Instatistics andimage processing, tosmooth adata set is to create an approximatingfunction that attempts to capture importantpatterns in the data, while leaving outnoise or other fine-scale structures/rapid phenomena. In smoothing, the data points of a signal are modified so individual points higher than the adjacent points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased, leading to a smoother signal.
Reducing noise by smoothing may aid in data analysis in two notable ways:
Many differentalgorithms are used in smoothing, most commonlybinning,kernels, andlocal weighted regression.[1]
Smoothing may be distinguished from the related and partially overlapping concept ofcurve fitting in the following ways:
In the case that the smoothed values can be written as alinear transformation of the observed values, the smoothing operation is known as alinear smoother; the matrix representing the transformation is known as asmoother matrix orhat matrix.[citation needed]
The operation of applying such a matrix transformation is calledconvolution. Thus the matrix is also called convolution matrix or aconvolution kernel. In the case of simple series of data points (rather than a multi-dimensional image), the convolution kernel is a one-dimensionalvector.
One of the most common algorithms is the "moving average", often used to try to capture important trends in repeatedstatistical surveys. Inimage processing andcomputer vision, smoothing ideas are used inscale space representations. The simplest smoothing algorithm is the "rectangular" or "unweighted sliding-average smooth". This method replaces each point in the signal with the average of "m" adjacent points, where "m" is a positive integer called the "smooth width". Usually m is an odd number. Thetriangular smooth is like therectangular smooth except that it implements a weighted smoothing function.[3]
Some specific smoothing and filter types, with their respective uses, pros and cons are:
| Algorithm | Overview and uses | Pros | Cons |
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| Additive smoothing | used to smoothcategorical data. | ||
| Butterworth filter | Slowerroll-off than aChebyshev Type I/Type II filter or anelliptic filter |
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| Chebyshev filter | Has a steeperroll-off and morepassbandripple (type I) orstopband ripple (type II) thanButterworth filters. |
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| Digital filter | Used on asampled,discrete-timesignal to reduce or enhance certain aspects of that signal | ||
| Elliptic filter | |||
| Exponential smoothing |
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| Kalman filter |
| Estimates of unknown variables it produces tend to be more accurate than those based on a single measurement alone, when assumptions are met. | Assumes and therefore requires knowledge of how the system generating the data-points advances in time and how the measurements are acquired. |
| Kernel smoother | The estimated function is smooth, and the level of smoothness is set by a single parameter. | ||
| Kolmogorov–Zurbenko filter |
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| Laplacian smoothing | algorithm to smooth apolygonal mesh.[5][6] | ||
| Local regression also known as "loess" or "lowess" | A generalization ofmoving average andpolynomial regression. Generalizes aSavitzky–Golay smoothing filter to non-regular sampling instances. |
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| Low-pass filter |
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| Moving average |
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| Ramer–Douglas–Peucker algorithm | decimates a curve composed of line segments to a similar curve with fewer points. | ||
| Savitzky–Golay smoothing filter | Based on the least-squares fitting of polynomials to segments of the data. A specific case ofLocal regression ("loess" or "lowess") when the sampling instances are regular. | ||
| Smoothing spline | |||
| Stretched grid method |
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