机器学习的算法学习

版权声明：未经同意窃取和转载我的内容，如果涉及到权益问题，后果自负！ https://blog.csdn.net/weixin_41605937/article/details/88956098

其中在Python中提供了函数自动计算的均值和标准差和偏执值和峰值的函数： mu, sigma, skew, kurtosis = calc_statistics(d)

#!/usr/bin/python
#  -*- coding:utf-8 -*-

import numpy as np
from scipy import stats
import math
import matplotlib as mpl
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm


def calc_statistics(x):
    n = x.shape[0]  # 样本个数

    # 手动计算
    m = 0
    m2 = 0
    m3 = 0
    m4 = 0
    for t in x:
        m += t
        m2 += t * t
        m3 += t ** 3
        m4 += t ** 4
    m /= n
    m2 /= n
    m3 /= n
    m4 /= n

    mu = m
    sigma = np.sqrt(m2 - mu * mu)
    skew = (m3 - 3 * mu * m2 + 2 * mu ** 3) / sigma ** 3
    kurtosis = (m4 - 4 * mu * m3 + 6 * mu * mu * m2 - 4 * mu ** 3 * mu + mu ** 4) / sigma ** 4 - 3
    print('手动计算均值、标准差、偏度、峰度：', mu, sigma, skew, kurtosis)

    # 使用系统函数验证
    mu = np.mean(x, axis=0)
    sigma = np.std(x, axis=0)
    skew = stats.skew(x)
    kurtosis = stats.kurtosis(x)
    return mu, sigma, skew, kurtosis


if __name__ == '__main__':
    d = np.random.randn(100000)
    print(d)
    mu, sigma, skew, kurtosis = calc_statistics(d)
    print('函数库计算均值、标准差、偏度、峰度：', mu, sigma, skew, kurtosis)
    # 一维直方图
    mpl.rcParams[u'font.sans-serif'] = 'SimHei'
    mpl.rcParams[u'axes.unicode_minus'] = False
    y1, x1, dummy = plt.hist(d, bins=50, normed=True, color='g', alpha=0.75)
    t = np.arange(x1.min(), x1.max(), 0.05)
    y = np.exp(-t ** 2 / 2) / math.sqrt(2 * math.pi)
    plt.plot(t, y, 'r-', lw=2)
    plt.title(u'高斯分布，样本个数：%d' % d.shape[0])
    plt.grid(True)
    plt.show()

    d = np.random.randn(100000, 2)
    mu, sigma, skew, kurtosis = calc_statistics(d)
    print('函数库计算均值、标准差、偏度、峰度：', mu, sigma, skew, kurtosis)

    # 二维图像
    N = 30
    density, edges = np.histogramdd(d, bins=[N, N])
    print('样本总数：', np.sum(density))
    density /= density.max()
    x = y = np.arange(N)
    t = np.meshgrid(x, y)
    fig = plt.figure(facecolor='w')
    ax = fig.add_subplot(111, projection='3d')
    ax.scatter(t[0], t[1], density, c='r', s=15 * density, marker='o', depthshade=True)
    ax.plot_surface(t[0], t[1], density, cmap=cm.Accent, rstride=2, cstride=2, alpha=0.9, lw=0.75)
    ax.set_xlabel(u'X')
    ax.set_ylabel(u'Y')
    ax.set_zlabel(u'Z')
    plt.title(u'二元高斯分布，样本个数：%d' % d.shape[0], fontsize=20)
    plt.tight_layout(0.1)
    plt.show()

#!/usr/bin/python
#  -*- coding:utf-8 -*-

import numpy as np
from scipy import stats
import matplotlib as mpl
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm

mpl.rcParams['axes.unicode_minus'] = False
mpl.rcParams['font.sans-serif'] = 'SimHei'

if __name__ == '__main__':
    x1, x2 = np.mgrid[-5:5:51j, -5:5:51j]
    x = np.stack((x1, x2), axis=2)

    plt.figure(figsize=(9, 8), facecolor='w')
    sigma = (np.identity(2), np.diag((3,3)), np.diag((2,5)), np.array(((2,1), (2,5))))
    for i in np.arange(4):
        ax = plt.subplot(2, 2, i+1, projection='3d')
        norm = stats.multivariate_normal((0, 0), sigma[i])
        y = norm.pdf(x)
        ax.plot_surface(x1, x2, y, cmap=cm.Accent, rstride=2, cstride=2, alpha=0.9, lw=0.3)
        ax.set_xlabel(u'X')
        ax.set_ylabel(u'Y')
        ax.set_zlabel(u'Z')
    plt.suptitle(u'二元高斯分布方差比较', fontsize=18)
    plt.tight_layout(1.5)
    plt.show()

# -*- coding:utf-8 -*-
# /usr/bin/python

import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from scipy.special import gamma
from scipy.special import factorial

mpl.rcParams['axes.unicode_minus'] = False
mpl.rcParams['font.sans-serif'] = 'SimHei'

if __name__ == '__main__':
    N = 5
    x = np.linspace(0, N, 50)
    y = gamma(x + 1)
    plt.figure(facecolor='w')
    plt.plot(x, y, 'r-', x, y, 'm*', lw=2)
    z = np.arange(0, N + 1)
    f = factorial(z, exact=True)  # 阶乘
    print(f)
    plt.plot(z, f, 'go', markersize=8)
    plt.grid(b=True)
    plt.xlim(-0.1, N + 0.1)
    plt.ylim(0.5, np.max(y) * 1.05)
    plt.xlabel(u'X', fontsize=15)
    plt.ylabel(u'Gamma(X) - 阶乘', fontsize=15)
    plt.title(u'阶乘和Gamma函数', fontsize=16)
    plt.show()

# !/usr/bin/python
# -*- coding:utf-8 -*-

import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt


def triangle_wave(size, T):
    t = np.linspace(-1, 1, size, endpoint=False)
    # where
    # y = np.where(t < 0, -t, 0)
    # y = np.where(t >= 0, t, y)
    y = np.abs(t)
    y = np.tile(y, T) - 0.5
    x = np.linspace(0, 2 * np.pi * T, size * T, endpoint=False)
    return x, y


def sawtooth_wave(size, T):
    t = np.linspace(-1, 1, size)
    y = np.tile(t, T)
    x = np.linspace(0, 2 * np.pi * T, size * T, endpoint=False)
    return x, y


def triangle_wave2(size, T):
    x, y = sawtooth_wave(size, T)
    return x, np.abs(y)


def non_zero(f):
    f1 = np.real(f)
    f2 = np.imag(f)
    eps = 1e-4
    return f1[(f1 > eps) | (f1 < -eps)], f2[(f2 > eps) | (f2 < -eps)]


if __name__ == "__main__":
    mpl.rcParams['font.sans-serif'] = [u'simHei']
    mpl.rcParams['axes.unicode_minus'] = False
    np.set_printoptions(suppress=True)

    x = np.linspace(0, 2 * np.pi, 16, endpoint=False)
    print('时域采样值：', x)
    y = np.sin(2 * x) + np.sin(3 * x + np.pi / 4) + np.sin(5 * x)
    # y = np.sin(x)

    N = len(x)
    print('采样点个数：', N)
    print('\n原始信号：', y)
    f = np.fft.fft(y)
    print('\n频域信号：', f / N)
    a = np.abs(f / N)
    print('\n频率强度：', a)

    iy = np.fft.ifft(f)
    print('\n逆傅里叶变换恢复信号：', iy)
    print('\n虚部：', np.imag(iy))
    print('\n实部：', np.real(iy))
    print('\n恢复信号与原始信号是否相同：', np.allclose(np.real(iy), y))

    plt.subplot(211)
    plt.plot(x, y, 'go-', lw=2)
    plt.title(u'时域信号', fontsize=15)
    plt.grid(True)
    plt.subplot(212)
    w = np.arange(N) * 2 * np.pi / N
    print(u'频率采样值：', w)
    plt.stem(w, a, linefmt='r-', markerfmt='ro')
    plt.title(u'频域信号', fontsize=15)
    plt.grid(True)
    plt.show()

    # 三角/锯齿波
    x, y = triangle_wave(20, 5)
    # x, y = sawtooth_wave(20, 5)
    N = len(y)
    f = np.fft.fft(y)
    # print '原始频域信号：', np.real(f), np.imag(f)
    print('原始频域信号：', non_zero(f))
    a = np.abs(f / N)

    # np.real_if_close
    f_real = np.real(f)
    eps = 0.3 * f_real.max()
    print(eps)
    f_real[(f_real < eps) & (f_real > -eps)] = 0
    f_imag = np.imag(f)
    eps = 0.3 * f_imag.max()
    print(eps)
    f_imag[(f_imag < eps) & (f_imag > -eps)] = 0
    f1 = f_real + f_imag * 1j
    y1 = np.fft.ifft(f1)
    y1 = np.real(y1)
    # print '恢复频域信号：', np.real(f1), np.imag(f1)
    print('恢复频域信号：', non_zero(f1))

    plt.figure(figsize=(8, 8), facecolor='w')
    plt.subplot(311)
    plt.plot(x, y, 'g-', lw=2)
    plt.title(u'三角波', fontsize=15)
    plt.grid(True)
    plt.subplot(312)
    w = np.arange(N) * 2 * np.pi / N
    plt.stem(w, a, linefmt='r-', markerfmt='ro')
    plt.title(u'频域信号', fontsize=15)
    plt.grid(True)
    plt.subplot(313)
    plt.plot(x, y1, 'b-', lw=2, markersize=4)
    plt.title(u'三角波恢复信号', fontsize=15)
    plt.grid(True)
    plt.tight_layout(1.5, rect=[0, 0.04, 1, 0.96])
    plt.suptitle(u'快速傅里叶变换FFT与频域滤波', fontsize=17)
    plt.show()