sirt: lq_fit – R documentation

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lq_fit

Fit of a L_q Regression Model

Description

Fits a regression model in the L_q norm (also labeled as the L_p norm). In more detail, the optimization function ∑_i | y_i - x_i β | ^p is optimized. The nondifferentiable function is approximated by a differentiable approximation, i.e., we use |x| \approx √{x^2 + \varepsilon } . The power p can also be estimated by using est_pow=TRUE, see Giacalone, Panarello and Mattera (2018). The algorithm iterates between estimating regression coefficients and the estimation of power values. The estimation of the power based on a vector of residuals e can be conducted using the function lq_fit_estimate_power.

Using the L_q norm in the regression is equivalent to assuming an expontial power function for residuals (Giacalone et al., 2018). The density function and a simulation function is provided by dexppow and rexppow, respectively. See also the normalp package.

Usage

lq_fit(y, X, w=NULL, pow=2, eps=0.001, beta_init=NULL, est_pow=FALSE, optimizer="optim",
    eps_vec=10^seq(0,-10, by=-.5), conv=1e-4, miter=20, lower_pow=.1, upper_pow=5)

lq_fit_estimate_power(e, pow_init=2, lower_pow=.1, upper_pow=10)

dexppow(x, mu=0, sigmap=1, pow=2, log=FALSE)

rexppow(n, mu=0, sigmap=1, pow=2, xbound=100, xdiff=.01)

Arguments

`y`	Dependent variable
`X`	Design matrix
`w`	Optional vector of weights
`pow`	Power p in L_q norm
`est_pow`	Logical indicating whether power should be estimated
`eps`	Parameter governing the differentiable approximation
`e`	Vector of resiuals
`pow_init`	Initial value of power
`beta_init`	Initial vector
`optimizer`	Can be `"optim"` or `"nlminb"`.
`eps_vec`	Vector with decreasing \varepsilon values used in optimization
`conv`	Convergence criterion
`miter`	Maximum number of iterations
`lower_pow`	Lower bound for estimated power
`upper_pow`	Upper bound for estimated power
`x`	Vector
`mu`	Location parameter
`sigmap`	Scale parameter
`log`	Logical indicating whether the logarithm should be provided
`n`	Sample size
`xbound`	Lower and upper bound for density approximation
`xdiff`	Grid width for density approximation

Value

List with following several entries

`coefficients`	Vector of coefficients
`res_optim`	Results of optimization
`...`	More values

References

Giacalone, M., Panarello, D., & Mattera, R. (2018). Multicollinearity in regression: an efficiency comparison between $L_p$-norm and least squares estimators. Quality & Quantity, 52(4), 1831-1859. doi: 10.1007/s11135-017-0571-y

Examples

#############################################################################
# EXAMPLE 1: Small simulated example with fixed power
#############################################################################

set.seed(98)
N <- 300
x1 <- stats::rnorm(N)
x2 <- stats::rnorm(N)
par1 <- c(1,.5,-.7)
y <- par1[1]+par1[2]*x1+par1[3]*x2 + stats::rnorm(N)
X <- cbind(1,x1,x2)

#- lm function in stats
mod1 <- stats::lm.fit(y=y, x=X)

#- use lq_fit function
mod2 <- sirt::lq_fit( y=y, X=X, pow=2, eps=1e-4)
mod1$coefficients
mod2$coefficients

## Not run: 
#############################################################################
# EXAMPLE 2: Example with estimated power values
#############################################################################

#*** simulate regression model with residuals from the exponential power distribution
#*** using a power of .30
set.seed(918)
N <- 2000
X <- cbind( 1, c(rep(1,N), rep(0,N)) )
e <- sirt::rexppow(n=2*N, pow=.3, xdiff=.01, xbound=200)
y <- X %*% c(1,.5) + e

#*** estimate model
mod <- sirt::lq_fit( y=y, X=X, est_pow=TRUE, lower_pow=.1)
mod1 <- stats::lm( y ~ 0 + X )
mod$coefficients
mod$pow
mod1$coefficients

## End(Not run)

sirt

Supplementary Item Response Theory Models

v3.10-118

GPL (>= 2)

Authors

Alexander Robitzsch [aut,cre] (<https://orcid.org/0000-0002-8226-3132>)

Initial release

2021-09-22 17:45:34