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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. In technical analysis of financial data, a weighted moving average WMA has the specific meaning of weights that decrease in arithmetical progression.
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. An exponential moving average EMA , also known as an exponentially weighted moving average EWMA ,  is a first-order infinite impulse response filter that applies weighting factors which decrease exponentially.
The weighting for each older datum decreases exponentially, never reaching zero. The graph at right shows an example of the weight decrease. S 1 may be initialized in a number of different ways, most commonly by setting S 1 to Y 1 as shown above, though other techniques exist, such as setting S 1 to an average of the first 4 or 5 observations. Whatever is done for S 1 it assumes something about values prior to the available data and is necessarily in error.
In view of this, the early results should be regarded as unreliable until the iterations have had time to converge. This is sometimes called a 'spin-up' interval. One way to assess when it can be regarded as reliable is to consider the required accuracy of the result. Sometimes with very small alpha, this can mean little of the result is useful. This is analogous to the problem of using a convolution filter such as a weighted average with a very long window.
This formulation is according to Hunter This formula can also be expressed in technical analysis terms as follows, showing how the EMA steps towards the latest datum, but only by a proportion of the difference each time :. It can also be calculated recursively without introducing the error when initializing the first estimate n starts from 1 :.
This is an infinite sum with decreasing terms. The question of how far back to go for an initial value depends, in the worst case, on the data. Large price values in old data will affect the total even if their weighting is very small.
If prices have small variations then just the weighting can be considered. The power formula above gives a starting value for a particular day, after which the successive days formula shown first can be applied. The weight omitted by stopping after k terms is. Taking derivatives of both sides with respect to x gives:.
The sum of the weights of all the terms i. Both of these sums can be derived by using the formula for the sum of a geometric series. In addition to the mean, we may also be interested in the variance and in the standard deviation to evaluate the statistical significance of a deviation from the mean. EWMVar can be computed easily along with the moving average. We can then use the standard score to normalize data with respect to the moving average and variance.
This algorithm is based on Welford's algorithm for computing the variance. Some computer performance metrics, e. An example of a coefficient giving bigger weight to the current reading, and smaller weight to the older readings is. For example, a minute average L of a process queue length Q , measured every 5 seconds time difference is 5 seconds , is computed as. Other weighting systems are used occasionally — for example, in share trading a volume weighting will weight each time period in proportion to its trading volume.
A further weighting, used by actuaries, is Spencer's Point Moving Average  a central moving average. Outside the world of finance, weighted running means have many forms and applications. Each weighting function or "kernel" has its own characteristics.
In engineering and science the frequency and phase response of the filter is often of primary importance in understanding the desired and undesired distortions that a particular filter will apply to the data.
A mean does not just "smooth" the data. A mean is a form of low-pass filter. The effects of the particular filter used should be understood in order to make an appropriate choice. On this point, the French version of this article discusses the spectral effects of 3 kinds of means cumulative, exponential, Gaussian.
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:.
For larger values of n , the median can be efficiently computed by updating an indexable skiplist. Statistically, the moving average is optimal for recovering the underlying trend of the time series when the fluctuations about the trend are 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 distributed , then the moving median is statistically optimal.
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. In a moving average regression model , a variable of interest is assumed to be a weighted moving average of unobserved independent error terms; the weights in the moving average are parameters to be estimated.
Those two concepts are often confused due to their name, but while they share many similarities, they represent distinct methods and are used in very different contexts. From Wikipedia, the free encyclopedia. Type of statistical measure over subsets of a dataset. For the statistical process, see Moving average model.
For other uses, see Moving average disambiguation. Main article: Exponential smoothing. Further information: EWMA chart. Main article: Moving-average model. This article includes a list of general references , but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. February Learn how and when to remove this template message. Archived from the original on Retrieved University of Cambridge. Retrieved 19 December Accessed Outline Index.
Descriptive statistics. This would be a case of the indicator being too fast for your own good. When you want a moving average that is smoother and slower to respond to price action, then a longer period SMA is the best way to go. This would work well when looking at longer time frames, as it could give you an idea of the overall trend. Although it is slow to respond to the price action, it could possibly save you from many fake outs.
The downside is that it might delay you too long, and you might miss out on a good entry price or the trade altogether. However, it has a hard shell to protect itself, and similarly, using SMAs would help you avoid getting caught up in fakeouts. On the other hand, the hare is quick, like the EMA. With moving averages in general, the longer the time period, the slower it is to react to price movement. Because of this, the exponential moving average is typically considered more appropriate for short-term trading.
The same attributes that make the EMA more suited for short-term trading limit its effectiveness when it comes to longer-term trading. The SMA, with its slower lag, tends to smooth price action over time, making it a good trend indicator, allowing it to remain long when the price is above the SMA and short when the price is below the SMA. Many traders plot several different moving averages to give them both sides of the story.
SMA and it's speed. The EMA moves much faster and it changes its direction earlier than the SMA. The EMA gives more weight to the most recent price action which. In case we are talking of Simple Moving Average, all prices of the time period in question are equal in value. Exponential Moving Average and Linear Weighted. Simple Moving Average is represented as a line and is calculated based on the arithmetic means of the previous price values. The bigger the period (the number.