netresponse: Functional Network Analysis

# Copyright (C) 2010-2012 Leo Lahti Contact: Leo Lahti <leo.lahti@iki.fi> This
# program is free software; you can redistribute it and/or modify it under the
# terms of the GNU General Public License as published by the Free Software
# Foundation; either version 2, or (at your option) any later version.  This
# program is distributed in the hope that it will be useful, but WITHOUT ANY
# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A
# PARTICULAR PURPOSE.  See the GNU General Public License for more details.


info.criterion <- function(nparams, nlog, logp, criterion = "BIC") {
    
    # Calculate various information criteria
    
    if (criterion == "AIC") {
        # Akaike IC
        return(AIC(nparams, nlog, logp))
    } else if (criterion == "BIC") {
        # Bayesian IC
        return(BIC(nparams, nlog, logp))
    } else if (criterion == "AICc") {
        # Akaike for linear models, finite sample
        return(AICc(nparams, nlog, logp))
    }
    
}




BIC <- function(nparams, nlog, logp) {
    
    # Calculate Bayesian Information Criterion (BIC)
    
    # NOTE: Original formulation (Schwartz) assumed that data is iid and likelihood
    # is in exponential family.  However, BIC is later derived with considerably
    # loosened assumptions, see e.g.  Cavanaugh et al.: Generalizing the Derivation
    # of the Schwarz Information Criterion it seems that the assumptions listed in
    # Section 3 will quarantee the validity of BIC for mixtures of exponential family
    # distributions; confirm.
    
    # negative free energy is lower bound for log(P(D|H)) logp = -cost
    
    nparams * nlog - 2 * logp
    
}

AIC <- function(nparams, nlog, logp) {
    
    # Calculate Akaike Information Criterion (AIC) Note: nlog not used here but
    # included for compatibility
    
    # negative free energy is lower bound for log(P(D|H)) logp = -cost
    
    2 * (nparams - logp)
    
}

AICc <- function(nparams, nlog, logp) {
    
    # Calculate Akaike Information Criterion correction for finite sample size (AICc)
    
    # AICc is AIC with a correction for finite sample sizes, giving a greater penalty
    # for extra parameters. Burnham & Anderson (2002) strongly recommend using AICc,
    # rather than AIC, if n is small or k is large. Since AICc converges to AIC as n
    # gets large, AICc generally should be employed. Using AIC, instead of AICc, when
    # n is not many times larger than k2, increases the probability of selecting
    # models that have too many parameters, i.e. of overfitting. The probability of
    # AIC overfitting can be substantial, in some cases. Brockwell & Davis (p. 273)
    # advise using AICc as the primary criterion in selecting the orders of an ARMA
    # model for time series. McQuarrie & Tsai ground their high opinion of AICc on
    # extensive simulation work with regression and time series. AICc was first
    # proposed by Hurvich & Tsai (1989). Different derivations of it are given by
    # Brockwell & Davis, Burnham & Anderson, and Cavanaugh. All the derivations
    # assume a univariate linear model with normally-distributed errors (conditional
    # upon regressors); if that assumption does not hold, then the formula for AICc
    # will usually change. Further discussion of this, with examples of other
    # assumptions, is given by Burnham & Anderson (2002, ch.7). Note that when all
    # the models in the candidate set have the same k, then AICc and AIC will give
    # identical (relative) valuations. In that situation, then, AIC can always be
    # used.
    
    # negative free energy is lower bound for log(P(D|H)) logp = -cost
    
    n <- exp(nlog)
    
    AIC(nparams, nlog, logp) + 2 * nparams(nparams + 1)/(n - nparams - 1)
    
}




# AIC.c <- cmpfun(AIC) AICc.c <- cmpfun(AICc) BIC.c <- cmpfun(BIC)