Forecasting

Abstract

Since the beginning of civilizations, the ability to predict future events has been one of the most important abilities and capacities of the human mind, greatly assisting in its survival. The ability to foretell the future has always been a major source of power. On the other hand, the example of Cassandra, the ancient princess who could clearly see and prophesize catastrophic near-future events but was dismissed as insane by her people, underscores the importance of the fact that the forecaster must not only be able to make accurate forecasts, but also convince others of the accuracy of her/his forecasts. In today’s world, the ability to accurately forecast near-term as well as medium-term events such as demand for existing or new products is among the most crucial capacities of an enterprise. In general, the use of forecasts falls under one of three major types: (a) economic forecasts, which attempt to measure and predict macro-economic quantities such as business cycles, inflation rates, money supply and currency exchange rates, (b) technological forecasts, whose main purpose is to predict imminent and upcoming technological break-through and innovation, and to a lesser degree market penetration of completely new products and (c) demand forecasts, whose main purpose is to predict short and medium term sales of existing products, whose sales’ history exists and is accurately recorded.

Exercises

• 1 Show that the forecasts produced by the Single Exponential Smoothing method minimize the discounted cost criterion $S^{\prime} = \sum\limits_{j = 0}^{\infty } {(1 - a)^{j + 1} (d_{t - j} - F_{t} )^{2}}.$

• 2 Show that for the two-variable function φ defined as φ(a, b) = ||at + be–d||² where t = [t ₁ t ₂ … t _n ]^T, d = [d ₁ d ₂ … d _n ]^T, and e = [1 1 … 1]^T are given n-dimensional column vectors, the unique point (a*,b*) where ∇φ(a*, b* ) = 0 is its global minimizer.

• 3 Two methods for predicting weekly sales for a product gave the following results

   

  Period	Method 1	Method 2
  1	80	80
  2	83	82
  3	87	84
  4	90	86
  5	81	86
  6	82	83
  7	82	82
  8	85	82

  The actual demand observed was the following:

   

  Period	Actual demand
  1	83
  2	86
  3	85
  4	89
  5	85
  6	84
  7	84
  8	83

  Determine the MAPD _i and MSE _i value of the two methods for i = 1,…,8, as well as the Tracking Signal S _i of the two methods. Based on this information, determine whether it is possible to safely use one method or the other.

• 4 Implement the SES and DES formulae on a spreadsheet program, and use it to determine the optimal parameter a in the SES method giving the best MAPD₈ error on the time-series of the previous exercise.

• 5 Implement the Levinson-Durbin algorithm.

  • (a) Test the implementation with the following values of p = 2,3,5 on the following time-series:

     

    Period i	d i
    1	20
    2	18
    3	17
    4	17
    5	19
    6	22
    7	20
    8	20
    9	19
    10	18
    11	19
    12	22
    13	20
    14	21
    15	22
    16	24
    17	23
    18	25
    19	27
    20	26

  • (b) Test the same implementation on the differentiated time-series Δd _i  = d _i –d _i−1

• 6 For the time-series d _i of Exercise 5, compute the best estimate for the value d ₂₁ using the additive model for time-series decomposition of Sect. 2.2.1. To estimate trend in the data, use the centered-moving-average method with parameter k = 6. Assume that the seasonality length s = 6.

• 7 Assume demand for some good is a martingale process where d _t+1 = d _t  + R _t where the R _t are independent random variables normally distributed, with zero mean, and variance σ _t  = √t. Which of the forecast methods discussed so far would give––when optimized in its parameters– the best results in the mean square error sense?