2.2.3. Exponential Moving Average

The CMA weights all the data points for each half-hour period equally regardless of how long ago they were received. For static fleet behaviour, this approach may be appropriate; however, in many cases, there are likely to be the changes in how vehicles within the fleet operate over time. The CMA would be slow to adapt to any such changes, which would be of concern for averages constructed over a significant period and thus representing large sets of data points. One method to combat this issue is to use an exponential moving average (EMA) in which the weighting of historical data points decays over time, and more recent data points have a greater influence on the current average, as shown in Equation (4), for a vehicle *v* in the half-hour period defined by *d* and *hh*.

$$EMA\_{\mathbb{H}}(\upsilon, d, h\mathbb{h}) = (a\_{\upsilon} - EMA\_{n-1}(\upsilon, d, h\mathbb{h})) \ast \left(\frac{2}{N+1}\right) + EMA\_{n-1}(\upsilon, d, h\mathbb{h}) \tag{4}$$

The parameter *N* determines the weighting given to the most recent data point, a setting of *N* = 1 applies a 100% weighting, whereas larger values of *N* reduce the weighting. In this work, a value of *N* = 20 was used, thus applying a 9.52% weighting to the most recent data point. It should be noted that this increased weighting in comparison to CMA (for averages of 10 or more data points) also has a potentially negative consequence in emphasising outliers in the data that are not representative of sustained changes in behaviour. As for CMA, a vehicle was predicted to be available for a given half-hour period if the associated average was greater than 0.5.
