机器学习基础Scipy（科学计算库） 手把手手把手
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0.导语
Scipy是一个用于数学、科学、工程领域的常用软件包，可以处理插值、积分、优化、图像处理、常微分方程数值解的求解、信号处理等问题。它用于有效计算Numpy矩阵，使Numpy和Scipy协同工作，高效解决问题。
Scipy是由针对特定任务的子模块组成：
| 模块名 | 应用领域 |
|---|---|
| scipy.cluster | 向量计算/Kmeans |
| scipy.constants | 物理和数学常量 |
| scipy.fftpack | 傅立叶变换 |
| scipy.integrate | 积分程序 |
| scipy.interpolate | 插值 |
| scipy.io | 数据输入输出 |
| scipy.linalg | 线性代数程序 |
| scipy.ndimage | n维图像包 |
| scipy.odr | 正交距离回归 |
| scipy.optimize | 优化 |
| scipy.signal | 信号处理 |
| scipy.sparse | 稀疏矩阵 |
| scipy.spatial | 空间数据结构和算法 |
| scipy.special | 一些特殊的数学函数 |
| scipy.stats | 统计 |
备注：本文代码可以在github下载
https://github.com/fengdu78/Data-Science-Notes/tree/master/4.scipy
1.SciPy-数值计算库
import numpy as np
import pylab as pl
import matplotlib as mpl
mpl.rcParams['font.sans-serif'] = ['SimHei']
import scipy
scipy.__version__#查看版本
'1.0.0'
常数和特殊函数
from scipy import constants as C
print (C.c) # 真空中的光速
print (C.h) # 普朗克常数
299792458.0
6.62607004e-34
C.physical_constants["electron mass"]
(9.10938356e-31, 'kg', 1.1e-38)
# 1英里等于多少米, 1英寸等于多少米, 1克等于多少千克, 1磅等于多少千克
print(C.mile)
print(C.inch)
print(C.gram)
print(C.pound)
1609.3439999999998
0.0254
0.001
0.45359236999999997
import scipy.special as S
print (1 + 1e-20)
print (np.log(1+1e-20))
print (S.log1p(1e-20))
1.0
0.0
1e-20
m = np.linspace(0.1, 0.9, 4)
u = np.linspace(-10, 10, 200)
results = S.ellipj(u[:, None], m[None, :])
print([y.shape for y in results])
[(200, 4), (200, 4), (200, 4), (200, 4)]
#%figonly=使用广播计算得到的`ellipj()`返回值
fig, axes = pl.subplots(2, 2, figsize=(12, 4))
labels = ["$sn$", "$cn$", "$dn$", "$\\phi$"]
for ax, y, label in zip(axes.ravel(), results, labels):
ax.plot(u, y)
ax.set_ylabel(label)
ax.margins(0, 0.1)
axes[1, 1].legend(["$m={:g}$".format(m_) for m_ in m], loc="best", ncol=2);
2.拟合与优化-optimize
非线性方程组求解
import pylab as pl
import numpy as np
import matplotlib as mpl
mpl.rcParams['font.sans-serif'] = ['SimHei']
from math import sin, cos
from scipy import optimize
def f(x): #❶
x0, x1, x2 = x.tolist() #❷
return [
5*x1+3,
4*x0*x0 - 2*sin(x1*x2),
x1*x2 - 1.5
]
# f计算方程组的误差，[1,1,1]是未知数的初始值
result = optimize.fsolve(f, [1,1,1]) #❸
print (result)
print (f(result))
[-0.70622057 -0.6 -2.5 ]
[0.0, -9.126033262418787e-14, 5.329070518200751e-15]
def j(x): #❶
x0, x1, x2 = x.tolist()
return [[0, 5, 0],
[8 * x0, -2 * x2 * cos(x1 * x2), -2 * x1 * cos(x1 * x2)],
[0, x2, x1]]
result = optimize.fsolve(f, [1, 1, 1], fprime=j) #❷
print(result)
print(f(result))
[-0.70622057 -0.6 -2.5 ]
[0.0, -9.126033262418787e-14, 5.329070518200751e-15]
最小二乘拟合
import numpy as np
from scipy import optimize
X = np.array([ 8.19, 2.72, 6.39, 8.71, 4.7 , 2.66, 3.78])
Y = np.array([ 7.01, 2.78, 6.47, 6.71, 4.1 , 4.23, 4.05])
def residuals(p): #❶
"计算以p为参数的直线和原始数据之间的误差"
k, b = p
return Y - (k*X + b)
# leastsq使得residuals()的输出数组的平方和最小，参数的初始值为[1,0]
r = optimize.leastsq(residuals, [1, 0]) #❷
k, b = r[0]
print ("k =",k, "b =",b)
k = 0.6134953491930442 b = 1.794092543259387
#%figonly=最小化正方形面积之和（左），误差曲面（右）
scale_k = 1.0
scale_b = 10.0
scale_error = 1000.0
def S(k, b):
"计算直线y=k*x+b和原始数据X、Y的误差的平方和"
error = np.zeros(k.shape)
for x, y in zip(X, Y):
error += (y - (k * x + b)) ** 2
return error
ks, bs = np.mgrid[k - scale_k:k + scale_k:40j, b - scale_b:b + scale_b:40j]
error = S(ks, bs) / scale_error
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.patches import Rectangle
fig = pl.figure(figsize=(12, 5))
ax1 = pl.subplot(121)
ax1.plot(X, Y, "o")
X0 = np.linspace(2, 10, 3)
Y0 = k*X0 + b
ax1.plot(X0, Y0)
for x, y in zip(X, Y):
y2 = k*x+b
rect = Rectangle((x,y), abs(y-y2), y2-y, facecolor="red", alpha=0.2)
ax1.add_patch(rect)
ax1.set_aspect("equal")
ax2 = fig.add_subplot(122, projection='3d')
ax2.plot_surface(
ks, bs / scale_b, error, rstride=3, cstride=3, cmap="jet", alpha=0.5)
ax2.scatter([k], [b / scale_b], [S(k, b) / scale_error], c="r", s=20)
ax2.set_xlabel("$k$")
ax2.set_ylabel("$b$")
ax2.set_zlabel("$error$");
#%fig=带噪声的正弦波拟合
def func(x, p): #❶
"""
数据拟合所用的函数: A*sin(2*pi*k*x + theta)
"""
A, k, theta = p
return A * np.sin(2 * np.pi * k * x + theta)
def residuals(p, y, x): #❷
"""
实验数据x, y和拟合函数之间的差，p为拟合需要找到的系数
"""
return y - func(x, p)
x = np.linspace(0, 2 * np.pi, 100)
A, k, theta = 10, 0.34, np.pi / 6 # 真实数据的函数参数
y0 = func(x, [A, k, theta]) # 真实数据
# 加入噪声之后的实验数据
np.random.seed(0)
y1 = y0 + 2 * np.random.randn(len(x)) #❸
p0 = [7, 0.40, 0] # 第一次猜测的函数拟合参数
# 调用leastsq进行数据拟合
# residuals为计算误差的函数
# p0为拟合参数的初始值
# args为需要拟合的实验数据
plsq = optimize.leastsq(residuals, p0, args=(y1, x)) #❹
print(u"真实参数:", [A, k, theta])
print(u"拟合参数", plsq[0]) # 实验数据拟合后的参数
pl.plot(x, y1, "o", label=u"带噪声的实验数据")
pl.plot(x, y0, label=u"真实数据")
pl.plot(x, func(x, plsq[0]), label=u"拟合数据")
pl.legend(loc="best")
真实参数: [10, 0.34, 0.5235987755982988]
拟合参数 [10.25218748 0.3423992 0.50817423]
def func2(x, A, k, theta):
return A*np.sin(2*np.pi*k*x+theta)
popt, _ = optimize.curve_fit(func2, x, y1, p0=p0)
print (popt)
[10.25218748 0.3423992 0.50817425]
popt, _ = optimize.curve_fit(func2, x, y1, p0=[10, 1, 0])
print(u"真实参数:", [A, k, theta])
print(u"拟合参数", popt)
真实参数: [10, 0.34, 0.5235987755982988]
拟合参数 [ 0.71093469 1.02074585 -0.12776742]
计算函数局域最小值
def target_function(x, y):
return (1 - x)**2 + 100 * (y - x**2)**2
class TargetFunction(object):
def __init__(self):
self.f_points = []
self.fprime_points = []
self.fhess_points = []
def f(self, p):
x, y = p.tolist()
z = target_function(x, y)
self.f_points.append((x, y))
return z
def fprime(self, p):
x, y = p.tolist()
self.fprime_points.append((x, y))
dx = -2 + 2 * x - 400 * x * (y - x**2)
dy = 200 * y - 200 * x**2
return np.array([dx, dy])
def fhess(self, p):
x, y = p.tolist()
self.fhess_points.append((x, y))
return np.array([[2 * (600 * x**2 - 200 * y + 1), -400 * x],
[-400 * x, 200]])
def fmin_demo(method):
target = TargetFunction()
init_point = (-1, -1)
res = optimize.minimize(
target.f,
init_point,
method=method,
jac=target.fprime,
hess=target.fhess)
return res, [
np.array(points) for points in (target.f_points, target.fprime_points,
target.fhess_points)
]
methods = ("Nelder-Mead", "Powell", "CG", "BFGS", "Newton-CG", "L-BFGS-B")
for method in methods:
res, (f_points, fprime_points, fhess_points) = fmin_demo(method)
print(
"{:12s}: min={:12g}, f count={:3d}, fprime count={:3d}, fhess count={:3d}"
.format(method, float(res["fun"]), len(f_points), len(fprime_points),
len(fhess_points)))
Nelder-Mead : min= 5.30934e-10, f count=125, fprime count= 0, fhess count= 0
Powell : min= 0, f count= 52, fprime count= 0, fhess count= 0
CG : min= 9.63056e-21, f count= 39, fprime count= 39, fhess count= 0
BFGS : min= 1.84992e-16, f count= 40, fprime count= 40, fhess count= 0
Newton-CG : min= 5.22666e-10, f count= 60, fprime count= 97, fhess count= 38
L-BFGS-B : min= 6.5215e-15, f count= 33, fprime count= 33, fhess count= 0
#%figonly=各种优化算法的搜索路径
def draw_fmin_demo(f_points, fprime_points, ax):
xmin, xmax = -3, 3
ymin, ymax = -3, 3
Y, X = np.ogrid[ymin:ymax:500j,xmin:xmax:500j]
Z = np.log10(target_function(X, Y))
zmin, zmax = np.min(Z), np.max(Z)
ax.imshow(Z, extent=(xmin,xmax,ymin,ymax), origin="bottom", aspect="auto", cmap="gray")
ax.plot(f_points[:,0], f_points[:,1], lw=1)
ax.scatter(f_points[:,0], f_points[:,1], c=range(len(f_points)), s=50, linewidths=0)
if len(fprime_points):
ax.scatter(fprime_points[:, 0], fprime_points[:, 1], marker="x", color="w", alpha=0.5)
ax.set_xlim(xmin, xmax)
ax.set_ylim(ymin, ymax)
fig, axes = pl.subplots(2, 3, figsize=(9, 6))
methods = ("Nelder-Mead", "Powell", "CG", "BFGS", "Newton-CG", "L-BFGS-B")
for ax, method in zip(axes.ravel(), methods):
res, (f_points, fprime_points, fhess_points) = fmin_demo(method)
draw_fmin_demo(f_points, fprime_points, ax)
ax.set_aspect("equal")
ax.set_title(method)
计算全域最小值
def func(x, p):
A, k, theta = p
return A*np.sin(2*np.pi*k*x+theta)
def func_error(p, y, x):
return np.sum((y - func(x, p))**2)
x = np.linspace(0, 2*np.pi, 100)
A, k, theta = 10, 0.34, np.pi/6
y0 = func(x, [A, k, theta])
np.random.seed(0)
y1 = y0 + 2 * np.random.randn(len(x))
result = optimize.basinhopping(func_error, (1, 1, 1),
niter = 10,
minimizer_kwargs={"method":"L-BFGS-B",
"args":(y1, x)})
print (result.x)
[10.25218676 -0.34239909 2.63341581]
#%figonly=用`basinhopping()`拟合正弦波
pl.plot(x, y1, "o", label=u"带噪声的实验数据")
pl.plot(x, y0, label=u"真实数据")
pl.plot(x, func(x, result.x), label=u"拟合数据")
pl.legend(loc="best");
3.线性代数-linalg
解线性方程组
import pylab as pl
import numpy as np
from scipy import linalg
import matplotlib as mpl
mpl.rcParams['font.sans-serif'] = ['SimHei']
import numpy as np
from scipy import linalg
m, n = 500, 50
A = np.random.rand(m, m)
B = np.random.rand(m, n)
X1 = linalg.solve(A, B)
X2 = np.dot(linalg.inv(A), B)
print (np.allclose(X1, X2))
%timeit linalg.solve(A, B)
%timeit np.dot(linalg.inv(A), B)
True
5.38 ms ± 120 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
8.14 ms ± 994 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
luf = linalg.lu_factor(A)
X3 = linalg.lu_solve(luf, B)
np.allclose(X1, X3)
True
M, N = 1000, 100
np.random.seed(0)
A = np.random.rand(M, M)
B = np.random.rand(M, N)
Ai = linalg.inv(A)
luf = linalg.lu_factor(A)
%timeit linalg.inv(A)
%timeit np.dot(Ai, B)
%timeit linalg.lu_factor(A)
%timeit linalg.lu_solve(luf, B)
50.6 ms ± 1.94 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
3.49 ms ± 306 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
20.1 ms ± 1.42 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
4.49 ms ± 65 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
最小二乘解
from numpy.lib.stride_tricks import as_strided
def make_data(m, n, noise_scale): #❶
np.random.seed(42)
x = np.random.standard_normal(m)
h = np.random.standard_normal(n)
y = np.convolve(x, h)
yn = y + np.random.standard_normal(len(y)) * noise_scale * np.max(y)
return x, yn, h
def solve_h(x, y, n): #❷
X = as_strided(
x, shape=(len(x) - n + 1, n), strides=(x.itemsize, x.itemsize)) #❸
Y = y[n - 1:len(x)] #❹
h = linalg.lstsq(X, Y) #❺
return h[0][::-1] #❻
x, yn, h = make_data(1000, 100, 0.4)
H1 = solve_h(x, yn, 120)
H2 = solve_h(x, yn, 80)
print("Average error of H1:", np.mean(np.abs(h[:100] - h)))
print("Average error of H2:", np.mean(np.abs(h[:80] - H2)))
Average error of H1: 0.0
Average error of H2: 0.2958422158342371
#%figonly=实际的系统参数与最小二乘解的比较
fig, (ax1, ax2) = pl.subplots(2, 1, figsize=(6, 4))
ax1.plot(h, linewidth=2, label=u"实际的系统参数")
ax1.plot(H1, linewidth=2, label=u"最小二乘解H1", alpha=0.7)
ax1.legend(loc="best", ncol=2)
ax1.set_xlim(0, len(H1))
ax2.plot(h, linewidth=2, label=u"实际的系统参数")
ax2.plot(H2, linewidth=2, label=u"最小二乘解H2", alpha=0.7)
ax2.legend(loc="best", ncol=2)
ax2.set_xlim(0, len(H1));
特征值和特征向量
A = np.array([[1, -0.3], [-0.1, 0.9]])
evalues, evectors = linalg.eig(A)
print(evalues)
print(evectors)
[1.13027756+0.j 0.76972244+0.j]
[[ 0.91724574 0.79325185]
[-0.3983218 0.60889368]]
#%figonly=线性变换将蓝色箭头变换为红色箭头
points = np.array([[0, 1.0], [1.0, 0], [1, 1]])
def draw_arrows(points, **kw):
props = dict(color="blue", arrowstyle="->")
props.update(kw)
for x, y in points:
pl.annotate("",
xy=(x, y), xycoords='data',
xytext=(0, 0), textcoords='data',
arrowprops=props)
draw_arrows(points)
draw_arrows(np.dot(A, points.T).T, color="red")
draw_arrows(evectors.T, alpha=0.7, linewidth=2)
draw_arrows(np.dot(A, evectors).T, color="red", alpha=0.7, linewidth=2)
ax = pl.gca()
ax.set_aspect("equal")
ax.set_xlim(-0.5, 1.1)
ax.set_ylim(-0.5, 1.1);
np.random.seed(42)
t = np.random.uniform(0, 2*np.pi, 60)
alpha = 0.4
a = 0.5
b = 1.0
x = 1.0 + a*np.cos(t)*np.cos(alpha) - b*np.sin(t)*np.sin(alpha)
y = 1.0 + a*np.cos(t)*np.sin(alpha) - b*np.sin(t)*np.cos(alpha)
x += np.random.normal(0, 0.05, size=len(x))
y += np.random.normal(0, 0.05, size=len(y))
D = np.c_[x**2, x*y, y**2, x, y, np.ones_like(x)]
A = np.dot(D.T, D)
C = np.zeros((6, 6))
C[[0, 1, 2], [2, 1, 0]] = 2, -1, 2
evalues, evectors = linalg.eig(A, C) #❶
evectors = np.real(evectors)
err = np.mean(np.dot(D, evectors)**2, 0) #❷
p = evectors[:, np.argmin(err) ] #❸
print (p)
[-0.55214278 0.5580915 -0.23809922 0.54584559 -0.08350449 -0.14852803]
#%figonly=用广义特征向量计算的拟合椭圆
def ellipse(p, x, y):
a, b, c, d, e, f = p
return a*x**2 + b*x*y + c*y**2 + d*x + e*y + f
X, Y = np.mgrid[0:2:100j, 0:2:100j]
Z = ellipse(p, X, Y)
pl.plot(x, y, "ro", alpha=0.5)
pl.contour(X, Y, Z, levels=[0]);
奇异值分解-SVD
r, g, b = np.rollaxis(pl.imread("vinci_target.png"), 2).astype(float)
img = 0.2989 * r + 0.5870 * g + 0.1140 * b
img.shape
(505, 375)
U, s, Vh = linalg.svd(img)
print(U.shape)
print(s.shape)
print(Vh.shape)
(505, 505)
(375,)
(375, 375)
#%fig=按从大到小排列的奇异值
pl.semilogy(s, lw=3);
def composite(U, s, Vh, n):
return np.dot(U[:, :n], s[:n, np.newaxis] * Vh[:n, :])
print (np.allclose(img, composite(U, s, Vh, len(s))))
True
#%fig=原始图像、使用10、20、50个向量合成的图像（从左到右）
img10 = composite(U, s, Vh, 10)
img20 = composite(U, s, Vh, 20)
img50 = composite(U, s, Vh, 50)
%array_image img; img10; img20; img50
UsageError: Line magic function `%array_image` not found.
pl.imshow(img)
pl.imshow(img10)
pl.imshow(img20)
pl.imshow(img50)
4.统计-stats
import numpy as np
import pylab as pl
from scipy import stats
import matplotlib.pyplot as plt
import matplotlib as mpl
mpl.rcParams['font.sans-serif'] = ['SimHei']
连续概率分布
from scipy import stats
[k for k, v in stats.__dict__.items() if isinstance(v, stats.rv_continuous)]
['ksone',
'kstwobign',
'norm',
'alpha',
'anglit',
'arcsine',
'beta',
'betaprime',
'bradford',
'burr',
'burr12',
'fisk',
'cauchy',
'chi',
'chi2',
'cosine',
'dgamma',
'dweibull',
'expon',
'exponnorm',
'exponweib',
'exponpow',
'fatiguelife',
'foldcauchy',
'f',
'foldnorm',
'weibull_min',
'weibull_max',
'frechet_r',
'frechet_l',
'genlogistic',
'genpareto',
'genexpon',
'genextreme',
'gamma',
'erlang',
'gengamma',
'genhalflogistic',
'gompertz',
'gumbel_r',
'gumbel_l',
'halfcauchy',
'halflogistic',
'halfnorm',
'hypsecant',
'gausshyper',
'invgamma',
'invgauss',
'invweibull',
'johnsonsb',
'johnsonsu',
'laplace',
'levy',
'levy_l',
'levy_stable',
'logistic',
'loggamma',
'loglaplace',
'lognorm',
'gilbrat',
'maxwell',
'mielke',
'kappa4',
'kappa3',
'nakagami',
'ncx2',
'ncf',
't',
'nct',
'pareto',
'lomax',
'pearson3',
'powerlaw',
'powerlognorm',
'powernorm',
'rdist',
'rayleigh',
'reciprocal',
'rice',
'recipinvgauss',
'semicircular',
'skewnorm',
'trapz',
'triang',
'truncexpon',
'truncnorm',
'tukeylambda',
'uniform',
'vonmises',
'vonmises_line',
'wald',
'wrapcauchy',
'gennorm',
'halfgennorm',
'crystalball',
'argus']
stats.norm.stats()
(array(0.), array(1.))
X = stats.norm(loc=1.0, scale=2.0)
X.stats()
(array(1.), array(4.))
x = X.rvs(size=10000) # 对随机变量取10000个值
np.mean(x), np.var(x) # 期望值和方差
(1.0048352738823323, 3.9372117720073554)
stats.norm.fit(x) # 得到随机序列期望值和标准差
(1.0048352738823323, 1.984240855341749)
pdf, t = np.histogram(x, bins=100, normed=True) #❶
t = (t[:-1] + t[1:]) * 0.5 #❷
cdf = np.cumsum(pdf) * (t[1] - t[0]) #❸
p_error = pdf - X.pdf(t)
c_error = cdf - X.cdf(t)
print ("max pdf error: {}, max cdf error: {}".format(
np.abs(p_error).max(),
np.abs(c_error).max()))
max pdf error: 0.018998755595167102, max cdf error: 0.018503342378306975
#%figonly=正态分布的概率密度函数（左）和累积分布函数（右）
fig, (ax1, ax2) = pl.subplots(1, 2, figsize=(7, 2))
ax1.plot(t, pdf, label=u"统计值")
ax1.plot(t, X.pdf(t), label=u"理论值", alpha=0.6)
ax1.legend(loc="best")
ax2.plot(t, cdf)
ax2.plot(t, X.cdf(t), alpha=0.6);
print(stats.gamma.stats(1.0))
print(stats.gamma.stats(2.0))
(array(1.), array(1.))
(array(2.), array(2.))
stats.gamma.stats(2.0, scale=2)
(array(4.), array(8.))
x = stats.gamma.rvs(2, scale=2, size=4)
x
array([4.40563983, 6.17699951, 3.65503843, 3.28052152])
stats.gamma.pdf(x, 2, scale=2)
array([0.12169605, 0.07037188, 0.14694352, 0.15904745])
X = stats.gamma(2, scale=2)
X.pdf(x)
array([0.12169605, 0.07037188, 0.14694352, 0.15904745])
离散概率分布
x = range(1, 7)
p = (0.4, 0.2, 0.1, 0.1, 0.1, 0.1)
dice = stats.rv_discrete(values=(x, p))
dice.rvs(size=20)
array([2, 5, 2, 6, 1, 6, 6, 5, 3, 1, 5, 2, 1, 1, 1, 1, 1, 2, 1, 6])
np.random.seed(42)
samples = dice.rvs(size=(20000, 50))
samples_mean = np.mean(samples, axis=1)
核密度估计
#%fig=核密度估计能更准确地表示随机变量的概率密度函数
_, bins, step = pl.hist(
samples_mean, bins=100, normed=True, histtype="step", label=u"直方图统计")
kde = stats.kde.gaussian_kde(samples_mean)
x = np.linspace(bins[0], bins[-1], 100)
pl.plot(x, kde(x), label=u"核密度估计")
mean, std = stats.norm.fit(samples_mean)
pl.plot(x, stats.norm(mean, std).pdf(x), alpha=0.8, label=u"正态分布拟合")
pl.legend()
#%fig=`bw_method`参数越大核密度估计曲线越平滑
for bw in [0.2, 0.3, 0.6, 1.0]:
kde = stats.gaussian_kde([-1, 0, 1], bw_method=bw)
x = np.linspace(-5, 5, 1000)
y = kde(x)
pl.plot(x, y, lw=2, label="bw={}".format(bw), alpha=0.6)
pl.legend(loc="best");
二项、泊松、伽玛分布
stats.binom.pmf(range(6), 5, 1/6.0)
array([4.01877572e-01, 4.01877572e-01, 1.60751029e-01, 3.21502058e-02,
3.21502058e-03, 1.28600823e-04])
#%fig=当n足够大时二项分布和泊松分布近似相等
lambda_ = 10.0
x = np.arange(20)
n1, n2 = 100, 1000
y_binom_n1 = stats.binom.pmf(x, n1, lambda_ / n1)
y_binom_n2 = stats.binom.pmf(x, n2, lambda_ / n2)
y_poisson = stats.poisson.pmf(x, lambda_)
print(np.max(np.abs(y_binom_n1 - y_poisson)))
print(np.max(np.abs(y_binom_n2 - y_poisson)))
#%hide
fig, (ax1, ax2) = pl.subplots(1, 2, figsize=(7.5, 2.5))
ax1.plot(x, y_binom_n1, label=u"binom", lw=2)
ax1.plot(x, y_poisson, label=u"poisson", lw=2, color="red")
ax2.plot(x, y_binom_n2, label=u"binom", lw=2)
ax2.plot(x, y_poisson, label=u"poisson", lw=2, color="red")
for n, ax in zip((n1, n2), (ax1, ax2)):
ax.set_xlabel(u"次数")
ax.set_ylabel(u"概率")
ax.set_title("n={}".format(n))
ax.legend()
fig.subplots_adjust(0.1, 0.15, 0.95, 0.90, 0.2, 0.1)
0.00675531110335309
0.0006301754049777564
#%fig=模拟泊松分布
np.random.seed(42)
def sim_poisson(lambda_, time):
t = np.random.uniform(0, time, size=lambda_ * time) #❶
count, time_edges = np.histogram(t, bins=time, range=(0, time)) #❷
dist, count_edges = np.histogram(
count, bins=20, range=(0, 20), density=True) #❸
x = count_edges[:-1]
poisson = stats.poisson.pmf(x, lambda_)
return x, poisson, dist
lambda_ = 10
times = 1000, 50000
x1, poisson1, dist1 = sim_poisson(lambda_, times[0])
x2, poisson2, dist2 = sim_poisson(lambda_, times[1])
max_error1 = np.max(np.abs(dist1 - poisson1))
max_error2 = np.max(np.abs(dist2 - poisson2))
print("time={}, max_error={}".format(times[0], max_error1))
print("time={}, max_error={}".format(times[1], max_error2))
#%hide
fig, (ax1, ax2) = pl.subplots(1, 2, figsize=(7.5, 2.5))
ax1.plot(x1, dist1, "-o", lw=2, label=u"统计结果")
ax1.plot(x1, poisson1, "->", lw=2, label=u"泊松分布", color="red", alpha=0.6)
ax2.plot(x2, dist2, "-o", lw=2, label=u"统计结果")
ax2.plot(x2, poisson2, "->", lw=2, label=u"泊松分布", color="red", alpha=0.6)
for ax, time in zip((ax1, ax2), times):
ax.set_xlabel(u"次数")
ax.set_ylabel(u"概率")
ax.set_title(u"time = {}".format(time))
ax.legend(loc="lower center")
fig.subplots_adjust(0.1, 0.15, 0.95, 0.90, 0.2, 0.1)
time=1000, max_error=0.01964230201602718
time=50000, max_error=0.001798012894964722
#%fig=模拟伽玛分布
def sim_gamma(lambda_, time, k):
t = np.random.uniform(0, time, size=lambda_ * time) #❶
t.sort() #❷
interval = t[k:] - t[:-k] #❸
dist, interval_edges = np.histogram(interval, bins=100, density=True) #❹
x = (interval_edges[1:] + interval_edges[:-1])/2 #❺
gamma = stats.gamma.pdf(x, k, scale=1.0/lambda_) #❺
return x, gamma, dist
lambda_ = 10
time = 1000
ks = 1, 2
x1, gamma1, dist1 = sim_gamma(lambda_, time, ks[0])
x2, gamma2, dist2 = sim_gamma(lambda_, time, ks[1])
#%hide
fig, (ax1, ax2) = pl.subplots(1, 2, figsize=(7.5, 2.5))
ax1.plot(x1, dist1, lw=2, label=u"统计结果")
ax1.plot(x1, gamma1, lw=2, label=u"伽玛分布", color="red", alpha=0.6)
ax2.plot(x2, dist2, lw=2, label=u"统计结果")
ax2.plot(x2, gamma2, lw=2, label=u"伽玛分布", color="red", alpha=0.6)
for ax, k in zip((ax1, ax2), ks):
ax.set_xlabel(u"时间间隔")
ax.set_ylabel(u"概率密度")
ax.set_title(u"k = {}".format(k))
ax.legend(loc="upper right")
fig.subplots_adjust(0.1, 0.15, 0.95, 0.90, 0.2, 0.1);
T = 100000
A_count = int(T / 5)
B_count = int(T / 10)
A_time = np.random.uniform(0, T, A_count) #❶
B_time = np.random.uniform(0, T, B_count)
bus_time = np.concatenate((A_time, B_time)) #❷
bus_time.sort()
N = 200000
passenger_time = np.random.uniform(bus_time[0], bus_time[-1], N) #❸
idx = np.searchsorted(bus_time, passenger_time) #❹
np.mean(bus_time[idx] - passenger_time) * 60 #❺
202.3388747879705
np.mean(np.diff(bus_time)) * 60
199.99833251643057
#%figonly=观察者偏差
import matplotlib.gridspec as gridspec
pl.figure(figsize=(7.5, 3))
G = gridspec.GridSpec(10, 1)
ax1 = pl.subplot(G[:2, 0])
ax2 = pl.subplot(G[3:, 0])
ax1.vlines(bus_time[:10], 0, 1, lw=2, color="blue", label=u"公交车")
ptime = np.random.uniform(bus_time[0], bus_time[9], 100)
ax1.vlines(ptime, 0, 1, lw=1, color="red", alpha=0.2, label=u"乘客")
ax1.legend()
count, bins = np.histogram(passenger_time, bins=bus_time)
ax2.plot(np.diff(bins), count, ".", alpha=0.3, rasterized=True)
ax2.set_xlabel(u"公交车的时间间隔")
ax2.set_ylabel(u"等待人数");
from scipy import integrate
t = 10.0 / 3 # 两辆公交车之间的平均时间间隔
bus_interval = stats.gamma(1, scale=t)
n, _ = integrate.quad(lambda x: 0.5 * x * x * bus_interval.pdf(x), 0, 1000)
d, _ = integrate.quad(lambda x: x * bus_interval.pdf(x), 0, 1000)
n / d * 60
200.0
学生 t-分布与 t 检验
#%fig=模拟学生t-分布
mu = 0.0
n = 10
samples = stats.norm(mu).rvs(size=(100000, n)) #❶
t_samples = (np.mean(samples, axis=1) - mu) / np.std(
samples, ddof=1, axis=1) * n**0.5 #❷
sample_dist, x = np.histogram(t_samples, bins=100, density=True) #❸
x = 0.5 * (x[:-1] + x[1:])
t_dist = stats.t(n - 1).pdf(x)
print("max error:", np.max(np.abs(sample_dist - t_dist)))
#%hide
pl.plot(x, sample_dist, lw=2, label=u"样本分布")
pl.plot(x, t_dist, lw=2, alpha=0.6, label=u"t分布")
pl.xlim(-5, 5)
pl.legend(loc="best")
max error: 0.006832108369761447
#%figonly=当`df`增大，学生t-分布趋向于正态分布
fig, (ax1, ax2) = pl.subplots(1, 2, figsize=(7.5, 2.5))
ax1.plot(x, stats.t(6-1).pdf(x), label=u"df=5", lw=2)
ax1.plot(x, stats.t(40-1).pdf(x), label=u"df=39", lw=2, alpha=0.6)
ax1.plot(x, stats.norm.pdf(x), "k--", label=u"norm")
ax1.legend()
ax2.plot(x, stats.t(6-1).sf(x), label=u"df=5", lw=2)
ax2.plot(x, stats.t(40-1).sf(x), label=u"df=39", lw=2, alpha=0.6)
ax2.plot(x, stats.norm.sf(x), "k--", label=u"norm")
ax2.legend();
n = 30
np.random.seed(42)
s = stats.norm.rvs(loc=1, scale=0.8, size=n)
t = (np.mean(s) - 0.5) / (np.std(s, ddof=1) / np.sqrt(n))
print (t, stats.ttest_1samp(s, 0.5))
2.658584340882224 Ttest_1sampResult(statistic=2.658584340882224, pvalue=0.01263770225709123)
print ((np.mean(s) - 1) / (np.std(s, ddof=1) / np.sqrt(n)))
print (stats.ttest_1samp(s, 1), stats.ttest_1samp(s, 0.9))
-1.1450173670383303
Ttest_1sampResult(statistic=-1.1450173670383303, pvalue=0.26156414618801477) Ttest_1sampResult(statistic=-0.3842970254542196, pvalue=0.7035619103425202)
#%fig=红色部分为`ttest_1samp()`计算的p值
x = np.linspace(-5, 5, 500)
y = stats.t(n-1).pdf(x)
plt.plot(x, y, lw=2)
t, p = stats.ttest_1samp(s, 0.5)
mask = x > np.abs(t)
plt.fill_between(x[mask], y[mask], color="red", alpha=0.5)
mask = x < -np.abs(t)
plt.fill_between(x[mask], y[mask], color="red", alpha=0.5)
plt.axhline(color="k", lw=0.5)
plt.xlim(-5, 5);
from scipy import integrate
x = np.linspace(-10, 10, 100000)
y = stats.t(n-1).pdf(x)
mask = x >= np.abs(t)
integrate.trapz(y[mask], x[mask])*2
0.012633433707685974
m = 200000
mean = 0.5
r = stats.norm.rvs(loc=mean, scale=0.8, size=(m, n))
ts = (np.mean(s) - mean) / (np.std(s, ddof=1) / np.sqrt(n))
tr = (np.mean(r, axis=1) - mean) / (np.std(r, ddof=1, axis=1) / np.sqrt(n))
np.mean(np.abs(tr) > np.abs(ts))
0.012695
np.random.seed(42)
s1 = stats.norm.rvs(loc=1, scale=1.0, size=20)
s2 = stats.norm.rvs(loc=1.5, scale=0.5, size=20)
s3 = stats.norm.rvs(loc=1.5, scale=0.5, size=25)
print (stats.ttest_ind(s1, s2, equal_var=False)) #❶
print (stats.ttest_ind(s2, s3, equal_var=True)) #❷
Ttest_indResult(statistic=-2.2391470627176755, pvalue=0.033250866086743665)
Ttest_indResult(statistic=-0.5946698521856172, pvalue=0.5551805875810539)
卡方分布和卡方检验
#%fig=使用随机数验证卡方分布
a = np.random.normal(size=(300000, 4))
cs = np.sum(a**2, axis=1)
sample_dist, bins = np.histogram(cs, bins=100, range=(0, 20), density=True)
x = 0.5 * (bins[:-1] + bins[1:])
chi2_dist = stats.chi2.pdf(x, 4)
print("max error:", np.max(np.abs(sample_dist - chi2_dist)))
#%hide
pl.plot(x, sample_dist, lw=2, label=u"样本分布")
pl.plot(x, chi2_dist, lw=2, alpha=0.6, label=u"$\\chi ^{2}$分布")
pl.legend(loc="best")
max error: 0.0030732520533635066
#%fig=模拟卡方分布
repeat_count = 60000
n, k = 100, 5
np.random.seed(42)
ball_ids = np.random.randint(0, k, size=(repeat_count, n)) #❶
counts = np.apply_along_axis(np.bincount, 1, ball_ids, minlength=k) #❷
cs2 = np.sum((counts - n/k)**2.0/(n/k), axis=1) #❸
k = stats.kde.gaussian_kde(cs2) #❹
x = np.linspace(0, 10, 200)
pl.plot(x, stats.chi2.pdf(x, 4), lw=2, label=u"$\\chi ^{2}$分布")
pl.plot(x, k(x), lw=2, color="red", alpha=0.6, label=u"样本分布")
pl.legend(loc="best")
pl.xlim(0, 10);
def choose_balls(probabilities, size):
r = stats.rv_discrete(values=(range(len(probabilities)), probabilities))
s = r.rvs(size=size)
counts = np.bincount(s)
return counts
np.random.seed(42)
counts1 = choose_balls([0.18, 0.24, 0.25, 0.16, 0.17], 400)
counts2 = choose_balls([0.2]*5, 400)
print(counts1)
print(counts2)
[80 93 97 64 66]
[89 76 79 71 85]
chi1, p1 = stats.chisquare(counts1)
chi2, p2 = stats.chisquare(counts2)
print ("chi1 =", chi1, "p1 =", p1)
print ("chi2 =", chi2, "p2 =", p2)
chi1 = 11.375 p1 = 0.022657601239769634
chi2 = 2.55 p2 = 0.6357054527037017
#%figonly=卡方检验计算的概率为阴影部分的面积
x = np.linspace(0, 30, 200)
CHI2 = stats.chi2(4)
pl.plot(x, CHI2.pdf(x), "k", lw=2)
pl.vlines(chi1, 0, CHI2.pdf(chi1))
pl.vlines(chi2, 0, CHI2.pdf(chi2))
pl.fill_between(x[x>chi1], 0, CHI2.pdf(x[x>chi1]), color="red", alpha=1.0)
pl.fill_between(x[x>chi2], 0, CHI2.pdf(x[x>chi2]), color="green", alpha=0.5)
pl.text(chi1, 0.015, r"$\\chi^2_1$", fontsize=14)
pl.text(chi2, 0.015, r"$\\chi^2_2$", fontsize=14)
pl.ylim(0, 0.2)
pl.xlim(0, 20);
table = [[43, 9], [44, 4]]
chi2, p, dof, expected = stats.chi2_contingency(table)
print(chi2)
print(p)
1.0724852071005921
0.300384770390566
stats.fisher_exact(table)
(0.43434343434343436, 0.23915695682224306)
5.数值积分-integrate
import pylab as pl
import numpy as np
from scipy import integrate
from scipy.integrate import odeint
import matplotlib as mpl
mpl.rcParams['font.sans-serif'] = ['SimHei']
球的体积
def half_circle(x):
return (1-x**2)**0.5
N = 10000
x = np.linspace(-1, 1, N)
dx = x[1] - x[0]
y = half_circle(x)
2 * dx * np.sum(y) # 面积的两倍
3.1415893269307373
np.trapz(y, x) * 2 # 面积的两倍
3.1415893269315975
from scipy import integrate
pi_half, err = integrate.quad(half_circle, -1, 1)
pi_half * 2
3.141592653589797
def half_sphere(x, y):
return (1-x**2-y**2)**0.5
volume, error = integrate.dblquad(half_sphere, -1, 1,
lambda x:-half_circle(x),
lambda x:half_circle(x))
print (volume, error, np.pi*4/3/2)
2.094395102393199 1.0002356720661965e-09 2.0943951023931953
解常微分方程组
#%fig=洛伦茨吸引子：微小的初值差别也会显著地影响运动轨迹
from scipy.integrate import odeint
import numpy as np
def lorenz(w, t, p, r, b): #❶
# 给出位置矢量w，和三个参数p, r, b计算出
# dx/dt, dy/dt, dz/dt的值
x, y, z = w.tolist()
# 直接与lorenz的计算公式对应
return p*(y-x), x*(r-z)-y, x*y-b*z
t = np.arange(0, 30, 0.02) # 创建时间点
# 调用ode对lorenz进行求解, 用两个不同的初始值
track1 = odeint(lorenz, (0.0, 1.00, 0.0), t, args=(10.0, 28.0, 3.0)) #❷
track2 = odeint(lorenz, (0.0, 1.01, 0.0), t, args=(10.0, 28.0, 3.0)) #❸
#%hide
from mpl_toolkits.mplot3d import Axes3D
fig = pl.figure()
ax = Axes3D(fig)
ax.plot(track1[:,0], track1[:,1], track1[:,2], lw=1)
ax.plot(track2[:,0], track2[:,1], track2[:,2], lw=1);
ode 类
def mass_spring_damper(xu, t, m, k, b, F):
x, u = xu.tolist()
dx = u
du = (F - k*x - b*u)/m
return dx, du
#%fig=滑块的速度和位移曲线
m, b, k, F = 1.0, 10.0, 20.0, 1.0
init_status = 0.0, 0.0
args = m, k, b, F
t = np.arange(0, 2, 0.01)
result = odeint(mass_spring_damper, init_status, t, args)
#%hide
fig, (ax1, ax2) = pl.subplots(2, 1)
ax1.plot(t, result[:, 0], label=u"位移")
ax1.legend()
ax2.plot(t, result[:, 1], label=u"速度")
ax2.legend();
from scipy.integrate import ode
class MassSpringDamper(object): #❶
def __init__(self, m, k, b, F):
self.m, self.k, self.b, self.F = m, k, b, F
def f(self, t, xu):
x, u = xu.tolist()
dx = u
du = (self.F - self.k*x - self.b*u)/self.m
return [dx, du]
system = MassSpringDamper(m=m, k=k, b=b, F=F)
init_status = 0.0, 0.0
dt = 0.01
r = ode(system.f) #❷
r.set_integrator('vode', method='bdf')
r.set_initial_value(init_status, 0)
t = []
result2 = [init_status]
while r.successful() and r.t + dt < 2: #❸
r.integrate(r.t + dt)
t.append(r.t)
result2.append(r.y)
result2 = np.array(result2)
np.allclose(result, result2)
True
class PID(object):
def __init__(self, kp, ki, kd, dt):
self.kp, self.ki, self.kd, self.dt = kp, ki, kd, dt
self.last_error = None
self.status = 0.0
def update(self, error):
p = self.kp * error
i = self.ki * self.status
if self.last_error is None:
d = 0.0
else:
d = self.kd * (error - self.last_error) / self.dt
self.status += error * self.dt
self.last_error = error
return p + i + d
#%fig=使用PID控制器让滑块停在位移为1.0处
def pid_control_system(kp, ki, kd, dt, target=1.0):
system = MassSpringDamper(m=m, k=k, b=b, F=0.0)
pid = PID(kp, ki, kd, dt)
init_status = 0.0, 0.0
r = ode(system.f)
r.set_integrator('vode', method='bdf')
r.set_initial_value(init_status, 0)
t = [0]
result = [init_status]
F_arr = [0]
while r.successful() and r.t + dt < 2.0:
r.integrate(r.t + dt)
err = target - r.y[0] #❶
F = pid.update(err) #❷
system.F = F #❸
t.append(r.t)
result.append(r.y)
F_arr.append(F)
result = np.array(result)
t = np.array(t)
F_arr = np.array(F_arr)
return t, F_arr, result
t, F_arr, result = pid_control_system(50.0, 100.0, 10.0, 0.001)
print(u"控制力的终值:", F_arr[-1])
#%hide
fig, (ax1, ax2, ax3) = pl.subplots(3, 1, figsize=(6, 6))
ax1.plot(t, result[:, 0], label=u"位移")
ax1.legend(loc="best")
ax2.plot(t, result[:, 1], label=u"速度")
ax2.legend(loc="best")
ax3.plot(t, F_arr, label=u"控制力")
ax3.legend(loc="best")
控制力的终值: 19.943404699515057
%%time
from scipy import optimize
def eval_func(k):
kp, ki, kd = k
t, F_arr, result = pid_control_system(kp, ki, kd, 0.01)
return np.sum(np.abs(result[:, 0] - 1.0))
kwargs = {
"method": "L-BFGS-B",
"bounds": [(10, 200), (10, 100), (1, 100)],
"options": {
"approx_grad": True
}
}
opt_k = optimize.basinhopping(
eval_func, (10, 10, 10), niter=10, minimizer_kwargs=kwargs)
print(opt_k.x)
[56.67106149 99.97434757 1.33963577]
Wall time: 55.1 s
#%fig=优化PID的参数降低控制响应时间
kp, ki, kd = opt_k.x
t, F_arr, result = pid_control_system(kp, ki, kd, 0.01)
idx = np.argmin(np.abs(t - 0.5))
x, u = result[idx]
print ("t={}, x={:g}, u={:g}".format(t[idx], x, u))
#%hide
fig, (ax1, ax2, ax3) = pl.subplots(3, 1, figsize=(6, 6))
ax1.plot(t, result[:, 0], label=u"位移")
ax1.legend(loc="best")
ax2.plot(t, result[:, 1], label=u"速度")
ax2.legend(loc="best")
ax3.plot(t, F_arr, label=u"控制力")
ax3.legend(loc="best");
t=0.5000000000000002, x=1.07098, u=0.315352
6.信号处理-signal
import pylab as pl
import numpy as np
from scipy import signal
import matplotlib as mpl
mpl.rcParams['font.sans-serif'] = ['SimHei']
中值滤波
#%fig=使用中值滤波剔除瞬间噪声
t = np.arange(0, 20, 0.1)
x = np.sin(t)
x[np.random.randint(0, len(t), 20)] += np.random.standard_normal(20)*0.6 #❶
x2 = signal.medfilt(x, 5) #❷
x3 = signal.order_filter(x, np.ones(5), 2)
print (np.all(x2 == x3))
pl.plot(t, x, label=u"带噪声的信号")
pl.plot(t, x2 + 0.5, alpha=0.6, label=u"中值滤波之后的信号")
pl.legend(loc="best");
True
滤波器设计
sampling_rate = 8000.0
# 设计一个带通滤波器：
# 通带为0.2*4000 - 0.5*4000
# 阻带为<0.1*4000, >0.6*4000
# 通带增益的最大衰减值为2dB
# 阻带的最小衰减值为40dB
b, a = signal.iirdesign([0.2, 0.5], [0.1, 0.6], 2, 40) #❶
# 使用freq计算滤波器的频率响应
w, h = signal.freqz(b, a) #❷
# 计算增益
power = 20*np.log10(np.clip(np.abs(h), 1e-8, 1e100)) #❸
freq = w / np.pi * sampling_rate / 2
#%fig=用频率扫描波测量的频率响应
# 产生2秒钟的取样频率为sampling_rate Hz的频率扫描信号
# 开始频率为0， 结束频率为sampling_rate/2
t = np.arange(0, 2, 1/sampling_rate) #❶
sweep = signal.chirp(t, f0=0, t1=2, f1=sampling_rate/2) #❷
# 对频率扫描信号进行滤波
out = signal.lfilter(b, a, sweep) #❸
# 将波形转换为能量
out = 20*np.log10(np.abs(out)) #❹
# 找到所有局部最大值的下标
index = signal.argrelmax(out, order=3) #❺
# 绘制滤波之后的波形的增益
pl.figure(figsize=(8, 2.5))
pl.plot(freq, power, label=u"带通IIR滤波器的频率响应")
pl.plot(t[index]/2.0*4000, out[index], label=u"频率扫描波测量的频谱", alpha=0.6) #❻
pl.legend(loc="best")
#%hide
pl.title(u"频率扫描波测量的滤波器频谱")
pl.ylim(-100,20)
pl.ylabel(u"增益(dB)")
pl.xlabel(u"频率(Hz)");
连续时间线性系统
#%fig=系统的阶跃响应和正弦波响应
m, b, k = 1.0, 10, 20
numerator = [1]
denominator = [m, b, k]
plant = signal.lti(numerator, denominator) #❶
t = np.arange(0, 2, 0.01)
_, x_step = plant.step(T=t) #❷
_, x_sin, _ = signal.lsim(plant, U=np.sin(np.pi * t), T=t) #❸
#%hide
pl.plot(t, x_step, label=u"阶跃响应")
pl.plot(t, x_sin, label=u"正弦波响应")
pl.legend(loc="best")
pl.xlabel(u"时间（秒）")
pl.ylabel(u"位移（米）")
Text(0,0.5,'位移（米）')
7.插值-interpolate
import numpy as np
import pylab as pl
from scipy import interpolate
import matplotlib as mpl
mpl.rcParams['font.sans-serif'] = ['SimHei']
一维插值
WARNING：高次
interp1d()插值的运算量很大，因此对于点数较多的数据，建议使用后面介绍的UnivariateSpline()。
#%fig=`interp1d`的各阶插值
from scipy import interpolate
x = np.linspace(0, 10, 11)
y = np.sin(x)
xnew = np.linspace(0, 10, 101)
pl.plot(x, y, 'ro')
for kind in ['nearest', 'zero', 'slinear', 'quadratic']:
f = interpolate.interp1d(x, y, kind=kind) #❶
ynew = f(xnew) #❷
pl.plot(xnew, ynew, label=str(kind))
pl.legend(loc='lower right')
外推和 Spline 拟合
#%fig=使用UnivariateSpline进行插值：外推（上），数据拟合（下）
x1 = np.linspace(0, 10, 20)
y1 = np.sin(x1)
sx1 = np.linspace(0, 12, 100)
sy1 = interpolate.UnivariateSpline(x1, y1, s=0)(sx1) #❶
x2 = np.linspace(0, 20, 200)
y2 = np.sin(x2) + np.random.standard_normal(len(x2)) * 0.2
sx2 = np.linspace(0, 20, 2000)
spline2 = interpolate.UnivariateSpline(x2, y2, s=8) #❷
sy2 = spline2(sx2)
pl.figure(figsize=(8, 5))
pl.subplot(211)
pl.plot(x1, y1, ".", label=u"数据点")
pl.plot(sx1, sy1, label=u"spline曲线")
pl.legend()
pl.subplot(212)
pl.plot(x2, y2, ".", label=u"数据点")
pl.plot(sx2, sy2, linewidth=2, label=u"spline曲线")
pl.plot(x2, np.sin(x2), label=u"无噪声曲线")
pl.legend()
print(np.array_str(spline2.roots(), precision=3))
[ 0.053 3.151 6.36 9.386 12.603 15.619 18.929]
#%fig=计算Spline与水平线的交点
def roots_at(self, v): #❶
coeff = self.get_coeffs()
coeff -= v
try:
root = self.roots()
return root
finally:
coeff += v
interpolate.UnivariateSpline.roots_at = roots_at #❷
pl.plot(sx2, sy2, linewidth=2, label=u"spline曲线")
ax = pl.gca()
for level in [0.5, 0.75, -0.5, -0.75]:
ax.axhline(level, ls=":", color="k")
xr = spline2.roots_at(level) #❸
pl.plot(xr, spline2(xr), "ro")
参数插值
#%fig=使用参数插值连接二维平面上的点
x = [
4.913, 4.913, 4.918, 4.938, 4.955, 4.949, 4.911, 4.848, 4.864, 4.893,
4.935, 4.981, 5.01, 5.021
]
y = [
5.2785, 5.2875, 5.291, 5.289, 5.28, 5.26, 5.245, 5.245, 5.2615, 5.278,
5.2775, 5.261, 5.245, 5.241
]
pl.plot(x, y, "o")
for s in (0, 1e-4):
tck, t = interpolate.splprep([x, y], s=s) #❶
xi, yi = interpolate.splev(np.linspace(t[0], t[-1], 200), tck) #❷
pl.plot(xi, yi, lw=2, label=u"s=%g" % s)
pl.legend()
单调插值
import numpy as np
import matplotlib.pyplot as plt
from scipy import interpolate
x = np.arange(0, 2 * np.pi + np.pi / 4, 2 * np.pi / 8)
y = np.sin(x)
tck = interpolate.splrep(x, y, s=0)
xnew = np.arange(0, 2 * np.pi, np.pi / 50)
ynew = interpolate.splev(xnew, tck, der=0)
plt.figure()
plt.plot(x, y, 'x', xnew, ynew, xnew, np.sin(xnew), x, y, 'b')
plt.legend(['Linear', 'Cubic Spline', 'True'])
plt.axis([-0.05, 6.33, -1.05, 1.05])
plt.title('三次样条插值')
plt.show()
多维插值
#%fig=使用interp2d类进行二维插值
def func(x, y): #❶
return (x + y) * np.exp(-5.0 * (x**2 + y**2))
# X-Y轴分为15*15的网格
y, x = np.mgrid[-1:1:15j, -1:1:15j] #❷
fvals = func(x, y) # 计算每个网格点上的函数值
# 二维插值
newfunc = interpolate.interp2d(x, y, fvals, kind='cubic') #❸
# 计算100*100的网格上的插值
xnew = np.linspace(-1, 1, 100)
ynew = np.linspace(-1, 1, 100)
fnew = newfunc(xnew, ynew) #❹
#%hide
pl.subplot(121)
pl.imshow(
fvals,
extent=[-1, 1, -1, 1],
cmap=pl.cm.jet,
interpolation='nearest',
origin="lower")
pl.title("fvals")
pl.subplot(122)
pl.imshow(
fnew,
extent=[-1, 1, -1, 1],
cmap=pl.cm.jet,
interpolation='nearest',
origin="lower")
pl.title("fnew")
pl.show()
griddata
WARNING
griddata()使用欧几里得距离计算插值。如果 K 维空间中每个维度的取值范围相差较大，则应先将数据正规化，然后使用griddata()进行插值运算。
#%fig=使用gridata进行二维插值
# 计算随机N个点的坐标，以及这些点对应的函数值
N = 200
np.random.seed(42)
x = np.random.uniform(-1, 1, N)
y = np.random.uniform(-1, 1, N)
z = func(x, y)
yg, xg = np.mgrid[-1:1:100j, -1:1:100j]
xi = np.c_[xg.ravel(), yg.ravel()]
methods = 'nearest', 'linear', 'cubic'
zgs = [
interpolate.griddata((x, y), z, xi, method=method).reshape(100, 100)
for method in methods
]
#%hide
fig, axes = pl.subplots(1, 3, figsize=(11.5, 3.5))
for ax, method, zg in zip(axes, methods, zgs):
ax.imshow(
zg,
extent=[-1, 1, -1, 1],
cmap=pl.cm.jet,
interpolation='nearest',
origin="lower")
ax.set_xlabel(method)
ax.scatter(x, y, c=z)
径向基函数插值
#%fig=一维RBF插值
from scipy.interpolate import Rbf
x1 = np.array([-1, 0, 2.0, 1.0])
y1 = np.array([1.0, 0.3, -0.5, 0.8])
funcs = ['multiquadric', 'gaussian', 'linear']
nx = np.linspace(-3, 4, 100)
rbfs = [Rbf(x1, y1, function=fname) for fname in funcs] #❶
rbf_ys = [rbf(nx) for rbf in rbfs] #❷
#%hide
pl.plot(x1, y1, "o")
for fname, ny in zip(funcs, rbf_ys):
pl.plot(nx, ny, label=fname, lw=2)
pl.ylim(-1.0, 1.5)
pl.legend()
for fname, rbf in zip(funcs, rbfs):
print (fname, rbf.nodes)
multiquadric [-0.88822885 2.17654513 1.42877511 -2.67919021]
gaussian [ 1.00321945 -0.02345964 -0.65441716 0.91375159]
linear [-0.26666667 0.6 0.73333333 -0.9 ]
#%fig=二维径向基函数插值
rbfs = [Rbf(x, y, z, function=fname) for fname in funcs]
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