Fast Negative Binomial Regression with Variance Calculation (C++ Backend)

Fits the same mean/dispersion-parameterized negative-binomial regression as fast_neg_bin_cpp (see that page, and fast_dnbinom_mu_vec_cpp, for the full model) and additionally computes the full variance-covariance matrix of c(beta, log(theta)).

Usage

fast_neg_bin_with_var_cpp(
  X,
  y,
  warm_start_params = NULL,
  smart_cold_start = FALSE,
  maxit = 1000L,
  eps_f = 1e-08,
  eps_g = 1e-06,
  fixed_idx = NULL,
  fixed_values = NULL,
  optimization_alg = "lbfgs",
  warm_start_fisher_info = NULL,
  estimate_only = FALSE
)

Arguments

X: A numeric matrix of predictors, \(n \times p\).
y: A numeric (integer-valued) vector of non-negative observed counts, length \(n\).
warm_start_params: Optional starting values for coefficients and dispersion. 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.
maxit: Maximum number of optimizer iterations.
eps_f: Convergence tolerance on the objective (log-likelihood) value.
eps_g: Convergence tolerance on the gradient norm.
fixed_idx: Optional integer indices (into the c(beta, log(theta)) 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: "lbfgs" (default) or "newton_raphson".
warm_start_fisher_info: Optional initial Fisher Information matrix to warm-start curvature information.

Value

A list containing the following components:

b: A numeric vector of the obtained Poisson regression coefficients.
hess_fisher_info_matrix: The Fisher information matrix.
neg_ll: The negative log-likelihood at the final iteration.
converged: A logical value indicating whether the final gradient norm was below the convergence tolerance (gradient_norm < tol); uniform across the "lbfgs"/"newton_raphson" optimizers.
num_iter: The number of optimizer iterations performed.
hit_iteration_cap: A logical value, mutually exclusive with converged: TRUE iff the optimizer exhausted maxit iterations without meeting the gradient-norm convergence criterion.
gradient_norm: The norm of the gradient at the returned parameters.

A list with components b (\(\hat\beta\)), theta_hat (\(\hat\theta\)), logLik, vcov (the full (p + 1) x (p + 1) parameter variance-covariance matrix), converged, iterations, and hess_fisher_info_matrix (the working-weights curvature matrix vcov was inverted from).

Details

Variance computation. The working-weights curvature matrix res.XtWX (over all p + 1 parameters) 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. A rank-deficient or near-singular design (after restricting to free parameters) will therefore produce numerically unstable or NaN variances rather than a graceful fallback.

Examples

X = matrix(rnorm(100), 10, 10)
y = rpois(10, 2)
fast_neg_bin_with_var_cpp(X, y)
#> $b
#>  [1]  -5.5535274   6.4091499   2.6718152   2.6447782  -0.7615799   5.2675376
#>  [7]   0.5837481  -8.7495596 -17.2772539 -10.2864877
#> 
#> $theta_hat
#> [1] 1094981920
#> 
#> $logLik
#> [1] -15.18427
#> 
#> $converged
#> [1] TRUE
#> 
#> $num_iter
#> [1] 76
#> 
#> $hit_iteration_cap
#> [1] FALSE
#> 
#> $gradient_norm
#> [1] 0.02793997
#> 
#> $min_eigenvalue_information
#> [1] NaN
#> 
#> $dispersion_at_poisson_boundary
#> [1] FALSE
#> 
#> $hess_fisher_info_matrix
#>                [,1]          [,2]          [,3]          [,4]          [,5]
#>  [1,]  1.408460e+01 -1.685258e+00  2.505326e+00 -2.436189e+00  1.137067e+01
#>  [2,] -1.685258e+00  8.569259e+00 -3.513348e+00 -9.310828e+00 -1.030916e+01
#>  [3,]  2.505326e+00 -3.513348e+00  4.705939e+01 -7.600849e+00  2.890151e+00
#>  [4,] -2.436189e+00 -9.310828e+00 -7.600849e+00  3.682673e+01  2.279544e+01
#>  [5,]  1.137067e+01 -1.030916e+01  2.890151e+00  2.279544e+01  7.072506e+01
#>  [6,]  1.938622e-01  9.859119e+00  7.163678e+00 -1.038931e+01 -1.807038e+00
#>  [7,]  7.500484e+00 -8.914326e-01  2.652133e+01 -2.199734e+01 -3.506756e-01
#>  [8,] -4.764506e+00  1.032350e+01  2.143270e+01 -1.460305e+01  1.151428e+00
#>  [9,] -7.280647e+00 -1.609342e+00 -2.370612e-02  1.315614e+01  1.467770e+00
#> [10,]  6.761537e+00  2.641441e+00 -7.357906e+00 -1.530877e+01 -1.786101e+01
#> [11,]  2.140978e-11 -2.463651e-13  3.619509e-11 -2.185004e-11 -3.817267e-11
#>                [,6]          [,7]          [,8]          [,9]         [,10]
#>  [1,]  1.938622e-01  7.500484e+00 -4.764506e+00 -7.280647e+00  6.761537e+00
#>  [2,]  9.859119e+00 -8.914326e-01  1.032350e+01 -1.609342e+00  2.641441e+00
#>  [3,]  7.163678e+00  2.652133e+01  2.143270e+01 -2.370612e-02 -7.357906e+00
#>  [4,] -1.038931e+01 -2.199734e+01 -1.460305e+01  1.315614e+01 -1.530877e+01
#>  [5,] -1.807038e+00 -3.506756e-01  1.151428e+00  1.467770e+00 -1.786101e+01
#>  [6,]  2.464169e+01  8.738664e+00  2.038722e+01  6.664992e+00 -1.049122e+01
#>  [7,]  8.738664e+00  4.452260e+01  1.639023e+01 -1.253240e+01  1.160082e+01
#>  [8,]  2.038722e+01  1.639023e+01  4.108681e+01 -3.435348e+00 -7.851036e+00
#>  [9,]  6.664992e+00 -1.253240e+01 -3.435348e+00  2.031220e+01 -2.222669e+01
#> [10,] -1.049122e+01  1.160082e+01 -7.851036e+00 -2.222669e+01  3.380938e+01
#> [11,]  1.799285e-11  4.121886e-11  2.942159e-11  5.678107e-12 -1.768595e-11
#>               [,11]
#>  [1,]  2.140978e-11
#>  [2,] -2.463651e-13
#>  [3,]  3.619509e-11
#>  [4,] -2.185004e-11
#>  [5,] -3.817267e-11
#>  [6,]  1.799285e-11
#>  [7,]  4.121886e-11
#>  [8,]  2.942159e-11
#>  [9,]  5.678107e-12
#> [10,] -1.768595e-11
#> [11,]  9.398988e-06
#> 
#> $vcov
#>                [,1]          [,2]          [,3]          [,4]          [,5]
#>  [1,]  1.829840e+00 -7.011402e+00 -9.410939e-01 -1.402299e+00  2.226417e-01
#>  [2,] -7.011402e+00  6.448245e+01  5.729446e+00  9.943168e+00 -9.679961e-01
#>  [3,] -9.410939e-01  5.729446e+00  6.567335e-01  9.763596e-01 -1.165606e-01
#>  [4,] -1.402299e+00  9.943168e+00  9.763596e-01  1.739573e+00 -2.420944e-01
#>  [5,]  2.226417e-01 -9.679961e-01 -1.165606e-01 -2.420944e-01  9.447547e-02
#>  [6,]  4.374795e-01 -1.504654e+01 -9.461926e-01 -1.877397e+00  3.709398e-02
#>  [7,] -1.726978e+00  1.637389e+01  1.399961e+00  2.537158e+00 -2.608159e-01
#>  [8,]  3.841848e+00 -2.476228e+01 -2.571085e+00 -4.214534e+00  5.610673e-01
#>  [9,]  5.751735e+00 -3.104477e+01 -3.486902e+00 -5.715806e+00  9.577810e-01
#> [10,]  4.861399e+00 -3.484455e+01 -3.399322e+00 -5.813569e+00  8.069721e-01
#> [11,] -2.700666e-06  2.479803e-06  9.913580e-07  9.295910e-07 -6.421313e-09
#>                [,6]          [,7]          [,8]          [,9]         [,10]
#>  [1,]  4.374795e-01 -1.726978e+00  3.841848e+00  5.751735e+00  4.861399e+00
#>  [2,] -1.504654e+01  1.637389e+01 -2.476228e+01 -3.104477e+01 -3.484455e+01
#>  [3,] -9.461926e-01  1.399961e+00 -2.571085e+00 -3.486902e+00 -3.399322e+00
#>  [4,] -1.877397e+00  2.537158e+00 -4.214534e+00 -5.715806e+00 -5.813569e+00
#>  [5,]  3.709398e-02 -2.608159e-01  5.610673e-01  9.577810e-01  8.069721e-01
#>  [6,]  5.060475e+00 -3.899040e+00  4.345709e+00  4.235991e+00  6.753729e+00
#>  [7,] -3.899040e+00  4.230201e+00 -6.230792e+00 -7.791962e+00 -8.848956e+00
#>  [8,]  4.345709e+00 -6.230792e+00  1.091158e+01  1.483929e+01  1.477061e+01
#>  [9,]  4.235991e+00 -7.791962e+00  1.483929e+01  2.142895e+01  1.995581e+01
#> [10,]  6.753729e+00 -8.848956e+00  1.477061e+01  1.995581e+01  2.051496e+01
#> [11,]  1.996668e-06  2.840109e-07 -3.339906e-06 -5.669579e-06 -2.945407e-06
#>               [,11]
#>  [1,] -2.700666e-06
#>  [2,]  2.479803e-06
#>  [3,]  9.913580e-07
#>  [4,]  9.295910e-07
#>  [5,] -6.421313e-09
#>  [6,]  1.996668e-06
#>  [7,]  2.840109e-07
#>  [8,] -3.339906e-06
#>  [9,] -5.669579e-06
#> [10,] -2.945407e-06
#> [11,]  1.063944e+05
#>