double
gsl_cdf_lognormal_Qinv (const double Q, const double zeta, const double sigma)
{
double x, u;
if (Q == 0.0)
{
return GSL_POSINF;
}
else if (Q == 1.0)
{
return 0.0;
}
u = gsl_cdf_ugaussian_Qinv (Q);
x = exp (zeta + sigma * u);
return x;
}
void test_sparse_als_zero_order_only(TestFixture_T *pFix, gconstpointer pg) {
int n_features = pFix->X->n;
int k = 0;
ffm_param param = {.n_iter = 1,
.warm_start = true,
.ignore_w = true,
.init_sigma = 0.1,
.SOLVER = SOLVER_ALS,
.TASK = TASK_REGRESSION};
ffm_coef *coef = alloc_fm_coef(n_features, k, true);
param.init_lambda_w = 0;
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
// g_assert_cmpfloat(4466.666666, ==, coef->w_0);
g_assert_cmpfloat(fabs(4466.666666 - coef->w_0), <, 1e-6);
free_ffm_coef(coef);
}
void test_sparse_als_first_order_only(TestFixture_T *pFix, gconstpointer pg) {
int n_features = pFix->X->n;
int k = 0;
ffm_param param = {.n_iter = 1,
.warm_start = true,
.ignore_w_0 = true,
.init_sigma = 0.1,
.SOLVER = SOLVER_ALS,
.TASK = TASK_REGRESSION};
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
coef->w_0 = 0;
param.init_lambda_w = 0;
ffm_vector_set(coef->w, 0, 10);
ffm_vector_set(coef->w, 1, 20);
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
// hand calculated results 1660.57142857 -11.87755102
g_assert_cmpfloat(fabs(1660.57142857 - ffm_vector_get(coef->w, 0)), <, 1e-8);
g_assert_cmpfloat(fabs(-11.87755102 - ffm_vector_get(coef->w, 1)), <, 1e-8);
free_ffm_coef(coef);
}
void test_sparse_als_second_order_only(TestFixture_T *pFix, gconstpointer pg) {
int n_features = pFix->X->n;
int k = 1;
ffm_param param = {.n_iter = 1,
.warm_start = true,
.ignore_w_0 = true,
.ignore_w = true,
.init_sigma = 0.1,
.SOLVER = SOLVER_ALS,
.TASK = TASK_REGRESSION};
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
coef->w_0 = 0;
param.init_lambda_w = 0;
param.init_lambda_V = 0;
ffm_matrix_set(coef->V, 0, 0, 300);
ffm_matrix_set(coef->V, 0, 1, 400);
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
// hand calculated results 0.79866412 400.
g_assert_cmpfloat(fabs(0.79866412 - ffm_matrix_get(coef->V, 0, 0)), <, 1e-8);
g_assert_cmpfloat(fabs(400 - ffm_matrix_get(coef->V, 0, 1)), <, 1e-8);
free_ffm_coef(coef);
}
void test_sparse_als_all_interactions(TestFixture_T *pFix, gconstpointer pg) {
int n_features = pFix->X->n;
int k = 1;
ffm_param param = {.n_iter = 1,
.warm_start = true,
.ignore_w_0 = false,
.ignore_w = false,
.init_sigma = 0.1,
.SOLVER = SOLVER_ALS,
.TASK = TASK_REGRESSION};
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
coef->w_0 = 0;
ffm_vector_set(coef->w, 0, 10);
ffm_vector_set(coef->w, 1, 20);
ffm_matrix_set(coef->V, 0, 0, 300);
ffm_matrix_set(coef->V, 0, 1, 400);
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
// hand calculated results checked with libfm
g_assert_cmpfloat(fabs(-1755643.33333 - coef->w_0), <, 1e-5);
g_assert_cmpfloat(fabs(-191459.71428571 - ffm_vector_get(coef->w, 0)), <,
1e-6);
g_assert_cmpfloat(fabs(30791.91836735 - ffm_vector_get(coef->w, 1)), <, 1e-6);
g_assert_cmpfloat(fabs(253.89744249 - ffm_matrix_get(coef->V, 0, 0)), <,
1e-6);
g_assert_cmpfloat(fabs(400 - ffm_matrix_get(coef->V, 0, 1)), <, 1e-6);
param.n_iter = 99;
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
g_assert_cmpfloat(fabs(210911.940403 - coef->w_0), <, 1e-7);
g_assert_cmpfloat(fabs(-322970.68313639 - ffm_vector_get(coef->w, 0)), <,
1e-6);
g_assert_cmpfloat(fabs(51927.60978978 - ffm_vector_get(coef->w, 1)), <, 1e-6);
g_assert_cmpfloat(fabs(94.76612018 - ffm_matrix_get(coef->V, 0, 0)), <, 1e-6);
g_assert_cmpfloat(fabs(400 - ffm_matrix_get(coef->V, 0, 1)), <, 1e-6);
free_ffm_coef(coef);
}
void test_sparse_als_first_order_interactions(TestFixture_T *pFix,
gconstpointer pg) {
ffm_vector *y_pred = ffm_vector_calloc(5);
int n_features = pFix->X->n;
int k = 0;
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
ffm_param param = {.n_iter = 500,
.init_sigma = 0.1,
.SOLVER = SOLVER_ALS,
.TASK = TASK_REGRESSION};
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
sparse_predict(coef, pFix->X, y_pred);
/* reference values from sklearn LinearRegression
y_pred: [ 321.05084746 346.6779661 -40.15254237 321.05084746
790.37288136]
coef: [ 69.6779661 152.16949153]
mse: 3134.91525424 */
g_assert_cmpfloat(fabs(321.05084746 - ffm_vector_get(y_pred, 0)), <, 1e-6);
g_assert_cmpfloat(fabs(346.6779661 - ffm_vector_get(y_pred, 1)), <, 1e-6);
g_assert_cmpfloat(fabs(-40.15254237 - ffm_vector_get(y_pred, 2)), <, 1e-6);
g_assert_cmpfloat(fabs(321.05084746 - ffm_vector_get(y_pred, 3)), <, 1e-6);
g_assert_cmpfloat(fabs(790.37288136 - ffm_vector_get(y_pred, 4)), <, 1e-6);
ffm_vector_free(y_pred);
free_ffm_coef(coef);
}
void test_sparse_als_second_interactions(TestFixture_T *pFix,
gconstpointer pg) {
ffm_vector *y_pred = ffm_vector_calloc(5);
int n_features = pFix->X->n;
int k = 2;
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
ffm_param param = {.n_iter = 1000, .init_sigma = 0.1, .SOLVER = SOLVER_ALS};
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
sparse_predict(coef, pFix->X, y_pred);
/* reference values from sklearn LinearRegression
y_pred: [ 298. 266. 29. 298. 848.]
coeff: [ 9. 2. 40.]
mse: 4.53374139449e-27 */
g_assert_cmpfloat(fabs(298 - ffm_vector_get(y_pred, 0)), <, 1e-4);
g_assert_cmpfloat(fabs(266 - ffm_vector_get(y_pred, 1)), <, 1e-4);
g_assert_cmpfloat(fabs(29 - ffm_vector_get(y_pred, 2)), <, 1e-3);
g_assert_cmpfloat(fabs(298 - ffm_vector_get(y_pred, 3)), <, 1e-4);
g_assert_cmpfloat(fabs(848.0 - ffm_vector_get(y_pred, 4)), <, 1e-4);
ffm_vector_free(y_pred);
free_ffm_coef(coef);
}
void test_sparse_mcmc_second_interactions(TestFixture_T *pFix,
gconstpointer pg) {
int n_features = pFix->X->n;
int n_samples = pFix->X->m;
int k = 2;
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
ffm_vector *y_pred = ffm_vector_calloc(n_samples);
ffm_param param = {.n_iter = 100,
.init_sigma = 0.1,
.SOLVER = SOLVER_MCMC,
.TASK = TASK_REGRESSION,
.rng_seed = 1234};
sparse_fit(coef, pFix->X, pFix->X, pFix->y, y_pred, param);
g_assert_cmpfloat(ffm_r2_score(pFix->y, y_pred), >, .98);
ffm_vector_free(y_pred);
free_ffm_coef(coef);
}
void test_sparse_mcmc_second_interactions_classification(TestFixture_T *pFix,
gconstpointer pg) {
int n_features = pFix->X->n;
int n_samples = pFix->X->m;
int k = 2;
ffm_vector_make_labels(pFix->y);
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
ffm_vector *y_pred = ffm_vector_calloc(n_samples);
ffm_param param = {.n_iter = 10,
.init_sigma = 0.1,
.SOLVER = SOLVER_MCMC,
.TASK = TASK_CLASSIFICATION};
sparse_fit(coef, pFix->X, pFix->X, pFix->y, y_pred, param);
g_assert_cmpfloat(ffm_vector_accuracy(pFix->y, y_pred), >=, .98);
ffm_vector_free(y_pred);
free_ffm_coef(coef);
}
void test_train_test_of_different_size(TestFixture_T *pFix, gconstpointer pg) {
int n_features = pFix->X->n;
int k = 2;
int n_samples_short = 3;
int m = n_samples_short;
int n = n_features;
cs *X = cs_spalloc(m, n, m * n, 1, 1); /* create triplet identity matrix */
cs_entry(X, 0, 0, 6);
cs_entry(X, 0, 1, 1);
cs_entry(X, 1, 0, 2);
cs_entry(X, 1, 1, 3);
cs_entry(X, 2, 0, 3);
cs *X_csc = cs_compress(X); /* A = compressed-column form of T */
cs *X_t = cs_transpose(X_csc, 1);
cs_spfree(X);
ffm_vector *y = ffm_vector_calloc(n_samples_short);
// y [ 298 266 29 298 848 ]
y->data[0] = 298;
y->data[1] = 266;
y->data[2] = 29;
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
ffm_vector *y_pred = ffm_vector_calloc(n_samples_short);
ffm_param param = {.n_iter = 20, .init_sigma = 0.01};
// test: train > test
param.SOLVER = SOLVER_ALS;
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
sparse_predict(coef, X_csc, y_pred);
param.TASK = TASK_CLASSIFICATION;
sparse_fit(coef, pFix->X, NULL, pFix->y, NULL, param);
sparse_predict(coef, X_csc, y_pred);
param.SOLVER = SOLVER_MCMC;
param.TASK = TASK_CLASSIFICATION;
sparse_fit(coef, pFix->X, X_csc, pFix->y, y_pred, param);
param.TASK = TASK_REGRESSION;
sparse_fit(coef, pFix->X, X_csc, pFix->y, y_pred, param);
// test: train < test
param.SOLVER = SOLVER_MCMC;
param.TASK = TASK_CLASSIFICATION;
sparse_fit(coef, X_csc, pFix->X, y_pred, pFix->y, param);
param.TASK = TASK_REGRESSION;
sparse_fit(coef, X_csc, pFix->X, y_pred, pFix->y, param);
param.SOLVER = SOLVER_ALS;
sparse_fit(coef, X_csc, NULL, y_pred, NULL, param);
sparse_predict(coef, pFix->X, pFix->y);
param.TASK = TASK_CLASSIFICATION;
sparse_fit(coef, X_csc, NULL, y_pred, NULL, param);
sparse_predict(coef, pFix->X, pFix->y);
ffm_vector_free(y_pred);
free_ffm_coef(coef);
cs_spfree(X_t);
cs_spfree(X_csc);
}
void test_sparse_als_generated_data(void) {
int n_features = 10;
int n_samples = 100;
int k = 2;
TestFixture_T *data = makeTestFixture(124, n_samples, n_features, k);
ffm_vector *y_pred = ffm_vector_calloc(n_samples);
ffm_coef *coef = alloc_fm_coef(n_features, k, false);
ffm_param param = {.n_iter = 50, .init_sigma = 0.01, .SOLVER = SOLVER_ALS};
param.init_lambda_w = 23.5;
param.init_lambda_V = 23.5;
sparse_fit(coef, data->X, NULL, data->y, NULL, param);
sparse_predict(coef, data->X, y_pred);
g_assert_cmpfloat(ffm_r2_score(data->y, y_pred), >, 0.85);
ffm_vector_free(y_pred);
free_ffm_coef(coef);
TestFixtureDestructor(data, NULL);
}
void test_hyerparameter_sampling(void) {
ffm_rng *rng = ffm_rng_seed(12345);
int n_features = 20;
int n_samples = 150;
int k = 1; // don't just change k, the rank is hard coded in the test
// (ffm_vector_get(coef->lambda_V, 0);)
int n_replication = 40;
int n_draws = 1000;
ffm_vector *alpha_rep = ffm_vector_calloc(n_replication);
ffm_vector *lambda_w_rep = ffm_vector_calloc(n_replication);
ffm_vector *lambda_V_rep = ffm_vector_calloc(n_replication);
ffm_vector *mu_w_rep = ffm_vector_calloc(n_replication);
ffm_vector *mu_V_rep = ffm_vector_calloc(n_replication);
ffm_vector *err = ffm_vector_alloc(n_samples);
for (int j = 0; j < n_replication; j++) {
TestFixture_T *data = makeTestFixture(124, n_samples, n_features, k);
ffm_coef *coef = data->coef;
sparse_predict(coef, data->X, err);
ffm_vector_scale(err, -1);
ffm_vector_add(err, data->y);
// make sure that distribution is converged bevore selecting
// reference / init values
for (int l = 0; l < 50; l++) sample_hyper_parameter(coef, err, rng);
double alpha_init = coef->alpha;
double lambda_w_init = coef->lambda_w;
double lambda_V_init = ffm_vector_get(coef->lambda_V, 0);
double mu_w_init = coef->mu_w;
double mu_V_init = ffm_vector_get(coef->mu_V, 0);
double alpha_count = 0;
double lambda_w_count = 0, lambda_V_count = 0;
double mu_w_count = 0, mu_V_count = 0;
for (int l = 0; l < n_draws; l++) {
sample_hyper_parameter(coef, err, rng);
if (alpha_init > coef->alpha) alpha_count++;
if (lambda_w_init > coef->lambda_w) lambda_w_count++;
if (lambda_V_init > ffm_vector_get(coef->lambda_V, 0)) lambda_V_count++;
if (mu_w_init > coef->mu_w) mu_w_count++;
if (mu_V_init > ffm_vector_get(coef->mu_V, 0)) mu_V_count++;
}
ffm_vector_set(alpha_rep, j, alpha_count / (n_draws + 1));
ffm_vector_set(lambda_w_rep, j, lambda_w_count / (n_draws + 1));
ffm_vector_set(lambda_V_rep, j, lambda_V_count / (n_draws + 1));
ffm_vector_set(mu_w_rep, j, mu_w_count / (n_draws + 1));
ffm_vector_set(mu_V_rep, j, mu_V_count / (n_draws + 1));
TestFixtureDestructor(data, NULL);
}
double chi_alpha = 0;
for (int i = 0; i < n_replication; i++)
chi_alpha +=
ffm_pow_2(gsl_cdf_ugaussian_Qinv(ffm_vector_get(alpha_rep, i)));
g_assert_cmpfloat(gsl_ran_chisq_pdf(chi_alpha, n_replication), <, .05);
double chi_lambda_w = 0;
for (int i = 0; i < n_replication; i++)
chi_lambda_w +=
ffm_pow_2(gsl_cdf_ugaussian_Qinv(ffm_vector_get(lambda_w_rep, i)));
g_assert_cmpfloat(gsl_ran_chisq_pdf(chi_lambda_w, n_replication), <, .05);
double chi_lambda_V = 0;
for (int i = 0; i < n_replication; i++)
chi_lambda_V +=
ffm_pow_2(gsl_cdf_ugaussian_Qinv(ffm_vector_get(lambda_V_rep, i)));
g_assert_cmpfloat(gsl_ran_chisq_pdf(chi_lambda_V, n_replication), <, .05);
double chi_mu_w = 0;
for (int i = 0; i < n_replication; i++)
chi_mu_w += ffm_pow_2(gsl_cdf_ugaussian_Qinv(ffm_vector_get(mu_w_rep, i)));
g_assert_cmpfloat(gsl_ran_chisq_pdf(chi_mu_w, n_replication), <, .05);
double chi_mu_V = 0;
for (int i = 0; i < n_replication; i++)
chi_mu_V += ffm_pow_2(gsl_cdf_ugaussian_Qinv(ffm_vector_get(mu_V_rep, i)));
g_assert_cmpfloat(gsl_ran_chisq_pdf(chi_mu_V, n_replication), <, .05);
ffm_vector_free_all(alpha_rep, lambda_w_rep, lambda_V_rep, mu_w_rep, mu_V_rep,
err);
ffm_rng_free(rng);
}
double
gsl_cdf_gaussian_Qinv (const double Q, const double sigma)
{
return sigma * gsl_cdf_ugaussian_Qinv (Q);
}
double
gsl_cdf_gamma_Qinv (const double Q, const double a, const double b)
{
double x;
if (Q == 1.0)
{
return 0.0;
}
else if (Q == 0.0)
{
return GSL_POSINF;
}
/* Consider, small, large and intermediate cases separately. The
boundaries at 0.05 and 0.95 have not been optimised, but seem ok
for an initial approximation. */
if (Q < 0.05)
{
double x0 = -log (Q) + gsl_sf_lngamma (a);
x = x0;
}
else if (Q > 0.95)
{
double x0 = exp ((gsl_sf_lngamma (a) + log1p (-Q)) / a);
x = x0;
}
else
{
double xg = gsl_cdf_ugaussian_Qinv (Q);
double x0 = (xg < -0.5*sqrt (a)) ? a : sqrt (a) * xg + a;
x = x0;
}
/* Use Lagrange's interpolation for E(x)/phi(x0) to work backwards
to an improved value of x (Abramowitz & Stegun, 3.6.6)
where E(x)=P-integ(phi(u),u,x0,x) and phi(u) is the pdf.
*/
{
double lambda, dQ, phi;
unsigned int n = 0;
start:
dQ = Q - gsl_cdf_gamma_Q (x, a, 1.0);
phi = gsl_ran_gamma_pdf (x, a, 1.0);
if (dQ == 0.0 || n++ > 32)
goto end;
lambda = -dQ / GSL_MAX (2 * fabs (dQ / x), phi);
{
double step0 = lambda;
double step1 = -((a - 1) / x - 1) * lambda * lambda / 4.0;
double step = step0;
if (fabs (step1) < 0.5 * fabs (step0))
step += step1;
if (x + step > 0)
x += step;
else
{
x /= 2.0;
}
if (fabs (step0) > 1e-10 * x)
goto start;
}
}
end:
return b * x;
}
double
gsl_cdf_tdist_Qinv (const double Q, const double nu)
{
double x, qtail;
if (Q == 0.0)
{
return GSL_POSINF;
}
else if (Q == 1.0)
{
return GSL_NEGINF;
}
if (nu == 1.0)
{
x = tan (M_PI * (0.5 - Q));
return x;
}
else if (nu == 2.0)
{
x = (1 - 2 * Q) / sqrt (2 * Q * (1 - Q));
return x;
}
qtail = (Q < 0.5) ? Q : 1 - Q;
if (sqrt (M_PI * nu / 2) * qtail > pow (0.05, nu / 2))
{
double xg = gsl_cdf_ugaussian_Qinv (Q);
x = inv_cornish_fisher (xg, nu);
}
else
{
/* Use an asymptotic expansion of the tail of integral */
double beta = gsl_sf_beta (0.5, nu / 2);
if (Q < 0.5)
{
x = sqrt (nu) * pow (beta * nu * Q, -1.0 / nu);
}
else
{
x = -sqrt (nu) * pow (beta * nu * (1 - Q), -1.0 / nu);
}
/* Correct nu -> nu/(1+nu/x^2) in the leading term to account
for higher order terms. This avoids overestimating x, which
makes the iteration unstable due to the rapidly decreasing
tails of the distribution. */
x /= sqrt (1 + nu / (x * x));
}
{
double dQ, phi;
unsigned int n = 0;
start:
dQ = Q - gsl_cdf_tdist_Q (x, nu);
phi = gsl_ran_tdist_pdf (x, nu);
if (dQ == 0.0 || n++ > 32)
goto end;
{
double lambda = - dQ / phi;
double step0 = lambda;
double step1 = ((nu + 1) * x / (x * x + nu)) * (lambda * lambda / 4.0);
double step = step0;
if (fabs (step1) < fabs (step0))
{
step += step1;
}
if (Q < 0.5 && x + step < 0)
x /= 2;
else if (Q > 0.5 && x + step > 0)
x /= 2;
else
x += step;
if (fabs (step) > 1e-10 * fabs (x))
goto start;
}
}
end:
return x;
}
