15.2 Normal-Id Generalized Linear Model (Linear Regression)
Stan also supplies a single function for a generalized linear lodel with normal likelihood and identity link function, i.e. a function for a linear regression. This provides a more efficient implementation of linear regression than a manually written regression in terms of a normal likelihood and matrix multiplication.
15.2.1 Probability Distribution Function
If \(x\in \mathbb{R}^{n\cdot m}, \alpha \in \mathbb{R}^n, \beta\in \mathbb{R}^m, \sigma\in \mathbb{R}^+\), then for \(y \in \mathbb{R}^n\), \[ \text{NormalIdGLM}(y|x, \alpha, \beta, \sigma) = \prod_{1\leq i \leq n}\text{Normal}(y_i|\alpha_i + x_i\cdot \beta, \sigma). \]
15.2.2 Sampling Statement
y ~
normal_id_glm
(x, alpha, beta, sigma)
Increment target log probability density with normal_id_glm_lpdf(y | x, alpha, beta, sigma)
dropping constant additive terms.
15.2.3 Stan Functions
real
normal_id_glm_lpdf
(real y | matrix x, real alpha, vector beta, real sigma)
The log normal probability density of y
given location alpha + x * beta
and scale sigma
, where the same intercept alpha
, scale sigma
and dependent variable value y
are used for all observations. The number of columns of x
needs to match the size of the coefficient vector beta
. If x
and y
are data (not parameters) this function can be executed on a GPU.
real
normal_id_glm_lpdf
(real y | matrix x, vector alpha, vector beta, real sigma)
The log normal probability density of y
given location alpha + x * beta
and scale sigma
, where the same scale sigma
and dependent variable valuey
are used for all observations and an intercept alpha
is used that is allowed to vary by observation. The number of rows of the independent variable matrix x
needs to match the size of the intercept vector alpha
and the number of columns of x
needs to match the size of the coefficient vector beta
. If x
and y
are data (not parameters) this function can be executed on a GPU.
real
normal_id_glm_lpdf
(vector y | row_vector x, real alpha, vector beta, real sigma)
The log normal probability density of y
given location alpha + x * beta
and scale sigma
, where the same intercept alpha
, scale sigma
and independent variable values x
are used for all observations. The number of columns of x
needs to match the size of the coefficient vector beta
.
real
normal_id_glm_lpdf
(vector y | row_vector x, vector alpha, vector beta, real sigma)
The log normal probability density of y
given location alpha + x * beta
and scale sigma
, where the same scale sigma
and independent variable values x
are used for all observations and an intercept alpha
is used that is allowed to vary by observation. The size of the dependent variable vector y
needs to match the size of the intercept vector alpha
and the number of columns of x
needs to match the size of the coefficient vector beta
.
real
normal_id_glm_lpdf
(vector y | matrix x, real alpha, vector beta, real sigma)
The log normal probability density of y
given location alpha + x * beta
and scale sigma
, where the same intercept alpha
and scale sigma
is used for all observations. The number of rows of the independent variable matrix x
needs to match the size of the dependent variable vector y
and the number of columns of x
needs to match the size of the coefficient vector beta
. If x
and y
are data (not parameters) this function can be executed on a GPU.
real
normal_id_glm_lpdf
(vector y | matrix x, vector alpha, vector beta, real sigma)
The log normal probability density of y
given location alpha + x * beta
and scale sigma
, where the same scale sigma
is used for all observations and an intercept alpha
is used that is allowed to vary by observation. The number of rows of the independent variable matrix x
needs to match the size of the dependent variable vector y
and the size of the intercept vector alpha. The number of columns of x
needs to match the size of the coefficient vector beta
. If x
and y
are data (not parameters) this function can be executed on a GPU.