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 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 and inverted via a plain matrix inverse (.inverse(), not a rank-aware pseudo-inverse as used by, e.g., fast_adjacent_category_logit_with_var_cpp) before being expanded back to the full (p + 1) x (p + 1) size as vcov; std_errs is sqrt(diag(vcov)). Because this uses a plain inverse, a rank-deficient or near-singular X (after restricting to free parameters) will produce numerically unstable or NaN standard errors rather than a graceful fallback — callers should ensure X is full rank on the free parameters (e.g. via the package's shared drop_linearly_dependent_cols() preprocessing) before calling this function if that is not already guaranteed.

Examples

X = matrix(rnorm(100), 10, 10)
y = runif(10)
fast_beta_regression_with_var_cpp(X, y)
#> $coefficients
#>  [1]  0.1404631  0.4523441  0.4706064  0.8154016 -0.2878395 -0.6651487
#>  [7] -0.1563096  0.3101339  2.7315070  0.3901471
#> 
#> $phi
#> [1] 1.387335e+12
#> 
#> $neg_loglik
#> [1] -138.4185
#> 
#> $vcov
#>                [,1]          [,2]          [,3]          [,4]          [,5]
#>  [1,] -2.432272e-12 -1.594200e-12 -2.292889e-12  5.542899e-12 -5.265807e-12
#>  [2,] -1.594200e-12  2.651966e-13 -9.546776e-13  3.591789e-12 -2.898714e-12
#>  [3,] -2.292889e-12 -9.546776e-13 -8.143168e-13  5.814217e-12 -5.057101e-12
#>  [4,]  5.542899e-12  3.591789e-12  5.814217e-12 -7.049834e-12  6.104040e-12
#>  [5,] -5.265807e-12 -2.898714e-12 -5.057101e-12  6.104040e-12 -4.171920e-12
#>  [6,]  6.937944e-12  3.456799e-12  5.754786e-12 -1.356846e-11  1.217487e-11
#>  [7,] -3.730488e-12 -2.005776e-12 -4.595015e-12  4.521445e-12 -3.700393e-12
#>  [8,] -8.142827e-12 -3.819782e-12 -6.989001e-12  1.480505e-11 -1.271826e-11
#>  [9,] -1.064552e-11 -4.358952e-12 -8.532547e-12  2.009427e-11 -1.727815e-11
#> [10,] -4.170627e-12 -2.126162e-12 -3.867059e-12  6.941165e-12 -5.833520e-12
#> [11,]  3.054574e-06  1.497290e-06  2.813415e-06 -4.673664e-06  3.832003e-06
#>                [,6]          [,7]          [,8]          [,9]         [,10]
#>  [1,]  6.937944e-12 -3.730488e-12 -8.142827e-12 -1.064552e-11 -4.170627e-12
#>  [2,]  3.456799e-12 -2.005776e-12 -3.819782e-12 -4.358952e-12 -2.126162e-12
#>  [3,]  5.754786e-12 -4.595015e-12 -6.989001e-12 -8.532547e-12 -3.867059e-12
#>  [4,] -1.356846e-11  4.521445e-12  1.480505e-11  2.009427e-11  6.941165e-12
#>  [5,]  1.217487e-11 -3.700393e-12 -1.271826e-11 -1.727815e-11 -5.833520e-12
#>  [6,] -1.655401e-11  9.140006e-12  1.987781e-11  2.512249e-11  1.030907e-11
#>  [7,]  9.140006e-12 -7.593776e-13 -9.424455e-12 -1.258573e-11 -4.076583e-12
#>  [8,]  1.987781e-11 -9.424455e-12 -2.210596e-11 -2.878315e-11 -1.130793e-11
#>  [9,]  2.512249e-11 -1.258573e-11 -2.878315e-11 -3.388207e-11 -1.428772e-11
#> [10,]  1.030907e-11 -4.076583e-12 -1.130793e-11 -1.428772e-11 -5.253806e-12
#> [11,] -7.283302e-06  2.699988e-06  7.869743e-06  1.022158e-05  3.792228e-06
#>               [,11]
#>  [1,]  3.054574e-06
#>  [2,]  1.497290e-06
#>  [3,]  2.813415e-06
#>  [4,] -4.673664e-06
#>  [5,]  3.832003e-06
#>  [6,] -7.283302e-06
#>  [7,]  2.699988e-06
#>  [8,]  7.869743e-06
#>  [9,]  1.022158e-05
#> [10,]  3.792228e-06
#> [11,] -1.216187e+00
#> 
#> $std_errs
#>  [1]          NaN 5.149724e-07          NaN          NaN          NaN
#>  [6]          NaN          NaN          NaN          NaN          NaN
#> [11]          NaN
#> 
#> $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,]  4.994758e+12  7.865334e+11 -1.082724e+12  9.700346e+11  2.025222e+12
#>  [2,]  7.865334e+11  2.355959e+12  3.918558e+10 -8.846065e+11  2.552485e+10
#>  [3,] -1.082724e+12  3.918558e+10  2.317478e+12 -7.896130e+11 -6.097834e+11
#>  [4,]  9.700346e+11 -8.846065e+11 -7.896130e+11  2.259217e+12  1.465497e+12
#>  [5,]  2.025222e+12  2.552485e+10 -6.097834e+11  1.465497e+12  2.035003e+12
#>  [6,]  7.839869e+11 -3.968795e+10  2.498100e+11 -5.054867e+11 -1.717808e+11
#>  [7,] -1.763221e+11 -2.109047e+11  9.101545e+11  1.108282e+11 -4.944108e+10
#>  [8,] -1.157129e+12 -1.053941e+12 -3.890328e+11  1.910187e+11 -6.162292e+11
#>  [9,]  7.076543e+11 -2.458972e+11 -1.883183e+11 -4.384314e+10  3.012147e+11
#> [10,] -2.291513e+12  2.117587e+11 -5.541052e+10  1.070313e+11 -1.253308e+12
#> [11,] -1.098245e+05 -1.032828e+04  5.465138e+04 -6.919683e+04  4.264785e+04
#>                [,6]          [,7]          [,8]          [,9]         [,10]
#>  [1,]  7.839869e+11 -1.763221e+11 -1.157129e+12  7.076543e+11 -2.291513e+12
#>  [2,] -3.968795e+10 -2.109047e+11 -1.053941e+12 -2.458972e+11  2.117587e+11
#>  [3,]  2.498100e+11  9.101545e+11 -3.890328e+11 -1.883183e+11 -5.541052e+10
#>  [4,] -5.054867e+11  1.108282e+11  1.910187e+11 -4.384314e+10  1.070313e+11
#>  [5,] -1.717808e+11 -4.944108e+10 -6.162292e+11  3.012147e+11 -1.253308e+12
#>  [6,]  1.493552e+12 -2.809201e+10  6.370060e+11  4.916500e+11 -9.982104e+11
#>  [7,] -2.809201e+10  1.008676e+12 -1.553938e+11  3.153111e+10 -7.705534e+11
#>  [8,]  6.370060e+11 -1.553938e+11  2.195104e+12 -1.432496e+11 -2.953051e+11
#>  [9,]  4.916500e+11  3.153111e+10 -1.432496e+11  6.985410e+11 -1.345994e+12
#> [10,] -9.982104e+11 -7.705534e+11 -2.953051e+11 -1.345994e+12  6.091420e+12
#> [11,]  3.419044e+04  8.561741e+04  1.401742e+05  2.926163e+04  5.454706e+04
#>               [,11]
#>  [1,] -1.098245e+05
#>  [2,] -1.032828e+04
#>  [3,]  5.465138e+04
#>  [4,] -6.919683e+04
#>  [5,]  4.264785e+04
#>  [6,]  3.419044e+04
#>  [7,]  8.561741e+04
#>  [8,]  1.401742e+05
#>  [9,]  2.926163e+04
#> [10,]  5.454706e+04
#> [11,]  7.243058e-01
#>

Usage

Arguments

Value

Details

See also

Examples