def kde_demo1():
"""KDEDEMO1 Demonstrate the smoothing parameter impact on KDE.
KDEDEMO1 shows the true density (dotted) compared to KDE based on 7
observations (solid) and their individual kernels (dashed) for 3
different values of the smoothing parameter, hs.
Examples
--------
>>> kde_demo1()
"""
x = np.linspace(-4, 4, 101)
x0 = x / 2.0
data = np.random.normal(loc=0, scale=1.0, size=7)
kernel = Kernel('gauss')
hs = kernel.hns(data)
h_vec = [hs / 2, hs, 2 * hs]
for ix, h in enumerate(h_vec):
plt.figure(ix)
kde = KDE(data, hs=h, kernel=kernel)
f2 = kde(x, output='plot', title='h_s = {0:2.2f}'.format(float(h)),
ylab='Density')
f2.plot('k-')
plt.plot(x, st.norm.pdf(x, 0, 1), 'k:')
n = len(data)
plt.plot(data, np.zeros(data.shape), 'bx')
y = kernel(x0) / (n * h * kernel.norm_factor(d=1, n=n))
for i in range(n):
plt.plot(data[i] + x0 * h, y, 'b--')
plt.plot([data[i], data[i]], [0, np.max(y)], 'b')
plt.axis([min(x), max(x), 0, 0.5])
def kde_demo5(N=500):
"""Demonstrate that the improved Sheather-Jones plug-in (hisj) is superior
for 2D multimodal distributions
KDEDEMO5 shows that the improved Sheather-Jones plug-in smoothing is better
compared to normal reference rules (in this case the hns)
Examples
--------
>>> kde_demo5()
"""
data = np.hstack((st.norm.rvs(loc=5, scale=1, size=(2, N,)),
st.norm.rvs(loc=-5, scale=1, size=(2, N,))))
kde = KDE(data, kernel=Kernel('gauss', 'hns'))
f = kde(output='plot', plotflag=1,
title='Ordinary KDE, hns={0:s}'.format(str(list(kde.hs))))
kde1 = KDE(data, kernel=Kernel('gauss', 'hisj'))
f1 = kde1(output='plot', plotflag=1,
title='Ordinary KDE, hisj={0:s}'.format(str(list(kde1.hs))))
plt.figure(0)
plt.clf()
f.plot()
plt.plot(data[0], data[1], '.')
plt.figure(1)
plt.clf()
f1.plot()
plt.plot(data[0], data[1], '.')
def kde_demo4(N=50):
"""Demonstrate that the improved Sheather-Jones plug-in (hisj) is superior
for 1D multimodal distributions
KDEDEMO4 shows that the improved Sheather-Jones plug-in smoothing is a
better compared to normal reference rules (in this case the hns)
Examples
--------
>>> kde_demo4()
"""
data = np.hstack((st.norm.rvs(loc=5, scale=1, size=(N,)),
st.norm.rvs(loc=-5, scale=1, size=(N,))))
# x = np.linspace(1.5e-3, 5, 55)
kde = KDE(data, kernel=Kernel('gauss', 'hns'))
f = kde(output='plot', title='Ordinary KDE', plotflag=1)
kde1 = KDE(data, kernel=Kernel('gauss', 'hisj'))
f1 = kde1(output='plot', label='Ordinary KDE', plotflag=1)
plt.figure(0)
f.plot('r', label='hns={0}'.format(kde.hs))
# plt.figure(2)
f1.plot('b', label='hisj={0}'.format(kde1.hs))
x = np.linspace(-9, 9)
plt.plot(x, (st.norm.pdf(x, loc=-5, scale=1) +
st.norm.pdf(x, loc=5, scale=1)) / 2, 'k:',
label='True density')
plt.legend()
def kreg_demo1(hs=None, fast=True, fun='hisj'):
"""Compare KRegression to KernelReg from statsmodels.nonparametric
Examples
--------
>>> kreg_demo1()
"""
N = 100
# ei = np.random.normal(loc=0, scale=0.075, size=(N,))
ei = np.array([
-0.08508516, 0.10462496, 0.07694448, -0.03080661, 0.05777525,
0.06096313, -0.16572389, 0.01838912, -0.06251845, -0.09186784,
-0.04304887, -0.13365788, -0.0185279, -0.07289167, 0.02319097,
0.06887854, -0.08938374, -0.15181813, 0.03307712, 0.08523183,
-0.0378058, -0.06312874, 0.01485772, 0.06307944, -0.0632959,
0.18963205, 0.0369126, -0.01485447, 0.04037722, 0.0085057,
-0.06912903, 0.02073998, 0.1174351, 0.17599277, -0.06842139,
0.12587608, 0.07698113, -0.0032394, -0.12045792, -0.03132877,
0.05047314, 0.02013453, 0.04080741, 0.00158392, 0.10237899,
-0.09069682, 0.09242174, -0.15445323, 0.09190278, 0.07138498,
0.03002497, 0.02495252, 0.01286942, 0.06449978, 0.03031802,
0.11754861, -0.02322272, 0.00455867, -0.02132251, 0.09119446,
-0.03210086, -0.06509545, 0.07306443, 0.04330647, 0.078111,
-0.04146907, 0.05705476, 0.02492201, -0.03200572, -0.02859788,
-0.05893749, 0.00089538, 0.0432551, 0.04001474, 0.04888828,
-0.17708392, 0.16478644, 0.1171006, 0.11664846, 0.01410477,
-0.12458953, -0.11692081, 0.0413047, -0.09292439, -0.07042327,
0.14119701, -0.05114335, 0.04994696, -0.09520663, 0.04829406,
-0.01603065, -0.1933216, 0.19352763, 0.11819496, 0.04567619,
-0.08348306, 0.00812816, -0.00908206, 0.14528945, 0.02901065])
x = np.linspace(0, 1, N)
va_1 = 0.3 ** 2
va_2 = 0.7 ** 2
y0 = np.exp(-x ** 2 / (2 * va_1)) + 1.3 * np.exp(-(x - 1) ** 2 / (2 * va_2))
y = y0 + ei
kernel = Kernel('gauss', fun=fun)
hopt = kernel.hisj(x)
kreg = KRegression(
x, y, p=0, hs=hs, kernel=kernel, xmin=-2 * hopt, xmax=1 + 2 * hopt)
if fast:
kreg.__call__ = kreg.eval_grid_fast
f = kreg(x, output='plot', title='Kernel regression', plotflag=1)
plt.figure(0)
f.plot(label='p=0')
kreg.p = 1
f1 = kreg(x, output='plot', title='Kernel regression', plotflag=1)
f1.plot(label='p=1')
# print(f1.data)
plt.plot(x, y, '.', label='data')
plt.plot(x, y0, 'k', label='True model')
from statsmodels.nonparametric.kernel_regression import KernelReg
kreg2 = KernelReg(y, x, ('c'))
y2 = kreg2.fit(x)
plt.plot(x, y2[0], 'm', label='statsmodel')
plt.legend()
def kde_demo3():
"""Demonstrate the difference between transformation and ordinary-KDE in 2D
KDEDEMO3 shows that the transformation KDE is a better estimate for
Rayleigh distributed data around 0 than the ordinary KDE.
Examples
--------
>>> kde_demo3()
"""
data = st.rayleigh.rvs(scale=1, size=(2, 300))
# x = np.linspace(1.5e-3, 5, 55)
kde = KDE(data)
f = kde(output='plot', title='Ordinary KDE', plotflag=1)
plt.figure(0)
f.plot()
plt.plot(data[0], data[1], '.')
# plotnorm((data).^(L2)) % gives a straight line => L2 = 0.5 reasonable
hs = Kernel('gauss').get_smoothing(data**0.5)
tkde = TKDE(data, hs=hs, L2=0.5)
ft = tkde.eval_grid_fast(
output='plot', title='Transformation KDE', plotflag=1)
plt.figure(1)
ft.plot()
plt.plot(data[0], data[1], '.')
plt.figure(0)
def kde_demo2():
"""Demonstrate the difference between transformation- and ordinary-KDE.
KDEDEMO2 shows that the transformation KDE is a better estimate for
Rayleigh distributed data around 0 than the ordinary KDE.
Examples
--------
>>> kde_demo2()
"""
data = st.rayleigh.rvs(scale=1, size=300)
x = np.linspace(1.5e-2, 5, 55)
kde = KDE(data)
f = kde(output='plot', title='Ordinary KDE (hs={0:})'.format(kde.hs))
plt.figure(0)
f.plot()
plt.plot(x, st.rayleigh.pdf(x, scale=1), ':')
# plotnorm((data).^(L2)) # gives a straight line => L2 = 0.5 reasonable
hs = Kernel('gauss').get_smoothing(data**0.5)
tkde = TKDE(data, hs=hs, L2=0.5)
ft = tkde(x, output='plot',
title='Transformation KDE (hs={0:})'.format(tkde.tkde.hs))
plt.figure(1)
ft.plot()
plt.plot(x, st.rayleigh.pdf(x, scale=1), ':')
plt.figure(0)