PerRegMod (version 4.4.3)

lm_per: Fitting periodic coefficients regression model by using LSE

Description

lm_per() function gives the least squares estimation of parameters, intercept \(\mu_s\), slope \(\boldsymbol{\beta}_s\), and standard deviation \(\sigma_s\), of a periodic coefficients regression model using LSE_Reg_per and sd_estimation_for_each_s functions. \(\widehat{\boldsymbol{\vartheta}}=\left(X^{'}X\right)^{-1}X^{'} Y\) where \(X= \left[\begin{array}{ccccccccccc} &\mathbf{X}^1_{1}&0&\ldots & 0& &\mathbf{X}^p_{1}&0&\ldots & 0 \\ & 0&\mathbf{X}^1_{2} &\ldots &0 & &0&\mathbf{X}^p_{2} &\ldots &0\\ \textbf{I}_{S}\otimes \mathbf{1}_{m} &0&0& \ddots&\vdots&\ldots&0& 0&\ddots&\vdots \\& 0 &0&0 &\mathbf{X}^1_{S}& &0 &0&0 &\mathbf{X}^p_{S} \end{array}\right]\ \),

\( \mathbf{X}^j_{s}=\left(x^j_{s},...,x^j_{s+(m-1)S}\right)^{'}\), \(Y=(\mathbf{Y}_1^{'},...,\mathbf{Y}_S^{'})^{'}\), \(\mathbf{Y}_{s} =(y_{s},...,y_{(m-1)S+s})^{'}\), \(\mathbf{\epsilon}=(\mathbf{\epsilon}_{1}^{'},...,\mathbf{\epsilon}_{S}^{'})^{'}\), \(\mathbf{\epsilon}_{s} =(\varepsilon_{s},...,\varepsilon_{(m-1)S+s})^{'}\), \(\mathbf{1}_{m}\) is a vector of ones of dimension \(m\), \(\textbf{I}_{S}\) is the identity matrix of dimension \(S\), \(\otimes\) denotes the Kronecker product, and \(\boldsymbol{\vartheta} =\left(\boldsymbol{\mu}^{'} ,{\boldsymbol{\beta}}^{'}\right)^{'}\) with \(\boldsymbol{\mu}=(\mu_1,...,\mu_S)^{'}\) and \(\boldsymbol{\beta}=(\beta^1_{1},...,\beta^1_{S};...;\beta^p_{1},...,\beta^p_{S})^{'}\).

Usage

lm_per(x,y,s)

Value

Residuals

the residuals, that is response minus fitted values

Coefficients

a named vector of coefficients

Root mean square error

The root mean square error

Arguments

x

A list of independent variables with dimension \(p\).

y

A response variable.

s

A period of the regression model.

Examples

set.seed(6)
n=400
s=4
x1=rnorm(n,0,1.5)
x2=rnorm(n,0,0.9)
x3=rnorm(n,0,2)
x4=rnorm(n,0,1.9)
y=rnorm(n,0,2.5)
x=list(x1,x2,x3,x4)
lm_per(x,y,s)