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It significantly speeds up 1D, [16] 2D, [17] and 3D [18] convolution. If one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available. [ 19 ]
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of () into a discrete-time Fourier transform (DTFT), which generally entails a type of distortion called aliasing. Choice of an appropriate sample-rate (see Nyquist rate) is the key to minimizing that distortion.
The "moving average filter" is a trivial example of a Savitzky–Golay filter that is commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles. Each subset of the data set is fit with a straight horizontal line as opposed to a higher order polynomial.
Convolutional networks can provide an improved forecasting performance when there are multiple similar time series to learn from. [143] CNNs can also be applied to further tasks in time series analysis (e.g., time series classification [144] or quantile forecasting [145]).
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
Above: A plot of a series of 100 random numbers concealing a sine function. Below: The sine function revealed in a correlogram produced by autocorrelation. Visual comparison of convolution, cross-correlation, and autocorrelation.
The key idea is to smooth a bit, by taking the convolution of with a mollifier. The latter is just a bump function with a very small support and whose integral is 1. {\displaystyle 1.} Such a mollifier can be obtained, for example, by taking the bump function Φ {\displaystyle \Phi } from the previous section and performing appropriate scalings.