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Smoothing

Fitting an approximating function to data

Summary

Fitting an approximating function to data

Simple exponential smoothing example. Raw data: mean daily temperatures at the Paris-Montsouris weather station (France) from 1960/01/01 to 1960/02/29. Smoothed data with alpha factor = 0.1.

In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise 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:

Help uncover more meaningful information from the underlying data, such as trends.
Provide analyses that are both flexible and robust.

Many different algorithms are used in smoothing, most commonly binning, kernels, and local weighted regression.

Compared to curve fitting

Smoothing may be distinguished from the related and partially overlapping concept of curve fitting in the following ways:

curve fitting often involves the use of an explicit function form for the result, whereas the immediate results from smoothing are the "smoothed" values with no later use made of a functional form if there is one;
the aim of smoothing is to give a general idea of relatively slow changes of value with little attention paid to the close matching of data values, while curve fitting concentrates on achieving as close a match as possible.
smoothing methods often have an associated tuning parameter which is used to control the extent of smoothing. Curve fitting will adjust any number of parameters of the function to obtain the 'best' fit.

Linear smoothers

In the case that the smoothed values can be written as a linear transformation of the observed values, the smoothing operation is known as a linear smoother; the matrix representing the transformation is known as a smoother matrix or hat matrix.

The operation of applying such a matrix transformation is called convolution. Thus the matrix is also called convolution matrix or a convolution kernel. In the case of simple series of data points (rather than a multi-dimensional image), the convolution kernel is a one-dimensional vector.

Algorithms

One of the most common algorithms is the "moving average", often used to try to capture important trends in repeated statistical surveys. In image processing and computer vision, smoothing ideas are used in scale 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. The triangular smooth is like the rectangular smooth except that it implements a weighted smoothing function.

Some specific smoothing and filter types, with their respective uses, pros and cons are:

Algorithm	Overview and uses	Pros	Cons
Additive smoothing	used to smooth categorical data.
Butterworth filter	Slower roll-off than a Chebyshev Type I/Type II filter or an elliptic filter
Chebyshev filter	Has a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters.
Digital filter	Used on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal
Elliptic filter
Exponential smoothing
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.
Kernel smoother		The estimated function is smooth, and the level of smoothness is set by a single parameter.
Kolmogorov–Zurbenko filter
Laplacian smoothing	algorithm to smooth a polygonal mesh.{{citation	first = Leonard R.	issue = 5
Local regression also known as "loess" or "lowess"	A generalization of moving average and polynomial regression.
Low-pass filter
Moving average
Ramer–Douglas–Peucker algorithm	decimates a curve composed of line segments to a similar curve with fewer points.
Savitzky–Golay smoothing filter
Smoothing spline
Stretched grid method

References

Irizarry, Rafael A.. "Chapter 28 Smoothing {{!}} Introduction to Data Science".
Simonoff, Jeffrey S. (1998) ''[https://www.springer.com/gp/book/9780387947167 Smoothing Methods in Statistics]'', 2nd edition. Springer {{ISBN. 978-0387947167 {{page needed. (June 2012)
O'Haver, T.. (January 2012). "Smoothing". terpconnect.umd.edu.
Easton, V. J.; & McColl, J. H. (1997)[http://www.stats.gla.ac.uk/steps/glossary/time_series.html "Time series"], STEPS Statistics Glossary

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

curve-fitting statistical-charts-and-diagrams time-series image-processing

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