Fitting Linear Models with Hyperbolic Errors

Description

Fits linear models with hyperbolic errors. Can be used to carry out linear regression for data exhibiting heavy tails and skewness. Displays the histogram, log-histogram (both with fitted error distribution), Q-Q plot and residuals vs. fitted values plot for the fitted linear model.

Usage

hyperblm(formula, data, subset, weights, na.action,
         xx = FALSE, y = FALSE, contrasts = NULL,
         offset, method = "Nelder-Mead",
         startMethod = "Nelder-Mead", startStarts = "BN",
         paramStart = NULL,
         maxiter = 100, tolerance = 0.0001,
         controlBFGS = list(maxit = 1000),
         controlNM = list(maxit = 10000),
         maxitNLM = 10000,
         controlCO = list(), silent = TRUE, ...)

## S3 method for class 'hyperblm'
print(x, digits = max(3, getOption("digits")-3), ...)

## S3 method for class 'hyperblm'
coef(object, ...)

## S3 method for class 'hyperblm'
plot(x, breaks = "FD",
                        plotTitles = c("Residuals vs Fitted Values",
                                       "Histogram of residuals",
                                       "Log-Histogram of residuals",
                                       "Q-Q Plot"),
                        ...)

Arguments

Details

Models for hyperblm are specified symbolically. A typical model has the form response ~ terms where response is the (numeric) response vector and terms is a series of terms which specifies a linear predictor for response. A terms specification of the form first + second indicates all the terms in first together with all the terms in second with duplicates removed. A specification of the form first:second indicates the set of terms obtained by taking the interactions of all terms in first with all terms in second. The specification first*second indicates the cross of first and second. This is the same as first + second + first:second.

If the formula includes an offset, this is evaluated and subtracted from the response.

If response is a matrix a linear model is fitted separately by least-squares to each column of the matrix.

See model.matrix for some further details. The terms in the formula will be re-ordered so that main effects come first, followed by the interactions, all second-order, all third-order and so on.

A formula has an implied intercept term. To remove this use either y ~ x - 1 or y ~ 0 + x. See formula for more details of allowed formulae.

Non-NULL weights can be used to indicate that different observations have different variances (with the values in weights being inversely proportional to the variances); or equivalently, when the elements of weights are positive integers w_i, that each response y_i is the mean of w_i unit-weight observations (including the case that there are w_i observations equal to y_i and the data have been summarized).

hyperblm calls the lower level function hyperblmFit for the actual numerical computations.

All of weights, subset and offset are evaluated in the same way as variables in formula, that is first in data and then in the environment of formula.

hyperblmFit uses a two-stage alternating optimization routine. The quality of parameter start values (especially the error distribution parameters) is crucial to the routine's convergence. The user can specify the start values via the paramStart argument, otherwise the function finds reliable start values by calling the hyperbFitStand function.

startMethod in the argument list is the optimization method for function hyperbFitStandStart which finds the start values for function hyperbFitStand. It is set to "Nelder-Mead" by default due to the robustness of this optimizer. The "BFGS" method is also implemented as it is relatively fast to converge. Since "BFGS" method is a quasi-Newton method it will not as robust and for some data will not achieve convergence.

startStarts is the method used to find the start values for function hyperbFitStandStart which includes:

• "BN"A method from Barndorff-Nielsen (1977) based on estimates of psi and gamma the absolute slopes of the left and right asymptotes to the log density function

• "FN"Based on a fitted normal distribution as it is a limit of the hyperbolic distribution

• "SL"Based on a fitted skew-Laplace distribution for which the log density has the form of two straight line with absolute slopes 1/alpha, 1/beta

• "MoM"A method of moment approach

• "US"User specified

method is the method used in stage one of the two-stage alternating optimization routine. As the startMethod, it is set to "Nelder-Mead" by default. Besides "BFGS","nlm" is also implemented as a alternative. Since BFGS method is a quasi-Newton method it will not as robust and for some data will not achieve convergence.

If the maximum of the ratio the change of the individual coefficients is smaller than tolerance then the routine assumes convergence, otherwise if the alternating iteration number exceeds maxiter with the maximum of the ratio the change of the individual coefficients larger than tolerance, the routine is considered not to have converged.

Value

hyperblm returns an object of class "hyperblm" which is a list containing:

Author(s)

David Scott d.scott@auckland.ac.nz, Xinxing Li xli053@aucklanduni.ac.nz

References

Barndorff-Nielsen, O. (1977) Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Prause, K. (1999). The generalized hyperbolic models: Estimation, financial derivatives and risk measurement. PhD Thesis, Mathematics Faculty, University of Freiburg.

Trendall, Richard (2005). hypReg: A Function for Fitting a Linear Regression Model in R with Hyperbolic Error. Masters Thesis, Statistics Faculty, University of Auckland.

Paolella, Marc S. (2007). Intermediate Probability: A Computational Approach. pp. 415 -Chichester: Wiley.

Scott, David J. and W<c3><bc>rtz, Diethelm and Chalabi, Yohan, (2011). Fitting the Hyperbolic Distribution with R: A Case Study of Optimization Techniques. In preparation.

Stryhn, H. and Christensen, J. (2003). Confidence intervals by the profile likelihood method, with applications in veterinary epidemiology. ISVEE X.

See Also

print.hyperblm prints the regression result in a table. coef.hyperblm obtains the regression coefficients and error distribution parameters of the fitted model. summary.hyperblm obtains a summary output of class hyperblm object. print.summary.hyperblm prints the summary output in a table. plot.hyperblm obtains a residual vs fitted value plot, a histgram of residuals with error distribution density curve on top, a histgram of log residuals with error distribution error density curve on top and a QQ plot. hyperblmFit, optim, nlm, constrOptim, hist, hyperbFitStand, hyperbFitStandStart.

Examples

## stackloss data example

 airflow <- stackloss[, 1]
 temperature <- stackloss[, 2]
 acid <- stackloss[, 3]
 stack <- stackloss[, 4]

 hyperblm.fit <- hyperblm(stack ~ airflow + temperature + acid,
                          tolerance = 1e-11)

 coef.hyperblm(hyperblm.fit)
 plot.hyperblm(hyperblm.fit, breaks = 20)
 summary.hyperblm(hyperblm.fit, hessian = FALSE)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(GeneralizedHyperbolic)
Loading required package: DistributionUtils
Loading required package: RUnit
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/GeneralizedHyperbolic/hyperblm.Rd_%03d_medium.png", width=480, height=480)
> ### Name: hyperblm
> ### Title: Fitting Linear Models with Hyperbolic Errors
> ### Aliases: hyperblm print.hyperblm coef.hyperblm plot.hyperblm
> 
> ### ** Examples
> 
> ## stackloss data example
> 
>  airflow <- stackloss[, 1]
>  temperature <- stackloss[, 2]
>  acid <- stackloss[, 3]
>  stack <- stackloss[, 4]
> 
>  hyperblm.fit <- hyperblm(stack ~ airflow + temperature + acid,
+                           tolerance = 1e-11)
> 
>  coef.hyperblm(hyperblm.fit)
$coefficients
[1] -36.9799824   0.8129461   0.7648236  -0.1246500

$distributionParams
[1] -1.0715709  0.1904949  0.5529651  0.1737802

>  plot.hyperblm(hyperblm.fit, breaks = 20)
>  summary.hyperblm(hyperblm.fit, hessian = FALSE)

Call:
hyperblm(formula = stack ~ airflow + temperature + acid, tolerance = 1e-11)

Data:      (Intercept) airflow temperature acid 
Parameter estimates:
             Estimate Std. Error t value  Pr(>|t|)    
(Intercept) -36.97998   13.91268 -2.6580  0.018734 *  
airflow       0.81295    0.08635  9.4145 1.962e-07 ***
temperature   0.76482    0.22756  3.3610  0.004661 ** 
acid         -0.12465    0.14274 -0.8733  0.397237    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Parameters for hyperbolic error distribution:
     mu    delta    alpha     beta  
-1.0716   0.1905   0.5530   0.1738  
Likelihood:         
Method:             Nelder-Mead 
Convergence code:   0 
Iterations:         6 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>