def find_opt_h(x_list, eps):
N = len(x_list)
X = x_list
shift = min(X)
X_shifted = [x - shift for x in X]
scale = 1 / max(X_shifted)
X_shifted_scale = [x * scale for x in X_shifted]
sigma = stdev(X_shifted_scale)
phi6 = (-15 / (16 * math.sqrt(math.pi))) * math.pow(sigma, -7)
phi8 = (105 / (32 * math.sqrt(math.pi))) * math.pow(sigma, -9)
g1 = (-6 / (math.sqrt(2 * math.pi) * phi6 * N))**(1 / 7)
g2 = (30 / (math.sqrt(2 * math.pi) * phi8 * N))**(1 / 9)
D4 = FUDD(N, N, X_shifted_scale, X_shifted_scale, g1, 4, eps)
D4.evaluate()
phi4 = sum(D4.pD) / (N - 1)
D6 = FUDD(N, N, X_shifted_scale, X_shifted_scale, g2, 6, eps)
D6.evaluate()
phi6 = sum(D6.pD) / (N - 1)
constant1 = (1 / (2 * math.sqrt(math.pi) * N))**(1 / 5)
constant2 = (-6 * math.sqrt(2) * phi4 / phi6)**(1 / 7)
h_initial = constant1 * phi4**(-1 / 5)
h = ls(
fast_h_fun,
h_initial,
bounds=(0, np.inf),
ftol=1e-14,
xtol=1e-14,
verbose=1,
args=(N, X_shifted_scale, constant1, constant2, eps),
)
h = float(h.x) / scale
print(h)
sortX = sorted(X_shifted_scale)
D0 = FUDD(N, N, sortX, sortX, h, 0, eps)
D0.evaluate()
return h, D0.pD
def _get_opt_h(x_list, eps):
N = len(x_list)
X_shifted_scale, scale = _get_scale_list(x_list)
sigma = std(X_shifted_scale)
phi6 = (-15 / (16 * sqrt(pi))) * power(sigma, -7)
phi8 = (105 / (32 * sqrt(pi))) * power(sigma, -9)
g1 = (-6 / (sqrt(2 * pi) * phi6 * N))**(1 / 7)
g2 = (30 / (sqrt(2 * pi) * phi8 * N))**(1 / 9)
D4 = FUDD(N, N, X_shifted_scale, X_shifted_scale, g1, 4, eps)
D6 = FUDD(N, N, X_shifted_scale, X_shifted_scale, g2, 6, eps)
D4.evaluate()
D6.evaluate()
phi4 = sum(D4.pD) / (N - 1)
phi6 = sum(D6.pD) / (N - 1)
constant1 = (1 / (2 * sqrt(pi) * N))**(1 / 5)
constant2 = (-6 * sqrt(2) * phi4 / phi6)**(1 / 7)
h_initial = constant1 * phi4**(-1 / 5)
h = ls(
_fast_h_fun,
h_initial,
bounds=(0, inf),
ftol=1e-14,
xtol=1e-14,
verbose=0,
args=(N, X_shifted_scale, constant1, constant2, eps),
)
h = float(h.x) / scale
return h
def fast_h_fun(h, N, X, c1, c2, eps):
lam = c2 * h**(5 / 7)
D4 = FUDD(N, N, X, X, float(lam), 4, eps)
D4.evaluate()
phi4 = sum(D4.pD) / (N - 1)
return h - c1 * phi4**(-1 / 5)