| qfratio-package {qfratio} | R Documentation |
qfratio: Moments and Distributions of Ratios of Quadratic Forms
Description
This package is for evaluating moments of ratios (and products) of quadratic forms in normal variables, specifically using recursive algorithms developed by Bao et al. (2013) and Hillier et al. (2014) (see also Smith, 1989, 1993; Hillier et al., 2009). It also provides some functions to evaluate distribution, quantile, and probability density functions of simple ratios of quadratic forms in normal variables using several algorithms. It was originally developed as a supplement to Watanabe (2023) for evaluating average evolvability measures in evolutionary quantitative genetics, but can be used for a broader class of statistics.
Details
The primary front-end functions of this package are
qfrm() and qfmrm() for evaluating moments of
ratios of quadratic forms. These pass arguments to one of the several
“internal” (though exported) functions which do actual calculations,
depending on the argument matrices and exponents. In addition, there are
a few functions to calculate moments of products
of quadratic forms (integer exponents only; qfpm).
There are many internal functions for calculating coefficients in
power-series expansion of generating functions for these moments
(d1_i, d2_ij, d3_ijk,
dtil2_pq) using “super-short” recursions
(Bao and Kan, 2013; Hillier et al. 2014). Some of these coefficients are
related to the top-order zonal and invariant
polynomials of matrix arguments.
The package also has some functions to evaluate distribution, quantile, and
density functions of simple ratios of quadratic forms: pqfr(),
qqfr(), and dqfr().
See package vignettes (vignette("qfratio") and
vignette("qfratio_distr")) for more details.
The DESCRIPTION file:
| Package: | qfratio |
| Type: | Package |
| Title: | Moments and Distributions of Ratios of Quadratic Forms Using Recursion |
| Version: | 1.1.1 |
| Date: | 2024-02-08 |
| Authors@R: | c(person("Junya", "Watanabe", email = "Junya.Watanabe@uab.cat", role = c("aut", "cre", "cph"), comment = c(ORCID = "0000-0002-9810-5286")), person("Patrick", "Alken", role = "cph", comment = "Author of bundled C codes from GSL"), person("Brian", "Gough", role = "cph", comment = "Author of bundled C codes from GSL"), person("Pavel", "Holoborodko", role = "cph", comment = "Author of bundled C codes from GSL"), person("Gerard", "Jungman", role = "cph", comment = "Author of bundled C codes from GSL"), person("Reid", "Priedhorsky", role = "cph", comment = "Author of bundled C codes from GSL"), person("Free Software Foundation, Inc.", role = "cph", comment = "Copyright holder of some bundled scripts")) |
| Description: | Evaluates moments of ratios (and products) of quadratic forms in normal variables, specifically using recursive algorithms developed by Bao and Kan (2013) <doi:10.1016/j.jmva.2013.03.002> and Hillier et al. (2014) <doi:10.1017/S0266466613000364>. Also provides distribution, quantile, and probability density functions of simple ratios of quadratic forms in normal variables with several algorithms. Originally developed as a supplement to Watanabe (2023) <doi:10.1007/s00285-023-01930-8> for evaluating average evolvability measures in evolutionary quantitative genetics, but can be used for a broader class of statistics. Generating functions for these moments are also closely related to the top-order zonal and invariant polynomials of matrix arguments. |
| License: | GPL (>= 3) |
| URL: | https://github.com/watanabe-j/qfratio |
| BugReports: | https://github.com/watanabe-j/qfratio/issues |
| Imports: | Rcpp, MASS, stats |
| LinkingTo: | Rcpp, RcppEigen |
| Suggests: | mvtnorm, CompQuadForm, graphics, testthat (>= 3.0.0), knitr, rmarkdown |
| Encoding: | UTF-8 |
| Roxygen: | list(markdown = TRUE) |
| RoxygenNote: | 7.2.3 |
| Config/testthat/edition: | 3 |
| VignetteBuilder: | knitr, rmarkdown |
| Author: | Junya Watanabe [aut, cre, cph] (<https://orcid.org/0000-0002-9810-5286>), Patrick Alken [cph] (Author of bundled C codes from GSL), Brian Gough [cph] (Author of bundled C codes from GSL), Pavel Holoborodko [cph] (Author of bundled C codes from GSL), Gerard Jungman [cph] (Author of bundled C codes from GSL), Reid Priedhorsky [cph] (Author of bundled C codes from GSL), Free Software Foundation, Inc. [cph] (Copyright holder of some bundled scripts) |
| Maintainer: | Junya Watanabe <Junya.Watanabe@uab.cat> |
Index of help topics:
KiK Matrix square root and generalized inverse
S_fromUL Make covariance matrix from eigenstructure
a1_pk Recursion for a_{p,k}
d1_i Coefficients in polynomial expansion of
generating function-single matrix
d2_ij Coefficients in polynomial expansion of
generating function-for ratios with two
matrices
d3_ijk Coefficients in polynomial expansion of
generating function-for ratios with three
matrices
dqfr Probability distribution of ratio of quadratic
forms
dtil2_pq Coefficients in polynomial expansion of
generating function-for products
hgs Calculate hypergeometric series
hyperg_1F1_vec_b Internal C++ wrappers for GSL
is_diagonal Is this matrix diagonal?
iseq Are these vectors equal?
new_qfrm Construct qfrm object
p_A1B1_Ed Internal C++ functions
print.qfrm Methods for qfrm and qfpm objects
qfmrm Moment of multiple ratio of quadratic forms in
normal variables
qfpm Moment of (product of) quadratic forms in
normal variables
qfratio-package qfratio: Moments and Distributions of Ratios of
Quadratic Forms
qfrm Moment of ratio of quadratic forms in normal
variables
range_qfr Get range of ratio of quadratic forms
rqfr Monte Carlo sampling of ratio/product of
quadratic forms
sum_counterdiag Summing up counter-diagonal elements
tr Matrix trace function
Author/Maintainer
Junya Watanabe Junya.Watanabe@uab.cat
References
Bao, Y. and Kan, R. (2013) On the moments of ratios of quadratic forms in normal random variables. Journal of Multivariate Analysis, 117, 229–245. doi:10.1016/j.jmva.2013.03.002.
Hillier, G., Kan, R. and Wang, X. (2009) Computationally efficient recursions for top-order invariant polynomials with applications. Econometric Theory, 25, 211–242. doi:10.1017/S0266466608090075.
Hillier, G., Kan, R. and Wang, X. (2014) Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors. Econometric Theory, 30, 436–473. doi:10.1017/S0266466613000364.
Smith, M. D. (1989) On the expectation of a ratio of quadratic forms in normal variables. Journal of Multivariate Analysis, 31, 244–257. doi:10.1016/0047-259X(89)90065-1.
Smith, M. D. (1993) Expectations of ratios of quadratic forms in normal variables: evaluating some top-order invariant polynomials. Australian Journal of Statistics, 35, 271–282. doi:10.1111/j.1467-842X.1993.tb01335.x.
Watanabe, J. (2023) Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing G matrices. Journal of Mathematical Biology, 86, 95. doi:10.1007/s00285-023-01930-8.
See Also
qfrm: Moment of simple ratio of quadratic forms
qfmrm: Moment of multiple ratio of quadratic forms
qfpm: Moment of product of quadratic forms
rqfr: Monte Carlo sampling of ratio/product of
quadratic forms
dqfr: Probability distribution of simple ratio of
quadratic forms
Examples
## Symmetric matrices
nv <- 4
A <- diag(nv:1)
B <- diag(sqrt(1:nv))
D <- diag((1:nv)^2 / nv)
mu <- nv:1 / nv
Sigma <- matrix(0.5, nv, nv)
diag(Sigma) <- 1
## Expectation of (x^T A x)^2 / (x^T x)^2 where x ~ N(0, I)
qfrm(A, p = 2)
## And a Monte Carlo mean of the same
mean(rqfr(1000, A = A, p = 2))
## Expectation of (x^T A x)^1/2 / (x^T x)^1/2 where x ~ N(0, I)
(res1 <- qfrm(A, p = 1/2))
plot(res1)
## A Monte Carlo mean
mean(rqfr(1000, A = A, p = 1/2))
## (x^T A x)^2 / (x^T B x)^3 where x ~ N(mu, Sigma)
(res2 <- qfrm(A, B, p = 2, q = 3, mu = mu, Sigma = Sigma))
plot(res2)
## A Monte Carlo mean
mean(rqfr(1000, A = A, B = B, p = 2, q = 3, mu = mu, Sigma = Sigma))
## Expectation of (x^T A x)^2 / (x^T B x) (x^T x) where x ~ N(0, I)
(res3 <- qfmrm(A, B, p = 2, q = 1, r = 1))
plot(res3)
## A Monte Carlo mean
mean(rqfmr(1000, A = A, B = B, p = 2, q = 1, r = 1))
## Expectation of (x^T A x)^2 where x ~ N(0, I)
qfm_Ap_int(A, 2)
## A Monte Carlo mean
mean(rqfp(1000, A = A, p = 2, q = 0, r = 0))
## Expectation of (x^T A x) (x^T B x) (x^T D x) where x ~ N(mu, I)
qfpm_ABDpqr_int(A, B, D, 1, 1, 1, mu = mu)
## A Monte Carlo mean
mean(rqfp(1000, A = A, B = B, D = D, p = 1, q = 1, r = 1, mu = mu))
## Distribution and quantile functions,
## and density of (x^T A x) / (x^T B x)
quantiles <- 0:nv + 0.5
(probs <- pqfr(quantiles, A, B))
qqfr(probs, A, B) # p = 1 yields maximum of ratio
dqfr(quantiles, A, B)