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. 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.[1] Many different algorithms are used in smoothing.

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 edit

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.[citation needed]

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 edit

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.[2]

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
  • More linear phase response in the passband than Chebyshev Type I/Type II and elliptic filters can achieve.
  • Designed to have a frequency response as flat as possible in the passband.
  • requires a higher order to implement a particular stopband specification
Chebyshev filter Has a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters.
  • Minimizes the error between the idealized and the actual filter characteristic over the range of the filter
Digital filter Used on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal
Elliptic filter
Exponential smoothing
  • Used to reduce irregularities (random fluctuations) in time series data, thus providing a clearer view of the true underlying behaviour of the series.
  • Also, provides an effective means of predicting future values of the time series (forecasting).[3]
Kalman filter Estimates of unknown variables it produces tend to be more accurate than those based on a single measurement alone
Kernel smoother
  • used to estimate a real valued function as the weighted average of neighboring observed data.
  • most appropriate when the dimension of the predictor is low (p < 3), for example for data visualization.
The estimated function is smooth, and the level of smoothness is set by a single parameter.
Kolmogorov–Zurbenko filter
  • robust and nearly optimal
  • performs well in a missing data environment, especially in multidimensional time and space where missing data can cause problems arising from spatial sparseness
  • the two parameters each have clear interpretations so that it can be easily adopted by specialists in different areas
  • Software implementations for time series, longitudinal and spatial data have been developed in the popular statistical package R, which facilitate the use of the KZ filter and its extensions in different areas.
Laplacian smoothing algorithm to smooth a polygonal mesh.[4][5]
Local regression also known as "loess" or "lowess" a generalization of moving average and polynomial regression.
  • 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
  • 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.
  • 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.
Low-pass filter
Moving average
  • A calculation to analyze data points by creating a series of averages of different subsets of the full data set.
  • a smoothing technique used to make the long term trends of a time series clearer.[3]
  • the first element of the moving average is obtained by taking the average of the initial fixed subset of the number series
  • commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles.
  • has been adjusted to allow for seasonal or cyclical components of a time series
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
Smoothing spline
Stretched grid method
  • a numerical technique for finding approximate solutions of various mathematical and engineering problems that can be related to an elastic grid behavior
  • meteorologists use the stretched grid method for weather prediction
  • engineers use the stretched grid method to design tents and other tensile structures.

See also edit

References edit

  1. ^ Simonoff, Jeffrey S. (1998) Smoothing Methods in Statistics, 2nd edition. Springer ISBN 978-0387947167 [page needed]
  2. ^ O'Haver, T. (January 2012). "Smoothing".
  3. ^ a b Easton, V. J.; & McColl, J. H. (1997)"Time series", STEPS Statistics Glossary
  4. ^ Herrmann, Leonard R. (1976), "Laplacian-isoparametric grid generation scheme", Journal of the Engineering Mechanics Division, 102 (5): 749–756, doi:10.1061/JMCEA3.0002158.
  5. ^ Sorkine, O., Cohen-Or, D., Lipman, Y., Alexa, M., Rössl, C., Seidel, H.-P. (2004). "Laplacian Surface Editing". Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. SGP '04. Nice, France: ACM. pp. 175–184. doi:10.1145/1057432.1057456. ISBN 3-905673-13-4. S2CID 1980978.{{cite book}}: CS1 maint: multiple names: authors list (link)

Further reading edit

  • Hastie, T.J. and Tibshirani, R.J. (1990), Generalized Additive Models, New York: Chapman and Hall.