Smoothing with trend

Demand may exhibit additive trend components that, in a static case, could be represented by the following demand model

images

where B is the level and T is the trend. Looking at the demand model, linear regression seems a natural candidate to estimate these two parameters. However, level and trend might change over time, suggesting the opportunity of a dynamic demand model and an adaptation of exponential smoothing. Given estimates images and images of level and trend, respectively, at the end of time bucket t, after observing Yt, the demand forecast with horizon h is

images

Note that, unlike simple moving average and exponential smoothing, the forecast horizon does play a role here, as it multiplies trend. The following adaptation of exponential smoothing is known as Holt’s linear method:

images
images

Here αβ are two smoothing coefficients in the range [0, 1]. Comparing Eq. (11.20) against simple exponential smoothing, we note that we cannot update images solely on the basis of images as the two values are not directly comparable when trend is involved. In fact, a forecast for Yt, based on the information set up to t − 1, would be

images

Then, by applying the same error correction logic of Eq. (11.17), we should adapt the estimate of level as follows:

images

which leads to Eq. (11.20)Equation (11.21) uses a smoothing coefficient β to update the estimate of trend. In this case, the new information consists of the difference between the two last estimates of level, images. We might wonder if we should not use the growth in demand, Yt − Yt−1, in updating trend estimates. Indeed, this is a possible alternative; however, the difference in demand might be oversensitive to noise, whereas the difference in estimated levels is more stable. This might be a disadvantage, however, when trend changes, since adaptation could be too slow. This can be countered by using a larger coefficient β, which is sensible since the level estimates are by themselves noise smoothers.

We may appreciate the flexibility of the exponential smoothing framework, which can be easily and intuitively adapted to different demand models. However, a careful look at Eq. (11.19) suggests a potential danger when h is large: We should not extrapolate a trend too much in the future. In particular, with a negative trend, this could even result in a negative demand forecast. The above formulas apply to an additive trend, but when demand is low and trend is negative, a model with multiplicative trend should be adopted. In such a model, the demand forecast is

images

and a decreasing trend is represented by images; this corresponds to a percentage decrease in demand, and it prevents negative forecasts. We will not give update formulas for multiplicative trend, but they can be found in the specialized literature.

Exponential smoothing with trend presents the same initialization issues that we have encountered with stationary demand. Given a fit sample, there are alternative initialization methods:

  • Linear regression on the fit sample
  • Heuristic approaches based on time series decomposition

Whatever approach we take, the minimal fit sample consists of two demand observations. From a mathematical perspective, it is impossible to estimate two parameters with just one observation; a simpler view is that with one observation there is no way to estimate a trend. Also note that, if we select this minimal fit sample, we will just fit a line passing through two points. On this fit sample, errors will be zero, because with two parameters and two observations we are able to find an exact fit. This reinforces the view that errors should not be computed on the fit sample.


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