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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.
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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
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Smoothing may be used in two important ways that can aid in data analysis (1) by being able to extract more information from the data as long as the assumption of smoothing is reasonable and (2) by being able to provide analyses that are both flexible and robust. Many different
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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
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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
97:
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.
41:
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.
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one of the chief attractions of this method is that the data analyst is not required to specify a global function of any form to fit a model to the data, only to fit segments of the data.
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increased computation. Because it is so computationally intensive, LOESS would have been practically impossible to use in the era when least squares regression was being developed.
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fitting simple models to localized subsets of the data to build up a function that describes the deterministic part of the variation in the data, point by point
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performs well in a missing data environment, especially in multidimensional time and space where missing data can cause problems arising from spatial sparseness
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Used to reduce irregularities (random fluctuations) in time series data, thus providing a clearer view of the true underlying behaviour of the series.
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133:. In the case of simple series of data points (rather than a multi-dimensional image), the convolution kernel is a one-dimensional
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for finding approximate solutions of various mathematical and engineering problems that can be related to an elastic grid behavior
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Software implementations for time series, longitudinal and spatial data have been developed in the popular statistical package
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the first element of the moving average is obtained by taking the average of the initial fixed subset of the number series
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the two parameters each have clear interpretations so that it can be easily adopted by specialists in different areas
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Estimates of unknown variables it produces tend to be more accurate than those based on a single measurement alone
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Minimizes the error between the idealized and the actual filter characteristic over the range of the filter
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Also, provides an effective means of predicting future values of the time series (forecasting).
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This article is about a type of statistical technique for handling data. For other uses, see
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The estimated function is smooth, and the level of smoothness is set by a single parameter.
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Some specific smoothing and filter types, with their respective uses, pros and cons are:
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data to smooth out short-term fluctuations and highlight longer-term trends or cycles.
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Smoothing may be distinguished from the related and partially overlapping concept of
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Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on
Geometry Processing
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a smoothing technique used to make the long term trends of a time series clearer.
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has been adjusted to allow for seasonal or cyclical components of a time series
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based on the least-squares fitting of polynomials to segments of the data
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a curve composed of line segments to a similar curve with fewer points.
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meteorologists use the stretched grid method for weather prediction
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Used for continuous time realization and discrete time realization.
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engineers use the stretched grid method to design tents and other
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The operation of applying such a matrix transformation is called
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than
Chebyshev Type I/Type II and elliptic filters can achieve.
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of the observed values, the smoothing operation is known as a
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A calculation to analyze data points by creating a series of
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Uses a series of measurements observed over time, containing
149:", often used to try to capture important trends in repeated
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114:; the matrix representing the transformation is known as a
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signals with frequencies higher than the cutoff frequency.
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except that it implements a weighted smoothing function.
129:. Thus the matrix is also called convolution matrix or a
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In the case that the smoothed values can be written as a
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as the weighted average of neighboring observed data.
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to reduce or enhance certain aspects of that signal
256:requires a higher order to implement a particular
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900:. SGP '04. Nice, France: ACM. pp. 175–184.
859:"Laplacian-isoparametric grid generation scheme"
863:Journal of the Engineering Mechanics Division
936:: CS1 maint: multiple names: authors list (
422:< 3), for example for data visualization.
821:
414:most appropriate when the dimension of the
954:Hastie, T.J. and Tibshirani, R.J. (1990),
598:of different subsets of the full data set.
145:One of the most common algorithms is the "
843:Easton, V. J.; & McColl, J. H. (1997)
27:Fitting an approximating function to data
856:
36:
888:Sorkine, O., Cohen-Or, D., Lipman, Y.,
745:Numerical smoothing and differentiation
384:and other inaccuracies by estimating a
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388:over the variables for each timeframe.
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249:as flat as possible in the passband.
892:, Rössl, C., Seidel, H.-P. (2004).
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65:that attempts to capture important
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513:also known as "loess" or "lowess"
238:More linear phase response in the
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998:
834:
977:Statistical charts and diagrams
802:Smoothing Methods in Statistics
645:Savitzky–Golay smoothing filter
628:Ramer–Douglas–Peucker algorithm
407:used to estimate a real valued
69:in the data, while leaving out
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850:
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386:joint probability distribution
161:, smoothing ideas are used in
61:is to create an approximating
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958:, New York: Chapman and Hall.
857:Herrmann, Leonard R. (1976),
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770:Statistical signal processing
740:Graph cuts in computer vision
735:Filtering (signal processing)
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822:O'Haver, T. (January 2012).
799:Simonoff, Jeffrey S. (1998)
230:Type I/Type II filter or an
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956:Generalized Additive Models
894:"Laplacian Surface Editing"
847:, STEPS Statistics Glossary
777:, used in computer graphics
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463:is a positive, odd integer.
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435:Kolmogorov–Zurbenko filter
32:Smoothing (disambiguation)
29:
730:Edge preserving smoothing
470:robust and nearly optimal
805:, 2nd edition. Springer
303:Contains ripples in the
906:10.1145/1057432.1057456
85:in the following ways:
78:are used in smoothing.
875:10.1061/JMCEA3.0002158
565:lower than a selected
497:algorithm to smooth a
287:ripple (type II) than
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755:Scatterplot smoothing
676:Stretched grid method
522:polynomial regression
353:Exponential smoothing
108:linear transformation
40:
516:a generalization of
828:terpconnect.umd.edu
775:Subdivision surface
684:numerical technique
607:commonly used with
493:Laplacian smoothing
289:Butterworth filters
245:Designed to have a
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151:statistical surveys
694:tensile structures
247:frequency response
218:Butterworth filter
200:Additive smoothing
188:Overview and uses
180:
172:rectangular smooth
131:convolution kernel
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455:filter of length
447:Uses a series of
382:statistical noise
168:triangular smooth
16:(Redirected from
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987:Image processing
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760:Smoothing spline
663:Smoothing spline
567:cutoff frequency
511:Local regression
268:Chebyshev filter
206:categorical data
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155:image processing
102:Linear smoothers
51:image processing
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781:Window function
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547:Low-pass filter
443:low-pass filter
401:Kernel smoother
340:Elliptic filter
232:elliptic filter
204:used to smooth
159:computer vision
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116:smoother matrix
112:linear smoother
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557:that passes
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283:(type I) or
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170:is like the
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982:Time series
824:"Smoothing"
750:Scale space
715:Convolution
609:time series
163:scale space
127:convolution
966:Categories
787:References
765:Smoothness
571:attenuates
449:iterations
441:A type of
319:Used on a
185:Algorithm
141:Algorithms
120:hat matrix
76:algorithms
47:statistics
932:cite book
890:Alexa, M.
633:decimates
563:frequency
416:predictor
276:and more
228:Chebyshev
94:possible.
709:See also
596:averages
459:, where
418:is low (
409:function
305:passband
285:stopband
278:passband
274:roll-off
258:stopband
240:passband
224:roll-off
67:patterns
63:function
59:data set
18:Smoothed
924:1980978
561:with a
559:signals
321:sampled
226:than a
222:Slower
922:
912:
809:
555:filter
328:signal
281:ripple
135:vector
55:smooth
920:S2CID
451:of a
194:Cons
191:Pros
153:. In
71:noise
53:, to
938:link
910:ISBN
807:ISBN
569:and
520:and
157:and
90:one;
49:and
902:doi
871:doi
867:102
118:or
45:In
968::
934:}}
930:{{
918:.
908:.
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836:^
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208:.
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926:.
904::
878:.
873::
830:.
696:.
481:R
461:m
457:m
420:p
307:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
