library("broadcast")
# variances ====
vc <- datasets::ability.cov$cov
X <- matrix(rnorm(100), 100, ncol(vc))
solve(vc)
#> general picture blocks maze reading
#> general 0.082259001 -0.0312406436 -7.750932e-03 -0.013309494 -2.061705e-02
#> picture -0.031240644 0.2369906996 -2.484938e-02 0.017844845 8.603286e-04
#> blocks -0.007750932 -0.0248493822 1.344272e-02 -0.012544830 -3.802671e-05
#> maze -0.013309494 0.0178448450 -1.254483e-02 0.101625400 5.508423e-03
#> reading -0.020617049 0.0008603286 -3.802671e-05 0.005508423 5.713620e-02
#> vocab -0.002420800 0.0019394999 -1.157864e-03 -0.002857265 -2.406969e-02
#> vocab
#> general -0.002420800
#> picture 0.001939500
#> blocks -0.001157864
#> maze -0.002857265
#> reading -0.024069692
#> vocab 0.020323179
cinv(vc) # faster than `solve()`, but only works on positive definite matrices
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.082259001 -0.0312406436 -7.750932e-03 -0.013309494 -2.061705e-02
#> [2,] -0.031240644 0.2369906996 -2.484938e-02 0.017844845 8.603286e-04
#> [3,] -0.007750932 -0.0248493822 1.344272e-02 -0.012544830 -3.802671e-05
#> [4,] -0.013309494 0.0178448450 -1.254483e-02 0.101625400 5.508423e-03
#> [5,] -0.020617049 0.0008603286 -3.802671e-05 0.005508423 5.713620e-02
#> [6,] -0.002420800 0.0019394999 -1.157864e-03 -0.002857265 -2.406969e-02
#> [,6]
#> [1,] -0.002420800
#> [2,] 0.001939500
#> [3,] -0.001157864
#> [4,] -0.002857265
#> [5,] -0.024069692
#> [6,] 0.020323179
all(round(solve(vc), 6) == round(cinv(vc), 6)) # they're the same
#> [1] TRUE
sd_lc(X, vc)
#> [1] 7.3978139 8.5562241 23.4813202 37.0200876 1.8527980 23.2205426
#> [7] 27.3223905 6.5229828 43.8069255 50.1674944 35.5987186 6.0915392
#> [13] 21.4330611 47.6309275 46.1383939 29.0764456 8.3933124 54.0515286
#> [19] 39.0448275 56.4971659 51.3489457 24.3651302 8.3340032 38.8741622
#> [25] 4.8848476 50.9637868 7.0462990 9.5544009 39.1757830 44.7270516
#> [31] 19.6361143 76.1580167 13.3409997 40.6154667 1.4984483 9.9031805
#> [37] 30.2642762 13.6425174 56.2417052 51.1165565 20.2224610 64.0632474
#> [43] 17.7065577 68.8625997 32.8261753 2.4319083 57.5230301 4.3774075
#> [49] 12.5240579 5.6114562 11.0791800 26.1515801 23.3726756 70.5666741
#> [55] 3.0389326 17.7327386 50.9374994 32.7689507 8.9913872 32.1303183
#> [61] 10.5939434 20.7785952 8.6814126 38.9682782 19.3492788 10.7994349
#> [67] 6.5609018 11.2200944 17.4845933 38.2725564 43.0540951 12.7915536
#> [73] 2.7722921 25.4486733 21.3140400 34.5052373 35.8189716 23.3139035
#> [79] 68.7793517 8.9706483 26.1242820 0.2557657 18.6770648 23.6536179
#> [85] 25.4251194 51.6149125 59.9393520 2.9612636 66.7549812 30.9224210
#> [91] 94.8526480 1.6316646 4.2035132 21.8754098 27.7988063 4.0590443
#> [97] 17.2121908 30.7945865 19.4061724 29.9342595
# ecumprob() ====
sim <- rnbinom(10 * 5000L, mu = 3, size = 2) |> matrix(10, 5000)
y <- sample(0:9)
# vector:
pnbinom(y[1], mu = 3, size = 2) # real probability
#> [1] 0.9697669
ecumprob(y[1], sim[1, , drop = TRUE]) # approximation
#> [1] 0.9752
# matrix:
pnbinom(y, mu = 3, size = 2) # real probability
#> [1] 0.9697669 0.7667200 0.9294561 0.6630400 0.3520000 0.5248000 0.8413696
#> [8] 0.9536426 0.8936243 0.1600000
ecumprob(y, sim) # approximation
#> [1] 0.9752 0.7574 0.9284 0.6634 0.3616 0.5336 0.8432 0.9582 0.8928 0.1562
# data.frame:
pnbinom(y, mu = 3, size = 2) # real probability
#> [1] 0.9697669 0.7667200 0.9294561 0.6630400 0.3520000 0.5248000 0.8413696
#> [8] 0.9536426 0.8936243 0.1600000
ecumprob(y, as.data.frame(sim)) # approximation
#> [1] 0.9752 0.7574 0.9284 0.6634 0.3616 0.5336 0.8432 0.9582 0.8928 0.1562linear_algebra_stats
Simple Linear Algebra Functions for Statistics
Description
‘broadcast’ provides some simple Linear Algebra Functions for Statistics:
cinv()
sd_lc()
ecumprob()
Usage
cinv(x)
sd_lc(X, vc, bad_rp = NaN)
ecumprob(y, sim, eps = 0)
Arguments
x
|
a real symmetric positive-definite square matrix. |
X
|
a numeric (or logical) matrix of multipliers/constants |
vc
|
the variance-covariance matrix for the (correlated) random variables. |
bad_rp
|
if vc is not a Positive (semi-) Definite matrix, give here the value to replace bad standard deviations with. |
y
|
values to estimate the cumulative probability for. |
sim
|
a matrix (or data.frame) with at least 500 columns of simulated values. If sim is given as a dimensionless vector, it will be treated as a matrix with 1 row and length(sim) columns, and this will be noted with a message.
|
eps
|
a non-negative numeric scaler smaller than 0.1, giving the cut-off value for probabilities. Probabilities smaller than eps will be replaced with eps, and probabilities larger than 1 - eps will be replaced with 1 - eps. Set eps = 0 to disable probability trimming.
|
Details
cinv()
cinv() computes the Choleski inverse of a real symmetric positive-definite square matrix.
sd_lc()
Given the linear combination X %*% b, where:
-
Xis a matrix of multipliers/constants; -
bis a vector of (correlated) random variables; -
vcis the symmetric variance-covariance matrix forb;
sd_lc(X, vc) computes the standard deviations for the linear combination X %*% b, without making needless copies.
sd_lc(X, vc) will use much less memory than a base ‘R’ approach.
sd_lc(X, vc) will usually be faster than a base ‘R’ approach (depending on the Linear Algebra Library used for base ‘R’).
ecumprob()
The ecumprod(y, sim) function takes a matrix (or data.frame) of simulated values sim, and for each row i (after broadcasting), estimates the cumulative distribution function of sim[i, ], and returns the cumulative probability for y[i].
In terms of statistics, it is equivalent to the following operation for each index i:
ecdf(sim[i,])(y[i])
However, ecumprob() is much faster, and supports NAs/NaNs.
In terms of linear algebra, it is equivalent to the following broadcasted operation:
rowMeans(sim <= y)
where y and sim are broadcaster arrays.
However, ecumprob() is much more memory-efficient, supports a data.frame for sim, and has statistical safety checks.
Value
For cinv():
A matrix.
For sd_lc():
A vector of standard deviations.
For ecumprob():
A vector of cumulative probabilities.
If for any observation i (after broadcasting,) y[i] is NA/NaN or any of sim[i,] is NA/NaN, the result for i will be NA.
If zero-length y or sim is given, a zero-length numeric vector is returned.
References
John A. Rice (2007), Mathematical Statistics and Data Analysis (6th Edition)
See Also
chol, chol2inv