Fast Beta Regression with Variance Calculation (C++ Backend)

Fits the same beta regression model as fast_beta_regression_cpp (see that page for the full model, parameterization, and optimizer contract) and additionally computes the variance-covariance matrix and standard errors of the fitted parameters, via the same working-weights (\(X^\top W X\)) curvature matrix documented there.

Usage

fast_beta_regression_with_var_cpp(
  X,
  y,
  warm_start_beta = NULL,
  smart_cold_start = TRUE,
  start_phi = 10,
  compute_std_errs = TRUE,
  fixed_idx = NULL,
  fixed_values = NULL,
  optimization_alg = "lbfgs",
  warm_start_fisher_info = NULL
)

Arguments

X: A numeric matrix of predictors, \(n \times p\).
y: A numeric vector of responses, strictly in \((0, 1)\).
warm_start_beta: Optional starting values for coefficients. If provided, smart_cold_start is ignored.
smart_cold_start: Logical. If TRUE, use an initial OLS-based guess when starting from scratch (a "cold start") with no prior knowledge. This is ignored if a warm start is provided.
start_phi: Starting value for the precision parameter \(\phi\) (natural scale).
compute_std_errs: Deprecated; standard errors are always computed by this entry point regardless of this argument's value.
fixed_idx: Optional integer indices (into the c(beta, log(phi)) parameter layout) of parameters to hold fixed rather than estimate.
fixed_values: Optional values to fix the parameters named by fixed_idx at.
optimization_alg: Optimization algorithm; see fast_beta_regression_cpp.

Value

A list containing the following components:

coefficients: A numeric vector of the obtained Poisson regression coefficients.
phi: The estimated precision parameter phi.
vcov: The variance-covariance matrix.
neg_ll: The negative log-likelihood at the final iteration.
converged: A logical value indicating whether the algorithm converged.

A list with components coefficients (\(\hat\beta\)), phi (\(\hat\phi\)), neg_loglik, vcov (the full (p + 1) x (p + 1) parameter variance-covariance matrix), std_errs (sqrt(diag(vcov))), converged (logical), and information_invertible (whether the free information block passed the invertibility check), and fisher_information (the working-weights curvature matrix vcov was inverted from).

Details

Variance computation. The fit's working-information matrix (fit.XtWX, over all p + 1 parameters c(beta, log(phi))) is restricted to the free (non-fixed_idx) parameters. A FullPivLU invertibility check guards the same plain matrix inverse used previously, so invertible fits retain their existing covariance values. A singular or non-finite free block returns a NaN covariance and standard errors; callers should check is.finite(vcov) before using its entries.

Examples

X = matrix(rnorm(100), 10, 10)
y = runif(10)
fast_beta_regression_with_var_cpp(X, y)
#> $coefficients
#>  [1]  -55.83039 -111.85338  254.11466   36.35758 -275.89588  -53.31728
#>  [7]  111.83331  -76.94184 -407.27657  -60.45378
#> 
#> $phi
#> [1] 228619673178
#> 
#> $neg_loglik
#> [1] -127.2674
#> 
#> $vcov
#>                [,1]          [,2]          [,3]          [,4]          [,5]
#>  [1,] -6.217188e-08 -1.248597e-07  2.864423e-07  4.073676e-08 -3.110189e-07
#>  [2,] -1.248597e-07 -2.507377e-07  5.752312e-07  8.180236e-08 -6.245863e-07
#>  [3,]  2.864423e-07  5.752312e-07 -1.319566e-06 -1.876419e-07  1.432787e-06
#>  [4,]  4.073676e-08  8.180236e-08 -1.876419e-07 -2.667153e-08  2.037452e-07
#>  [5,] -3.110189e-07 -6.245863e-07  1.432787e-06  2.037452e-07 -1.555712e-06
#>  [6,] -6.024826e-08 -1.209776e-07  2.775224e-07  3.946146e-08 -3.013304e-07
#>  [7,]  1.254513e-07  2.519368e-07 -5.779072e-07 -8.217701e-08  6.274909e-07
#>  [8,] -8.397318e-08 -1.686409e-07  3.868780e-07  5.501074e-08 -4.200829e-07
#>  [9,] -4.585204e-07 -9.208006e-07  2.112308e-06  3.003768e-07 -2.293543e-06
#> [10,] -6.726702e-08 -1.350847e-07  3.098896e-07  4.406141e-08 -3.364842e-07
#> [11,]  1.653353e-04  3.320560e-04 -7.617187e-04 -1.083037e-04  8.271053e-04
#>                [,6]          [,7]          [,8]          [,9]         [,10]
#>  [1,] -6.024826e-08  1.254513e-07 -8.397318e-08 -4.585204e-07 -6.726702e-08
#>  [2,] -1.209776e-07  2.519368e-07 -1.686409e-07 -9.208006e-07 -1.350847e-07
#>  [3,]  2.775224e-07 -5.779072e-07  3.868780e-07  2.112308e-06  3.098896e-07
#>  [4,]  3.946146e-08 -8.217701e-08  5.501074e-08  3.003768e-07  4.406141e-08
#>  [5,] -3.013304e-07  6.274909e-07 -4.200829e-07 -2.293543e-06 -3.364842e-07
#>  [6,] -5.835047e-08  1.215464e-07 -8.137923e-08 -4.442483e-07 -6.517760e-08
#>  [7,]  1.215464e-07 -2.530775e-07  1.694346e-07  9.250918e-07  1.357184e-07
#>  [8,] -8.137923e-08  1.694346e-07 -1.133869e-07 -6.192952e-07 -9.083899e-08
#>  [9,] -4.442483e-07  9.250918e-07 -6.192952e-07 -3.381287e-06 -4.960604e-07
#> [10,] -6.517760e-08  1.357184e-07 -9.083899e-08 -4.960604e-07 -7.276489e-08
#> [11,]  1.602565e-04 -3.335670e-04  2.232100e-04  1.219330e-03  1.788385e-04
#>               [,11]
#>  [1,]  0.0001653353
#>  [2,]  0.0003320560
#>  [3,] -0.0007617187
#>  [4,] -0.0001083037
#>  [5,]  0.0008271053
#>  [6,]  0.0001602565
#>  [7,] -0.0003335670
#>  [8,]  0.0002232100
#>  [9,]  0.0012193295
#> [10,]  0.0001788385
#> [11,] -0.2197364468
#> 
#> $std_errs
#>  [1] NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
#> 
#> $information_invertible
#> [1] TRUE
#> 
#> $converged
#> [1] FALSE
#> 
#> $num_iter
#> [1] 1000
#> 
#> $hit_iteration_cap
#> [1] FALSE
#> 
#> $gradient_norm
#> [1] NaN
#> 
#> $min_eigenvalue_information
#> [1] NaN
#> 
#> $fisher_information
#>                [,1]          [,2]          [,3]          [,4]          [,5]
#>  [1,]  8.672470e+11 -2.190599e+11  4.321716e+10 -2.219666e+11 -2.084816e+11
#>  [2,] -2.190599e+11  2.442975e+11  1.474870e+10 -3.787606e+10 -1.517448e+10
#>  [3,]  4.321716e+10  1.474870e+10  7.378529e+11 -2.761832e+11 -6.011291e+10
#>  [4,] -2.219666e+11 -3.787606e+10 -2.761832e+11  3.720351e+11  1.540186e+11
#>  [5,] -2.084816e+11 -1.517448e+10 -6.011291e+10  1.540186e+11  3.205095e+11
#>  [6,]  2.868647e+11 -1.015707e+11 -1.728329e+10 -1.716611e+09 -6.985003e+10
#>  [7,] -7.194255e+10 -9.873166e+10 -7.108883e+10  5.833609e+10  9.139662e+10
#>  [8,] -8.503299e+10 -1.953403e+10 -1.567645e+11  8.462209e+10  1.155524e+11
#>  [9,]  5.239667e+10 -1.988048e+10  4.524140e+11 -2.216654e+11 -2.006659e+11
#> [10,] -2.355436e+10 -7.422379e+10  1.800844e+11  1.282500e+11  3.283893e+10
#> [11,] -1.635101e+04  5.503996e+02 -1.003498e+04  9.450008e+03  9.116706e+03
#>                [,6]          [,7]          [,8]          [,9]         [,10]
#>  [1,]  2.868647e+11 -7.194255e+10 -8.503299e+10  5.239667e+10 -2.355436e+10
#>  [2,] -1.015707e+11 -9.873166e+10 -1.953403e+10 -1.988048e+10 -7.422379e+10
#>  [3,] -1.728329e+10 -7.108883e+10 -1.567645e+11  4.524140e+11  1.800844e+11
#>  [4,] -1.716611e+09  5.833609e+10  8.462209e+10 -2.216654e+11  1.282500e+11
#>  [5,] -6.985003e+10  9.139662e+10  1.155524e+11 -2.006659e+11  3.283893e+10
#>  [6,]  2.184145e+11 -3.800770e+10  3.449403e+10 -1.689430e+10 -2.271004e+10
#>  [7,] -3.800770e+10  1.502179e+11  4.526684e+10 -2.869832e+10  1.308288e+10
#>  [8,]  3.449403e+10  4.526684e+10  1.754539e+11 -1.506423e+11 -1.743443e+11
#>  [9,] -1.689430e+10 -2.869832e+10 -1.506423e+11  4.058813e+11  9.153157e+10
#> [10,] -2.271004e+10  1.308288e+10 -1.743443e+11  9.153157e+10  4.906945e+11
#> [11,] -8.968773e+03 -2.111567e+03  1.651672e+04 -1.048676e+04  1.083451e+03
#>               [,11]
#>  [1,] -16351.014499
#>  [2,]    550.399596
#>  [3,] -10034.976598
#>  [4,]   9450.008105
#>  [5,]   9116.705913
#>  [6,]  -8968.772674
#>  [7,]  -2111.566718
#>  [8,]  16516.720797
#>  [9,] -10486.764827
#> [10,]   1083.451445
#> [11,]      4.554915
#>

Usage

Arguments

Value

Details

See also

Examples