Editing Moving average (section)

==Moving median==
From a statistical point of view, the moving average, when used to estimate the underlying trend in a time series, is susceptible to rare events such as rapid shocks or other anomalies. A more robust estimate of the trend is the '''simple moving median''' over ''n'' time points:
<math display="block">\widetilde{p}_\text{SM} = \text{Median}( p_M, p_{M-1}, \ldots, p_{M-n+1} )</math>
where the [[median]] is found by, for example, sorting the values inside the brackets and finding the value in the middle.  For larger values of ''n'', the median can be efficiently computed by updating an [[Skip list#Indexable skiplist|indexable skiplist]].<ref>{{Cite web | url=http://code.activestate.com/recipes/576930/ |title = Efficient Running Median using an Indexable Skiplist « Python recipes « ActiveState Code}}</ref>

Statistically, the moving average is optimal for recovering the underlying trend of the time series when the fluctuations about the trend are [[normal distribution|normally distributed]]. However, the normal distribution does not place high probability on very large deviations from the trend which explains why such deviations will have a disproportionately large effect on the trend estimate. It can be shown that if the fluctuations are instead assumed to be [[Laplace distribution|Laplace distributed]], then the moving median is statistically optimal.<ref>G.R. Arce, "Nonlinear Signal Processing: A Statistical Approach", Wiley:New Jersey, US, 2005.</ref> For a given variance, the Laplace distribution places higher probability on rare events than does the normal, which explains why the moving median tolerates shocks better than the moving mean.

When the simple moving median above is central, the smoothing is identical to the [[median filter]] which has applications in, for example, image signal processing. The Moving Median is a more robust alternative to the Moving Average when it comes to estimating the underlying trend in a time series. While the Moving Average is optimal for recovering the trend if the fluctuations around the trend are normally distributed, it is susceptible to the impact of rare events such as rapid shocks or anomalies. In contrast, the Moving Median, which is found by sorting the values inside the time window and finding the value in the middle, is more resistant to the impact of such rare events. This is because, for a given variance, the Laplace distribution, which the Moving Median assumes, places higher probability on rare events than the normal distribution that the Moving Average assumes. As a result, the Moving Median provides a more reliable and stable estimate of the underlying trend even when the time series is affected by large deviations from the trend. Additionally, the Moving Median smoothing is identical to the Median Filter, which has various applications in image signal processing.