Tracking signal

One form of tracking signal is the ratio of the cumulative sum of forecast errors (the deviations between the estimated forecasts and the actual values) to the mean absolute deviation.[1] The formula for this tracking signal is:

${\displaystyle {\text{Tracking signal}}={\frac {\Sigma (a_{t}-f_{t})}{\text{MAD}}}}$

where a_t is the actual value of the quantity being forecast, and f_t is the forecast. MAD is the mean absolute deviation. The formula for the MAD is:

${\displaystyle {\text{MAD}}={\frac {\Sigma \left|a_{t}-f_{t}\right|}{n}}}$

where n is the number of periods. Plugging this in, the entire formula for tracking signal is:

${\displaystyle {\text{Tracking signal}}={\frac {\Sigma (a_{t}-f_{t})}{{\frac {1}{n}}\Sigma \left|a_{t}-f_{t}\right|}}}$

Another proposed tracking signal was developed by Trigg (1964). In this model, e_t is the observed error in period t and |e_t| is the absolute value of the observed error. The smoothed values of the error and the absolute error are given by:

${\displaystyle E_{t}=\beta e_{t}+(1-\beta )E_{t-1}}$

${\displaystyle M_{t}=\beta |e_{t}|+(1-\beta )M_{t-1}}$

Then the tracking signal is the ratio:

${\displaystyle T_{t}=\left|{\frac {E_{t}}{M_{t}}}\right|}$

If no significant bias is present in the forecast, then the smoothed error E_t should be small compared to the smoothed absolute error M_t. Therefore, a large tracking signal value indicates a bias in the forecast. For example, with a β of 0.1, a value of T_t greater than .51 indicates nonrandom errors. The tracking signal also can be used directly as a variable smoothing constant.[2]

There have also been proposed methods for adjusting the smoothing constants used in forecasting methods based on some measure of prior performance of the forecasting model. One such approach is suggested by Trigg and Leach (1967), which requires the calculation of the tracking signal. The tracking signal is then used as the value of the smoothing constant for the next forecast. The idea is that when the tracking signal is large, it suggests that the time series has undergone a shift; a larger value of the smoothing constant should be more responsive to a sudden shift in the underlying signal.[3]

• Calculating demand forecast accuracy
• Demand forecasting

• Alstrom, P., Madsen, P. (1996) "Tracking signals in inventory control systems: A simulation study", International Journal of Production Economics, 45 (1-3), 293–302, doi:10.1016/0925-5273(95)00120-4
• Nahmias, Steven (2005) Production & Operations Analysis, Fifth Edition, McGraw-Hill. ISBN 0-07-123837-9
• Trigg, D.W. (1964) "Monitoring a forecasting system". Operational Research Quarterly, 15, 271–274.
• Trigg, D.W. and Leach, A.G. (1967). "Exponential smoothing with an adaptive response rate". Operational Research Quarterly, 18 (1), 53–59
• Mita Montero, J David (1973). "Análise de Sistemas de Previsão - Amortecimento Exponencial". Tese de Mestrado de Engenharia Industrial PUC-RJ,Brasil. Aplicação Industrial de Tracking Signal.

• Tracking signal in forecasting by Dr Muhammad Al-Salamah
• Tracking Signal:A Measure of Forecast Accuracy by Tyler Hedin, Brigham Young University (Powerpoint)