Linear regression

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In statistics, linear regression is a method of estimating the conditional expected value of one variable y given the values of some other variable or variables x. The variable of interest, y, is conventionally called the "dependent variable". The terms "endogenous variable" and "output variable" are also used. The other variables x are called the "independent variables". The terms "exogenous variables" and "input variables" are also used. The dependent and independent variables may be scalars or vectors. If the independent variable is a vector, one speaks of multiple linear regression.

The term independent variable suggests that its value can be chosen at will, and the dependent variable is an effect, i.e., causally dependent on the independent variable, as in a stimulus-response model. Although many linear regression models are formulated as models of cause and effect, the direction of causation may just as well go the other way, or indeed there need not be any causal relation at all.

Regression, in general, is the problem of estimating a conditional expected value. Linear regression is called "linear" because the relation of the dependent to the independent variables is a linear function of some parameters. Regression models which are not a linear function of the parameters are called nonlinear regression models. A neural network is an example of a nonlinear regression model.

Still more generally, regression may be viewed as a special case of density estimation. The joint distribution of the dependent and independent variables can be constructed from the conditional distribution of the dependent variable and the marginal distribution of the independent variables. In some problems, it is convenient to work in the other direction: from the joint distribution, the conditional distribution of the dependent variable can be derived.

Contents

Historical remarks

The earliest form of linear regression was the method of least squares, which was published by Legendre in 1805, and by Gauss in 1809. The term "least squares" is from Legendre's term, moindres quarrés. However, Gauss claimed that he had known the method since 1795.

Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the sun. Euler had worked on the same problem (1748) without success. Gauss published a further development of the theory of least squares in 1821, including a version of the Gauss-Markov theorem.

The term "reversion" was used in the nineteenth century to describe a biological phenomenon, namely that the progeny of exceptional individuals tend on average to be less exceptional than their parents, and more like their more distant ancestors. Francis Galton studied this phenomenon, and applied the slightly misleading term "regression towards mediocrity" to it (parents of exceptional individuals also tend on average to be less exceptional than their children). For Galton, regression had only this biological meaning, but his work (1877, 1885) was extended by Karl Pearson and Udny Yule to a more general statistical context (1897, 1903). In the work of Pearson and Yule, the joint distribution of the dependent and independent variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925. Fisher assumed that the conditional distribution of the dependent variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.

Statement of the linear regression model

A linear regression model is typically stated in the form

<math> y = \alpha + \beta x + \varepsilon. \, <math>

The right hand side may take other forms, but generally comprises a linear combination of the parameters, here denoted α and β. The term ε represents the unpredicted or unexplained variation in the dependent variable; it is conventionally called the "error" whether it is really a measurement error or not. The error term is conventionally assumed to have expected value equal to zero, as a nonzero expected value could be absorbed into α. See also errors and residuals in statistics; the difference between an error and a residual is also dealt with below. It is also assumed that <math>\varepsilon<math> is independent of <math>x<math>.

An equivalent formulation which explicitly shows the linear regression as a model of conditional expectation is

<math> \mbox{E}(y|x) = \alpha + \beta x \, <math>

with the conditional distribution of y given x essentially the same as the distribution of the error term.

A linear regression model need not be affine, let alone linear, in the independent variables x. For example,

<math>y = \alpha + \beta x + \gamma x^2 + \varepsilon<math>

is a linear regression model, for the right-hand side is a linear combination of the parameters α, β, and γ. In this case it is useful to think of x2 as a new independent variable, formed by modifying the original variable x. Indeed, any linear combination of functions f(x), g(x), h(x), ..., is linear regression model, so long as these functions do not have any free parameters (otherwise the model is generally a nonlinear regression model). The least-squares estimates of α, β, and γ are linear in the response variable y, and nonlinear in x (they are nonlinear in x even if the γ and α terms are absent; if only β were present then doubling all observed x values would multiply the least-squares estimate of β by 1/2).

Parameter estimation

Often in linear regression problems statisticians rely on the Gauss-Markov assumptions:

  • The random errors εi have expected value 0.
  • The random errors εi are uncorrelated (this is weaker than an assumption of probabilistic independence).
  • The random errors εi are "homoscedastic", i.e., they all have the same variance.

(See also Gauss-Markov theorem. That result says that under the assumptions above, least-squares estimators are in a certain sense optimal.)

Sometimes stronger assumptions are relied on:

If xi is a vector we can take the product βxi to be a "dot product".

A statistician will usually estimate the unobservable values of the parameters α and β by the method of least squares, which consists of finding the values of a and b that minimize the sum of squares of the residuals

<math>e_i = y_i - (a + bx_i). \,<math>

Those values of a and b are the "least-squares estimates." The residuals may be regarded as estimates of the errors; see also errors and residuals in statistics.

Notice that, whereas the errors are independent, the residuals cannot be independent because the use of least-squares estimates implies that the sum of the residuals must be 0, and the dot product of the vector of residuals with the vector of x-values must be 0, i.e., we must have

<math>e_1 + \cdots + e_n = 0 \,<math>

and

<math>e_1 x_1 + \cdots + e_n x_n = 0. \,<math>

These two linear constraints imply that the vector of residuals must lie within a certain (n − 2)-dimensional subspace of Rn; hence we say that there are "n − 2 degrees of freedom for error". If one assumes the errors are normally distributed and independent, then it can be shown to follow that 1) the sum of squares of residuals

<math>e_1^2 + \cdots + e_n^2 \,<math>

is distributed as

<math>\sigma^2 \chi^2_{n - 2}, \,<math>

i.e., the sum of squares divided by the error-variance σ2, has a chi-square distribution with n − 2 degrees of freedom, and 2) the sum of squares of residuals is actually probabilistically independent of the estimates a, b of the parameters α and β.

These facts make it possible to use Student's t-distribution with n − 2 degrees of freedom (so named in honor of the pseudonymous "Student") to find confidence intervals for α and β.

Denote by capital Y the column vector whose ith entry is yi, and by capital X the n*2 matrix whose second column contains the xi as its ith entry, and whose first column contains n 1s. Let ε be the column vector containing the errors εi. Let δ and d be respectively the 2*1 column vector containing α and β and the 2*1 column vector containing the estimates a and b. Then the model can be written as

<math>Y = X \delta + \varepsilon \,<math>

where ε is normally distributed with expected value 0 (i.e., a column vector of 0s) and variance σ2 In, where In is the n*n identity matrix. The matrix Xd (where (remember) d is the vector of estimates) is then the orthogonal projection of Y onto the column space of X.

Then it can be shown that

<math>d = (X' X)^{-1}\; X' Y \,<math>

(where X′ is the transpose of X) and the sum of squares of residuals is

<math>Y' (I_n - X (X' X)^{-1} X')\, Y.<math>

The fact that the matrix X(XX)−1X′ is a symmetric idempotent matrix is incessantly relied on both in computations and in proofs of theorems. The linearity of d as a function of the vector Y, expressed above by saying d = (XX)−1 XY, is the reason why this is called "linear" regression. Nonlinear regression uses nonlinear methods of estimation.

The matrix In − X (XX)−1 X′ that appears above is a symmetric idempotent matrix of rank n − 2. Here is an example of the use of that fact in the theory of linear regression. The finite-dimensional spectral theorem of linear algebra says that any real symmetric matrix M can be diagonalized by an orthogonal matrix G, i.e., the matrix GMG is a diagonal matrix. If the matrix M is also idempotent, then the diagonal entries in GMG must be idempotent numbers. Only two real numbers are idempotent: 0 and 1. So In − X(XX) 1X′, after diagonalization, has n − 2 0s and two 1s on the diagonal. That is most of the work in showing that the sum of squares of residuals has a chi-square distribution with n−2 degrees of freedom.

Regression parameters can also be estimated by Bayesian methods. This has the advantages that

  • confidence intervals can be produced for parameter estimates without the use of asymptotic approximations,
  • prior information can be incorporated into the analysis.

Suppose that in the linear regression

<math> y = \alpha + \beta x + \varepsilon \, <math>

we know from domain knowledge that alpha can only take one of the values {−1, +1} but we do not which. We can build this information into the analysis by choosing a prior for alpha which is a discrete distribution with a probability of 0.5 on −1 and 0.5 on +1. The posterior for alpha will also be a discrete distribution on {−1, +1}, but the probability weights will change to reflect the evidence from the data.

Robust regression

A useful alternative to linear regression is robust regression in which mean absolute error is minimized instead of mean squared error as in linear regression. Robust regression is computationally much more intensive than linear regression and is somewhat more difficult to implement as well.

Robust regression usually means linear regression with robust (Huber-White) standard errors (e.g. relaxing the assumption of homoskedasticity).

An alternative approach to robust regression is to replace the normal distribution in the error term with a Student's t-distribution.

Summarizing the data

We sum the observations, the squares of the Ys and Xs and the products XY to obtain the following quantities.

<math>S_X = x_1 + x_2 + \cdots + x_n \,<math>

and <math>S_Y<math> similarly.

<math>S_{XX} = x_1^2 + x_2^2 + \cdots + x_n^2 \,<math>

and SYY similarly.

<math>S_{XY} = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n. \,<math>

Estimating beta

We use the summary statistics above to calculate b, the estimate of β.

<math>b = {n S_{XY} - S_X S_Y \over n S_{XX} - S_X S_X}. \,<math>

Estimating alpha

We use the estimate of β and the other statistics to estimate α by:

<math>a = {S_Y - b S_X \over n}. \,<math>

A consequence of this estimate is that the regression line will always pass through the "center" <math>(\bar{x},\bar{y}) = (S_X/n, S_Y/n)<math>.

Displaying the residuals

The first method of displaying the residuals use the histogram or cumulative distribution to depict the similarity (or lack thereof) to a normal distribution. Non-normality suggests that the model may not be a good summary description of the data.

We plot the residuals,

<math>e_i=y_i - a - bx_i \,<math>

against the independent variable, x. There should be no discernible trend or pattern if the model is satisfactory for these data. Some of the possible problems are:

  • Residuals increase (or decrease) as the independent variable increases —indicates mistakes in the calculations. Find the mistakes and correct them.
  • Residuals first rise and then fall (or first fall and then rise) —indicates that the appropriate model is (at least) quadratic. Adding a quadratic term (and then possibly higher) to the model may be appropriate. See polynomial regression.
  • One residual is much larger than the others and opposite in sign —suggests that there is one unusual observation which is distorting the fit.
    • Verify its value before publishing or
    • Eliminate it, document your decision to do so, and recalculate the statistics.
Studentized residuals can be used in outlier detection.
  • The vertical spread of the residuals increases as the independent variable increases (funnel-shaped plot) —indicates that the homoscedasticity assumption is violated (i.e. there is heteroskedasticity: the variability of the response depends on the value of x.

Ancillary statistics

The sum of squared deviations can be partitioned as in ANOVA to indicate what part of the dispersion of the dependent variable is explained by the independent variable.

The correlation coefficient, r, can be calculated by

<math>r = {n S_{XY} - S_X S_Y \over \sqrt{(n S_{XX} - S_X^2) (n S_{YY} - S_Y^2)}}. \,<math>

This statistic is a measure of how well a straight line describes the data. Values near zero suggest that the model is ineffective. r2 is frequently interpreted as the fraction of the variability explained by the independent variable, X.

Multiple linear regression

Linear regression can be extended to functions of two or more variables, for example

<math>Z = a X + b Y + c +\varepsilon. \,<math>

Here X and Y are independent variables. The values of the parameters a, b and c are estimated by the method of least squares, that minimize the sum of squares of the residuals

<math>S_r = \sum_{i=1}^n (a x_i + b y_i + c - z_i)^2. \,<math>

Take derivatives with respect to a, b and c in <math>S_r<math> and set equal to zero. This leads to a system of three linear equations i.e. the normal equations to find estimates of parameters a, b and c:

<math>
 \begin{bmatrix}
   \sum_{i=1}^n x_i^2 & \sum_{i=1}^n x_i y_i & \sum_{i=1}^n x_i \\
   \sum_{i=1}^n x_i y_i & \sum_{i=1}^n y_i^2 & \sum_{i=1}^n y_i \\
   \sum_{i=1}^n x_i & \sum_{i=1}^n y_i & n \\
 \end{bmatrix}

\times

 \begin{bmatrix}
   a \\
   b \\
   c
 \end{bmatrix}

=

 \begin{bmatrix}
   \sum_{i=1}^n x_i z_i \\
   \sum_{i=1}^n y_i z_i \\
   \sum_{i=1}^n z_i
 \end{bmatrix}

<math>

References

Historical

  • A.M. Legendre. Nouvelles méthodes pour la détermination des orbites des comètes (1805). "Sur la Méthode des moindres quarrés" appears as an appendix.
  • C.F. Gauss. Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum. (1809)
  • C.F. Gauss. Theoria combinationis observationum erroribus minimis obnoxiae. (1821/1823)
  • Charles Darwin. The Variation of Animals and Plants under Domestication. (1869) (Chapter XIII describes what was known about reversion in Galton's time. Darwin uses the term "reversion".)
  • Francis Galton. "Typical laws of heredity", Nature 15 (1877), 492-495, 512-514, 532-533. (Galton uses the term "reversion" in this paper, which discusses the size of peas.)
  • Francis Galton. Presidential address, Section H, Anthropology. (1885) (Galton uses the term "regression" in this paper, which discusses the height of humans.)
  • Francis Galton. "Regression Towards Mediocrity in Hereditary Stature," Journal of the Anthropological Institute, 15:246-263 (1886). (Facsimile at: [1] (http://www.mugu.com/galton/essays/1880-1889/galton-1886-jaigi-regression-stature.pdf))
  • G. Udny Yule. "On the Theory of Correlation", J. Royal Statist. Soc., 1897, p. 812-54.
  • Karl Pearson, G. U. Yule, Norman Blanchard, and Alice Lee. "The Law of Ancestral Heredity", Biometrika (1903)
  • R.A. Fisher. "The goodness of fit of regression formulae, and the distribution of regression coefficients", J. Royal Statist. Soc., 85, 597-612 (1922)

Modern theory

  • Draper, N.R. and Smith, H. Applied Regression Analysis Wiley Series in Probability and Statistics (1998)

Modern practice

External links

fr:Régression linéaire it:Regressione lineare nl:Lineaire regressie pl:Regresja liniowa

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