It is a common scenario: A practitioner has sales data for the past several months and wants to forecast next month’s sales volume. This type of forecasting can help manufacturers and distributors ensure they have enough product to meet customer demands. But how is this forecasting done? Statistical analysis software offers two ways to plot the data in order to make a forecast: 1) a linear trend model or 2) a quadratic trend model. It is important for practitioners to understand both methods, as each can be beneficial, depending on the type of process being analyzed.
The following explanations use the sample data shown in Table 1.
Table 1: Sample Volume Data
Month  Volume 
Jan. 2009  98 
Feb. 2009  105 
March 2009  116 
April 2009  119 
May 2009  135 
June 2009  156 
July 2009  177 
Aug. 2009  208 
To begin, use statistical analysis software to create a time series plot with a linear trend analysis (Figure 1).
The software will generate a fitted line using the equation Y_{t} = 71.43 + (15.1 x t). The t represents the time period during which each data point was collected – i.e., the first time period is 1, the second is 2 and so on. Hence, if someone wants to know the fitted value for January 2009, it is 71.43 +15.1*(1) = 86.53.
Table 2: Fitted Values for Past Months
Month  Volume  Fitted Value  t 
Jan. 2009  98  86.53  1 
Feb. 2009  105  101.63  2 
March 2009  116  116.73  3 
April 2009  119  131.83  4 
May 2009  135  146.93  5 
June 2009  156  162.03  6 
July 2009  177  177.13  7 
Aug. 2009  208  192.23  8 
To forecast for September 2009, the practitioner would get 207.33 (71.43 + (15.1 x 9)).
But how does the software get the equation Y_{t} = 71.43 + (15.1 x t)? It is nothing but linear regression. If practitioners used the linear regression function in their statistical analysis software instead, using volume for Y and the t (1, 2, 3, 4, etc.) for X they would get the same equation:
Regression Analysis: Volume versus t
The regression equation is
Volume = 71.4 + 15.1 t
Predictor Coef SE Coef T P
Constant 71.429 8.626 8.28 0.000
t 15.071 1.708 8.82 0.000
S = 11.0701 RSq = 92.8% RSq(adj) = 91.7%
Analysis of Variance
Source DF SS MS F P
Regression 1 9540.2 9540.2 77.85 0.000
Residual Error 6 735.3 122.5
Total 7 10275.5
Another potentially confusing element of the linear trend plot is the forecast accuracy measures: MAD, MAPE and MSD. These are used to determine how well the trend will accurately predict the future volume.
MAD stands for mean absolute deviation, which is the average of the absolute deviations. An absolute deviation is the absolute value of the actual data minus the fitted value (Table 3).
Table 3: Sample Data Including Absolute Deviation
Month  Volume  Fitted Value  t  Absolute Deviation 
Jan. 2009  98  86.53  1  11.47 
Feb. 2009  105  101.63  2  3.37 
March 2009  116  116.73  3  0.73 
April 2009  119  131.83  4  12.83 
May 2009  135  146.93  5  11.93 
June 2009  156  162.03  6  6.03 
July 2009  177  177.13  7  0.13 
Aug. 2009  208  192.23  8  15.77 
Sum  62.26  
n  8  
MAD  7.7825 
The best fitted line should have zero MAD; the larger the MAD, the worse the model. For instance, instead of using linear regression to generate a forecast, a practitioner might base the forecast on last month’s volume. This generates a different MAD value (Table 4).
Table 4: Forecast Using Last Month’s Volume
Month  Volume  Fitted Value  t  Absolute Deviation 
Jan. 2009  98  80  1  18 
Feb. 2009  105  98  2  7 
March 2009  116  105  3  11 
April 2009  119  116  4  3 
May 2009  135  119  5  16 
June 2009  156  135  6  21 
July 2009  177  156  7  21 
Aug. 2009  208  177  8  31 
Sum  128  
n  8  
MAD  16 
The MAD value allows the practitioner to conclude that the model generated by linear regression is better than the model generated by last month’s volume.
MSD
The linear trend plot also uses the accuracy measure MSD, which stands for mean square deviation. It is very similar to MAD, but instead of summing the absolute deviations, this measure sums up the squared deviations (Table 5).
Table 5: Sample Data Including Squared Deviations
Month  Volume  Fitted Value  t  Absolute Deviation  Squared Deviations 
Jan. 2009  98  86.5  1  11.47  132.25 
Feb. 2009  105  101.571429  2  3.37  11.75510204 
March 2009  116  116.642857  3  0.73  0.413265306 
April 2009  119  131.714286  4  12.83  161.6530612 
May 2009  135  146.785714  5  11.93  138.9030612 
June 2009  156  161.857143  6  6.03  34.30612245 
July 2009  177  176.928571  7  0.13  0.005102041 
Aug. 2009  208  192  8  15.77  256 
Sum  735.2857143  
n  8  
MSD  91.91071429 
So what is the difference? MSD weights large deviations more heavily because it takes the square of the deviations. In general, MSD is preferred over MAD because there seems to be more theoretical support for it.
MAPE
The third accuracy measure is MAPE, or mean absolute percentage error. It is calculated by taking the absolute deviation and dividing it by the data (the volume in this case) to get the percent error (Table 6).
Table 6: Sample Data Including Absolute Percent Error
Month  Volume  Fitted Value  t  Absolute Deviation  Absolute Percent Error 
Jan. 2009  98  86.5  1  11.5  11.73469388 
Feb. 2009  105  101.571429  2  3.428571429  3.265306122 
March 2009  116  116.642857  3  0.642857143  0.554187192 
April 2009  119  131.714286  4  12.71428571  10.68427371 
May 2009  135  146.785714  5  11.78571429  8.73015873 
June 2009  156  161.857143  6  5.8571428571  3.754578755 
July 2009  177  176.928571  7  0.071428571  0.040355125 
Aug. 2009  208  192  8  16  7.692307692 
Sum  46.4558612  
n  8  
MAPE  5.80698265 
MAPE is typically used less often than MAD and MSE.
If the practitioner suspects the trend in volume is quadratic rather than linear (meaning the volume is increasing at a faster rate than it would with linear proportion), they would create a plot with a quadratic trend in their statistical analysis software (Figure 2).
It is no surprise that this model is better than the linear model from a MAPE, MAD and MSD perspective because it is a more complex model, requiring more terms. However, it is generally impossible to say which is the correct model to use. The decision requires a judgment call based on the practitioner’s understanding of the process.
The equation used with the quadratic trend is Y_{t} = 101.61 – (3.04 x t) + (2.012 x t^{2}), Once again, this equation is reached through regression analysis. To complete this regression using statistical analysis software, the practitioner first needs to square the t series (Table 7).
Table 7: Sample Data with T Value Squared
Month  Volume  t  t^{2} 
Jan. 2009  98  1  1 
Feb. 2009  105  2  4 
March 2009  116  3  9 
April 2009  119  4  16 
May 2009  135  5  25 
June 2009  156  6  36 
July 2009  177  7  49 
Aug. 2009  208  8  64 
Next, the practitioner performs a multiple regression of the volume on t and t^{2}. The result is:
Regression Analysis: Volume versus t, t sqr
The regression equation is
Volume = 102 – 3.04 t + 2.01 t sqr
Predictor Coef SE Coef T P
Constant 101.607 4.638 21.91 0.000
t 3.036 2.365 1.28 0.256
t sqr 2.0119 0.2565 7.84 0.001
S = 3.32451 RSq = 99.5% RSq(adj) = 99.2%
Analysis of Variance
Source DF SS MS F P
Regression 2 10220.2 5110.1 462.35 0.000
Residual Error 5 55.3 11.1
Total 7 10275.5
Source DF Seq SS
t 1 9540.2
t sqr 1 680.0
The equation is the same as the one generated when creating the quadratic trend analysis plot.


Comments
how to get 71.43 and 15.1 ?
Nice summary! Thanks for sharing your insights.
Well done.
Excellent and easy explanation.