Анализ статистических данных в MINITAB - Моделирование временного ряда (Time Series Forecasting)
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Chapter 13
Time Series Forecasting
Topics to be covered in this chapter:
Time Series Plot
Trend Analysis
Seasonal Patterns
Decomposition
Autocorrelation
Moving Average Models
Exponential Smoothing Models
Time Series Plot
Example 13.1 of PBS discusses monthly retail sales of General Merchandise Stores beginning January
1992 and ending May 2002 (125 months). The data are provided in EG13_001.MTW. Minitab can be
used to plot the monthly retail sales by selecting
Stat h Time Series h Time Series Plot
from the menu. This command plots measurement data on the y-axis versus time data on the x-axis.
Minitab assumes that the y-axis data occurred in the order that the values appear in the column, in
equally spaced time intervals. If the data did not occur that way, you may want to use
Graph h Plot
to plot the y-axis data versus a date/time column on the x-axis.
In the dialog box, Graph variables defines the variables to be used in each graph. Under Y en-
ter a column containing the observations on the graph. The x-axis is automatically the time axis. The
x-axis time scales can be labeled with values from a date/time column, called a date/time stamp, or with
time units chosen in the dialog box: index units, calendar units, or clock units.
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Click on the Annotation button to add a title to the plot. You can also change the default start
time, suppress time unit scales, or tell Minitab to use a different time interval by clicking on the Options
button. For each time unit axis shown in the Options sub-dialog box, check the box to show the time
scale, or uncheck the box to suppress the axis. This is useful for getting rid of cluttered axes. For the
retail sales data, we uncheck the box to suppress the month axis:
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Since January 1992, overall sales have gradually increased and a distinct pattern repeats itself
approximately every 12 months.
Trend Analysis
Example 13.2 of PBS discusses monthly retail sales of General Merchandise Stores beginning January
1992 and ending May 2002 (125 months). The pattern of increasing growth in the time series plot of
the retail sales data is an example of a linear trend. Minitab can be used to estimate the linear trend.
Select
Stat h Time Series h Trend Analysis
from the menu. This command fits a general trend model to time series data and provides forecasts.
You can choose among linear, quadratic, exponential, and S-curve models. To forecast future sales,
check Generate forecasts and enter a number in Number of forecasts. In the Starting from origin box,
enter a positive integer to specify a starting point for the forecasts. If you leave this space blank, Mini-
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tab generates forecasts from the end of the data. For the retail sales data, we enter Sales as the Variable
and under Model Type, choose Linear. Minitab estimates the linear trend to be
Y
t
= 18736.5 + 145.53 t
where t is the number of months elapsed beginning with the first month of the time series.
Trend Analysis
Data Sales
Length 125.000
NMissing 0
Fitted Trend Equation
Yt = 18736.5 + 145.531*t
Accuracy Measures
MAPE: 12.4750
MAD: 3756.77
MSD: 36632916
The trend analysis includes a graphic display plot as shown below. To turn the graphics display on (or
off), click on the Results button in the trend analysis dialog box.
We can also use regression techniques to fit a linear model to the above data. This gives additional out-
put, such as the
value for the model. First, select
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Calc h Make Patterned Data h Simple Set of Numbers
from the menu to create a variable x (or t), where x is the number of months
elapsed, beginning
with the first month of the time series. That is, x = 1 corresponds to January 1992, x = 2 corresponds to
February 1992, etc. Next, select
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Stat h Regression h Regression
from the menu. Enter Sales as the response variable and x as the predictor variable and click OK.
Regression Analysis: Sales versus x
The regression equation is
Sales = 18736 + 146 x
Predictor Coef SE Coef T P
Constant 18736 1098 17.06 0.000
x 145.53 15.12 9.62 0.000
S = 6102 R-Sq = 42.9% R-Sq(adj) = 42.5%
Analysis of Variance
Source DF SS MS F P
Regression 1 3446933715 3446933715 92.59 0.000
Residual Error 123 4579114499 37228573
Total 124 8026048215
The trend-only model ignores the seasonal variation in the retail sales time series. Notice that the
value for the trend-only model is 42.9% or 0.429.
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Case 13.1 of PBS discusses the sales of DVD players since the introduction of the DVD format
in March 1997. At the end of June 2002, nearly 33 million DVD players had been sold in the U.S. with
over 18,000 titles available in the DVD format. The Consumers Electronic Association tracks monthly
sales of DVD players. The data are provided in CA13_001.MTW. Select Stat h Time Series h Time
Series Plot from the menu to plot the DVD sales data.
The pattern of increasing growth in this plot is an example of an exponential trend. Minitab
can be used to estimate the exponential trend. Select Stat h Time Series h Trend Analysis from the
menu. In the dialog box, enter Sales as the Variable. Under Model Type, choose Exponential growth.
As shown in the trend analysis, Minitab estimates the exponential trend to be
)07137.1(6.29523
t
t
Y =
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where t is the number of months elapsed, beginning with the first month of the time series. Since
1.07137 is equal to e
0.06894
, this is equivalent to the model given in Case 13.1 of PBS.
Trend Analysis
Data Sales
Length 63.0000
NMissing 0
Fitted Trend Equation
Yt = 29523.6*(1.07137**t)
Accuracy Measures
MAPE: 40.6388
MAD: 224482
MSD: 129023909665
The graphic display plot illustrates the exponential growth together with the raw data.
Seasonal Patterns
A trend equation may be a good description of the long run behavior of the data, but we need to account
for short run phenomena like seasonal variation to improve the accuracy of our forecasts. As in Exam-
ple 13.3 of PBS, we can use indicator variables to add the seasonal pattern to the trend model for the
monthly retail sales data. First, select
Calc h Make Patterned Data h Simple Set of Numbers from
the menu to create a new variable Month that takes the value 1 for each January, 2 for each February,
…. and 12 for each December in the data set. Next select
Calc h Make Indicator Variables
from the menu. In the dialog box, enter Month under Indicator variables for, and specify 12 new col-
umns in the Store results in textbox. This will create 12 indicator variables, one for each month. Name
the indicator variables X1, X2,…,X12. Select
Stat h Regression h Regression from the menu and
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enter Sales as the response variable. Enter x and all 12 indicator variables as the predictor variables and
click OK. (Recall that we defined the variable x in the previous section as the number of months
elapsed beginning with the first month of the time series). We get the following output from Minitab:
Regression Analysis: Sales versus x, X1, ...
* X12 is highly correlated with other X variables
* X12 has been removed from the equation
The regression equation is
Sales = 37473 + 140 x - 24276 X1 - 23749 X2 - 20271 X3 - 20250 X4 - 18518 X5
- 19575 X6 - 20324 X7 - 18627 X8 - 20878 X9 - 18933 X10 - 13842 X11
Predictor Coef SE Coef T P
Constant 37472.7 343.4 109.14 0.000
x 140.130 2.393 58.57 0.000
X1 -24276.2 421.4 -57.61 0.000
X2 -23748.7 421.4 -56.36 0.000
X3 -20271.2 421.3 -48.11 0.000
X4 -20250.5 421.3 -48.07 0.000
X5 -18518.0 421.3 -43.96 0.000
X6 -19574.5 431.4 -45.37 0.000
X7 -20323.6 431.3 -47.12 0.000
X8 -18626.9 431.3 -43.19 0.000
X9 -20878.1 431.2 -48.42 0.000
X10 -18933.0 431.2 -43.91 0.000
X11 -13842.2 431.2 -32.10 0.000
S = 964.1 R-Sq = 98.7% R-Sq(adj) = 98.6%
Analysis of Variance
Source DF SS MS F P
Regression 12 7921942577 660161881 710.22 0.000
Residual Error 112 104105638 929515
Total 124 8026048215
Minitab automatically removes the last indicator variable from the equation because it is highly
correlated with the first 11 indicator variables. The
value for the trend-and-season model is 98.7%
which is a dramatic improvement over the trend-only model. Recall that the
value for the trend-
only model was 42.9%.
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Decomposition
Another approach to accounting for seasonal variation is to calculate an adjustment factor for each sea-
son. The trend is adjusted each particular season by multiplying it by the appropriate seasonality factor.
Using seasonality factors views the model as a trend component times a seasonal component.
Y = TREND
× SEASON
Example 13.4 of PBS calculates the seasonality factors for the retail sales data. Minitab can be used to
calculate seasonality factors by selecting
Stat h Time Series h Decomposition
from the menu. You can use decomposition to separate the time series into linear trend and seasonal
components. You can choose whether the seasonal component is additive or multiplicative with the
trend.
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In the dialog box, enter the column containing the time series under Variable, and a positive in-
teger as the Seasonal length. Since the retail sales data is monthly data, we use a seasonal length of 12.
Under Model Type choose Multiplicative, and under Model Components choose Seasonal only. By
default the first observation is in seasonal period one because Minitab assumes that the first data value
in the series corresponds to the first seasonal period. Enter a different number to specify a different
starting value. Check Generate forecasts if you want to generate forecasts.
Minitab calculates the following seasonality factors for the retail sales data:
Time Series Decomposition
Data Sales(NSA)
Length 125.000
NMissing 0
Seasonal Indices
Period Index
1 0.781569
2 0.805703
3 0.919066
4 0.930090
5 0.988080
6 0.958632
7 0.931024
8 0.984559
9 0.910419
10 0.982894
11 1.15661
12 1.65135
These seasonal indices (or seasonality factors) differ slightly from the indices in Example 13.4 of PBS.
This is because in Minitab the data is smoothed before the seasonal indices are found.
Time Series Forecasting 211
Autocorrelation
The residuals from a regression model that uses time as an explanatory variable should be examined for
signs of autocorrelation. Examples 13.7 and 13.8 of PBS examine the residuals that result from fitting
an exponential trend to the DVD player sales data. The data are provided in CA13_001.MTW. First,
select
Calc h Calculator from the menu and calculate log
e
(Sales). Name this column lnSales. Next,
select
Stat h Regression h Regression from the menu and regress lnSales on the predictor variable x,
where x is the number of months elapsed beginning with the first month of the time series. In the dialog
box, click on the Storage button and check Residuals to obtain the residuals from the exponential trend
model. Finally, select
Graph h Plot from the menu and plot the residuals versus x, i.e., in time order.
The pattern in the plot indicates positive autocorrelation among the residuals. An alternative plot for
detecting autocorrelation is a lagged residual plot. Select
Stat h Time Series h Lag
from the menu. In the dialog box, enter the column containing the variable that you want to lag under
Series. Under Store lags in, select the storage column for the lags and then specify the value for the lag.
To lag the residuals of the DVD Sales data, specify a lag of one and name the output column lag_RES.
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Since the lag selected is one, Minitab moves the row elements of a column down one row. There will
be one missing value symbol (*) at the top of the output column. The output column has the same
number of rows as the input column, so the last value from the input column is not lagged.
Next, select
Graph h Plot from the menu and plot the residuals versus the lagged residuals. The graph
on the following page shows a linear pattern with positive slope. This indicates that the residuals may
have positive autocorrelation
.