linear_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:

X is a matrix of multipliers/constants;
b is a vector of (correlated) random variables;
vc is the symmetric variance-covariance matrix for b;

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)

Examples

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] 49.0706755 65.8032271 76.4437582 39.8593463 13.1737286 24.3628771
#>   [7] 11.8459570 11.4448414  9.1628776  0.1546015  8.4631525 14.7537956
#>  [13]  6.8950235 42.1873647 49.0486180 41.6351654  0.8124381  7.4620321
#>  [19] 11.0627649 35.1679558  1.1229385 40.9219031  8.7348261 19.4231891
#>  [25] 55.2941143 22.2599487  2.9557715 27.7305067 21.0666640 86.2248366
#>  [31] 42.3849068 39.8681687 19.6657904 14.9414320 43.3934686 26.3381060
#>  [37]  3.9878743 18.1421002 56.0763015 40.0610014 25.1305299  3.1773472
#>  [43]  2.7688956  4.8272518  4.0477897 25.8295957  1.9236312 13.2302630
#>  [49] 42.6110916 11.2812626 58.0106502 39.3100611 46.7945818 21.1900023
#>  [55] 32.8478907 16.0152405  7.2167474  4.3743617 36.0288228 24.0423744
#>  [61] 87.1205360 13.5011296 12.0387070  8.3118710 19.9517694  8.8915565
#>  [67] 28.5677454 34.5421946  7.2926042 31.6007465 13.1324508 12.8711046
#>  [73]  4.8222093 20.6851882 30.9812484 19.3868736  7.5477604  4.0108589
#>  [79] 18.9728346 21.3791577 30.9258293 57.7142401  0.5482425 34.8329059
#>  [85]  4.8517703 46.6225518 44.5962330 19.9902029 21.2043312 52.0988058
#>  [91]  3.7301312 36.7045424 59.1991992 44.5727807  1.1103543 18.4224112
#>  [97] 25.2193899  5.8939379 30.8571234 29.9376412



# ecumprob() ====

sim <- rnbinom(10 * 1e4, mu = 3, size = 2) |> matrix(10, 1e4)
y <- sample(0:9)

# vector:
pnbinom(y[1], mu = 3, size = 2) # real probability
#> [1] 0.8936243
ecumprob(y[1], sim[1, , drop = TRUE]) # approximation
#> [1] 0.8927

# matrix:
cbind(
  real = pnbinom(y, mu = 3, size = 2), # real probability
  approx = ecumprob(y, sim) # approximation
)
#>            real approx
#>  [1,] 0.8936243 0.8927
#>  [2,] 0.9697669 0.9718
#>  [3,] 0.9536426 0.9555
#>  [4,] 0.3520000 0.3455
#>  [5,] 0.6630400 0.6679
#>  [6,] 0.9294561 0.9313
#>  [7,] 0.1600000 0.1615
#>  [8,] 0.7667200 0.7664
#>  [9,] 0.8413696 0.8451
#> [10,] 0.5248000 0.5246

# data.frame:
cbind(
  real = pnbinom(y, mu = 3, size = 2), # real probability
  approx = ecumprob(y, as.data.frame(sim)) # approximation
)
#>            real approx
#>  [1,] 0.8936243 0.8927
#>  [2,] 0.9697669 0.9718
#>  [3,] 0.9536426 0.9555
#>  [4,] 0.3520000 0.3455
#>  [5,] 0.6630400 0.6679
#>  [6,] 0.9294561 0.9313
#>  [7,] 0.1600000 0.1615
#>  [8,] 0.7667200 0.7664
#>  [9,] 0.8413696 0.8451
#> [10,] 0.5248000 0.5246