Smoothing

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. Many different algorithms are used in smoothing. 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.

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.

However, the terminology used across applications is mixed. For example, use of an interpolating spline fits a smooth curve exactly through the given data points and is sometimes called "smoothing".^{[citation needed]}

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.

See also

Specific smoothing and filter types

• Additive smoothing
• Butterworth filter
• Digital filter
• Kalman filter
• Kernel smoother
• Laplacian smoothing
• Stretched grid method
• Low-pass filter
• Recursive filter
• Savitzky–Golay smoothing filter
• Local regression also known as "loess" or "lowess"
• Smoothing spline
• Ramer–Douglas–Peucker algorithm
• Moving average
• Kolmogorov–Zurbenko_filter

Other

• Convolution
• Curve fitting
• Edge preserving smoothing
• Graph cuts in computer vision
• Numerical smoothing and differentiation
• Scale space
• Statistical signal processing
• Window function

External links

• Hastie, T.J. and Tibshirani, R.J. (1990), Generalized Additive Models, New York: Chapman and Hall.
• Chapter on data smoothing from the instruction manual for Wolfram Research's Mathematica