[Returnanalytics-commits] r3446 - pkg/PortfolioAnalytics/R
noreply at r-forge.r-project.org
noreply at r-forge.r-project.org
Sun Jun 29 14:50:14 CEST 2014
Author: rossbennett34
Date: 2014-06-29 14:50:14 +0200 (Sun, 29 Jun 2014)
New Revision: 3446
Added:
pkg/PortfolioAnalytics/R/EntropyProg.R
Log:
Adding EntropyProg from Meucci
Added: pkg/PortfolioAnalytics/R/EntropyProg.R
===================================================================
--- pkg/PortfolioAnalytics/R/EntropyProg.R (rev 0)
+++ pkg/PortfolioAnalytics/R/EntropyProg.R 2014-06-29 12:50:14 UTC (rev 3446)
@@ -0,0 +1,227 @@
+#' Entropy pooling program for blending views on scenarios with a prior scenario-probability distribution
+#'
+#' Entropy program will change the initial predictive distribution 'p' to a new set 'p_' that satisfies
+#' specified moment conditions but changes other propoerties of the new distribution the least by
+#' minimizing the relative entropy between the two distributions. Theoretical note: Relative Entropy (Kullback-Leibler information criterion KLIC) is an
+#' asymmetric measure.
+#'
+#' We retrieve a new set of probabilities for the joint-scenarios using the Entropy pooling method
+#' Of the many choices of 'p' that satisfy the views, we choose 'p' that minimize the entropy or distance of the new probability
+#' distribution to the prior joint-scenario probabilities
+#' We use Kullback-Leibler divergence or relative entropy dist(p,q): Sum across all scenarios [ p-t * ln( p-t / q-t ) ]
+#' Therefore we define solution as p* = argmin (choice of p ) [ sum across all scenarios: p-t * ln( p-t / q-t) ],
+#' such that 'p' satisfies views. The views modify the prior in a cohrent manner (minimizing distortion)
+#' We forumulate the stress tests of the baseline scenarios as linear constraints on yet-to-be defined probabilities
+#' Note that the numerical optimization acts on a very limited number of variables equal
+#' to the number of views. It does not act directly on the very large number of variables
+#' of interest, namely the probabilities of the Monte Carlo scenarios. This feature guarantees
+#' the numerical feasability of entropy optimization
+#' Note that new probabilities are generated in much the same way that the state-price density modifies
+#' objective probabilities of pay-offs to risk-neutral probabilities in contingent-claims asset pricing
+#'
+#' Compute posterior (=change of measure) with Entropy Pooling, as described in
+#'
+#' @param p a vector of initial probabilities based on prior (reference model, empirical distribution, etc.). Sum of 'p' must be 1
+#' @param Aeq matrix consisting of equality constraints (paired with argument 'beq'). Denoted as 'H' in the Meucci paper. (denoted as 'H' in the "Meucci - Flexible Views Theory & Practice" paper formlua 86 on page 22)
+#' @param beq vector corresponding to the matrix of equality constraints (paired with argument 'Aeq'). Denoted as 'h' in the Meucci paper
+#' @param A matrix consisting of inequality constraints (paired with argument 'b'). Denoted as 'F' in the Meucci paper
+#' @param b vector consisting of inequality constraints (paired with matrix A). Denoted as 'f' in the Meucci paper
+#'
+#' ' \deqn{ \tilde{p} \equiv argmin_{Fx \leq f, Hx \equiv h} \big\{ \sum_1^J x_{j} \big(ln \big( x_{j} \big) - ln \big( p_{j} \big) \big) \big\}
+#' \\ \ell \big(x, \lambda, \nu \big) \equiv x' \big(ln \big(x\big) - ln \big(p\big) \big) + \lambda' \big(Fx - f\big) + \nu' \big(Hx - h\big)}
+#' @return a list with
+#' p_ revised probabilities based on entropy pooling
+#' optimizationPerformance a list with status of optimization, value, number of iterations and sum of probabilities.
+#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com}
+#' @references
+#' A. Meucci - "Fully Flexible Views: Theory and Practice". See page 22 for illustration of numerical implementation
+#' Symmys site containing original MATLAB source code \url{http://www.symmys.com}
+#' NLOPT open-source optimization site containing background on algorithms \url{http://ab-initio.mit.edu/wiki/index.php/NLopt}
+#' We use the information-theoretic estimator of Kitamur and Stutzer (1997).
+#' Reversing 'p' and 'p_' leads to the empirical likelihood" estimator of Qin and Lawless (1994).
+#' See Robertson et al, "Forecasting Using Relative Entropy" (2002) for more theory
+#' @export
+EntropyProg = function( p , A = NULL , b = NULL , Aeq , beq )
+{
+ library( nloptr )
+
+ if( !length(b) ) A = matrix( ,nrow = 0, ncol = 0)
+ if( !length(b) ) b = matrix( ,nrow = 0, ncol = 0)
+ # count the number of constraints
+ K_ = nrow( A ) # K_ is the number of inequality constraints in the matrix-vector pair A-b
+ K = nrow( Aeq ) # K is the number of equality views in the matrix-vector pair Aeq-beq
+
+ # parameter checks
+ if ( K_ + K == 0 ) { stop( "at least one equality or inequality constraint must be specified")}
+ if ( ( ( .999999 < sum(p)) & (sum(p) < 1.00001) ) == FALSE ) { stop( "sum of probabilities from prior distribution must equal 1")}
+ if ( nrow(Aeq)!=nrow(beq) ) { stop( "number of inequality constraints in matrix Aeq must match number of elements in vector beq") }
+ if ( nrow(A)!=nrow(b) ) { stop( "number of equality constraints in matrix A must match number of elements in vector b") }
+
+ # calculate derivatives of constraint matrices
+ A_ = t( A )
+ b_ = t( b )
+ Aeq_ = t( Aeq )
+ beq_ = t( beq )
+
+ # starting guess for optimization search with length = to number of views
+ x0 = matrix( 0 , nrow = K_ + K , ncol = 1 )
+
+ if ( !K_ ) # equality constraints only
+ {
+ # define maximum likelihood, gradient, and hessian functions for unconstrained and constrained optimization
+ eval_f_list = function( v ) # cost function for unconstrained optimization (no inequality constraints)
+ {
+ x = exp( log(p) - 1 - Aeq_ %*% v )
+ x = apply( cbind( x , 10^-32 ) , 1 , max ) # robustification
+ # L is the Lagrangian dual function (without inequality constraints). See formula 88 on p. 22 in "Meucci - Fully Flexible Views - Theory and Practice (2010)"
+ # t(x) is the derivative x'
+ # Aeq_ is the derivative of the Aeq matrix of equality constraints (denoted as 'H in the paper)
+ # beq_ is the transpose of the vector associated with Aeq equality constraints
+ # L= x' * ( log(x) - log(p) + Aeq_ * v ) - beq_ * v
+ # 1xJ * ( Jx1 - Jx1 + JxN+1 * N+1x1 ) - 1xN+1 * N+1x1
+ L = t(x) %*% ( log(x) - log(p) + Aeq_ %*% v ) - beq_ %*% v
+ mL = -L # take negative values since we want to maximize
+
+ # evaluate gradient
+ gradient = beq - Aeq %*% x
+
+ # evaluate Hessian
+ # We comment this out for now -- to be used when we find an optimizer that can utilize the Hessian in addition to the gradient
+ # H = ( Aeq %*% (( x %*% ones(1,K) ) * Aeq_) ) # Hessian computed by Chen Qing, Lin Daimin, Meng Yanyan, Wang Weijun
+
+ return( list( objective = mL , gradient = gradient ) )
+ }
+
+ # setup unconstrained optimization
+ start = Sys.time()
+ opts = list( algorithm = "NLOPT_LD_LBFGS" , xtol_rel = 1.0e-6 ,
+ check_derivatives = TRUE , check_derivatives_print = "all" , print_level = 2 , maxeval = 1000 )
+ optimResult = nloptr(x0 = x0, eval_f = eval_f_list , opts = opts )
+ end = Sys.time()
+ print( c("Optimization completed in " , end - start )) ; rm( start ) ; rm( end )
+
+ if ( optimResult$status < 0 ) { print( c("Exit code " , optimResult$status ) ) ; stop( "Error: The optimizer did not converge" ) }
+
+ # return results of optimization
+ v = optimResult$solution
+ p_ = exp( log(p) - 1 - Aeq_ %*% v )
+ optimizationPerformance = list( converged = (optimResult$status > 0) , ml = optimResult$objective , iterations = optimResult$iterations , sumOfProbabilities = sum( p_ ) )
+ }else # case inequality constraints are specified
+ {
+ # setup variables for constrained optimization
+ InqMat = -diag( 1 , K_ + K ) # -1 * Identity Matrix with dimension equal to number of constraints
+ InqMat = InqMat[ -c( K_+1:nrow( InqMat ) ) , ] # drop rows corresponding to equality constraints
+ InqVec = matrix( 0 , K_ , 1 )
+
+ # define maximum likelihood, gradient, and hessian functions for constrained optimization
+ InqConstraint = function( x ) { return( InqMat %*% x ) } # function used to evalute A %*% x <= 0 or A %*% x <= c(0,0) if there is more than one inequality constraint
+
+ jacobian_constraint = function( x ) { return( InqMat ) }
+ # Jacobian of the constraints matrix. One row per constraint, one column per control parameter (x1,x2)
+ # Turns out the Jacobian of the constraints matrix is always equal to InqMat
+
+ nestedfunC = function( lv )
+ {
+ lv = as.matrix( lv )
+ l = lv[ 1:K_ , , drop = FALSE ] # inequality Lagrange multiplier
+ v = lv[ (K_+1):length(lv) , , drop = FALSE ] # equality lagrange multiplier
+ x = exp( log(p) - 1 - A_ %*% l - Aeq_ %*% v )
+ x = apply( cbind( x , 10^-32 ) , 1 , max )
+
+ # L is the cost function used for constrained optimization
+ # L is the Lagrangian dual function with inequality constraints and equality constraints
+ L = t(x) %*% ( log(x) - log(p) ) + t(l) %*% (A %*% x-b) + t(v) %*% (Aeq %*% x-beq)
+ objective = -L # take negative values since we want to maximize
+
+ # calculate the gradient
+ gradient = rbind( b - A%*%x , beq - Aeq %*% x )
+
+ # compute the Hessian (commented out since no R optimizer supports use of Hessian)
+ # Hessian computed by Chen Qing, Lin Daimin, Meng Yanyan, Wang Weijun
+ #onesToK_ = array( rep( 1 , K_ ) ) ;onesToK = array( rep( 1 , K ) )
+ #x = as.matrix( x )
+ #H11 = A %*% ((x %*% onesToK_) * A_) ; H12 = A %*% ((x %*% onesToK) * Aeq_)
+ #H21 = Aeq %*% ((x %*% onesToK_) * A_) ; H22 = Aeq %*% ((x %*% onesToK) * Aeq_)
+ #H1 = cbind( H11 , H12 ) ; H2 = cbind( H21 , H22 ) ; H = rbind( H1 , H2 ) # Hessian for constrained optimization
+ return( list( objective = objective , gradient = gradient ) )
+ }
+
+ # find minimum of constrained multivariate function
+ start = Sys.time()
+ # Note: other candidates for constrained optimization in library nloptr: NLOPT_LD_SLSQP, NLOPT_LD_MMA, NLOPT_LN_AUGLAG, NLOPT_LD_AUGLAG_EQ
+ # See NLOPT open-source site for more details: http://ab-initio.mit.edu/wiki/index.php/NLopt
+ local_opts <- list( algorithm = "NLOPT_LD_SLSQP", xtol_rel = 1.0e-6 ,
+ check_derivatives = TRUE , check_derivatives_print = "all" ,
+ eval_f = nestedfunC , eval_g_ineq = InqConstraint , eval_jac_g_ineq = jacobian_constraint )
+ optimResult = nloptr( x0 = x0 , eval_f = nestedfunC , eval_g_ineq = InqConstraint , eval_jac_g_ineq = jacobian_constraint ,
+ opts = list( algorithm = "NLOPT_LD_AUGLAG" , local_opts = local_opts ,
+ print_level = 2 , maxeval = 1000 ,
+ check_derivatives = TRUE , check_derivatives_print = "all" , xtol_rel = 1.0e-6 ) )
+ end = Sys.time()
+ print( c("Optimization completed in " , end - start )) ; rm( start ) ; rm( end )
+
+ if ( optimResult$status < 0 ) { print( c("Exit code " , optimResult$status ) ) ; stop( "Error: The optimizer did not converge" ) }
+
+ # return results of optimization
+ lv = matrix( optimResult$solution , ncol = 1 )
+ l = lv[ 1:K_ , , drop = FALSE ] # inequality Lagrange multipliers
+ v = lv[ (K_+1):nrow( lv ) , , drop = FALSE ] # equality Lagrange multipliers
+ p_ = exp( log(p) -1 - A_ %*% l - Aeq_ %*% v )
+ optimizationPerformance = list( converged = (optimResult$status > 0) , ml = optimResult$objective , iterations = optimResult$iterations , sumOfProbabilities = sum( p_ ) )
+ }
+
+ print( optimizationPerformance )
+
+ if ( sum( p_ ) < .999 ) { stop( "Sum or revised probabilities is less than 1!" ) }
+ if ( sum( p_ ) > 1.001 ) { stop( "Sum or revised probabilities is greater than 1!" ) }
+
+ return ( list ( p_ = p_ , optimizationPerformance = optimizationPerformance ) )
+}
+
+
+
+#' Generates histogram
+#'
+#' @param X a vector containing the data points
+#' @param p a vector containing the probabilities for each of the data points in X
+#' @param nBins expected number of Bins the data set is to be broken down into
+#' @param freq a boolean variable to indicate whether the graphic is a representation of frequencies
+#'
+#' @return a list with
+#' f the frequency for each midpoint
+#' x the midpoints of the nBins intervals
+#'
+#' @references
+#' \url{http://www.symmys.com}
+#' See Meucci script pHist.m used for plotting
+#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com} and Xavier Valls \email{flamejat@@gmail.com}
+
+pHist = function( X , p , nBins, freq = FALSE )
+{
+ if ( length( match.call() ) < 3 )
+ {
+ J = dim( X )[ 1 ]
+ nBins = round( 10 * log(J) )
+ }
+
+ dist = hist( x = X , breaks = nBins , plot = FALSE );
+ n = dist$counts
+ x = dist$breaks
+ D = x[2] - x[1]
+
+ N = length(x)
+ np = zeros(N , 1)
+
+ for (s in 1:N)
+ {
+ # The boolean Index is true is X is within the interval centered at x(s) and within a half-break distance
+ Index = ( X >= x[s] - D/2 ) & ( X <= x[s] + D/2 )
+ # np = new probabilities?
+ np[ s ] = sum( p[ Index ] )
+ f = np/D
+ }
+
+ plot( x , f , type = "h", main = "Portfolio return distribution")
+
+ return( list( f = f , x = x ) )
+}
More information about the Returnanalytics-commits
mailing list