def CalibDegOfFreedomMLFP(x, FP, maxdf, stepdf):
# Student t model
# MLFP for mu and sigma on a grid of degrees of freedom (df) the best fit
# corresponds to df which gives rise to the highest (log)likelihood L)
# INPUTS
# x :[vector](1 x t_end) empirical realizations
# FP :[vector](1 x t_end) flexible probabilities associated with vector x
# maxdf :[scalar] maximum value for nu to be checked
# stepdf :[scalar] step between consecutive values of nu to be checked
# OPS
# mu :[scalar] estimated location parameter
# sig2 :[scalar] estimated dispersion parameter
# nu :[scalar] best degrees of freedom nu
## Code
df = arange(1, maxdf + stepdf, stepdf)
Tol = 10**(-6)
l_ = len(df)
Mu = zeros((l_, 1))
Sigma2 = zeros((l_, 1))
L = zeros((l_, 1))
for i in range(l_):
Mu[i], Sigma2[i], _ = MaxLikelihoodFPLocDispT(x, FP, df[i], Tol, 1)
L[i] = FP @ log(
t.pdf((x - Mu[i]) / sqrt(Sigma2[i]), df[i]) / sqrt(Sigma2[i])).T
imax = np.argmax(L)
mu = Mu[imax]
sig2 = Sigma2[imax]
nu = df[imax]
return mu, sig2, nu
sigma2 = sigma_temp.T@sigma_temp # covariance matrix of dimension (i_ x i_)
epsi_temp = mvtrvs(zeros(sigma2.shape[0]), sigma2, nu, t_).T
epsi = diagflat(sqrt(diag(sigma2)))@epsi_temp + tile(mu, (1, t_)) # dataset of dimension (i_ x t_end) from a t() distribution
# -
# ## Set the Flexible Probability profile for MLFP estimation (exponential decay with half life 12 months)
lam = log(2) / 360
p = exp(-lam*arange(t_, 1 + -1, -1)).reshape(1,-1)
p = p /npsum(p)
# ## Compute MLFP estimators of location and dispersion from the sample
mu_MLFP, sigma2_MLFP, err1 = MaxLikelihoodFPLocDispT(epsi, p, nu, 10**-15, 1)
# ## Define the likelihood function
# +
mvt = mvd.MVT(array([0]),array([[1]]),df=nu)
mvtpdf = mvt.pdf
likelihood = lambda theta: npsum(p * np.real(log((mvtpdf((epsi - tile(theta[:i_], (1, t_))).T@diagflat(
1 / sqrt(reshape(theta[i_:i_*(1 + i_)], (i_, -1),'F').astype(np.complex128))))).astype(np.complex128).T)))
# -
# ## Compute the Hessian matrix
hessian, err2 = numHess(likelihood, r_[mu_MLFP[...,newaxis],sigma2_MLFP])
# ## Standardize the invariants
# +
nu_marg = 6
tauHL_prior = 252*3 # 3 years
# set FP
p_t = exp(-(log(2) / (tauHL_prior + round(10*(rand() - 0.5)))*abs(arange(t_, 1 + -1, -1)))).reshape(1,-1) # FP setting for every invariants separately
p_t = p_t / npsum(p_t)
mu = zeros((bonds_i_, 1))
sig2 = zeros((bonds_i_, 1))
epsi_t = zeros((epsi.shape))
u = zeros((epsi.shape))
for i in range(bonds_i_):
mu[i], sig2[i],_ = MaxLikelihoodFPLocDispT(epsi[[i],:], p_t, nu_marg, 10 ** -6, 1)
epsi_t[i, :] = (epsi[i, :] - mu[i]) / sqrt(sig2[i])
u[i, :] = tstu.cdf(epsi_t[i, :], nu_marg)
# -
# ## Estimate the correlation of the t-copula
# ## map observations into copula realizations
# +
nu = 5
c = zeros((u.shape))
for i in range(bonds_i_):
c[i,:] = tstu.ppf(u[i, :], nu)
# estimate the correlation matrix
[_, s2_hat] = FPmeancov(c, ones((1, t_)) / t_)
epsi = diff(y, 1, 1) # rate daily changes
# ## Maximum Likelihood with Flexible Probabilities (MLFP) Student t fit
# +
# degrees of freedom
nu = 5
# flexible probabilities (exponential decay half life 6 months)
lam = log(2) / 180
p = exp(-lam * arange(t_ - 1, 1 + -1, -1)).reshape(1, -1)
p = p / npsum(p)
# Fit
tolerance = 10**(-10)
mu_MLFP, sigma2_MLFP, _ = MaxLikelihoodFPLocDispT(epsi, p, nu, tolerance, 1)
# Student t mean and covariance
m_MLFP = mu_MLFP
s2_MLFP = nu / (nu - 2) * sigma2_MLFP
# -
# ## Create figures
# +
CM, C = ColorCodedFP(p, npmin(p), npmax(p), arange(0, 0.8, 0.005), 0, 1,
[1, 0])
f = figure()
# colormap(CM)
scatter(epsi[0], epsi[1], 10, c=C, marker='.',
# Flexible Probabilities
p = ConditionalFP(conditioner, prior)
# ## Fit the t copula
# ## estimate marginal distributions by fitting a Student t distribution via
# ## MLFP and recover the invariants' grades
u = zeros((i_, t_))
epsi = sort(
epsi, 1
) # We sort scenario in ascending order (in order to apply CopMargComb later)
for i in range(i_):
mu_nu = zeros(nu_)
sig2_nu = zeros(nu_)
like_nu = zeros(nu_)
for k in range(nu_):
nu_k = nu_vec[k]
mu_nu[k], sig2_nu[k], _ = MaxLikelihoodFPLocDispT(
epsi[[i], :], p, nu_k, 10**-6, 1)
epsi_t = (epsi[[i], :] - mu_nu[k]) / sqrt(sig2_nu[k])
like_nu[k] = npsum(
p * log(t.pdf(epsi_t, nu_k) / sqrt(sig2_nu[k]))) # likelihood
j_nu = argsort(like_nu)[::-1]
# take as estimates the parameters giving rise to the highest likelihood
nu_marg = max(nu_vec[j_nu[0]], 10)
mu_marg = mu_nu[j_nu[0]]
sig2_marg = sig2_nu[j_nu[0]]
u[i, :] = t.cdf((epsi[i, :] - mu_marg) / sqrt(sig2_marg), nu_marg)
# Map the grades into standard Student t realizations
epsi_tilde = zeros((i_, t_))
for i in range(i_):
epsi_tilde[i, :] = t.ppf(u[i, :], nu)
# fit the ellipsoid via MLFP
nu_MLFP = zeros((1, i_))
# flexible probabilities
lam = log(2) / tau_HL
p = exp((-lam * arange(t_, 1 + -1, -1))).reshape(1, -1)
p = p / npsum(p)
# estimate marginal distributions
for i in range(i_):
mu_nu = zeros((1, nu_))
sig2_nu = zeros((1, nu_))
like_nu = zeros((1, nu_))
for j in range(nu_):
nu = nu_vec[j]
mu_nu[0, j], sig2_nu[0, j], _ = MaxLikelihoodFPLocDispT(
epsi[[i], :], p, nu, 10**-6, 1)
epsi_t = (epsi[i, :] - mu_nu[0, j]) / sqrt(sig2_nu[0, j])
like_nu[0, j] = sum(
p[0] *
log(t.pdf(epsi_t, nu) / sqrt(sig2_nu[0, j]))) # Log-likelihood
j_nu = argsort(like_nu[0])[::-1]
nu_MLFP[0, i] = nu_vec[j_nu[0]]
mu_MLFP[0, i] = mu_nu[0, j_nu[0]]
sig2_MLFP[0, i] = sig2_nu[0, j_nu[
0]] # Take as estimates the one giving rise to the highest log-likelihood
# -
# ## Recover the time series of standardized uniform variables
u = zeros((i_, t_))
q2 = percentile(epsi, 75, axis=1, keepdims=True)
interq_range = q2 - q1
epsi_rescaled = epsi / tile(interq_range, (1, t_obs))
# STEP 1: Invariants grades
epsi_grid, u_grid, grades = CopMargSep(epsi_rescaled, p)
nu = 4
# STEP [1:] Marginal t
epsi_st = zeros(epsi.shape)
for i in range(i_):
epsi_st[i, :] = t.ppf(grades[i, :], nu)
# STEP 3: Fit ellipsoid (MLFP ellipsoid under Student t assumption)
Tol = 10**-6
mu_epsi, sigma2_epsi, _ = MaxLikelihoodFPLocDispT(epsi_st, p, nu, Tol, 1)
# STEP 4: Shrinkage (we don't shrink sigma2)
# STEP 5: Correlation
c2_hat = np.diagflat(diag(sigma2_epsi)**(-1 / 2)) @ sigma2_epsi @ np.diagflat(
diag(sigma2_epsi)**(-1 / 2))
# Rescale back the invariants'o the original size
epsi_grid = epsi_grid * tile(interq_range, (1, t_obs))
# -
# ## Marginal distributions: HFP distributions for epsi_HST and epsi_MVOU parametric VG distribution for epsi_VG
# +
marginals_grid = r_[epsi_grid[:4, :], shifted_epsi_grid_vg.reshape(1, -1)]
conditioner.TargetValue = np.atleast_2d(z_vix_star)
conditioner.Leeway = alpha
p = ConditionalFP(conditioner, prior)
# ## Estimate the marginal distributions
nu_marg_SPX = zeros(i_)
mu_marg_SPX = zeros(i_)
sig2_marg_SPX = zeros(i_)
for i in range(i_):
mu_nu = zeros(nu_)
sig2_nu = zeros(nu_)
like_nu = zeros(nu_)
for k in range(nu_):
nu = nu_vec[k]
mu_nu[k], sig2_nu[k], _ = MaxLikelihoodFPLocDispT(
epsi_SPX[[i], :], p, nu, 10**-6, 1)
epsi_t = (epsi_SPX[i, :] - mu_nu[k]) / sqrt(sig2_nu[k])
like_nu[k] = npsum(p * log(tstu.pdf(epsi_t, nu) / sqrt(sig2_nu[k])))
k_nu = argsort(like_nu)[::-1]
nu_marg_SPX[i] = max(nu_vec[k_nu[0]], 10)
mu_marg_SPX[i] = mu_nu[k_nu[0]]
sig2_marg_SPX[i] = sig2_nu[k_nu[0]]
# ## Compute the historical distribution of the invariants' copula
u_SPX = zeros((i_, t_))
for i in range(i_):
u_SPX[i, :] = tstu.cdf(
(epsi_SPX[i, :] - mu_marg_SPX[i]) / sqrt(sig2_marg_SPX[i]),
nu_marg_SPX[i])
conditioner.TargetValue = np.atleast_2d(z_vix_star)
conditioner.Leeway = alpha
p = ConditionalFP(conditioner, prior)
# marginal distribution fit
nu_marg = zeros(i_)
mu_marg = zeros(i_)
sig2_marg = zeros(i_)
for i in range(i_):
mu_nu = zeros(nu_)
sig2_nu = zeros(nu_)
like_nu = zeros((1, nu_))
for k in range(nu_):
nu = nu_vec[k]
mu_nu[k], sig2_nu[k], _ = MaxLikelihoodFPLocDispT(
epsi_stocks[[i], :], p, nu, 10**-6, 1)
epsi_t = (epsi_stocks[i, :] - mu_nu[k]) / sqrt(sig2_nu[k])
like_nu[0, k] = npsum(p * log(tstu.cdf(epsi_t, nu) / sqrt(sig2_nu[k])))
k_nu = argsort(like_nu[0])[::-1]
nu_marg[i] = max(nu_vec[k_nu[0]], 10)
mu_marg[i] = mu_nu[k_nu[0]]
sig2_marg[i] = sig2_nu[k_nu[0]]
# Realized marginals mapping into standard Student t realizations
u_stocks = zeros((i_, t_))
epsi_tilde_stocks = zeros((i_, t_))
for i in range(i_):
# u_stocks([i,:])=min((t.cdf((epsi_stocks[i,:]-mu_marg[i])/sqrt(sig2_marg[i]),nu_marg[i]),0.999))
u_stocks[i, :] = tstu.cdf(
(epsi_stocks[i, :] - mu_marg[i]) / sqrt(sig2_marg[i]), nu_marg[i])
StocksSPX.Dividends[25]) # Cisco Systems Inc x = x[[0], 1:] t_ = x.shape[1] # - # ## Set the Flexible Probabilities lam = log(2) / 800 p = exp((-lam * arange(t_, 1 + -1, -1))).reshape(1, -1) p = p / npsum(p) # FP-profile: exponential decay # ## Fit the data to a Cauchy distribution tol = 10**-6 nu = 1 mu, sigma2, _ = MaxLikelihoodFPLocDispT(dx, p, nu, tol, 1) sigma = sqrt(sigma2) # interquantile range corresponding to levels 1/4 and 3/4 mu = mu.squeeze() sigma2 = sigma2.squeeze() sigma = sigma.squeeze() # ## Initialize projection variables tau = 10 dt = 1 / 20 t_j = arange(0, tau + dt, dt) j_ = 15 # ## Simulate paths X = PathsCauchy(x[0, t_ - 1], mu, sigma, t_j, j_)
conditioner.Series = z_vix_cond.reshape(1, -1)
conditioner.TargetValue = np.atleast_2d(z_vix_star)
conditioner.Leeway = alpha
p = ConditionalFP(conditioner, prior)
# map invariants into student t realizations
nu_marg = r_[nu_marg, nu_marg_SPX]
mu_marg = r_[mu_marg, mu_marg_SPX]
sig2_marg = r_[sig2_marg, sig2_marg_SPX]
epsi_tilde = zeros((i_, t_))
for i in range(i_):
u = t.cdf((epsi[i, :] - mu_marg[i]) / sqrt(sig2_marg[i]), nu_marg[i])
epsi_tilde[i, :] = t.ppf(u, nu_joint)
# estimate joint correlation
_, sig2, _ = MaxLikelihoodFPLocDispT(epsi_tilde, p, nu_joint, 10**-6, 1)
c = np.diagflat(diag(sig2)**(-1 / 2)) @ sig2 @ np.diagflat(
diag(sig2)**(-1 / 2))
# replace the correlation block related to stocks with its low-rank-diagonal
# approximation
c_stocks, beta_stocks, *_ = FactorAnalysis(
c[i_SPX:i_SPX + i_stocks, i_SPX:i_SPX + i_stocks], array([[0]]), k_)
c_stocks, beta_stocks = np.real(c_stocks), np.real(beta_stocks)
c_SPX_stocks = c[:i_SPX, i_SPX:i_SPX + i_stocks]
c_SPX = c[:i_SPX, :i_SPX]
# -
# ## Perform Hybrid Monte-Carlo historical projection on the grades for each node path
Epsistocks_tilde_hor = zeros((i_stocks, U_stocks_hor.shape[2]))
sigma2 = 2 # square dispersion parameter sigma = sqrt(sigma2) threshold = 1e-4 last = 1 # - # ## Generate the observation of the Student t with 3 degree of freedom, location parameter 0 and dispersion parameter 2 Epsi_std = t.rvs(nu, size=(1, t_)) Epsi = mu + sigma * Epsi_std # Affine equivariance property x = linspace(npmin(Epsi_std), npmax(Epsi_std), t_ + 1) # ## Compute the Maximum Likelihood location and dispersion parameters p = (1 / t_) * ones((1, t_)) # probabilities mu_ML, sigma2_ML, _ = MaxLikelihoodFPLocDispT(Epsi, p, nu, threshold, last) # ## Compute the Maximum Likelihood pdf sigma_ML = sqrt(sigma2_ML) fML_eps = t.pdf((x - mu_ML) / sigma_ML, nu) # ## Compute the Maximum Likelihood cdf FML_eps = t.cdf((x - mu_ML) / sigma_ML, nu) # ## Compute the true pdf and cdf f_eps = t.pdf((x - mu) / sigma, nu) # ## Compute the true cdf