In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, several scatterplot smoothing methods are available to fit a function through the points of a
scatterplot
A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. ...
to best represent the relationship between the variables.
Scatterplots may be smoothed by fitting a line to the data points in a diagram. This line attempts to display the non-random component of the association between the variables in a 2D scatter plot. Smoothing attempts to separate the non-random behaviour in the data from the random fluctuations, removing or reducing these fluctuations, and allows prediction of the response based value of the
explanatory variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
.
[Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', OUP. (entry for "smoothing")]
Smoothing is normally accomplished by using any one of the techniques mentioned below.
* A straight line (
simple linear regression
In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the ''x'' and ...
)
* A
quadratic or a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
curve
*
Local regression
Local regression or local polynomial regression, also known as moving regression, is a generalization of the moving average and polynomial regression.
Its most common methods, initially developed for scatterplot smoothing, are LOESS (locally es ...
*
Smoothing spline
Smoothing splines are function estimates, \hat f(x), obtained from a set of noisy observations y_i of the target f(x_i), in order to balance a measure of goodness of fit of \hat f(x_i) to y_i with a derivative based measure of the smoothness of \ ...
s
The smoothing curve is chosen so as to provide the best fit in some sense, often defined as the fit that results in the minimum
sum of the squared errors (a
least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
criterion).
See also
*
Additive model
*
Generalized additive model In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear response variable depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth func ...
*
Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function (mathematics), function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena ...
References
{{DEFAULTSORT:Scatterplot Smoothing
Regression analysis
Statistical charts and diagrams