void
test_eval_zinb_dfdp
(void)
{
test_assert(eval_zinb_dfdp(1, .5, 0, 0) == 0.0);
test_assert(eval_zinb_dfdp(1, .5, 1, 0) == 0.0);
test_assert(fabs(eval_zinb_dfdp(2, .5, 1, 0)+3.5555555) < 1e-6);
test_assert(fabs(eval_zinb_dfdp(2, .3, 1, 0)+19.5896899) < 1e-6);
test_assert(fabs(eval_zinb_dfdp(2, .3, 9, 0)+176.3072092) < 1e-6);
test_assert(fabs(eval_zinb_dfdp(2, .3, 9, 1.7)+179.7765970) < 1e-6);
return;
}
zinb_par_t *
mle_zinb
(
int *x,
unsigned int nobs
)
{
tab_t *tab = tabulate(x, nobs);
double sum = 0.0;
unsigned int nona = 0;
for (size_t i = 0 ; i < tab->size ; i++) {
sum += tab->val[i]*tab->num[i];
nona += tab->num[i];
}
// Extract the number of all-zero observaions.
const unsigned int z0 = tab->val[0] == 0 ? tab->num[0] : 0;
double deficit[11] = {0,.1,.2,.3,.4,.5,.6,.7,.8,.9,1};
double init_a[12] = {-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1};
double init_p[12] = {0,0,0,0,0,0,0,0,0,0,0,.5};
// Deplete some 0s from the observations, compute alpha
// and p0 with standard negative binomial estimation
// and keep the values to be used as initial conditions.
for (size_t i = 0 ; i < 11 ; i++) {
if (tab->val[0] == 0) tab->num[0] = z0 * (1-deficit[i]);
double newmean = sum / nona / (1.0 - z0*deficit[i]/nobs);
double alpha = nb_est_alpha(tab);
init_a[i] = alpha;
init_p[i] = alpha / (alpha + newmean);
}
// Reset 'tab'.
if (tab->val[0] == 0) tab->num[0] = z0;
zinb_par_t *par = calloc(1, sizeof(zinb_par_t));
if (par == NULL) {
fprintf(stderr, "memory error: %s:%d\n", __FILE__, __LINE__);
return NULL;
}
// Try initial conditions. Number 12 is a safety in case
// all the rest failed during the first phase.
double max_loglik = -1.0/0.0;
for (size_t i = 0 ; i < 12 ; i++) {
if (init_a[i] < 0) continue; // Skip failures.
double a = init_a[i];
double p = init_p[i];
double grad;
unsigned int iter = 0;
double f = eval_zinb_f(a, p, nobs-z0, sum);
double g = eval_zinb_g(a, p, tab);
// Newton-Raphson iterations.
while ((grad = f*f+g*g) > sq(ZINM_TOL) && iter++ < ZINM_MAXITER) {
double dfda, dfdp, dgda, dgdp;
dfda = dgdp = eval_zinb_dfda(a, p, nobs-z0);
dfdp = eval_zinb_dfdp(a, p, nobs-z0, sum);
dgda = eval_zinb_dgda(a, p, tab);
double denom = dfdp*dgda - dfda*dgdp;
double da = (f*dgdp - g*dfdp) / denom;
double dp = (g*dfda - f*dgda) / denom;
// Maintain 'a' and 'p' in their domain of definition.
while (a+da < 0 || p+dp < 0 || p+dp > 1) {
da /= 2;
dp /= 2;
}
f = eval_zinb_f(a+da, p+dp, nobs-z0, sum);
g = eval_zinb_g(a+da, p+dp, tab);
// Backtrack if necessary.
for (int j = 0 ; j < ZINM_MAXITER && f*f+g*g > grad ; j++) {
da /= 2;
dp /= 2;
f = eval_zinb_f(a+da, p+dp, nobs-z0, sum);
g = eval_zinb_g(a+da, p+dp, tab);
}
a = a+da;
p = p+dp;
}
double pi = (nobs-z0) / (1-pow(p,a)) / nobs;
if (pi > 1) pi = 1.0;
if (pi < 0) pi = 0.0;
double loglik = ll_zinb(a, p, pi, tab);
if (loglik > max_loglik) {
max_loglik = loglik;
par->a = a;
par->pi = pi;
par->p = p;
}
}
free(tab);
return par;
}