see if this model is better. Graph / Time Series Plot / Multiple SALES and FITTED Does this model look a closer fit? It probably does but we need to compare the residuals. Time Series Plot of Sales, Fitted Variable Sales Fitted
75 70
Dat a
65 60 55 50
2
4
6
8
10 12 I ndex
14
16
18
20
6
5.6.9 Calculate the residuals from this model as RESI2 = SALES - FITTED 5.6.10 Compare residuals: (a), (b), (c)
Produce descriptive statistics of Sales and RESI1, RESI2.
Descriptive Statistics: Sales, RESI1, RESI2 Variable N N* Mean SE Mean Sales 20 0 60.50 1.66 RESI1 20 0 4.274E-15 0.274 RESI2 16 4 0.0547 0.209
Maximum 75.00 2.737 2.325
For both sets of errors produce boxplots and histograms with a normal plots. Graphs / Boxplots / Multiple Ys / Simple Boxplot of RESI 1 , RESI 2 3
2
Dat a
1
0
-1
-2
-3 RESI1
RESI2
Graphs / Histogram / With fit and groups Histogram of RESI 1 , RESI 2 Normal 8
Variable RESI 1 RESI 3
7
Mean 4.263256E-15 0.05469
6 Frequency
(d)
StDev Minimum Q1 Median Q3 7.40 48.00 54.00 60.00 65.00 1.227 -2.684 -0.916 -0.0449 0.550 0.835 -1.235 -0.344 0.0175 0.375
StDev N 1.227 20 0.8353 16
5 4 3 2 1 0
-3
-2
-1
0 Dat a
1
2
3
7
Carry out the K-S hypothesis test for normality: Stat / Basic Statistics / NormalityTest / Kolmogorov-Smirnov Probability Plot of RESI 1 Normal 99
Mean StDev N KS P-Value
95 90
4.263256E-15 1.227 20 0.120 > 0.150
80
Percent
70 60 50 40 30 20 10 5
1
-3
-2
-1
0 RESI 1
1
2
3
Probability Plot of RESI 2 Normal 99
Mean StDev N AD P-Value
95 90
0.05469 0.8353 16 0.455 0.234
80
Percent
70 60 50 40 30 20 10 5
1
-2
(e)
-1
0 RESI 2
1
2
Produce a time series plots of both sets of errors. Graph / Time Series Plot RESI1 RESI2
8
Time Series Plot of RESI1, RESI2 2.5
Variable RESI1 RESI2
2.0 1.5
Dat a
1.0 0.5 0.0 -0.5 -1.0 2
4
6
8
10 12 Index
14
16
18
20
Which set look preferable? The set from the moving average model is better. 5.6.11Assuming that the trend from the moving average model, AVER1, is increasing by 0.3 per quarter, what would be your forecasts for the four quarters of 2006? Compare with those produced in Task 5.6.3. Time 18 19 20 21 22 23 24
Trend 63.0 63.3 63.6 63.9 64.2 64.5 64.8
Seasonal +1.92 -1.14 +8.05 -8.83 +1.92 -1.14 +8.05
Forecast
55.07 66.12 63.36 72.85
£551 000 £661 000 £634 000 £729 000
9
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