lq_fit.Rd
Fits a regression model in the \(L_q\) norm (also labeled as the \(L_p\) norm).
In more detail,
the optimization function \( \sum_i | y_i - x_i \beta | ^p\) is optimized.
The nondifferentiable function is approximated by a differentiable approximation,
i.e., we use \(|x| \approx \sqrt{x^2 + \varepsilon } \). 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.
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)
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 |
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 |
List with following several entries
Vector of coefficients
Results of optimization
More values
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
############################################################################# # 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 if (FALSE) { ############################################################################# # 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 }