Making Sense of Time Series Forecasting

Making Sense of Time Series Forecasting

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.

Part 1 – Linear Trend

The following explanations use the sample data shown in Table 1.

Table 1: Sample Volume Data

Jan. 200998
Feb. 2009105
March 2009116
April 2009119
May 2009135
June 2009156
July 2009177
Aug. 2009208

To begin, use statistical analysis software to create a time series plot with a linear trend analysis (Figure 1).

Figure 1: Trend Analysis Plot for Volume – Linear Trend Model

Figure 1: Trend Analysis Plot for Volume – Linear Trend Model

The software will generate a fitted line using the equation Yt = 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

MonthVolumeFitted Valuet
Jan. 20099886.531
Feb. 2009105101.632
March 2009116116.733
April 2009119131.834
May 2009135146.935
June 2009156162.036
July 2009177177.137
Aug. 2009208192.238

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 Yt = 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:

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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 R-Sq = 92.8% R-Sq(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

MonthVolumeFitted ValuetAbsolute Deviation
Jan. 20099886.53111.47
Feb. 2009105101.6323.37
March 2009116116.7330.73
April 2009119131.83412.83
May 2009135146.93511.93
June 2009156162.0366.03
July 2009177177.1370.13
Aug. 2009208192.23815.77

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

MonthVolumeFitted ValuetAbsolute Deviation
Jan. 20099880118
Feb. 20091059827
March 2009116105311
April 200911911643
May 2009135119516
June 2009156135621
July 2009177156721
Aug. 2009208177831
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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.


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

MonthVolumeFitted ValuetAbsolute DeviationSquared Deviations
Jan. 20099886.5111.47132.25
Feb. 2009105101.57142923.3711.75510204
March 2009116116.64285730.730.413265306
April 2009119131.714286412.83161.6530612
May 2009135146.785714511.93138.9030612
June 2009156161.85714366.0334.30612245
July 2009177176.92857170.130.005102041
Aug. 2009208192815.77256

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.


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

MonthVolumeFitted ValuetAbsolute DeviationAbsolute Percent Error
Jan. 20099886.5111.511.73469388
Feb. 2009105101.57142923.4285714293.265306122
March 2009116116.64285730.6428571430.554187192
April 2009119131.714286412.7142857110.68427371
May 2009135146.785714511.785714298.73015873
June 2009156161.85714365.85714285713.754578755
July 2009177176.92857170.0714285710.040355125
Aug. 20092081928167.692307692
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MAPE is typically used less often than MAD and MSE.

Part 2 – Quadratic Trend

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).

Figure 2: Trend Analysis Plot for Volume – Quadratic Trend Model

Figure 2: Trend Analysis Plot for Volume – Quadratic Trend Model

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 Yt = 101.61 – (3.04 x t) + (2.012 x t2), 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

Jan. 20099811
Feb. 200910524
March 200911639
April 2009119416
May 2009135525
June 2009156636
July 2009177749
Aug. 2009208864

Next, the practitioner performs a multiple regression of the volume on t and t2. 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 R-Sq = 99.5% R-Sq(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 4

  1. peiling

    how to get 71.43 and 15.1 ?

  2. Carl Berardinelli

    Nice summary! Thanks for sharing your insights.

  3. Hemant

    Excellent and easy explanation.

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