%pylab inline

from scipy import interpolate

def f(x):
    return np.sin(x) - 0.3 * x

n_m = 10
x_m = arange(n_m)
y_m = f(x_m) + 0.15*randn(n_m) # simulated measured signal

x_real = linspace(0, n_m-1, 100)
y_real = f(x_real)

lin_interp = interpolate.interp1d(x_m, y_m)  # return linear interpolation function
y_est_lin = lin_interp(x_real)   # evaluate at desired points

cub_interp = interpolate.interp1d(x_m, y_m, kind='cubic')  # return cubic interpolation function
y_est_cub = cub_interp(x_real)

fig, ax = subplots(figsize=(10,5))

ax.plot(x_real, y_real, 'k:', label='actual')
ax.plot(x_m, y_m, 'bo', label='measured')
ax.plot(x_real, y_est_lin, 'g-', label='linear')
ax.plot(x_real, y_est_cub, 'r-', label='cubic')
legend(loc='lower left')

from scipy import stats

x = linspace(-5, 5, 100)
P = stats.norm()     # normal distribution

fig, ax = subplots(2,1, sharex=True)
ax[0].plot(x, P.pdf(x))
ax[0].set_title('probability density function')
ax[1].plot(x, P.cdf(x))
ax[1].set_title('cumulative density function')

s_sample = P.rvs(size=1000)
r_sample = randn(500)
h = hist(s_sample, bins=50)
h1 = hist(r_sample, bins=50, color='red', alpha=0.6)

t_stat, p_value = stats.ttest_ind(s_sample, r_sample)
print 't_stat:', t_stat
print 'p_value:', p_value

from scipy import stats

np = 50
x = linspace(-10, 20, np)

a0, b0 = 0.75, -3

y = a0 * x + b0
y_noise = y + randn(np)*2

resu = stats.linregress(x, y_noise)
a, b = resu[0:2]

y_fit = a*x + b

eq_lbl = '$y = %7.2f x %+7.2f$' % (a,b)

plot(x, y_noise, 'ko')
plot(x, y_fit, 'b-', label=eq_lbl)
legend(loc='upper left')

# same task using polynomial fit
a, b = polyfit(x, y_noise, 1)
y_fit = polyval((a,b), x)

eq_lbl = '$y = %7.2f x %+7.2f$' % (a,b)

plot(x, y_noise, 'ko')
plot(x, y_fit,'b-', label=eq_lbl)
legend(loc='upper left')

a0, b0, c0 = 0.5, -1.4, 8.0
y = a0*x**2 + b0*x + c0
y_noise = y + randn(np)*10
plot(x, y_noise, 'ko', mfc='none')  # mfc <-> markerfacecolor

a, b, c = polyfit(x, y_noise, 2)
eq_lbl = r'$%.2f x^2 %+.2f x %+.2f$' % (a, b, c)
y_fit = polyval((a,b,c), x)
plot(x, y_fit, 'b-', label=eq_lbl, lw=2)
legend(loc='upper left')

import numpy as np

x = linspace(-1, 1, 100)

pure = sin(15*x)
corrupt = pure +  0.25*randn(100)

# moving average
wgt = np.ones(5)
smooth = numpy.convolve(wgt/wgt.sum(), corrupt, mode='same')

fig, ax = subplots(figsize=(10,4))
ax.plot(x, pure, 'k-', label='original')
ax.plot(x, corrupt, label='corrupted')
ax.plot(x, smooth, 'r-', label='smoothed')
ax.set_ylim([-2,3])
legend()

length = 25
figure(figsize=(8,8))
subplot(221)
plot(np.hanning(length))
title('Hanning')
subplot(222)
plot(np.hamming(length))
title('Hamming')
subplot(223)
plot(np.bartlett(length))
title('Bartlett')
subplot(224)
plot(np.blackman(length))
title('Blackman')