TIME SERIES DECOMPOSITION

The general idea behind time series models is that the data-generating process consists of two components:

1. A pattern, which is the “regular” component and may be quite variable over time, but in a fairly predictable way
2. An error, which is the “irregular” and unpredictable component⁶

Some smoothing mechanism should be designed in order to filter errors and expose the underlying pattern. The simplest decomposition scheme we may adopt is

where  is a random variable with expected value 0. Additional assumptions, for the sake of statistical tractability, may concern independence and normality of these random shocks. Faced with such a simple decomposition, we could just take an average of past observations in order to come up with an estimate of μ, which is the expected value of Y_t. In real life, we do not have constant expected values, as market conditions may change over time. Hence, we could postulate a model like

Here μ_t could be a “slowly” varying function of time, associated with market cycles, on which fast swings due to noise are superimposed. Alternatively, we could think of μ_t as a stochastic process whose sample paths are piecewise constant. In other words, every now and then a shock arrives, possibly due to the introduction or withdrawal of similar products by competitors (or ourselves), and μ_t jumps to a new value. Hence, we should not only filter random errors but also track the variation of μ_t, adapting our estimate. Equation (11.8) may look like a sort of catchall, which could fit any process. However, it is typically much better to try and see more structure in the demand process, discerning predictable and unpredictable variability. The elements of predictable variability that we will be concerned.

• Trend, denoted by Θ_t
• Seasonality, denoted by S_t

Trend is quite familiar from linear regression, and, strictly speaking, it is related to the slope of a line. Such a line represent a tendency of demand, on which random shocks and possibly seasonality are superimposed. In the following text, the intercept and slope of this line will be denoted as B (the level or baseline demand) and T (the trend in the strict sense), respectively; hence, the equation of the line is

Seasonality is a form of predictable variability with precise regularity in timing. For instance, we know that in Italy more ice cream is consumed in summer than in winter, and that demand at a large retail store is larger on Saturdays than on Tuesdays. It is important to separate seasonality from the trend component of a time series; high sales of ice cream in July–August should not be mistaken for an increasing trend. An example of time series clearly featuring trend and seasonality is displayed in Fig. 11.1. There is an evident increasing trend, to which a periodic oscillatory pattern is superimposed. There is a little noise component, but we notice that demand peaks occur regularly in time, which makes this variability at least partially predictable.

Generally speaking, time series decomposition requires the specification of a functional form

Fig. 11.1 Time series featuring trend and seasonality.

depending on each component: trend, seasonality, and noise. In principle, we may come up with weird functional forms, but the two most common patterns are as follows:

• The additive decomposition
• The multiplicative decomposition

Within an additive decomposition, a possible assumption is that the noise term  is normally distributed, with zero mean. This is not a harmless assumption for a multiplicative scheme, as it results in negative values that make no sense when modeling demand. A possible alternative is assuming that  is lognormally distributed, with expected value 1. We recall from section 7.7.2 that a lognormal distribution results from taking the exponential of a normal variable, and it has positive support; hence, if we transform data by taking logarithms in Eq. (11.10), we obtain an additive decomposition with normal noise terms. It is also important to realize that the seasonal component is periodic

for some seasonality cycle s. If time buckets correspond to months, yearly seasonality corresponds to s = 12; if we partition a year in four quarters, then s = 4. The seasonal factor S_t in a multiplicative scheme tells us the extent to which the demand in time bucket t exceeds its long-run average, corrected by trend. For instance, if there is no trend, a multiplicative factor S_t = 1.2 tells that demand in time bucket t is 20% larger than the average. It is reasonable to normalize multiplicative seasonality factors in such a way that their average value is 1, so that they may be easily interpreted.

Example 11.2 Consider a year divided in four 3-month time buckets. We could associate S₁ with winter, S₂ with spring, S₃ with summer, and S₄ with autumn. Then, if things were completely static, we would have

Now assume that, in a multiplicative model, we have

What should the value of S₄ be? It is easy to see that we should have S₄ = 0.7.

Example 11.3 (Additive vs. multiplicative seasonal factors) The kind of seasonality we have observed in Fig. 11.1 is additive. In fact, the width of the oscillatory pattern does not change over time. We are just adding and subtracting periodic factors. In Fig. 11.2 we can still see a trend with the superimposition of a seasonal pattern. However, we notice that the width of the oscillations is increasing. In fact, the two following conditions have a remarkably different effect:

• Demand in August is 20 items above normal (Fig. 11.1)
• Demand in August is 20% above normal (Fig. 11.2)

In the case of an additive seasonality term, the average of the seasonal factors in a whole cycle should be 0. With a multiplicative seasonality and an increasing trend, the absolute increase in August demand, measured by items, is itself increasing, leading to wider and wider swings.

The impact of noise differs as well, if we consider additive rather than multiplicative shocks. The most sensible model can often be inferred by visual inspection of data. To fix ideas and keep the treatment to a reasonable size, we will always refer to the following hybrid scheme, because of its simplicity:

If there were no seasonality, the pattern would just be a line with intercept B and slope T. Since we assume , on the basis of estimates of parameters , , and , our demand forecast is

Here we use the notation , which is closer to the one used in linear regression, rather than F_t,h. We do so because, so far, it is not clear when and on the basis of which information the forecast is generated. Given a sample of demand observations, there are several ways to fit the parameters in this decomposition.

Fig. 11.2 Time series featuring trend and multiplicative rather than additive seasonality.

Example 11.4 Consider a sample of two seasonal cycles, consisting of four time buckets each; the seasonal cycle is s = 4, and we have demand observations Y_t, t = 1, 2, …, 8. To decompose the time series, we could solve the following optimization model:

The optimization is carried out with respect to level B, trend T, and the four seasonal factors that repeat over time. Note that seasonal factors are normalized, so that their average value is 1; furthermore, either level or trend could be negative (but not both), whereas seasonal factors are restricted to nonnegative values, otherwise negative demand could be predicted.

The optimization problem in the example looks much like a least-squares problem, but, given the nonlinearity in the constraints, it is not as easy to solve as the ordinary least-squares problems we encounter in linear regression. Alternative procedures, much simpler and heuristic in nature, have been proposed to decompose a time series. We will not pursue these approaches in detail, as they are based on the unrealistic idea that the parameters in the demand model are constant over time.

In fact, market conditions do change, and we must track variations in the underlying unknown parameters, updating their estimates when new information is received. Hence, we should modify the static decomposition scheme of Eq. (11.12) as follows:

Since we update estimates dynamically, now we make the information set, i.e., demand observations up to and including time bucket t, and the forecast horizon h explicit. In this scheme we have three time-varying estimates:

• The estimate of the level component  at the end of time bucket t; if there were no trend or seasonality, the “true” parameter B_t would be an average demand, possibly subject to slow variations over time.
• The estimate of the trend component  at the end of time bucket t, which is linked to the slope of a line. The slope can also change over time, as inversions in trend are a natural occurrence. Note that when forecasting demand Y_t+h at the end of time bucket t, the estimate of trend at time t should be multiplied by the forecast horizon h.
• The estimate of the multiplicative seasonality factor , linked to percentage up- and downswings. This is the estimate at the end of time bucket t of the seasonal factor that applies to all time buckets similar to t + h, i.e., to a specific season within the seasonal cycle. We clarify in Section 11.5.4 that such an estimate is obtained at time bucket t + h − s, where s is the length of the seasonal cycle, but we may immediately notice that at the end time bucket t we cannot using anything like , since this is a future estimate.

This dynamic decomposition leads to the heuristic time series approaches that are described in the following two sections.