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Moving average
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For other uses, see Moving average (disambiguation).
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An example of two moving average curves
In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating series of averages of different subsets 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, and cumulative, or weighted forms (described below).
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. For example, it is often used in technical analysis of financial data, like stock prices, returns or trading volumes. 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.
Contents
[hide]
1 Simple moving average
2 Cumulative moving average
3 Weighted moving average
4 Exponential moving average
4.1 Exponentially weighted moving variance and standard deviation
4.2 Modified moving average
4.3 Application to measuring computer performance
5 Other weightings
6 Moving median
7 Moving average regression model
8 See also
9 Notes and references
10 External links
Simple moving average[edit]
In financial applications a simple moving average (SMA) is the unweighted mean of the previous n data. 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 for a n-day sample of closing price is the mean of the previous n days' closing prices. If those prices are p M , p M − 1 , … , p M − ( n − 1 ) {\displaystyle p_{M},p_{M-1},\dots ,p_{M-(n-1)}} then the formula is
p ¯ SM = p M + p M − 1 + ⋯ + p M − ( n − 1 ) n = 1 n ∑ i = 0 n − 1 p M − i {\displaystyle {\begin{aligned}{\overline {p}}_{\text{SM}}&={\frac {p_{M}+p_{M-1}+\cdots +p_{M-(n-1)}}{n}}\\&={\frac {1}{n}}\sum _{i=0}^{n-1}p_{M-i}\end{aligned}}}
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 for this simple case,
p ¯ SM = p ¯ SM , prev + p M n − p M − n n {\displaystyle {\overline {p}}_{\text{SM}}={\overline {p}}_{{\text{SM}},{\text{prev}}}+{p_{M} \over n}-{p_{M-n} \over n}}
The period selected depends on the type of movement of interest, such as short, intermediate, or long-term. In financial terms moving-average levels can be interpreted as support in a falling market, or resistance in a rising market.
If the data used are not centered around the mean, a simple moving average lags behind the latest datum point by half the sample width. An SMA can also be disproportionately influenced by old datum points dropping out or new data coming in. One characteristic of the SMA is that if the data have 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 datum 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. 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.
Cumulative moving average[edit]
In a cumulative moving average, 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 point. 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 x 1 . … , x n {\displaystyle x_{1}.\ldots ,x_{n}} up to the current time:
CMA n = x 1 + ⋯ + x n n . {\displaystyle {\textit {CMA}}_{n}={{x_{1}+\cdots +x_{n}} \over n}\,.}
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 datum points every time a new datum point arrived. However, it is possible to simply update cumulative average as a new value, x n + 1 {\displaystyle x_{n+1}} becomes available, using the formula:
CMA n + 1 = x n + 1 + n ⋅ CMA n n + 1 {\displaystyle {\textit {CMA}}_{n+1}={{x_{n+1}+n\cdot {\textit {CMA}}_{n}} \over {n+1}}}
Thus the current cumulative average for a new datum point is equal to the previous cumulative average, times n, plus the latest datum point, all divided by the number of points received so far, n+1. When all of the datum points arrive (n = N), then the cumulative average will equal the final average. It is also possible to store a running total of the datum point as well as the number of points and dividing the total by the number of datum points to get the CMA each time a new datum point arrives.
The derivation of the cumulative average formula is straightforward. Using
x 1 + ⋯ + x n = n ⋅ CMA n {\displaystyle x_{1}+\cdots +x_{n}=n\cdot {\textit {CMA}}_{n}}
and similarly for n + 1, it is seen that
x n + 1 = ( x 1 + ⋯ + x n + 1 ) − ( x 1 + ⋯ + x n ) = ( n + 1 ) ⋅ CMA n + 1 − n ⋅ CMA n {\displaystyle {\begin{aligned}x_{n+1}&=(x_{1}+\cdots +x_{n+1})-(x_{1}+\cdots +x_{n})\\[6pt]&=(n+1)\cdot {\textit {CMA}}_{n+1}-n\cdot {\textit {CMA}}_{n}\end{aligned}}}
Solving this equation for CMA n + 1 {\displaystyle {\textit {CMA}}_{n+1}} results in:
CMA n + 1 = x n + 1 + n ⋅ CMA n n + 1 = CMA n + x n + 1 − CMA n n + 1 {\displaystyle {\begin{aligned}{\textit {CMA}}_{n+1}&={x_{n+1}+n\cdot {\textit {CMA}}_{n} \over {n+1}}\\[6pt]&={{\textit {CMA}}_{n}}+{{x_{n+1}-{\textit {CMA}}_{n}} \over {n+1}}\end{aligned}}}
Weighted moving average[edit]
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 moving average is the convolution of the datum points with a fixed weighting function. One application is removing pixelisation 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.[4] In an n-day WMA the latest day has weight n, the second latest n − 1, etc., down to one.
WMA M = n p M + ( n − 1 ) p M − 1 + ⋯ + 2 p ( M − n + 2 ) + p ( M − n + 1 ) n + ( n − 1 ) + ⋯ + 2 + 1 {\displaystyle {\text{WMA}}_{M}={np_{M}+(n-1)p_{M-1}+\cdots +2p_{(M-n+2)}+p_{(M-n+1)} \over n+(n-1)+\cdots +2+1}}
WMA weights n = 15
The denominator is a triangle number equal to n ( n + 1 ) 2 . {\displaystyle {\frac {n(n+1)}{2}}.} 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 WMAM+1 and WMAM is npM+1 − pM − ⋅⋅⋅ − pM−n+1. If we denote the sum pM + ⋅⋅⋅ + pM−n+1 by TotalM, then
Total M + 1 = Total M + p M + 1 − p M − n + 1 Numerator M + 1 = Numerator M + n p M + 1 − Total M WMA M + 1 = Numerator M + 1 n + ( n − 1 ) + ⋯ + 2 + 1 {\displaystyle {\begin{aligned}{\text{Total}}_{M+1}&={\text{Total}}_{M}+p_{M+1}-p_{M-n+1}\\[3pt]{\text{Numerator}}_{M+1}&={\text{Numerator}}_{M}+np_{M+1}-{\text{Total}}_{M}\\[3pt]{\text{WMA}}_{M+1}&={{\text{Numerator}}_{M+1} \over n+(n-1)+\cdots +2+1}\end{aligned}}}
The graph at the right shows how the weights decrease, from highest weight for the most recent datum points, down to zero. It can be compared to the weights in the exponential moving average which follows.
Exponential moving average[edit]
Further information: EWMA chart and Exponential smoothing
EMA weights N = 15
An exponential moving average (EMA), also known as an exponentially weighted moving average (EWMA),[5] is a type of 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.
The EMA for a series Y may be calculated recursively:
S t = { Y 1 , t = 1 α ⋅ Y t + ( 1 − α ) ⋅ S t − 1 , t > 1 {\displaystyle S_{t}={\begin{cases}Y_{1},&t=1\\\alpha \cdot Y_{t}+(1-\alpha )\cdot S_{t-1},&t>1\end{cases}}}
Where:
The coefficient α represents the degree of weighting decrease, a constant smoothing factor between 0 and 1. A higher α discounts older observations faster.
Yt is the value at a time period t.
St is the value of the EMA at any time period t.
S1 may be initialized in a number of different ways, most commonly by setting S1 to Y1 as shown above, though other techniques exist, such as setting S1 to an average of the first 4 or 5 observations. The importance of the S1 initialisations effect on the resultant moving average depends on α; smaller α values make the choice of S1 relatively more important than larger α values, since a higher α discounts older observations faster.
Whatever is done for S1 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. For example, if 3% accuracy is required, initialising with Y1 and taking data after five time constants (defined above) will ensure that the calculation has converged to within 3% (only <3% of Y1 will remain in 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 (1986).[6] By repeated application of this formula for different times, we can eventually write St as a weighted sum of the datum points Yt, as:
S t = α [ Y t − 1 + ( 1 − α ) Y t − 2 + ( 1 − α ) 2 Y t − 3 + ⋯ ⋯ + ( 1 − α ) k Y t − ( k + 1 ) ] + ( 1 − α ) k + 1 S t − ( k + 1 ) {\displaystyle {\begin{aligned}S_{t}=\alpha &\left[Y_{t-1}+(1-\alpha )Y_{t-2}+(1-\alpha )^{2}Y_{t-3}+\cdots \right.\\[6pt]&\left.\cdots +(1-\alpha )^{k}Y_{t-(k+1)}\right]+(1-\alpha )^{k+1}S_{t-(k+1)}\end{aligned}}}
for any suitable k ∈ {0, 1, 2, …} The weight of the general datum point Y t − i {\displaystyle Y_{t-i}} is α ( 1 − α ) i − 1 {\displaystyle \alpha \left(1-\alpha \right)^{i-1}} .
An alternate approach by Roberts (1959) uses Yt in lieu of Yt−1:[7]
S t , alternate = α ⋅ Y t + ( 1 − α ) ⋅ S t − 1 {\displaystyle S_{t,{\text{ alternate}}}=\alpha \cdot Y_{t}+(1-\alpha )\cdot S_{t-1}}
This formula can also be expressed in technical analysis terms as follows, showing how the EMA steps towards the latest datum point, but only by a proportion of the difference (each time):
EMA today = EMA yesterday + α [ price today − EMA yesterday ] {\displaystyle {\text{EMA}}_{\text{today}}={\text{EMA}}_{\text{yesterday}}+\alpha \left[{\text{price}}_{\text{today}}-{\text{EMA}}_{\text{yesterday}}\right]}
Expanding out EMA yesterday {\displaystyle {\text{EMA}}_{\text{yesterday}}} each time results in the following power series, showing how the weighting factor on each datum point p1, p2, etc., decreases exponentially:
EMA today = α [ p 1 + ( 1 − α ) p 2 + ( 1 − α ) 2 p 3 + ( 1 − α ) 3 p 4 + ⋯ ] {\displaystyle {\text{EMA}}_{\text{today}}={\alpha \left[p_{1}+(1-\alpha )p_{2}+(1-\alpha )^{2}p_{3}+(1-\alpha )^{3}p_{4}+\cdots \right]}}
where
p 1 {\displaystyle p_{1}} is price today {\displaystyle {\text{price}}_{\text{today}}}
p 2 {\displaystyle p_{2}} is price yesterday {\displaystyle {\text{price}}_{\text{yesterday}}}
and so on
EMA today = p 1 + ( 1 − α ) p 2 + ( 1 − α ) 2 p 3 + ( 1 − α ) 3 p 4 + ⋯ 1 + ( 1 − α ) + ( 1 − α ) 2 + ( 1 − α ) 3 + ⋯ , {\displaystyle {\text{EMA}}_{\text{today}}={\frac {p_{1}+(1-\alpha )p_{2}+(1-\alpha )^{2}p_{3}+(1-\alpha )^{3}p_{4}+\cdots }{1+(1-\alpha )+(1-\alpha )^{2}+(1-\alpha )^{3}+\cdots }},}
since 1 / α = 1 + ( 1 − α ) + ( 1 − α ) 2 + ⋯ {\displaystyle 1/\alpha =1+(1-\alpha )+(1-\alpha )^{2}+\cdots } .
This is an infinite sum with decreasing terms.
The N periods in an N-day EMA only specify the α factor. N is not a stopping point for the calculation in the way it is in an SMA or WMA. For sufficiently large N, the first N datum points in an EMA represent about 86% of the total weight in the calculation when α = 2 / ( N + 1 ) {\displaystyle \alpha =2/(N+1)} :[8]
α [ 1 + ( 1 − α ) + ( 1 − α ) 2 + ⋯ + ( 1 − α ) N ] α [ 1 + ( 1 − α ) + ( 1 − α ) 2 + ⋯ ] ) = 1 − ( 1 − 2 N + 1 ) N + 1 {\displaystyle {\frac {\alpha \left[1+(1-\alpha )+(1-\alpha )^{2}+\cdots +(1-\alpha )^{N}\right]}{\alpha \left[1+(1-\alpha )+(1-\alpha )^{2}+\cdots \right])}}=1-{\left(1-{2 \over N+1}\right)}^{N+1}}
i.e. lim N → ∞ [ 1 − ( 1 − 2 N + 1 ) N + 1 ] {\displaystyle \lim _{N\to \infty }\left[1-{\left(1-{2 \over N+1}\right)}^{N+1}\right]} simplified,[9] tends to 1 − e − 2 ≈ 0.8647 {\displaystyle 1-e^{-2}\approx 0.8647} .
The above discussion requires a bit of clarification. The sum of the weights of all the terms (i.e., infinite number of terms) in an exponential moving average is 1. The sum of the weights of N {\displaystyle N} terms is 1 − ( 1 − α ) N + 1 {\displaystyle 1-(1-\alpha )^{N+1}} . Both of these sums can be derived by using the formula for the sum of a geometric series. The weight omitted after N {\displaystyle N} terms is given by subtracting this from 1, and you get 1 − [ 1 − ( 1 − α ) N + 1 ] = ( 1 − α ) N + 1 {\displaystyle 1-\left[1-(1-\alpha )^{N+1}\right]=(1-\alpha )^{N+1}} (this is essentially the formula given below for the weight omitted). Note that there is no "accepted" value that should be chosen for α {\displaystyle \alpha } although there are some recommended values based on the application. In the above discussion, we have substituted a commonly used value for α = 2 / ( N + 1 ) {\displaystyle \alpha =2/(N+1)} in the formula for the weight of N {\displaystyle N} terms. Again, it is just a recommendation—not a requirement. If you make this substitution, and you make use of[10] lim n → ∞ ( 1 + a n ) n = e a {\displaystyle \lim _{n\to \infty }\left(1+{a \over n}\right)^{n}=e^{a}} , then you get the 0.8647 approximation. Intuitively, what this is telling us is that the weight after N {\displaystyle N} terms of an `` N {\displaystyle N} -period" exponential moving average converges to 0.8647.
The power formula above gives a starting value for a particular day, after which the successive days formula shown first can be applied. 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 weight omitted by stopping after k terms is
α [ ( 1 − α ) k + ( 1 − α ) k + 1 + ( 1 − α ) k + 2 + ⋯ ] , {\displaystyle \alpha \left[(1-\alpha )^{k}+(1-\alpha )^{k+1}+(1-\alpha )^{k+2}+\cdots \right],}
which is
α ( 1 − α ) k [ 1 + ( 1 − α ) + ( 1 − α ) 2 + ⋯ ] , {\displaystyle \alpha (1-\alpha )^{k}\left[1+(1-\alpha )+(1-\alpha )^{2}+\cdots \right],}
i.e. a fraction
weight omitted by stopping after k terms total weight = α [ ( 1 − α ) k + ( 1 − α ) k + 1 + ( 1 − α ) k + 2 + ⋯ ] α [ 1 + ( 1 − α ) + ( 1 − α ) 2 + ⋯ ] = α ( 1 − α ) k 1 1 − ( 1 − α ) α 1 − ( 1 − α ) = ( 1 − α ) k {\displaystyle {\begin{aligned}&{\frac {{\text{weight omitted by stopping after }}k{\text{ terms}}}{\text{total weight}}}\\[6pt]={}&{\frac {\alpha \left[(1-\alpha )^{k}+(1-\alpha )^{k+1}+(1-\alpha )^{k+2}+\cdots \right]}{\alpha \left[1+(1-\alpha )+(1-\alpha )^{2}+\cdots \right]}}\\[6pt]={}&{\frac {\alpha (1-\alpha )^{k}{\frac {1}{1-(1-\alpha )}}}{\frac {\alpha }{1-(1-\alpha )}}}\\[6pt]={}&(1-\alpha )^{k}\end{aligned}}}
out of the total weight.
For example, to have 99.9% of the weight, set above ratio equal to 0.1% and solve for k:
k = log  ( 0.001 ) log  ( 1 − α ) {\displaystyle k={\log(0.001) \over \log(1-\alpha )}}
terms should be used. Since log ( 1 − α ) {\displaystyle \log \,(1-\alpha )} approaches − 2 N + 1 {\displaystyle -2 \over N+1} as N increases,[11] this simplifies to approximately[12]
k ≈ 3.45 ( N + 1 ) {\displaystyle k\approx 3.45(N+1)\,}
for this example (99.9% weight).
Exponentially weighted moving variance and standard deviation[edit]
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. The starting values are EMA 1 = x 1 {\displaystyle {\text{EMA}}_{1}=x_{1}} and EMVar 1 = 0 {\displaystyle {\text{EMVar}}_{1}=0} , and we then compute the subsequent values using:[13]
δ = x i − EMA i − 1 EMA i = EMA i − 1 + α ⋅ δ EMVar i = ( 1 − α ) ⋅ ( EMVar i − 1 + α ⋅ δ 2 ) {\displaystyle {\begin{aligned}\delta &=x_{i}-{\text{EMA}}_{i-1}\\{\text{EMA}}_{i}&={\text{EMA}}_{i-1}+\alpha \cdot \delta \\{\text{EMVar}}_{i}&=\left(1-\alpha \right)\cdot \left({\text{EMVar}}_{i-1}+\alpha \cdot \delta ^{2}\right)\end{aligned}}}
From this, the exponentially weighted moving standard deviation can be computed as EWSD i = EMVar i {\displaystyle {\text{EWSD}}_{i}={\sqrt {{\text{EMVar}}_{i}}}} . 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.
Modified moving average[edit]
A modified moving average (MMA), running moving average (RMA), or smoothed moving average (SMMA) is defined as:
p ¯ M M , today = ( N − 1 ) p ¯ M M , yesterday + p N {\displaystyle {\overline {p}}_{MM,{\text{today}}}={\frac {(N-1){\overline {p}}_{MM,{\text{yesterday}}}+p}{N}}}
In short, this is an exponential moving average, with α = 1 / N {\displaystyle \alpha =1/N} .
Application to measuring computer performance[edit]
Some computer performance metrics, e.g. the average process queue length, or the average CPU utilization, use a form of exponential moving average.
S n = α ( t n − t n − 1 ) Y n + [ 1 − α ( t n − t n − 1 ) ] S n − 1 . {\displaystyle S_{n}=\alpha (t_{n}-t_{n-1})Y_{n}+\left[1-\alpha (t_{n}-t_{n-1})\right]S_{n-1}.}
Here α is defined as a function of time between two readings. An example of a coefficient giving bigger weight to the current reading, and smaller weight to the older readings is
α ( t n − t n − 1 ) = 1 − exp  ( − t n − t n − 1 W ⋅ 60 ) {\displaystyle \alpha (t_{n}-t_{n-1})=1-\exp \left({-{\frac {t_{n}-t_{n-1}}{W\cdot 60}}}\right)}
where exp() is the exponential function, time for readings tn is expressed in seconds, and W is the period of time in minutes over which the reading is said to be averaged (the mean lifetime of each reading in the average). Given the above definition of α, the moving average can be expressed as
S n = [ 1 − exp  ( − t n − t n − 1 W ⋅ 60 ) ] Y n + exp  ( − t n − t n − 1 W ⋅ 60 ) S n − 1 {\displaystyle S_{n}=\left[1-\exp \left(-{{t_{n}-t_{n-1}} \over {W\cdot 60}}\right)\right]Y_{n}+\exp \left(-{{t_{n}-t_{n-1}} \over {W\cdot 60}}\right)S_{n-1}}
For example, a 15-minute average L of a process queue length Q, measured every 5 seconds (time difference is 5 seconds), is computed as
L n = [ 1 − exp  ( − 5 15 ⋅ 60 ) ] Q n + e − 5 15 ⋅ 60 L n − 1 = [ 1 − exp  ( − 1 180 ) ] Q n + e − 1 180 L n − 1 = Q n + e − 1 180 ( L n − 1 − Q n ) {\displaystyle {\begin{aligned}L_{n}&=\left[1-\exp \left({-{\frac {5}{15\cdot 60}}}\right)\right]Q_{n}+e^{-{\frac {5}{15\cdot 60}}}L_{n-1}\\[6pt]&=\left[1-\exp \left({-{\frac {1}{180}}}\right)\right]Q_{n}+e^{-{\frac {1}{180}}}L_{n-1}\\[6pt]&=Q_{n}+e^{-{\frac {1}{180}}}\left(L_{n-1}-Q_{n}\right)\end{aligned}}}
Other weightings[edit]
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 15-Point Moving Average[14] (a central moving average). The symmetric weight coefficients are −3, −6, −5, 3, 21, 46, 67, 74, 67, 46, 21, 3, −5, −6, −3.
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).
Moving median[edit]
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:
p ~ SM = Median ( p M , p M − 1 , … , p M − n + 1 ) {\displaystyle {\widetilde {p}}_{\text{SM}}={\text{Median}}(p_{M},p_{M-1},\ldots ,p_{M-n+1})}
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 indexable skiplist.[15]
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.[16] 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.
Moving average regression model[edit]
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.
See also[edit]
Wikimedia Commons has media related to Moving_averages.
Exponential smoothing
Moving average convergence/divergence indicator
Window function
Moving average crossover
Rising moving average
Running total
Local regression
Kernel smoothing
Moving least squares
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (February 2010) (Learn how and when to remove this template message)
Notes and references[edit]
Jump up ^ Hydrologic Variability of the Cosumnes River Floodplain (Booth et al., San Francisco Estuary and Watershed Science, Volume 4, Issue 2, 2006)
Jump up ^ Statistical Analysis, Ya-lun Chou, Holt International, 1975, ISBN 0-03-089422-0, section 17.9.
Jump up ^ The derivation and properties of the simple central moving average are given in full at Savitzky–Golay filter
Jump up ^ "Weighted Moving Averages: The Basics". Investopedia.
Jump up ^ http://lorien.ncl.ac.uk/ming/filter/filewma.htm
Jump up ^ NIST/SEMATECH e-Handbook of Statistical Methods: Single Exponential Smoothing at the National Institute of Standards and Technology
Jump up ^ NIST/SEMATECH e-Handbook of Statistical Methods: EWMA Control Charts at the National Institute of Standards and Technology
Jump up ^ The denominator on the left-hand side should be unity, and the numerator will become the right-hand side (geometric series), α [ 1 − ( 1 − α ) N + 1 1 − ( 1 − α ) ] {\displaystyle \alpha \left[{1-(1-\alpha )^{N+1} \over 1-(1-\alpha )}\right]} .
Jump up ^ Because (1 + x/n)n tends to the limit ex for large n.
Jump up ^ See the following link for a proof.
Jump up ^ It means α → 0 {\displaystyle \alpha \to 0} , and the Taylor series of log  ( 1 − α ) = − α − α 2 / 2 − ⋯ {\displaystyle \log(1-\alpha )=-\alpha -\alpha ^{2}/2-\cdots } is equivalent to − α {\displaystyle -\alpha } .
Jump up ^ loge(0.001) / 2 = −3.45
Jump up ^ Schubert, E.; Weiler, M.; Kriegel, H. P. (2014). SigniTrend: scalable detection of emerging topics in textual streams by hashed significance thresholds. Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '14. pp. 871–880. ISBN 9781450329569. doi:10.1145/2623330.2623740.
Jump up ^ Spencer's 15-Point Moving Average — from Wolfram MathWorld
Jump up ^ http://code.activestate.com/recipes/576930/
Jump up ^ G.R. Arce, "Nonlinear Signal Processing: A Statistical Approach", Wiley:New Jersey, USA, 2005.
External links[edit]
Download Excel file to calculate SMA & EMA
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Miramar, Florida
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Miramar, Florida
City
City of Miramar
Motto: Beauty and Progress
Coordinates: 25°58′44″N 80°16′57″W / 25.97889°N 80.28250°W / 25.97889; -80.28250Coordinates: 25°58′44″N 80°16′57″W / 25.97889°N 80.28250°W / 25.97889; -80.28250
Country
United States
State
Florida
County
Broward
Incorporated
May 26, 1955
Government
• Type
Commission-Manager
• Mayor
Wayne M. Messam
• Vice Mayor
Winston F. Barnes,
• Commissioners
Maxwell B. Chambers, Yvette Colbourne and Darline B. Riggs
• City Manager
Kathleen Woods-Richardson
• City Clerk
Denise A. Gibbs
Area[1]
• City
31.28 sq mi (81.01 km2)
• Land
29.38 sq mi (76.10 km2)
• Water
1.90 sq mi (4.92 km2) 5.66%
Elevation
9 ft (2 m)
Population (2010)
• City
122,041
• Estimate (2016)[2]
138,449
• Rank
194th
• Density
4,712.36/sq mi (1,819.42/km2)
• Metro
5,762,717 (8th)
• Metro density
4,134.1/sq mi (1,596.2/km2)
Time zone
EST (UTC-5)
• Summer (DST)
EDT (UTC-4)
ZIP codes
33023, 33025, 33027, 33029
Area code(s)
754, 954
FIPS code
12-45975[3]
GNIS feature ID
0286974[4]
Website
www.miramarfl.gov
Miramar is a city in Broward County, Florida, United States. As of the 2010 census, the population was 122,041.[5] It is a principal city of the Miami metropolitan area, which was home to an estimated 6,012,331 people at the 2015 census.
Contents
[hide]
1 History
2 Geography
3 Demographics
4 Economy
4.1 Top employers
5 Education
5.1 Elementary schools
5.2 Public middle schools
5.3 Public high schools
5.4 Private schools
5.5 Trade schools
5.6 Higher education
6 Media
7 Notable residents
8 References
9 External links
History[edit]
Miramar was founded by A.L. Mailman to serve as a "bedroom community" for nearby Miami and Fort Lauderdale. Mr. Mailman bought the original property he was to develop from H.D. Perry, Sr. in 1953. He built 56 homes on the property that were inexpensive homes of concrete and flat roofs. These homes sold quickly because of the low cost of both the homes and the land, and the city of Miramar came into being.
The city was incorporated on May 26, 1955 and named for the Cuban city outside of Havana where Mailman had a summer home (Miramar translates to "Look at the Sea" in Spanish).[6] At the time of incorporation, the city had a population of less than two hundred people. With approximately 2.9 square miles land area, Miramar's original city boundaries were Southwest 64 Avenue on the east, University Drive on the west, the Dade County line on the south, and Pembroke Road on the north. On June 20, 1955, the city's first mayor (Robert Gordon) and city council were sworn in, all of which were appointed by the governor and served until January 1959, at which time the first municipal election was held. The city seal is inscribed with the motto "Beauty and Progress".
Mr. H.D. Perry Sr.'s part in Miramar did not cease with selling of the land to Mr. Mailman for development. He is recognized as one of the foremost pioneers in the history of Miramar. His character and civic-activities influenced not only the lives of early residents, but continues to the present-day, as evidenced by the schools and parks in the city which bear his family's name. Many long-time residents fondly recall the community barbecues hosted by Mr. and Mrs. Perry during those early years. Others are grateful to Mr. Perry for the lessons in animal husbandry, which he conducted for the benefit of Miramar's youth so that they could learn something of farm life.
The only major roads when Miramar was developed were U.S. 441 which was a two-lane road at that time, Hallandale Beach Boulevard to Southwest 66 Terrace and Pembroke Road which was a dirt road to University Drive. There were no other transportation routes of any kind supplying access to the new community. Miramar's early city fathers advocated the philosophy of planned and controlled growth. The city adopted a Comprehensive Land Use Plan in 1972 before cities and counties were mandated to do so. This provided the framework for the orderly development of future growth. A fact not widely known is that the residential City of Miramar is one of the largest cities in size in the State of Florida. With 2/3 of land not yet developed, the decision to guide and control the city's growth remains a wise one.
Geography[edit]
According to the United States Census Bureau, the city has a total area of 31.3 square miles (81.0 km2), of which 29.5 square miles (76.5 km2) is land and 1.8 square miles (4.6 km2) (5.66%) is water.[5]
The city is bordered by the following municipalities:
To the north:
Pembroke Pines
To the northeast:
Hollywood
To the east:
West Park
To the south:
Miami-Dade County
Demographics[edit]
Historical population
Census
Pop.
%±
1960 5,485 —
1970 23,997 337.5%
1980 32,813 36.7%
1990 40,663 23.9%
2000 72,739 78.9%
2010 122,041 67.8%
Est. 2016 138,449 [2] 13.4%
U.S. Decennial Census[7]
Miramar Demographics
2010 Census
Miramar
Broward County
Florida
Total population 122,041 1,748,066 18,801,310
Population, percent change, 2000 to 2010 +67.8% +7.7% +17.6%
Population density 4,134.1/sq mi 1,444.9/sq mi 350.6/sq mi
White or Caucasian (including White Hispanic) 41.0% 63.1% 75.0%
(Non-Hispanic White or Caucasian) 11.6% 43.5% 57.9%
Black or African-American 45.7% 26.7% 16.0%
Hispanic or Latino (of any race) 36.9% 25.1% 22.5%
Asian 5.2% 3.2% 2.4%
Native American or Native Alaskan 0.2% 0.3% 0.4%
Pacific Islander or Native Hawaiian 0.0% 0.1% 0.1%
Two or more races (Multiracial) 3.7% 2.9% 2.5%
Some Other Race 4.2% 3.7% 3.6%
As of 2010, there were 40,294 households, with 7.1% being vacant. As of 2000, 48.2% had children under the age of 18 living with them, 56.0% were married couples living together, 19.1% had a female householder with no husband present, and 19.1% were non-families. 14.3% of all households were made up of individuals and 3.3% had someone living alone who was 65 years of age or older. The average household size was 3.15 and the average family size was 3.48.
In 2000, the city's population was spread out with 31.0% under the age of 18, 8.6% from 18 to 24, 35.4% from 25 to 44, 18.7% from 45 to 64, and 6.3% who were 65 years of age or older. The median age was 32 years. For every 100 females there were 90.8 males. For every 100 females age 18 and over, there were 85.0 males.
In 2000, the median income for a household in the city was $50,289, and the median income for a family was $52,952. Males had a median income of $34,145 versus $28,283 for females. The per capita income for the city was $18,462. About 7.0% of families and 8.2% of the population were below the poverty line, including 9.8% of those under age 18 and 8.5% of those age 65 or over.
As of 2000, speakers of English as their first language accounted for 60.09% of the population, while Spanish made up 29.99%, French Creole 4.37%, French 2.13%, and Tagalog as a mother tongue was 0.50% of all residents.[8]
As of 2000, Miramar had the fifth highest percentage of Jamaican residents in the US, with 15.4% of the populace,[9] the fifty-eighth highest percentage of Colombian residents in the US, at 2.51% of the city's population,[10] and the forty-eighth highest percentage of Cuban residents in the US, at 8.77% of the city's population.[11] It also had the seventy-eighth most Dominicans in the US, at 1.98%,[12] while it had the thirty-first highest percentage of Haitians (tied with West Little River,) at 6% of all residents.[13] Miramar's Trinidad and Tobago community had the twelfth highest percentage of residents, which was at 1.2% (tying with Wheatley Heights, New York and Neptune City, New Jersey.)[14]
Economy[edit]
Spirit Airlines moved to Miramar from Eastpointe, Michigan in November 1999.[15] JL Audio and Arise Virtual Solutions are also headquartered in Miramar.
The Leadership in Energy & Environment Design in Miramar houses the Federal Bureau of Investigation (FBI) Miami field office and a General Services Administration (GSA) office; named after two FBI agents who died in the 1986 FBI Miami Shootout,[16] it is a 330,000 square feet (31,000 m2) Leadership in Energy & Environment Design (LEED) facility located on a 20-acre (8.1 ha) site. The FBI field office, previously in North Miami Beach, moved to Miramar on December 8, 2014.[17] The building was dedicated on April 10, 2015.[16]
Top employers[edit]
According to Miramar's 2010 Comprehensive Annual Financial Report,[18] the top employers in the city were:
#
Employer
# of Employees
1 City of Miramar 920
2 Premier Beverage 583
3 Comcast of South Florida 570
4 Humana Medical Plans 538
5 Royal Caribbean Cruises 479
6 Quest Diagnostics 469
7 Carnival 463
8 Caremark Miramar Pharmacy 450
9 Southern Wine & Spirits 420
10 Interactive Response Technologies 375
Education[edit]
Miramar is served by Broward County Public Schools.
Elementary schools[edit]
Annabel C. Perry Elementary School
Coconut Palm Elementary School
Coral Cove Elementary School
Dolphin Bay Elementary School
Fairway Elementary School
Miramar Elementary School
Sea Castle Elementary School
Silver Lakes Elementary School
Silver Shores Elementary School
Somerset Academy Miramar Elementary Charter School
Somerset Academy Central Miramar Elementary Charter School
Sunset Lakes Elementary School
Sunshine Elementary School
St. Stephen Catholic School
Snapper Creek Elementary School
Public middle schools[edit]
New Renaissance Middle School
Glades Middle School
Somerset Academy Miramar Middle Charter School
Somerset Academy Central Miramar Middle Charter School
Public high schools[edit]
Everglades High School
Miramar High School
Somerset Academy Central Miramar High Charter School
Private schools[edit]
Saint Stephen Catholic School
Saint Bartholomew Catholic School
Shakeria Catholic School
Trade schools[edit]
DeVry University
Chamberlain College of Nursing
University of Phoenix
Strayer University
Le Cordon Bleu
Higher education[edit]
Broward College (Miramar Town Center)
Broward College (Miramar West Center)
Florida International University (FIU) (Miramar West Center)
Nova Southeastern University (Miramar Campus)
Media[edit]
Miramar is a part of the Miami-Fort Lauderdale-Hollywood media market, which is the twelfth largest radio market[19] and the seventeenth largest television market[20] in the United States. Its primary daily newspapers are the South Florida-Sun Sentinel and The Miami Herald, and their Spanish-language counterparts El Sentinel and El Nuevo Herald. WTVJ, the Miami area's NBC owned and operated station and WSCV, the Telemundo station also owned by NBC shares their studios and administrative offices in Miramar.
Notable residents[edit]
Jon Beason, NFL football player
Daniel Braverman, NFL football player
David E. Canter, sports agent
Larry Gordon, NFL football player
Tracy Howard, NFL football player
Michael Mizrachi, professional poker player
Elvis Trujillo, jockey
Johnny Depp, actor
References[edit]
Jump up ^ "2016 U.S. Gazetteer Files". United States Census Bureau. Retrieved Jul 7, 2017.
^ Jump up to: a b "Population and Housing Unit Estimates". Retrieved June 9, 2017.
Jump up ^ "American FactFinder". United States Census Bureau. Archived from the original on 2013-09-11. Retrieved 2008-01-31.
Jump up ^ "US Board on Geographic Names". United States Geological Survey. 2007-10-25. Retrieved 2008-01-31.
^ Jump up to: a b "Geographic Identifiers: 2010 Demographic Profile Data (G001): Miramar city, Florida". U.S. Census Bureau, American Factfinder. Retrieved October 22, 2013.
Jump up ^ http://articles.sun-sentinel.com/1986-08-22/news/8602190930_1_mr-gordon-jewish-federation-synagogue
Jump up ^ "Census of Population and Housing". Census.gov. Archived from the original on May 12, 2015. Retrieved June 4, 2015.
Jump up ^ "MLA Data Center results for Miramar, FL". Modern Language Association. Retrieved 2007-09-24.
Jump up ^ "Ancestry Map of Jamaican Communities". Epodunk.com. Retrieved 2007-10-25.
Jump up ^ "Ancestry Map of Colombian Communities". Epodunk.com. Retrieved 2007-10-25.
Jump up ^ "Ancestry Map of Cuban Communities". Epodunk.com. Retrieved 2007-10-25.
Jump up ^ "Ancestry Map of Dominican Communities". Epodunk.com. Retrieved 2007-10-25.
Jump up ^ "Ancestry Map of Haitian Communities". Epodunk.com. Retrieved 2007-10-25.
Jump up ^ "Ancestry Map of Trinidadian & Tobagonian Communities". Epodunk.com. Retrieved 2007-10-25.
Jump up ^ "Spirit Airlines Honored as Good Corporate Citizen of the Year; Miramar Business Appreciation 2003." Business Wire. February 13, 2003. Retrieved on December 17, 2009.[dead link]
^ Jump up to: a b "FBI and GSA Dedicate the New Benjamin P. Grogan and Jerry L. Dove Federal Building." Federal Bureau of Investigation. April 10, 2015. Retrieved on June 9, 2015. "the Federal building at 2030 Southwest 145th Avenue in Miramar, Florida, as the 'Benjamin P. Grogan and Jerry L. Dove Federal Building'"
Jump up ^ "FBI Miami Division Moves to New Location." Federal Bureau of Investigation. December 8, 2014. Retrieved on June 9, 2015.
Jump up ^ City of Miramar CAFR Archived 2011-09-29 at the Wayback Machine.
Jump up ^ "Top 50 Radio Markets Ranked By Metro 12+ Population, Spring 2005". Northwestern University Media Management Center. Archived from the original on 2007-08-07. Retrieved 2007-09-24.
Jump up ^ "Top 50 TV markets ranked by households". Northwestern University Media Management Center. Archived from the original on 2007-08-07. Retrieved 2007-09-24.
External links[edit]
City of Miramar official website
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values: ميرامار (فلوريدا) – Arabic

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