beta0_logit, beta_logit = logistic.intercept_, logistic.coef_[0]
# conditional probability predicted from last observation
p_beta_logit = logistic.predict_proba(z_cubic[[-1], :])[0, 1]
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_lfm_executive_summary-implementation-step03): Perform generalized probabilistic inference
annualized_vol = np.sqrt(s2_xz[1, 1]) * np.sqrt(252)
p_base = np.ones(z_t.shape[0]) / z_t.shape[0]
mu_base = z_t @ p_base
z_ineq = -np.atleast_2d(z_t**2)
mu_view_ineq = -np.atleast_1d(sig_view**2 + mu_base**2)
z_eq = np.atleast_2d(z_t)
mu_view_eq = np.atleast_1d(mu_base)
p_upd = min_rel_entropy_sp(p_base,
z_ineq,
mu_view_ineq,
z_eq,
mu_view_eq,
normalize=False)
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_lfm_executive_summary-implementation-step04): Fit linear state-space model
h_t = fit_state_space(z_t, k_=1, p=p_upd)[0]
# ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_lfm_executive_summary-implementation-step05): Fit logistic model with Lasso penalty
C = 1 / lambda_lasso
logistic_lasso = LogisticRegression(penalty='l1',
C=C,
class_weight='balanced',
solver='liblinear',
random_state=1,
n_cum_trans, tau_hl_credit)
# constraints on the credit transition matrix via minimum relative entropy
# probability constraint
a = {}
a[0] = np.diagflat(np.ones((1, c_)), 1) -\
np.diagflat(np.ones((1, c_+1)), 0)
a[0] = a[0][:-1]
b = np.zeros((c_))
# monotonicity constraint (initialize)
a_eq = np.ones((1, c_ + 1))
b_eq = np.array([1])
# minimum relative entropy
p_credit = np.zeros((c_, c_ + 1))
for c in range(c_):
p_credit[c, :] = min_rel_entropy_sp(p_credit_prior[c, :], a[c], b, a_eq,
b_eq, False)
# update monotonicity constraint
a_temp = a.get(c).copy()
a_temp[c, :] = -a_temp[c, :]
a[c + 1] = a_temp.copy()
# default constraint
default_constraint = np.append(np.zeros(c_), 1).reshape(1, -1)
p_credit = np.r_[p_credit, default_constraint]
# -
# ## [Step 6](https://www.arpm.co/lab/redirect.php?permalink=s_checklist_scenariobased_step02-implementation-step06): Save databases
# +
dates = dates[1:]
return np.array([sk * sig**3 + 3 * mu * sig**2 + mu**3])
def mu_view_eq(mu, sig):
return np.array([mu, mu**2 + sig**2])
z_ineq_base = -v(x)[:1]
mu_ineq_base = -mu_view_ineq(mu_x_base, sig_x_base, sk_x_base)
z_eq_base = v(x)[1:]
mu_view_eq_base = mu_view_eq(mu_x_base, sig_x_base)
p_base = min_rel_entropy_sp(p_base_unif,
z_ineq_base,
mu_ineq_base,
z_eq_base,
mu_view_eq_base,
normalize=False)
# -
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_entropy_view-implementation-step03): Compute updated probabilities
# +
# Generate parameters specifying constraints for updated distribution
z_ineq = v(x)[:1]
mu_ineq = mu_view_ineq(-mu_x_base, sig_x_base, -sk_x_base)
z_eq = v(x)[1:]
mu_view_eq = mu_view_eq(-mu_x_base, sig_x_base)
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_fit_discrete_markov_chain-implementation-step02): Compute final transition matrix
# +
# probability constraint
a_eq = np.ones((1, c_+1))
b_eq = np.array([1])
# monotonicity constraint (initialize)
a_ineq = {}
a_ineq[0] = np.diagflat(np.ones((1, c_)), 1) -\
np.diagflat(np.ones((1, c_+1)), 0)
a_ineq[0] = a_ineq[0][:-1]
b_ineq = np.zeros((c_))
# relative entropy minimization
p = np.zeros((c_, c_+1))
for c in range(c_):
p[c, :] = min_rel_entropy_sp(p_[c, :],
a_ineq[c], b_ineq, a_eq, b_eq, False)
# update monotonicity constraint
a_temp = a_ineq.get(c).copy()
a_temp[c, :] = -a_temp[c, :]
a_ineq[c+1] = a_temp.copy()
# default constraint
p = np.r_[p, np.array([np.r_[np.zeros(7), 1]])]
# -
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_fit_discrete_markov_chain-implementation-step03): Compute cdf
# +
f = np.cumsum(p[r-1, :])
# -
z_i = z_eq[i]
covariance[i] = np.cov(z_i)
if np.linalg.matrix_rank(covariance[i]) > 1:
effrank[i] = eff_rank(np.corrcoef(z_i))
# -
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_views_cond_exp-implementation-step03): Compute updated probabilities
# +
i_san_check = np.where(effrank > 1)[0]
p_upd_i = np.zeros((j_, j_))
entropy = np.zeros(j_)
for i in range(j_):
if i in i_san_check:
p_upd_i[i] = min_rel_entropy_sp(p_base, None, None, z_eq[i],
mu_view_eq_c, normalize=False)
entropy[i] = p_upd_i[i] @ np.log(p_upd_i[i] / p_base)
p_upd_san = p_upd_i[i_san_check]
p_upd_ihat = p_upd_san[np.argmin(entropy[i_san_check])]
p_upd = p_upd_ihat[np.argsort(np.argsort(v(x)))]
# -
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_views_cond_exp-implementation-step04): Compute additive/multiplicative confidence-weighted probabilities
p_c_add = c * p_upd + (1 - c) * p_base
p_c_mul = p_upd ** c * p_base ** (1 - c) /\
np.sum(p_upd ** c * p_base ** (1 - c))
