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]  48.2544592  54.4871481  29.2948470  58.4651075  51.9000504  37.2946414
#>   [7]  40.0252442  30.9806890  10.4651803   4.5098208  18.4556678 118.1984575
#>  [13]  30.9785280   4.2218147  48.7257672   3.8350982  13.6478549  16.8464738
#>  [19]  14.5617050  15.6778036  18.2642797  57.0860369  50.4779093  33.5803748
#>  [25]   7.8182864  48.9596929  20.1718510  29.3492426  12.3651057  35.5752997
#>  [31]  16.7273135  66.9839655  22.0691344  23.6967639  22.5898203  10.5927278
#>  [37]   2.6633303   2.5746736  15.4836023   3.6537907  47.4352913  34.1369533
#>  [43]  25.7966501  17.9912683  26.4463014  79.8250731  13.6475285  13.5304931
#>  [49]  39.9674254  81.6359217  42.2547585   6.6373829   2.3187327  16.8613547
#>  [55]  22.8295513  41.5723850   9.9314439  16.2153181  11.3454051  12.5142923
#>  [61]  10.7689013  10.2963957  43.8797247  12.2690452  46.8058054  31.5504886
#>  [67]  11.6898774   7.3922440  23.5122294  22.1635504  26.7085001  33.3231809
#>  [73]  25.2109909   7.5772314  46.3662775  23.7979601   5.5902934  16.4327036
#>  [79]  19.7443232   0.8461569  16.6701866  28.8192204   6.6763255  14.1513391
#>  [85]   6.8439557  36.1030322  12.5108712   9.8673244  41.0876664  61.2117062
#>  [91]  13.0025039  37.6769788   2.6743123  33.3814862  33.9094007  65.1975323
#>  [97]  42.7461948  45.1901357  42.3767793   3.5578897



# 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] 0.76672
ecumprob(y[1], sim[1, , drop = TRUE]) # approximation
#> [1] 0.77

# matrix:
cbind(
  real = pnbinom(y, mu = 3, size = 2), # real probability
  approx = ecumprob(y, sim) # approximation
)
#>            real approx
#>  [1,] 0.7667200 0.7700
#>  [2,] 0.9536426 0.9527
#>  [3,] 0.1600000 0.1606
#>  [4,] 0.9697669 0.9693
#>  [5,] 0.6630400 0.6652
#>  [6,] 0.5248000 0.5284
#>  [7,] 0.9294561 0.9319
#>  [8,] 0.8413696 0.8376
#>  [9,] 0.3520000 0.3442
#> [10,] 0.8936243 0.8972

# data.frame:
cbind(
  real = pnbinom(y, mu = 3, size = 2), # real probability
  approx = ecumprob(y, as.data.frame(sim)) # approximation
)
#>            real approx
#>  [1,] 0.7667200 0.7700
#>  [2,] 0.9536426 0.9527
#>  [3,] 0.1600000 0.1606
#>  [4,] 0.9697669 0.9693
#>  [5,] 0.6630400 0.6652
#>  [6,] 0.5248000 0.5284
#>  [7,] 0.9294561 0.9319
#>  [8,] 0.8413696 0.8376
#>  [9,] 0.3520000 0.3442
#> [10,] 0.8936243 0.8972