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]  30.698031   4.357788  66.406404  12.913888  55.766713  38.673499
#>   [7]  12.717317   9.694643  28.722619  13.224355  33.786248  17.620245
#>  [13]  31.463111  39.200800  75.500716  47.762114  33.999742  69.964886
#>  [19]  56.693844   3.251399  12.668467  13.451583  23.168955  46.989004
#>  [25]  80.702642  15.138313  51.018509  54.942609  42.944538  18.037987
#>  [31]  24.960331  13.844087  20.882174   5.774570  10.311660  48.174602
#>  [37]  45.775125  15.018117   2.821280  26.988757  18.169083  49.187204
#>  [43]  12.865036   5.538522   0.699704  21.542838  28.542861  15.101262
#>  [49]  34.790119  38.025907  10.790279  79.953283  26.496744  41.710355
#>  [55]  69.214297  40.333599  10.233233  41.527494   4.371908  38.686342
#>  [61]   6.024879  44.714836  29.686667  21.585547  41.228308  35.514384
#>  [67]  21.068557  31.996302   3.341195  18.136852 106.370668  15.036687
#>  [73]   9.895431  14.862378  32.029046  49.826197   8.049538  23.417131
#>  [79]  39.186557  24.552408  56.451133  19.041888  13.661682  66.643353
#>  [85]  29.627287  44.412448   5.868856  32.843642  24.741527  66.915733
#>  [91]  39.515633  34.986596  86.466697  37.098539  11.146431  15.652303
#>  [97]  30.409921  70.437248   9.278315  16.234126



# 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] 0.8413696
ecumprob(y[1], sim[1, , drop = TRUE]) # approximation
#> [1] 0.8456

# matrix:
cbind(
  real = pnbinom(y, mu = 3, size = 2), # real probability
  approx = ecumprob(y, sim) # approximation
)
#>            real approx
#>  [1,] 0.8413696 0.8456
#>  [2,] 0.7667200 0.7607
#>  [3,] 0.1600000 0.1564
#>  [4,] 0.8936243 0.8943
#>  [5,] 0.5248000 0.5267
#>  [6,] 0.9294561 0.9294
#>  [7,] 0.3520000 0.3479
#>  [8,] 0.9536426 0.9554
#>  [9,] 0.6630400 0.6671
#> [10,] 0.9697669 0.9691

# data.frame:
cbind(
  real = pnbinom(y, mu = 3, size = 2), # real probability
  approx = ecumprob(y, as.data.frame(sim)) # approximation
)
#>            real approx
#>  [1,] 0.8413696 0.8456
#>  [2,] 0.7667200 0.7607
#>  [3,] 0.1600000 0.1564
#>  [4,] 0.8936243 0.8943
#>  [5,] 0.5248000 0.5267
#>  [6,] 0.9294561 0.9294
#>  [7,] 0.3520000 0.3479
#>  [8,] 0.9536426 0.9554
#>  [9,] 0.6630400 0.6671
#> [10,] 0.9697669 0.9691