Package 'expQ2'

Non-negative vector * exp of rate matrix

Description

v exp(Q) by uniformisation or scaling and squaring, or exp(Q) by scaling and squaring.

Package:	expQ2
Type:	Package
Version:	1.3
Date:	May 4, 2023
License:	CC0
LazyLoad:	yes

Note

This software package is based on the R package expQ developed by Chris Sherlock (https://github.com/ChrisGSherlock/expQ). This alternate version is maintained by scientists at the NOAA Fisheries Pacific Islands Fisheries Science Center and should be considered a fundamental research communication. The recommendations and conclusions presented here are those of the authors and this software should not be construed as official communication by NMFS, NOAA, or the U.S. Dept. of Commerce. In addition, reference to trade names does not imply endorsement by the National Marine Fisheries Service, NOAA. While the best efforts have been made to insure the highest quality, tools such as this are under constant development and are subject to change.

Author(s)

Chris Sherlock and Devin Johnson

Maintainer: Devin S. Johnson <[email protected]>

References

Sherlock, C. (2021). Direct statistical inference for finite Markov jump processes via the matrix exponential. Computational Statistics, 36(4), 2863-2887.

---

Exponentiate a rate matrix

Description

Exponentiates a whole rate matrix by making all elements non-negative, then scaling and squaring.

Usage

SS_exp_Q(Q, prec, renorm = TRUE)

Arguments

Q	Rate matrix (sparse or dense).
prec	Required precision - missing mass in the Poisson distribution.
renorm	Force elements of each row to sum to 1? Defaults to TRUE.

Value

exp(Q) A dense matrix.

Author(s)

Chris Sherlock

Examples

Qd <- matrix(nrow=2,ncol=2,data=c(-1,1,2,-2),byrow=TRUE); SS_exp_Q(Qd,1e-10)

library("Matrix")
d<-5; f<-0.3; ones<-rep(1,d)
Qs <- abs(rsparsematrix(d,d,f))
diag(Qs) <- 0
Qsum <- as.vector(Qs%*%ones)
diag(Qs) <- -Qsum
SS_exp_Q(Qs,1e-15)

## Not run: 
M <- matrix(nrow=2,ncol=2,data=c(1,1,2,2),byrow=TRUE); SS_exp_Q(M,1e-10)
M <- matrix(nrow=2,ncol=2,data=c(1,-1,-2,2),byrow=TRUE); SS_exp_Q(M,1e-10)
SS_exp_Q(Qs,1.5)
SS_exp_Q(Qs,-2.0)

## End(Not run)

---

Product of horizontal vector and exponential of a rate matrix

Description

Evaluates v Exp(Q) by making all elements of Q non-negative, then scaling down by a power of two, and finally using a mixture of squaring and vector-matrix products.

Usage

SS_v_exp_Q(v, Q, prec, renorm = TRUE, checks = TRUE)

Arguments

v	Non-negative horizontal vector (dense).
Q	Rate matrix (sparse or dense).
prec	Required precision - missing mass in the Poisson distribution.
renorm	Force elements of each row to sum to 1? Defaults to TRUE.
checks	Perform sanity checks on the arguments? Defaults to TRUE.

Value

v exp(Q) A dense horizontal vector.

Author(s)

Chris Sherlock

Examples

v<-runif(2); v<-v/sum(v)
Qd <- matrix(nrow=2,ncol=2,data=c(-1,1,2,-2),byrow=TRUE); SS_v_exp_Q(t(v),Qd,1e-10)

library("Matrix")
d<-5; f<-0.3; ones<-rep(1,d); v<-runif(d)
Qs <- abs(rsparsematrix(d,d,f))
diag(Qs) <- 0
Qsum <- as.vector(Qs%*%ones)
diag(Qs) <- -Qsum
SS_v_exp_Q(t(v),Qs,1e-15)

## Not run: 
v <- runif(2); 
M <- matrix(nrow=2,ncol=2,data=c(1,1,2,2),byrow=TRUE); SS_v_exp_Q(t(v),M,1e-10)
M <- matrix(nrow=2,ncol=2,data=c(1,-1,-2,2),byrow=TRUE); SS_v_exp_Q(t(v),M,1e-10)
d<-5; f<-0.3; ones<-rep(1,d); v<-runif(d)
Qs <- abs(rsparsematrix(d,d,f))
diag(Qs) <- 0; Qsum <- as.vector(Qs%*%ones); diag(Qs) <- -Qsum
SS_v_exp_Q(v,Qs,1e-15)
SS_v_exp_Q(t(v),Qs,1.5)
SS_v_exp_Q(t(v),Qs,-2.0)
v=-runif(d)
SS_v_exp_Q(t(v),Qs,1e-15)

## End(Not run)

---

Product of horizontal vector and exponential of rate matrix

Description

Evaluates v exp(Q) by making all elements of Q non-negative, then using the uniformisation method.

Usage

Unif_v_exp_Q(v, Q, prec, renorm = TRUE, t2 = TRUE, checks = TRUE)

Arguments

v	Non-negative horizontal vector (dense).
Q	Rate matrix (sparse or dense).
prec	Required precision - missing mass in the Poisson distribution.
renorm	Force elements of each row to sum to 1? Defaults to TRUE.
t2	Perform two-tailed truncation? Defaults to TRUE.
checks	Perform sanity checks on the arguments? Defaults to TRUE.

Value

v exp(Q) Dense horizontal vector.

Author(s)

Chris Sherlock

Examples

v <- runif(2); v <- v/sum(v)
Qd <- matrix(nrow=2,ncol=2,data=c(-1,1,2,-2),byrow=TRUE); Unif_v_exp_Q(t(v),Qd,1e-10)

library("Matrix")
d <- 5; f <- 0.3; ones <- rep(1,d); v <- runif(d)
Qs <- abs(rsparsematrix(d,d,f))
diag(Qs) <- 0
Qsum <- as.vector(Qs%*%ones)
diag(Qs) <- -Qsum
Unif_v_exp_Q(t(v),Qs,1e-15)

## Not run: 
v <- runif(2); 
M <- matrix(nrow=2,ncol=2,data=c(1,1,2,2),byrow=TRUE); Unif_v_exp_Q(t(v),M,1e-10)
M <- matrix(nrow=2,ncol=2,data=c(1,-1,-2,2),byrow=TRUE); Unif_v_exp_Q(t(v),M,1e-10)
d <- 5; f <- 0.3; ones <- rep(1,d); v <- runif(d)
Qs <- abs(rsparsematrix(d,d,f))
diag(Qs) <- 0; Qsum <- as.vector(Qs%*%ones); diag(Qs) <- -Qsum
Unif_v_exp_Q(v,Qs,1e-15)
Unif_v_exp_Q(t(v),Qs,1.5)
Unif_v_exp_Q(t(v),Qs,-2.0)
v <- -runif(d)
Unif_v_exp_Q(t(v),Qs,1e-15)

## End(Not run)

---

Product of horizontal vector and exponential of rate matrix

Description

Evaluates v exp(Q); automatically chooses between scaling and squaring or uniformisation.

Usage

v_exp_Q(v, Q, prec, renorm = TRUE, t2 = TRUE, checks = TRUE)

Arguments

v	Non-negative horizontal vector (dense).
Q	Rate matrix (sparse or dense).
prec	Required precision - missing mass in the Poisson distribution.
renorm	Force elements of each row to sum to 1? Defaults to TRUE.
t2	Perform two-tailed truncation? Defaults to TRUE.
checks	Perform sanity checks on the arguments? Defaults to TRUE.

Value

v exp(Q) Dense horizontal vector.

Author(s)

Chris Sherlock

Examples

v <- runif(2); v <- v/sum(v)
Qd <- matrix(nrow=2,ncol=2,data=c(-1,1,2,-2),byrow=TRUE); v_exp_Q(t(v),Qd,1e-10)

library("Matrix")
d <- 5; f=0.3; ones <- rep(1,d); v <- runif(d)
Qs <- abs(rsparsematrix(d,d,f))
diag(Qs) <- 0
Qsum <- as.vector(Qs%*%ones)
diag(Qs) <- -Qsum
v_exp_Q(t(v),Qs,1e-15)

## Not run: 
v <- runif(2); 
M <- matrix(nrow=2,ncol=2,data=c(1,1,2,2),byrow=TRUE); v_exp_Q(t(v),M,1e-10)
M <- matrix(nrow=2,ncol=2,data=c(1,-1,-2,2),byrow=TRUE); v_exp_Q(t(v),M,1e-10)
d <- 5; f=0.3; ones <- rep(1,d); v <- runif(d)
Qs <- abs(rsparsematrix(d,d,f))
diag(Qs) <- 0; Qsum <- as.vector(Qs%*%ones); diag(Qs) <- -Qsum
v_exp_Q(v,Qs,1e-15)
v_exp_Q(t(v),Qs,1.5)
v_exp_Q(t(v),Qs,-2.0)
v <- -runif(d)
v_exp_Q(t(v),Qs,1e-15)

## End(Not run)

---

Vector transpose * exponential of a rate matrix, all transposed.

Description

Evaluates [v' exp(Q)]'; automatically chooses between scaling and squaring or uniformisation.

Usage

vT_exp_Q(v, Q, prec, renorm = TRUE, t2 = TRUE, checks = TRUE)

Arguments

v	Non-negative vertical vector (dense).
Q	Rate matrix (sparse or dense).
prec	Required precision - missing mass in the Poisson distribution.
renorm	Force elements of each row to sum to 1? Defaults to TRUE.
t2	Perform two-tailed truncation? Defaults to TRUE.
checks	Perform sanity checks on the arguments? Defaults to TRUE.

Value

[v' exp(Q)]' Dense vertical vector.

Author(s)

Chris Sherlock

Examples

v <- runif(2); v <- v/sum(v)
Qd <- matrix(nrow=2,ncol=2,data=c(-1,1,2,-2),byrow=TRUE); vT_exp_Q(v,Qd,1e-10)

library("Matrix")
d <- 5; f <- 0.3; ones <- rep(1,d); v <- runif(d)
Qs <- abs(rsparsematrix(d,d,f))
diag(Qs) <- 0; Qsum <- as.vector(Qs%*%ones); diag(Qs) <- -Qsum
vT_exp_Q(v,Qs,1e-15)

## Not run: 
v <- runif(2); 
M <- matrix(nrow=2,ncol=2,data=c(1,1,2,2),byrow=TRUE); vT_exp_Q(v,M,1e-10)
M <- matrix(nrow=2,ncol=2,data=c(1,-1,-2,2),byrow=TRUE); vT_exp_Q(v,M,1e-10)
d <- 5; f <- 0.3; ones <- rep(1,d); v <- runif(d)
Qs=abs(rsparsematrix(d,d,f))
diag(Qs) <- 0; Qsum <- as.vector(Qs%*%ones); diag(Qs) <- -Qsum
vT_exp_Q(t(v),Qs,1e-15)
vT_exp_Q(v,Qs,1.5)
vT_exp_Q(v,Qs,-2.0)
v=-runif(d)
vT_exp_Q(v,Qs,1e-15)

## End(Not run)

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