e^{\ln(1-\alpha )}=e^{1-\lambda x}\}=\{1-\alpha \ln(1-\alpha )=1-\lambda x\}\\&=\{1-\alpha {\bar {q}}_{\alpha }(X)=x\}\\&=\{1-\alpha
Type of statistical measure over subsets of a dataset
For other uses, see Moving-average model and Moving average (disambiguation).
In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different selections of the full data set. It is also called a moving mean (MM)[1] or rolling mean and is a type of finite impulse response filter. Variations include: simple, cumulative, or weighted forms (described below).
A moving average filter is sometimes called a boxcar filter, especially when followed by decimation.
Given a series of numbers and a fixed subset size, the first element of the moving average is obtained by taking the average of the initial fixed subset of the number series. Then the subset is modified by "shifting forward"; that is, excluding the first number of the series and including the next value in the subset.
A moving average is commonly used with time series data to smooth out short-term fluctuations and highlight 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. It is also used in economics to examine gross domestic product, employment or other macroeconomic time series. Mathematically, a moving average is a type of convolution and so it can be viewed as an example of a low-pass filter used in signal processing. When used with non-time series data, a moving average filters higher frequency components without any specific connection to time, although typically some kind of ordering is implied. Viewed simplistically it can be regarded as smoothing the data.
In financial applications a simple moving average (SMA) is the unweighted mean of the previous data-points. However, in science and engineering, the mean is normally taken from an equal number of data on either side of a central value. This ensures that variations in the mean are aligned with the variations in the data rather than being shifted in time. An example of a simple equally weighted running mean is the mean over the last entries of a data-set containing entries. Let those data-points be . This could be closing prices of a stock. The mean over the last data-points (days in this example) is denoted as and calculated as:
When calculating the next mean with the same sampling width the range from to is considered. A new value comes into the sum and the oldest value drops out. This simplifies the calculations by reusing the previous mean .
This means that the moving average filter can be computed quite cheaply on real time data with a FIFO / circular bufferand only 3 arithmetic steps.During the initial filling of the FIFO / circular buffer the sampling window is equal to the data-set size thus and the average calculation is performed as a cumulative moving average.
The period selected () depends on the type of movement of interest, such as short, intermediate, or long-term.
If the data used are not centered around the mean, a simple moving average lags behind the latest datum by half the sample width. An SMA can also be disproportionately influenced by old data dropping out or new data coming in. One characteristic of the SMA is that if the data has a periodic fluctuation, then applying an SMA of that period will eliminate that variation (the average always containing one complete cycle). But a perfectly regular cycle is rarely encountered.[2]
For a number of applications, it is advantageous to avoid the shifting induced by using only "past" data. Hence a central moving average can be computed, using data equally spaced on either side of the point in the series where the mean is calculated.[3] This requires using an odd number of points in the sample window.
A major drawback of the SMA is that it lets through a significant amount of the signal shorter than the window length. Worse, it actually inverts it.[citation needed] This can lead to unexpected artifacts, such as peaks in the smoothed result appearing where there were troughs in the data. It also leads to the result being less smooth than expected since some of the higher frequencies are not properly removed.
Its frequency response is a type of low-pass filter called sinc-in-frequency.
In a cumulative average (CA), the data arrive in an ordered datum stream, and the user would like to get the average of all of the data up until the current datum. For example, an investor may want the average price of all of the stock transactions for a particular stock up until the current time. As each new transaction occurs, the average price at the time of the transaction can be calculated for all of the transactions up to that point using the cumulative average, typically an equally weighted average of the sequence of n values up to the current time:
The brute-force method to calculate this would be to store all of the data and calculate the sum and divide by the number of points every time a new datum arrived. However, it is possible to simply update cumulative average as a new value, becomes available, using the formula
Thus the current cumulative average for a new datum is equal to the previous cumulative average, times n, plus the latest datum, all divided by the number of points received so far, n+1. When all of the data arrive (n = N), then the cumulative average will equal the final average. It is also possible to store a running total of the data as well as the number of points and dividing the total by the number of points to get the CA each time a new datum arrives.
The derivation of the cumulative average formula is straightforward. Using
and similarly for n + 1, it is seen thatSolving this equation for results in
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.[citation needed]
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.[4] In an n-day WMA the latest day has weight n, the second latest , etc., down to one.
The denominator is a triangle number equal to 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 and is . If we denote the sum by , then
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.
Main article: Exponential smoothing
Further information: EWMA chart
An exponential moving average (EMA), also known as an exponentially weighted moving average (EWMA),[5] 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. This formulation is according to Hunter ().[6]
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[7] (a central moving average). Its symmetric weight coefficients are [−3, −6, −5, 3, 21, 46, 67, 74, 67, 46, 21, 3, −5, −6, −3], which factors as [1, 1, 1, 1]×[1, 1, 1, 1]×[1, 1, 1, 1, 1]×[−3, 3, 4, 3, −3]/ and leaves samples of any quadratic or cubic polynomial unchanged.[8][9]
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:
where the medianis 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 indexable skiplist.[10]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.[11] 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.
Main article: Moving-average model
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.
The directed Kullback–Leibler divergence in nats of ("approximating" distribution) from ('true' distribution) is given by
Among all continuous probability distributions with support[0, ∞) and mean μ, the exponential distribution with λ = 1/μ has the largest differential entropy. In other words, it is the maximum entropy probability distribution for a random variateX which is greater than or equal to zero and for which E[X] is fixed.[2]
Let X1, …, Xn be independent exponentially distributed random variables with rate parameters λ1, …, λn. Then
is also exponentially distributed, with parameterThis can be seen by considering the complementary cumulative distribution function:
The index of the variable which achieves the minimum is distributed according to the categorical distribution
A proof can be seen by letting . Then,
Note that
is not exponentially distributed, if X1, …, Xndo not all have parameter 0.[3]Let be independent and identically distributed exponential random variables with rate parameter λ. Let denote the corresponding order statistics. For , the joint moment of the order statistics and is given by
This can be seen by invoking the law of total expectation and the memoryless property:
Probability distribution
Not to be confused with the exponential family of probability distributions.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions.
The probability density function (pdf) of an exponential distribution is
Here λ > 0 is the parameter of the distribution, often called the rate parameter. The distribution is supported on the interval[0, ∞). If a random variableX has this distribution, we writeX ~ Exp(λ).
The exponential distribution exhibits infinite divisibility.
The cumulative distribution function is given by
The exponential distribution is sometimes parametrized in terms of the scale parameterβ = 1/λ, which is also the mean:
The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by
In light of the examples given below, this makes sense: if you receive phone calls at an average rate of 2 per hour, then you can expect to wait half an hour for every call.
The variance of X is given by
so the standard deviationis equal to the mean.The moments of X, for are given by
The central moments of X, for are given by
where!nis the subfactorialof nThe median of X is given by
where lnrefers to the natural logarithm. Thus the absolute differencebetween the mean and median isin accordance with the median-mean inequality.
An exponentially distributed random variable T obeys the relation
This can be seen by considering the complementary cumulative distribution function:
When T is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if T is conditioned on a failure to observe the event over some initial period of time s, the distribution of the remaining waiting time is the same as the original unconditional distribution. For example, if an event has not occurred after 30 seconds, the conditional probability that occurrence will take at least 10 more seconds is equal to the unconditional probability of observing the event more than 10 seconds after the initial time.
The exponential distribution and the geometric distribution are the only memoryless probability distributions.
The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant failure rate.
The quantile function (inverse cumulative distribution function) for Exp(λ) is
The quartiles are therefore:
And as a consequence the interquartile range is ln(3)/λ.
The conditional value at risk (CVaR) also known as the expected shortfall or superquantile for Exp(λ) is derived as follows:[1]
Main article: Buffered probability of exceedance
The buffered probability of exceedance is one minus the probability level at which the CVaR equals the threshold . It is derived as follows:[1]
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