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), information_invertible (whether its free information block passed the invertibility check), 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. A FullPivLU invertibility check guards the same plain matrix inverse used previously, so invertible fits retain their existing covariance values while a singular or non-finite free block returns a NaN covariance. Callers should check is.finite(vcov) before using its entries.

Examples

X = matrix(rnorm(100), 10, 10)
y = rpois(10, 2)
fast_neg_bin_with_var_cpp(X, y)
#> $b
#>  [1] -0.6046581 -0.9185522  3.4926759  3.7390077  0.9959934 -0.1752641
#>  [7]  2.6662493 -1.9580572  4.7232755 -5.1020716
#> 
#> $theta_hat
#> [1] 5945276
#> 
#> $logLik
#> [1] -11.66992
#> 
#> $converged
#> [1] TRUE
#> 
#> $num_iter
#> [1] 60
#> 
#> $hit_iteration_cap
#> [1] FALSE
#> 
#> $gradient_norm
#> [1] 0.02391691
#> 
#> $min_eigenvalue_information
#> [1] NaN
#> 
#> $dispersion_at_poisson_boundary
#> [1] FALSE
#> 
#> $information_invertible
#> [1] TRUE
#> 
#> $hess_fisher_info_matrix
#>                [,1]          [,2]          [,3]          [,4]          [,5]
#>  [1,]  2.294478e+01 -7.847226e+00 -2.816324e+00 -5.462212e+00  5.215446e+00
#>  [2,] -7.847226e+00  1.184595e+01 -4.722332e+00  9.066051e+00 -3.802423e+00
#>  [3,] -2.816324e+00 -4.722332e+00  1.429149e+01 -1.188275e+01  1.340130e-01
#>  [4,] -5.462212e+00  9.066051e+00 -1.188275e+01  2.995602e+01 -2.264010e+00
#>  [5,]  5.215446e+00 -3.802423e+00  1.340130e-01 -2.264010e+00  1.551828e+01
#>  [6,] -1.021573e+01  1.482848e+00 -3.408698e+00  7.489872e+00 -4.769824e+00
#>  [7,] -4.288844e+00  1.903619e+00 -1.692995e+00  2.181457e+00 -8.056131e+00
#>  [8,]  3.600625e+00 -6.706271e+00  4.831086e+00 -9.099955e-01  2.142658e-01
#>  [9,] -3.014193e+00  4.849642e+00  5.413902e+00 -6.627341e+00 -1.711735e+00
#> [10,] -1.569689e+01  1.221533e+01  3.446787e+00  1.052159e+01 -5.261873e+00
#> [11,]  9.371242e-09 -9.128390e-09  1.614789e-09 -4.592752e-09  2.356335e-10
#>                [,6]          [,7]          [,8]          [,9]         [,10]
#>  [1,] -1.021573e+01 -4.288844e+00  3.600625e+00 -3.014193e+00 -1.569689e+01
#>  [2,]  1.482848e+00  1.903619e+00 -6.706271e+00  4.849642e+00  1.221533e+01
#>  [3,] -3.408698e+00 -1.692995e+00  4.831086e+00  5.413902e+00  3.446787e+00
#>  [4,]  7.489872e+00  2.181457e+00 -9.099955e-01 -6.627341e+00  1.052159e+01
#>  [5,] -4.769824e+00 -8.056131e+00  2.142658e-01 -1.711735e+00 -5.261873e+00
#>  [6,]  3.020067e+01  7.656353e+00 -1.216376e+01 -4.833828e+00  8.428070e+00
#>  [7,]  7.656353e+00  9.226949e+00  1.501023e+00 -2.751527e+00  1.516851e+00
#>  [8,] -1.216376e+01  1.501023e+00  1.988648e+01 -2.626164e+00 -6.442747e+00
#>  [9,] -4.833828e+00 -2.751527e+00 -2.626164e+00  1.419431e+01  1.190579e+01
#> [10,]  8.428070e+00  1.516851e+00 -6.442747e+00  1.190579e+01  2.657836e+01
#> [11,] -5.992821e-09 -9.505400e-10  5.221635e-09 -6.462631e-09 -1.288458e-08
#>               [,11]
#>  [1,]  9.371242e-09
#>  [2,] -9.128390e-09
#>  [3,]  1.614789e-09
#>  [4,] -4.592752e-09
#>  [5,]  2.356335e-10
#>  [6,] -5.992821e-09
#>  [7,] -9.505400e-10
#>  [8,]  5.221635e-09
#>  [9,] -6.462631e-09
#> [10,] -1.288458e-08
#> [11,]  1.677425e-06
#> 
#> $vcov
#>                [,1]          [,2]          [,3]          [,4]          [,5]
#>  [1,]  1.538547e+02   6.611554102 -4.629346e+02 -5.759209e+02 -1.706233e+02
#>  [2,]  6.611554e+00   1.595904679 -1.902662e+01 -2.424586e+01 -7.283306e+00
#>  [3,] -4.629346e+02 -19.026622589  1.394627e+03  1.734387e+03  5.137759e+02
#>  [4,] -5.759209e+02 -24.245856391  1.734387e+03  2.157345e+03  6.391017e+02
#>  [5,] -1.706233e+02  -7.283305730  5.137759e+02  6.391017e+02  1.894835e+02
#>  [6,] -4.658580e+01  -1.340200015  1.406672e+02  1.747366e+02  5.169171e+01
#>  [7,] -3.897514e+02 -17.051117546  1.173442e+03  1.459826e+03  4.326737e+02
#>  [8,]  2.131783e+02   9.502643613 -6.417213e+02 -7.983792e+02 -2.365807e+02
#>  [9,] -7.941748e+02 -33.258348051  2.391567e+03  2.974801e+03  8.812779e+02
#> [10,]  7.865150e+02  32.394685425 -2.368834e+03 -2.946233e+03 -8.727399e+02
#> [11,] -6.348205e-05   0.001366067  4.244463e-04  2.060668e-04  1.719102e-04
#>                [,6]          [,7]          [,8]          [,9]         [,10]
#>  [1,] -4.658580e+01 -3.897514e+02  2.131783e+02 -7.941748e+02  7.865150e+02
#>  [2,] -1.340200e+00 -1.705112e+01  9.502644e+00 -3.325835e+01  3.239469e+01
#>  [3,]  1.406672e+02  1.173442e+03 -6.417213e+02  2.391567e+03 -2.368834e+03
#>  [4,]  1.747366e+02  1.459826e+03 -7.983792e+02  2.974801e+03 -2.946233e+03
#>  [5,]  5.169171e+01  4.326737e+02 -2.365807e+02  8.812779e+02 -8.727399e+02
#>  [6,]  1.455275e+01  1.178383e+02 -6.434055e+01  2.411094e+02 -2.390202e+02
#>  [7,]  1.178383e+02  9.885074e+02 -5.405997e+02  2.012896e+03 -1.993270e+03
#>  [8,] -6.434055e+01 -5.405997e+02  2.957928e+02 -1.100766e+03  1.090018e+03
#>  [9,]  2.411094e+02  2.012896e+03 -1.100766e+03  4.102267e+03 -4.062837e+03
#> [10,] -2.390202e+02 -1.993270e+03  1.090018e+03 -4.062837e+03  4.024130e+03
#> [11,]  7.369761e-04 -3.760330e-04  5.046174e-04  7.926552e-04 -9.239014e-04
#>               [,11]
#>  [1,] -6.348205e-05
#>  [2,]  1.366067e-03
#>  [3,]  4.244463e-04
#>  [4,]  2.060668e-04
#>  [5,]  1.719102e-04
#>  [6,]  7.369761e-04
#>  [7,] -3.760330e-04
#>  [8,]  5.046174e-04
#>  [9,]  7.926552e-04
#> [10,] -9.239014e-04
#> [11,]  5.961519e+05
#>

Usage

Arguments

Value

Details

See also

Examples