Misplaced Pages

#56943

69-406: Mathematically, a moving average is a type of convolution . Thus in signal processing it is viewed as a low-pass finite impulse response filter. Because the boxcar function outlines its filter coefficients, it is called a boxcar filter . It is sometimes followed by downsampling . Given a series of numbers and a fixed subset size, the first element of the moving average is obtained by taking

138-423: A finite impulse response ), a finite summation may be used: When a function g N {\displaystyle g_{_{N}}} is periodic, with period N , {\displaystyle N,} then for functions, f , {\displaystyle f,} such that f ∗ g N {\displaystyle f*g_{_{N}}} exists,

207-425: A continuous or discrete variable, convolution ( f ∗ g {\displaystyle f*g} ) differs from cross-correlation ( f ⋆ g {\displaystyle f\star g} ) only in that either f ( x ) {\displaystyle f(x)} or g ( x ) {\displaystyle g(x)} is reflected about the y-axis in convolution; thus it

276-467: A derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function ). See Convolution theorem for

345-403: A derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. The resulting waveform (not shown here) is the convolution of functions f {\displaystyle f} and g {\displaystyle g} . If f ( t ) {\displaystyle f(t)}

414-413: 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

483-446: A perfectly regular cycle is rarely encountered. 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. This requires using an odd number of points in the sample window. A major drawback of the SMA

552-402: A simple equally weighted running mean is the mean over the last k {\displaystyle k} entries of a data-set containing n {\displaystyle n} entries. Let those data-points be p 1 , p 2 , … , p n {\displaystyle p_{1},p_{2},\dots ,p_{n}} . This could be closing prices of

621-777: A stock. The mean over the last k {\displaystyle k} data-points (days in this example) is denoted as SMA k {\displaystyle {\textit {SMA}}_{k}} and calculated as: SMA k = p n − k + 1 + p n − k + 2 + ⋯ + p n k = 1 k ∑ i = n − k + 1 n p i {\displaystyle {\begin{aligned}{\textit {SMA}}_{k}&={\frac {p_{n-k+1}+p_{n-k+2}+\cdots +p_{n}}{k}}\\&={\frac {1}{k}}\sum _{i=n-k+1}^{n}p_{i}\end{aligned}}} When calculating

690-436: 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. Convolution In mathematics (in particular, functional analysis ), convolution

759-1344: Is n p M + 1 − p M − ⋯ − p M − n + 1 {\displaystyle np_{M+1}-p_{M}-\dots -p_{M-n+1}} . If we denote the sum p M + ⋯ + p M − n + 1 {\displaystyle p_{M}+\dots +p_{M-n+1}} by Total M {\displaystyle {\text{Total}}_{M}} , 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

SECTION 10

#1733085102057

828-418: Is a mathematical operation on two functions ( f {\displaystyle f} and g {\displaystyle g} ) that produces a third function ( f ∗ g {\displaystyle f*g} ). The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one

897-539: Is a triangle number equal to n ( n + 1 ) 2 . {\textstyle {\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 WMA M + 1 {\displaystyle {\text{WMA}}_{M+1}} and WMA M {\displaystyle {\text{WMA}}_{M}}

966-543: Is a unit impulse , the result of this process is simply g ( t ) {\displaystyle g(t)} . Formally: One of the earliest uses of the convolution integral appeared in D'Alembert 's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. Also, an expression of the type: is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series , which

1035-409: Is a cross-correlation of g ( − x ) {\displaystyle g(-x)} and f ( x ) {\displaystyle f(x)} , or f ( − x ) {\displaystyle f(-x)} and g ( x ) {\displaystyle g(x)} .  For complex-valued functions, the cross-correlation operator is the adjoint of

1104-533: 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

1173-585: Is according to Hunter (1986). 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 (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] / 320 ⁠ and leaves samples of any quadratic or cubic polynomial unchanged. Outside

1242-433: Is also compactly supported and continuous ( Hörmander 1983 , Chapter 1). More generally, if either function (say f ) is compactly supported and the other is locally integrable , then the convolution f ∗ g is well-defined and continuous. Convolution of f and g is also well defined when both functions are locally square integrable on R and supported on an interval of the form [ a , +∞) (or both supported on [−∞,

1311-427: 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. In the financial field, and more specifically in the analyses of financial data, a weighted moving average (WMA) has

1380-402: Is defined with the following integral. The ε {\displaystyle \varepsilon } environment [ x o − ε , x o + ε ] {\displaystyle [x_{o}-\varepsilon ,x_{o}+\varepsilon ]} around x o {\displaystyle x_{o}} defines the intensity of smoothing of

1449-560: Is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted toward the left (toward − ∞ {\displaystyle -\infty } ) by the amount of | t | {\displaystyle |t|} . For functions f {\displaystyle f} , g {\displaystyle g} supported on only [ 0 , ∞ ) {\displaystyle [0,\infty )} (i.e., zero for negative arguments),

SECTION 20

#1733085102057

1518-738: Is equivalent to ( f ∗ g ) ( t − t 0 ) {\displaystyle (f*g)(t-t_{0})} , but f ( t − t 0 ) ∗ g ( t − t 0 ) {\displaystyle f(t-t_{0})*g(t-t_{0})} is in fact equivalent to ( f ∗ g ) ( t − 2 t 0 ) {\displaystyle (f*g)(t-2t_{0})} . Given two functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) and respectively,

1587-512: Is itself a complex-valued function on R , defined by: and is well-defined only if f and g decay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blow-up in g at infinity can be easily offset by sufficiently rapid decay in f . The question of existence thus may involve different conditions on f and g : If f and g are compactly supported continuous functions , then their convolution exists, and

1656-586: Is known as a circular or cyclic convolution of f {\displaystyle f} and g {\displaystyle g} . And if the periodic summation above is replaced by f T {\displaystyle f_{T}} , the operation is called a periodic convolution of f T {\displaystyle f_{T}} and g T {\displaystyle g_{T}} . For complex-valued functions f {\displaystyle f} and g {\displaystyle g} defined on

1725-684: Is known as a circular convolution of f {\displaystyle f} and g . {\displaystyle g.} When the non-zero durations of both f {\displaystyle f} and g {\displaystyle g} are limited to the interval [ 0 , N − 1 ] , {\displaystyle [0,N-1],}   f ∗ g N {\displaystyle f*g_{_{N}}} reduces to these common forms : The notation f ∗ N g {\displaystyle f*_{N}g} for cyclic convolution denotes convolution over

1794-448: Is possible to simply update cumulative average as a new value, x n + 1 {\displaystyle x_{n+1}} becomes available, using the formula CA n + 1 = x n + 1 + n ⋅ CA n n + 1 . {\displaystyle {\textit {CA}}_{n+1}={{x_{n+1}+n\cdot {\textit {CA}}_{n}} \over {n+1}}.} Thus

1863-443: Is reflected about the y-axis and shifted. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity ). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation : for real-valued functions, of

1932-484: 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. Its frequency response is a type of low-pass filter called sinc-in-frequency . The continuous moving average

2001-421: Is the bilateral Laplace transform of ( f ∗ g ) ( t ) {\displaystyle (f*g)(t)} . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for

2070-620: Is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral , Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace , Jean-Baptiste Joseph Fourier , Siméon Denis Poisson , and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in German ), composition product , superposition integral , and Carson 's integral . Yet it appears as early as 1903, though

2139-405: Is used, because 2 ⋅ ε {\displaystyle 2\cdot \varepsilon } is the interval width for the integral. 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

Moving average - Misplaced Pages Continue

2208-391: The τ {\displaystyle \tau } -axis toward the right (toward + ∞ {\displaystyle +\infty } ) by the amount of t {\displaystyle t} , while if t {\displaystyle t} is a negative value, then g ( t − τ ) {\displaystyle g(t-\tau )}

2277-881: The cyclic group of integers modulo N . Circular convolution arises most often in the context of fast convolution with a fast Fourier transform (FFT) algorithm. In many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation. For example, convolution of digit sequences is the kernel operation in multiplication of multi-digit numbers, which can therefore be efficiently implemented with transform techniques ( Knuth 1997 , §4.3.3.C; von zur Gathen & Gerhard 2003 , §8.2). Eq.1 requires N arithmetic operations per output value and N operations for N outputs. That can be significantly reduced with any of several fast algorithms. Digital signal processing and other applications typically use fast convolution algorithms to reduce

2346-459: The discrete-time Fourier transform , can be defined on a circle and convolved by periodic convolution . (See row 18 at DTFT § Properties .) A discrete convolution can be defined for functions on the set of integers . Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra , and in the design and implementation of finite impulse response filters in signal processing. Computing

2415-408: The inverse of the convolution operation is known as deconvolution . The convolution of f {\displaystyle f} and g {\displaystyle g} is written f ∗ g {\displaystyle f*g} , denoting the operator with the symbol ∗ {\displaystyle *} . It is defined as the integral of the product of

2484-433: 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. 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

2553-494: The area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount t {\displaystyle t} . As t {\displaystyle t} changes, the weighting function g ( t − τ ) {\displaystyle g(t-\tau )} emphasizes different parts of

2622-406: 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

2691-493: The convolution is also periodic and identical to : The summation on k {\displaystyle k} is called a periodic summation of the function f . {\displaystyle f.} If g N {\displaystyle g_{_{N}}} is a periodic summation of another function, g , {\displaystyle g,} then f ∗ g N {\displaystyle f*g_{_{N}}}

2760-497: The convolution is also periodic and identical to: where t 0 {\displaystyle t_{0}} is an arbitrary choice. The summation is called a periodic summation of the function f {\displaystyle f} . When g T {\displaystyle g_{T}} is a periodic summation of another function, g {\displaystyle g} , then f ∗ g T {\displaystyle f*g_{T}}

2829-533: The convolution operation ( f ∗ g ) ( t ) {\displaystyle (f*g)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle G(s)} . More precisely, Let t = u + v {\displaystyle t=u+v} , then Note that F ( s ) ⋅ G ( s ) {\displaystyle F(s)\cdot G(s)}

Moving average - Misplaced Pages Continue

2898-420: The convolution operator. Convolution has applications that include probability , statistics , acoustics , spectroscopy , signal processing and image processing , geophysics , engineering , physics , computer vision and differential equations . The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures ). For example, periodic functions , such as

2967-522: The cost of the convolution to O( N log N ) complexity. The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the circular convolution theorem . Specifically, the circular convolution of two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT. Convolutions of the type defined above are then efficiently implemented using that technique in conjunction with zero-extension and/or discarding portions of

3036-401: 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

3105-402: The current time: CA n = x 1 + ⋯ + x n n . {\displaystyle {\textit {CA}}_{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 points every time a new datum arrived. However, it

3174-434: The data. In financial applications a simple moving average ( SMA ) is the unweighted mean of the previous k {\displaystyle k} 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

3243-553: The definition is rather unfamiliar in older uses. The operation: is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913. When a function g T {\displaystyle g_{T}} is periodic, with period T {\displaystyle T} , then for functions, f {\displaystyle f} , such that f ∗ g T {\displaystyle f*g_{T}} exists,

3312-636: The graph of the function. The continuous moving average of the function f {\displaystyle f} is defined as: A larger ε > 0 {\displaystyle \varepsilon >0} smoothes the source graph of the function (blue) f {\displaystyle f} more. The animations below show the moving average as animation in dependency of different values for ε > 0 {\displaystyle \varepsilon >0} . The fraction 1 2 ⋅ ε {\displaystyle {\frac {1}{2\cdot \varepsilon }}}

3381-400: The input function f ( τ ) {\displaystyle f(\tau )} ; If t {\displaystyle t} is a positive value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted along

3450-403: The integration limits can be truncated, resulting in: For the multi-dimensional formulation of convolution, see domain of definition (below). A common engineering notational convention is: which has to be interpreted carefully to avoid confusion. For instance, f ( t ) ∗ g ( t − t 0 ) {\displaystyle f(t)*g(t-t_{0})}

3519-443: The most computationally efficient method available. Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the overlap–save method and overlap–add method . A hybrid convolution method that combines block and FIR algorithms allows for a zero input-output latency that is useful for real-time convolution computations. The convolution of two complex-valued functions on R

SECTION 50

#1733085102057

3588-509: The moving average filter can be computed quite cheaply on real time data with a FIFO / circular buffer and only 3 arithmetic steps. During the initial filling of the FIFO / circular buffer the sampling window is equal to the data-set size thus k = n {\displaystyle k=n} and the average calculation is performed as a cumulative moving average . The period selected ( k {\displaystyle k} ) depends on

3657-462: 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

3726-473: The next mean SMA k , next {\displaystyle {\textit {SMA}}_{k,{\text{next}}}} with the same sampling width k {\displaystyle k} the range from n − k + 2 {\displaystyle n-k+2} to n + 1 {\displaystyle n+1} is considered. A new value p n + 1 {\displaystyle p_{n+1}} comes into

3795-520: 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. In a moving average regression model ,

3864-2209: The number of points to get the CA each time a new datum arrives. The derivation of the cumulative average formula is straightforward. Using x 1 + ⋯ + x n = n ⋅ CA n {\displaystyle x_{1}+\cdots +x_{n}=n\cdot {\textit {CA}}_{n}} and similarly for n + 1 , it is seen that x n + 1 = ( x 1 + ⋯ + x n + 1 ) − ( x 1 + ⋯ + x n ) {\displaystyle x_{n+1}=(x_{1}+\cdots +x_{n+1})-(x_{1}+\cdots +x_{n})} x n + 1 = ( n + 1 ) ⋅ CA n + 1 − n ⋅ CA n {\displaystyle x_{n+1}=(n+1)\cdot {\textit {CA}}_{n+1}-n\cdot {\textit {CA}}_{n}} Solving this equation for CA n + 1 {\displaystyle {\textit {CA}}_{n+1}} results in CA n + 1 = x n + 1 + n ⋅ CA n n + 1 = x n + 1 + ( n + 1 − 1 ) ⋅ CA n n + 1 = ( n + 1 ) ⋅ CA n + x n + 1 − CA n n + 1 = CA n + x n + 1 − CA n n + 1 {\displaystyle {\begin{aligned}{\textit {CA}}_{n+1}&={x_{n+1}+n\cdot {\textit {CA}}_{n} \over {n+1}}\\[6pt]&={x_{n+1}+(n+1-1)\cdot {\textit {CA}}_{n} \over {n+1}}\\[6pt]&={(n+1)\cdot {\textit {CA}}_{n}+x_{n+1}-{\textit {CA}}_{n} \over {n+1}}\\[6pt]&={{\textit {CA}}_{n}}+{{x_{n+1}-{\textit {CA}}_{n}} \over {n+1}}\end{aligned}}} A weighted average

3933-562: The output. Other fast convolution algorithms, such as the Schönhage–Strassen algorithm or the Mersenne transform, use fast Fourier transforms in other rings . The Winograd method is used as an alternative to the FFT. It significantly speeds up 1D, 2D, and 3D convolution. If one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not

4002-507: 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. This formulation

4071-558: The sequences are the coefficients of two polynomials , then the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences. This is known as the Cauchy product of the coefficients of the sequences. Thus when g has finite support in the set { − M , − M + 1 , … , M − 1 , M } {\displaystyle \{-M,-M+1,\dots ,M-1,M\}} (representing, for instance,

4140-406: The set Z {\displaystyle \mathbb {Z} } of integers, the discrete convolution of f {\displaystyle f} and g {\displaystyle g} is given by: or equivalently (see commutativity ) by: The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When

4209-776: The specific meaning of weights that decrease in arithmetical progression. In an n -day WMA the latest day has weight n , the second latest n − 1 {\displaystyle 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}} The denominator

SECTION 60

#1733085102057

4278-2212: The sum and the oldest value p n − k + 1 {\displaystyle p_{n-k+1}} drops out. This simplifies the calculations by reusing the previous mean SMA k , prev {\displaystyle {\textit {SMA}}_{k,{\text{prev}}}} . SMA k , next = 1 k ∑ i = n − k + 2 n + 1 p i = 1 k ( p n − k + 2 + p n − k + 3 + ⋯ + p n + p n + 1 ⏟ ∑ i = n − k + 2 n + 1 p i + p n − k + 1 − p n − k + 1 ⏟ = 0 ) = 1 k ( p n − k + 1 + p n − k + 2 + ⋯ + p n ) ⏟ = SMA k , prev − p n − k + 1 k + p n + 1 k = SMA k , prev + 1 k ( p n + 1 − p n − k + 1 ) {\displaystyle {\begin{aligned}{\textit {SMA}}_{k,{\text{next}}}&={\frac {1}{k}}\sum _{i=n-k+2}^{n+1}p_{i}\\&={\frac {1}{k}}{\Big (}\underbrace {p_{n-k+2}+p_{n-k+3}+\dots +p_{n}+p_{n+1}} _{\sum _{i=n-k+2}^{n+1}p_{i}}+\underbrace {p_{n-k+1}-p_{n-k+1}} _{=0}{\Big )}\\&=\underbrace {{\frac {1}{k}}{\Big (}p_{n-k+1}+p_{n-k+2}+\dots +p_{n}{\Big )}} _{={\textit {SMA}}_{k,{\text{prev}}}}-{\frac {p_{n-k+1}}{k}}+{\frac {p_{n+1}}{k}}\\&={\textit {SMA}}_{k,{\text{prev}}}+{\frac {1}{k}}{\Big (}p_{n+1}-p_{n-k+1}{\Big )}\end{aligned}}} This means that

4347-493: 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

4416-431: 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

4485-426: 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. 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

4554-399: The two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform : An equivalent definition is (see commutativity ): While the symbol t {\displaystyle t} is used above, it need not represent the time domain. At each t {\displaystyle t} , the convolution formula can be described as

4623-506: 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

4692-433: 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 . 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

4761-401: 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

#56943