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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
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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.
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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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125:. 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
475:, which facilitate the use of the KZ filter and its extensions in different areas.
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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
141:", often used to try to capture important trends in repeated
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106:; 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.
121:. 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
248:requires a higher order to implement a particular
892:. SGP '04. Nice, France: ACM. pp. 175–184.
851:"Laplacian-isoparametric grid generation scheme"
855:Journal of the Engineering Mechanics Division
8:
928:: CS1 maint: multiple names: authors list (
414:< 3), for example for data visualization.
406:most appropriate when the dimension of the
946:Hastie, T.J. and Tibshirani, R.J. (1990),
590:of different subsets of the full data set.
137:One of the most common algorithms is the "
835:Easton, V. J.; & McColl, J. H. (1997)
27:Fitting an approximating function to data
171:
880:Sorkine, O., Cohen-Or, D., Lipman, Y.,
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737:Numerical smoothing and differentiation
376:and other inaccuracies by estimating a
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380:over the variables for each timeframe.
7:
831:
829:
241:as flat as possible in the passband.
884:, Rössl, C., Seidel, H.-P. (2004).
57:that attempts to capture important
505:also known as "loess" or "lowess"
230:More linear phase response in the
25:
969:Statistical charts and diagrams
794:Smoothing Methods in Statistics
637:Savitzky–Golay smoothing filter
620:Ramer–Douglas–Peucker algorithm
399:used to estimate a real valued
61:in the data, while leaving out
378:joint probability distribution
153:, smoothing ideas are used in
53:is to create an approximating
1:
950:, New York: Chapman and Hall.
849:Herrmann, Leonard R. (1976),
762:Statistical signal processing
732:Graph cuts in computer vision
727:Filtering (signal processing)
814:O'Haver, T. (January 2012).
791:Simonoff, Jeffrey S. (1998)
222:Type I/Type II filter or an
948:Generalized Additive Models
886:"Laplacian Surface Editing"
839:, STEPS Statistics Glossary
769:, used in computer graphics
455:is a positive, odd integer.
995:
427:Kolmogorov–Zurbenko filter
32:Smoothing (disambiguation)
29:
722:Edge preserving smoothing
462:robust and nearly optimal
797:, 2nd edition. Springer
295:Contains ripples in the
898:10.1145/1057432.1057456
77:in the following ways:
70:are used in smoothing.
867:10.1061/JMCEA3.0002158
557:lower than a selected
489:algorithm to smooth a
279:ripple (type II) than
747:Scatterplot smoothing
668:Stretched grid method
514:polynomial regression
345:Exponential smoothing
100:linear transformation
508:a generalization of
820:terpconnect.umd.edu
767:Subdivision surface
676:numerical technique
599:commonly used with
485:Laplacian smoothing
281:Butterworth filters
237:Designed to have a
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143:statistical surveys
686:tensile structures
239:frequency response
210:Butterworth filter
192:Additive smoothing
180:Overview and uses
172:
164:rectangular smooth
123:convolution kernel
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447:filter of length
439:Uses a series of
374:statistical noise
160:triangular smooth
16:(Redirected from
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979:Image processing
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752:Smoothing spline
655:Smoothing spline
559:cutoff frequency
503:Local regression
260:Chebyshev filter
198:categorical data
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147:image processing
94:Linear smoothers
43:image processing
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941:Further reading
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773:Window function
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539:Low-pass filter
435:low-pass filter
393:Kernel smoother
332:Elliptic filter
224:elliptic filter
196:used to smooth
151:computer vision
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108:smoother matrix
104:linear smoother
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549:that passes
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275:(type I) or
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162:is like the
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974:Time series
816:"Smoothing"
742:Scale space
707:Convolution
601:time series
155:scale space
119:convolution
958:Categories
779:References
757:Smoothness
563:attenuates
441:iterations
433:A type of
311:Used on a
177:Algorithm
133:Algorithms
112:hat matrix
68:algorithms
39:statistics
924:cite book
882:Alexa, M.
625:decimates
555:frequency
408:predictor
268:and more
220:Chebyshev
86:possible.
701:See also
588:averages
451:, where
410:is low (
401:function
297:passband
277:stopband
270:passband
266:roll-off
250:stopband
232:passband
216:roll-off
59:patterns
55:function
51:data set
18:Smoothly
916:1980978
553:with a
551:signals
313:sampled
218:than a
214:Slower
914:
904:
801:
547:filter
320:signal
273:ripple
127:vector
47:smooth
912:S2CID
443:of a
186:Cons
183:Pros
145:. In
63:noise
45:, to
930:link
902:ISBN
799:ISBN
561:and
512:and
149:and
82:one;
41:and
894:doi
863:doi
859:102
110:or
37:In
960::
926:}}
922:{{
910:.
900:.
888:.
857:,
853:,
828:^
818:.
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545:A
516:.
493:.
315:,
283:.
200:.
129:.
114:.
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918:.
896::
870:.
865::
822:.
688:.
473:R
453:m
449:m
412:p
299:.
34:.
20:)
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