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A moving average series can be calculated for any time series, but is most often applied to stock Price s, Returns or trading volumes. Moving averages are used to smooth out short-term fluctuations, thus highlighting longer-term trends or cycles. The threshold between short-term and long-term depends on the application, and the parameters of the moving average will be set accordingly.

Mathematically, each of these moving averages is an example of a Convolution . These averages are also similar to the Low-pass Filter s used in Signal Processing .


SIMPLE MOVING AVERAGE


A simple moving average is the unweighted Mean of the previous ''n'' data points. For example, a 10-day simple moving average of closing price is the mean of the previous 10 days' closing prices. If those prices are p_1 to p_n then the formula is


:SMA = { p_1 + p_2 + \cdots + p_n \over n }

When calculating successive values, a new value comes into the sum and an old value drops out, meaning a full summation each time is unnecessary,

:SMA_{today} = SMA_{yesterday} + {p_1 \over n} - {p_{n+1} \over n}

In technical analysis there are various popular values for ''n'', like 10 days, 40 days, or 200 days. The period selected depends on the kind of movement one is concentrating on, such as short, intermediate, or long term. In any case moving average levels are interpreted as Support in a rising market, or Resistance in a falling market.

In all cases a moving average lags behind the latest price action, simply from the nature of its smoothing. An SMA can lag to an undesirable extent, and can be influenced too much by old prices dropping out of the average. This is addressed by giving extra weight to recent prices, as in the following two methods.


WEIGHTED MOVING AVERAGE


A weighted average is any average that has multiplying factors to give different weights to different data points. But in technical analysis a weighted moving average (WMA) has the specific meaning of weights which decrease arithmetically. In an ''n''-day WMA the latest day has weight ''n'', the second latest ''n-1'', etc, down to zero.

:WMA = { n p_1 + (n-1) p_2 + \cdots + 2 p_{n-1} + p_n \over n + (n-1) + \cdots + 2 + 1}

When calculating the WMA across successive values, it can be noted an amount p_2 to p_{n+1} drops out of the numerator each day. The WMA can thus be calculated starting with the above formula but then stepping successively with just additions and subtractions, not a full set of multiplications,

:Total_{today} = Total_{yesterday} + p_1 - p_{n+1}

:Numerator_{today} = Numerator_{yesterday} + n p_1 - Total_{yesterday}

:WMA_{today} = { Numerator_{today} \over n + (n-1) + \cdots + 2 + 1}

The denominator, incidentally, is a Triangle Number , and equals n(n+1)\over2

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


EXPONENTIAL MOVING AVERAGE


An exponential moving average (EMA), sometimes also called an '''exponentially weighted moving average''' (EWMA), applies weighting factors which decrease Exponentially . The weighting for each day decreases by a factor, or percentage, on the one before it. The graph at right shows an example of the decrease.

There are two ways to express this decrease. Firstly as a percentage whereby in say a 10% EMA, each weight is 10% less than the previous, ie. a multiplier factor f=0.90. Or secondly, and more commonly, as an ''n'' days where the factor is, by convention, f=1-{2\over{n+1}}. For instance n=19 is equivalent to the 10%. In either case the formula for calculating successive days is

:EMA_{today} = {p_1 + f imes EMA_{yesterday} \over 1 + f}

The full formula for a given day is


:EMA = { p_1 + f p_2 + f^2 p_3 + f^3 p_4 + \cdots \over 1 + f + f^2 + f^3 + \cdots }

In theory this is an Infinite Sum , but because ''f'' is less than 1, the terms become smaller and smaller, and can be ignored once small enough. The denominator approaches 1 \over {1-f}, and that value can be used instead of adding up the powers, provided one is using enough terms that the omitted portion is negligible.


The ''n'' in an ''n''-day EMA only specifies the ''f'' factor. It isn't a stopping point for the calculation in the way ''n'' is in an SMA or WMA. The first ''n'' days in an EMA do represent about 86% of the total weight in the calculation though.

The Power Formula above gives a starting value for a particular day, after which the successive days formula shown first can be applied. It's interesting to note one can get the same result as the powering by applying the successive formula starting from far enough back, with an initial ''EMAyesterday'' of 0.

Either way, the question of how far back to go for an initial value depends, in the worst case, on the data. If there are huge ''p'' price values in old data then they'll have an effect on the total even if their weighting is very small. If one assumes prices don't vary too wildly then just the weighting can be considered, and work out how much weight is omitted by stopping after say ''k'' terms. This is (f^k + f^{k+1} + \cdots), which is f^k imes (1 + f + f^2 \cdots), ie. a fraction f^k out of the total weight.

  • log(f) -> -2, thus log(f) -> -2/(n+1) -->

    • Thus if the aim was to have 99.9% of the weight then k={ \log 0.001 \over \log f} many terms should be used. And what's more it can be shown \log\,f approaches -2 \over n+1 as ''n'' increases, so this simplifies to (roughly) k=3.45 imes(n+1) for this example 99.9% weight.



OTHER WEIGHTINGS


Other weighting systems are used occasionally – for example, a volume weighting will weight each time period in proportion to its trading volume.

There are weighting systems designed using a combination of moving averages:
The DEMA indicator (Double Exponential Moving Average) and '''TEMA''' indicator (Triple Exponential Moving Average) are unique composites of a single exponential moving average, a double exponential moving average, and in the latter case a triple exponential moving average that provides less lag than either of the three components individually. They were originally introduced January 1994 by Patrick Mulloy.

The TRIX indicator uses a triple-EMA in its calculation. This ends up as just a certain set of weights on past data, and a set quite different to a plain EMA actually.


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