Linear Functions Family
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A Linear Functions Family is a function metamodel whose parameters can describe one or more linear functions.
- AKA: Linear Metamodel.
- Context:
- It can have parameters Slope and Origin.
- It can be a Statistical Model Family (e.g. Linear Statistical Model).
- It can be an input to: a Linear Regression Algorithm.
- It can range from being a Local Linear Model to being a Global Linear Model.
- It can be instantiated as: a Regressed Linear Model, a Linear Classifier.
- Example(s):
- [math]\displaystyle{ y_i = \beta_1 x_{i1} + \cdots + \beta_p x_{ip} + \varepsilon_i = x'_i\beta + \varepsilon_i, \qquad i = 1, \ldots, n, }[/math] where ′ denotes the transpose, so that xi′β is the inner product between vectors xi and β.
- [math]\displaystyle{ y = X\beta + \varepsilon, \, }[/math] where [math]\displaystyle{ y = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, \quad X = \begin{pmatrix} x'_1 \\ x'_2 \\ \vdots \\ x'_n \end{pmatrix} = \begin{pmatrix} x_{11} & \cdots & x_{1p} \\ x_{21} & \cdots & x_{2p} \\ \vdots & \ddots & \vdots \\ x_{n1} & \cdots & x_{np} \end{pmatrix}, \quad \beta = \begin{pmatrix} \beta_1 \\ \vdots \\ \beta_p \end{pmatrix}, \quad \varepsilon = \begin{pmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \vdots \\ \varepsilon_n \end{pmatrix}. }[/math]
- a Linear Probability Distribution.
- …
- Counter-Example(s):
- a Non-Linear Metamodel.
- a Logistic Regression Metamodel.
- a Classification Metamodel.
- a Linear Function, such as a Fitted Linear Function, because it does not have ungrounded model parameters.
- a Probabilistic Graphical Model.
- a Linear Program.
- See: Moving-Average Model, Autoregressive Model, Autoregressive-Moving Average Model.
References
2011
- (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Linear_regression#Introduction_to_linear_regression
- QUOTE: Given a data set [math]\displaystyle{ \{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n }[/math] of n statistical units, a linear regression model assumes that the relationship between the dependent variable yi and the p-vector of regressors xi is linear. This relationship is modeled through a so-called “disturbance term” εi — an unobserved random variable that adds noise to the linear relationship between the dependent variable and regressors. Thus the model takes the form :[math]\displaystyle{
y_i = \beta_1 x_{i1} + \cdots + \beta_p x_{ip} + \varepsilon_i
= x'_i\beta + \varepsilon_i,
\qquad i = 1, \ldots, n,
}[/math] where ′ denotes the transpose, so that xi′β is the inner product between vectors xi and β.
Often these n equations are stacked together and written in vector form as :[math]\displaystyle{ y = X\beta + \varepsilon, \, }[/math] where :[math]\displaystyle{ y = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, \quad X = \begin{pmatrix} x'_1 \\ x'_2 \\ \vdots \\ x'_n \end{pmatrix} = \begin{pmatrix} x_{11} & \cdots & x_{1p} \\ x_{21} & \cdots & x_{2p} \\ \vdots & \ddots & \vdots \\ x_{n1} & \cdots & x_{np} \end{pmatrix}, \quad \beta = \begin{pmatrix} \beta_1 \\ \vdots \\ \beta_p \end{pmatrix}, \quad \varepsilon = \begin{pmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \vdots \\ \varepsilon_n \end{pmatrix}. }[/math]
Some remarks on terminology and general use:
- [math]\displaystyle{ y_i\, }[/math] is called the regressand, endogenous variable, response variable, measured variable, or dependent variable (see dependent and independent variables.) The decision as to which variable in a data set is modeled as the dependent variable and which are modeled as the independent variables may be based on a presumption that the value of one of the variables is caused by, or directly influenced by the other variables. Alternatively, there may be an operational reason to model one of the variables in terms of the others, in which case there need be no presumption of causality.
- [math]\displaystyle{ x_i\, }[/math] are called regressors, exogenous variables, explanatory variables, covariates, input variables, predictor variables, or independent variables (see dependent and independent variables, but not to be confused with independent random variables). The matrix [math]\displaystyle{ X }[/math] is sometimes called the design matrix.
- Usually a constant is included as one of the regressors. For example we can take xi1 = 1 for i = 1, ..., n. The corresponding element of β is called the intercept. Many statistical inference procedures for linear models require an intercept to be present, so it is often included even if theoretical considerations suggest that its value should be zero.
- Sometimes one of the regressors can be a non-linear function of another regressor or of the data, as in polynomial regression and segmented regression. The model remains linear as long as it is linear in the parameter vector β.
- The regressors xi may be viewed either as random variables, which we simply observe, or they can be considered as predetermined fixed values which we can choose. Both interpretations may be appropriate in different cases, and they generally lead to the same estimation procedures; however different approaches to asymptotic analysis are used in these two situations.
- [math]\displaystyle{ \beta\, }[/math] is a p-dimensional parameter vector. Its elements are also called effects, or regression coefficients. Statistical estimation and inference in linear regression focuses on β.
- [math]\displaystyle{ \varepsilon_i\, }[/math] is called the error term, disturbance term, or noise. This variable captures all other factors which influence the dependent variable yi other than the regressors xi. The relationship between the error term and the regressors, for example whether they are correlated, is a crucial step in formulating a linear regression model, as it will determine the method to use for estimation.
- QUOTE: Given a data set [math]\displaystyle{ \{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n }[/math] of n statistical units, a linear regression model assumes that the relationship between the dependent variable yi and the p-vector of regressors xi is linear. This relationship is modeled through a so-called “disturbance term” εi — an unobserved random variable that adds noise to the linear relationship between the dependent variable and regressors. Thus the model takes the form :[math]\displaystyle{
y_i = \beta_1 x_{i1} + \cdots + \beta_p x_{ip} + \varepsilon_i
= x'_i\beta + \varepsilon_i,
\qquad i = 1, \ldots, n,
}[/math] where ′ denotes the transpose, so that xi′β is the inner product between vectors xi and β.
2006
- (Cox, 2006) ⇒ David R. Cox. (2006). “Principles of Statistical Inference." Cambridge University Press. ISBN:9780521685672
2003
- (Davison, 2003) ⇒ Anthony C. Davison. (2003). “Statistical Models." Cambridge University Press. ISBN:0521773393
1999
- (Zaiane, 1999) ⇒ Osmar Zaiane. (1999). “Glossary of Data Mining Terms." University of Alberta, Computing Science CMPUT-690: Principles of Knowledge Discovery in Databases.
- QUOTE: Linear Model:An analytical model that assumes linear relationships in the coefficients of the variables being studied.
1990
- (Lindstrom & Vorperian, 2005) ⇒ ⇒ Mary J. Lindstrom, and Houri K. Vorperian. (2005). “Nonlinear Mixed Effects Models.” In: (Everitt & Howell, 2005).
- QUOTE: … A second definition of a linear model is a model for which the derivative of the right-hand side of the model equation with respect to any individual parameter contains no parameters at all. Nonlinear models are most commonly used for data that show single …