Editing Moving average (section)

==Weighted moving average==
A weighted average is an average that has multiplying factors to give different weights to data at different positions in the sample window. Mathematically, the weighted moving average is the [[convolution]] of the data with a fixed weighting function. One application is removing [[pixelization]] from a digital graphical image. This is also known as [[Anti-aliasing]] {{Citation needed|date=February 2018}}

In the financial field, and more specifically in the analyses of financial data, a '''weighted moving average''' (WMA) has the specific meaning of weights that decrease in arithmetical progression.<ref>{{cite web|title=Weighted Moving Averages: The Basics |url=http://www.investopedia.com/articles/technical/060401.asp |publisher=Investopedia}}</ref> In an ''n''-day WMA the latest day has weight ''n'', the second latest <math>n-1</math>, etc., down to one.

<math display="block">\text{WMA}_{M} = { n p_{M} + (n-1) p_{M-1} + \cdots + 2 p_{((M-n)+2)} + p_{((M-n)+1)} \over n + (n-1) + \cdots + 2 + 1}</math>

[[Image:Weighted moving average weights N=15.svg|thumb|right|WMA weights ''n'' = 15]]

The denominator is a [[triangle number]] equal to <math display="inline">\frac{n(n + 1)}{2}.</math> In the more general case the denominator will always be the sum of the individual weights.

When calculating the WMA across successive values, the difference between the numerators of <math>\text{WMA}_{M+1}</math> and <math>\text{WMA}_{M}</math> is <math>np_{M+1} - p_{M} - \dots - p_{M-n+1}</math>. If we denote the sum <math>p_{M} + \dots + p_{M-n+1}</math> by <math>\text{Total}_{M}</math>, then

<math display="block">\begin{align}
    \text{Total}_{M+1} &= \text{Total}_M + p_{M+1} - p_{M-n+1} \\[3pt]
\text{Numerator}_{M+1} &= \text{Numerator}_M + n p_{M+1} - \text{Total}_M \\[3pt]
      \text{WMA}_{M+1} &= { \text{Numerator}_{M+1} \over n + (n-1) + \cdots + 2 + 1}
\end{align}</math>

The graph at the right shows how the weights decrease, from highest weight for the most recent data, down to zero.  It can be compared to the weights in the exponential moving average which follows.