人工智能数学基础学习PPT

学习视频：人工智能 -数学基础

1.简介

1.函数

2.极限

3.无穷小与无穷大

4.连续性与导数

5.偏导数

6.方向导数

7.梯度

2.微积分

1.微积分基本想法

2.微积分的解释

3.定积分

4.定积分性质

5.牛顿-莱布尼茨公式

3.泰勒公式与拉格朗日

1.泰勒公式

2.一点一世界

3.阶数的作用

4.阶乘的作用

5.拉格朗日乘子法

4.线性代数基础

1.行列式概述

2.矩阵与数据的关系

3.矩阵基本操作

4.矩阵的几种变换

5.矩阵的秩

6.内积与正交

5.特征值与矩阵分解

1.特征值与特征向量

2.特征空间与应用

3.SVD要解决的问题

4.特征值分解

5.SVD矩阵分解

6.随机变量

1.离散型随机变量

2.连续型随机变量

3.简单随机抽样

4.似然函数

5.极大似然估计

7.概率论基础

1.概率与频率

2.古典概型

3.条件概率

4.条件概率小例子独立性

5.独立性

6.二维离散型随机变量

7.二维连续型随机变量

8.边缘分布

9.期望

10.期望求解

11.马尔科夫不等式

12.切比雪夫不等式

13.后验概率估计

14.贝叶斯拼写纠错实例

15.垃圾邮件过滤实例

8.数据科学你得知道的几种分布

1.正态分布

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import matplotlib.style as style
from IPython.core.display import HTML#PLOTTING CONFIG
%matplotlib inline
style.use('fivethirtyeight')
plt.rcParams["figure.figsize"]=(14,7)
plt.figure(dpi=100)#PDF
plt.plot(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100))/ np.max(stats.norm.pdf(np.linspace(-3, 3, 100))),)
plt.fill_between(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100))/np.max(stats.norm. pdf(np.linspace(-3, 3, 100))),alpha=.15,)# CDF
plt.plot(np.linspace(-4,4,100),stats.norm.cdf(np.linspace(-4,4,100)),)# LEGEND
plt. text(x=-1.5, y=.7, s="pdf (normed)", rotation=65, alpha=.75, weight="bold", color="#008fd5")
plt.text(x=-.4,y=.5,s="cdf",rotation=55,alpha=.75,weight="bold", color="#fc4f30")# TICKS
plt.tick_params(axis = 'both',which= 'major', labelsize = 18)
plt.axhline(y=0,color='black',linewidth=1.3, alpha =.7)
# TITLE
plt.text(x=-5,y=1.25,s="Normal Distribution - Overview",fontsize=26,weight='bold',alpha=.75)
plt.text(x=-5,y=1.1,s="Depicted below are the normed probability density function (pdf) and the cumulative density\nfunction (cdf) of a normally distributed random variable y~ (μ, o), given μ = 0 and o = 1.",fontsize=19,alpha=.85)

均值

plt.figure(dpi=100)
#PDFMU=0
plt.plot(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100)),)
plt.fill_between(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100)),alpha=.15,)#PDFMU=2
plt.plot(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),loc=2),)
plt.fill_between(np.linspace(-4,4, 100),stats.norm.pdf(np.linspace(-4,4,100),loc=2),alpha=.15,)#PDFM=-2
plt.plot(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),loc=-2),)
plt.fill_between(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),loc=-2),alpha=.15,)# LEGEND
plt.text(x=-1,y=.35, s="$ \mu= 0$", rotation=65, alpha=.75, weight="bold", color="#008fd5")
plt.text(x=1, y=.35,s="$ \mu = 2$", rotation=65, alpha=.75, weight="bold", color="#fc4f30")
plt.text(x=-3,y=.35,s="$ \mu = -2$", rotation=65, alpha=.75, weight="bold", color="#e5ae38")# TICKS
plt.tick_params(axis='both',which ='major', labelsize = 18)
plt.axhline(y=0,color='black',linewidth=1.3, alpha =.7)# TITLE,
plt.text(x=-5,y=0.51,s="Normal Distribution - $ \mu $",fontsize=26,weight='bold',alpha=.75)
plt.text(x=-5,y= 0.45,s="Depicted below are three normally distributed random variables with varying $ \mu $. As one can easily\nsee the parameter $\mu$ shifts the distribution along the x-axis.",fontsize=19,alpha=.85)

标准差

plt.figure(dpi=100)#PDF SIGMA =1
plt.plot(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),scale=1),)
plt.fill_between(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),scale=1),alpha=.15,)#PDF SIGMA =2
plt.plot(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),scale=2),)
plt.fill_between(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),scale=2),alpha=.15,)#PDF SIGMA =0.5
plt.plot(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),scale=0.5),)
plt.fill_between(np.linspace(-4,4,100),stats.norm.pdf(np.linspace(-4,4,100),scale=0.5),alpha=.15,)# LEGEND
plt.text(x=-1.25, y=.3,s="$ \sigma = 1$", rotation=51, alpha=.75, weight="bold", color="#008fd5")
plt.text(x=-2.5,y=.13,s="$ \sigma = 2$", rotation=11, alpha=.75, weight="bold", color="#fc4f30")
plt. text(x=-0.75,y=.55, s="$ \sigma = 0.5$", rotation=75, alpha=.75, weight="bold", color="#e5ae38")# TICKS
plt.tick_params(axis='both',which='major', labelsize = 18)
plt.axhline(y=0,color='black',linewidth=1.3,alpha =.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-5,y=0.98,s="Normal Distribution - $ \sigma $",fontsize=26,weight='bold',alpha=.75)
plt.text(x=-5,y= 0.87,s="Depicted below are three normally distributed random variables with varying $\sigma $. As one can easily\nsee the parameter $\sigma$ “sharpens” the distribution (the smaller $ \sigma $ the sharper the function).",fontsize=19,alpha=.85)

from scipy.stats import norm#draw a single sample
print(norm.rvs(),end="\n\n")#draw l0 samples
print(norm.rvs(size=10),end="\n\n")#adjust mean (loc’)and standard deviation (scale’)
print(norm.rvs(loc=10,scale=0.1),end="\n\n")

from scipy.stats import norm
#additional imports for plotting purpose
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.rcParams["figure.figsize"] = (14, 7)#relative likelihood of x and y
x = -1
y = 2
print("pdf(x) = {} \npdf(y)= {}".format(norm.pdf(x), norm.pdf(y)))#continuous pdf for the plot
x_s=np.linspace(-3,3,50)
y_s = norm.pdf(x_s)
plt.scatter(x_s, y_s);

from scipy.stats import norm#probability ofxless or equal 0.3
print("P(X <0.3)={}".format(norm.cdf(0.3)))#probabilityofxin [-0.2,+0.2]
print("P(-0.2<X<0.2)={}".format(norm.cdf(0.2) - norm.cdf(-0.2)))

plt.figure(dpi=100)
##### COMPUTATION #####
#DECLARING THE "TRUE"PARAMETERS UNDERLYING THE SAMPLE
mu_real=10
sigma_real=2# DRAW A SAMPLE OF N=100O
np.random.seed(42)
sample = stats.norm.rvs(loc=mu_real, scale=sigma_real, size=1000)#ESTIMATE MU AND SIGMA
mu_est = np.mean(sample)
sigma_est = np.std(sample)
print("Estimated MU: {}\nEstimated SIGMA: {}".format(mu_est, sigma_est))#### PLOTTING ####
# SAMPLE DISTRIBUTION
plt.hist(sample, bins=50,density=True, alpha=.25)# TRUE CURVE
plt.plot (np.linspace(2, 18, 1000), norm.pdf(np.linspace(2, 18, 1000), loc=mu_real, scale=sigma_real))# ESTIMATED CURVE
plt.plot (np.linspace(2, 18, 1000), norm.pdf(np.linspace(2, 18, 1000), loc=np.mean(sample), scale=np.std(sample)))# LEGEND
plt.text(x=9.5,y=.1,s="sample", alpha=.75, weight="bold", color="#008fd5")
plt.text(x=7, y=.2, s="true distrubtion", rotation=55, alpha=.75, weight="bold", color="#fc4f30")
plt.text(x=5, y=.12, s="estimated distribution", rotation=55, alpha=.75, weight="bold", color="#e5ae38")# TICKS
plt.tick_params(axis ='both',which='major',labelsize= 18)
plt.axhline(y=0,color='black',linewidth= 1.3,alpha =.7)# TITLE
plt.text(x=0,y=0.3,s="Normal Distribution",fontsize=26,weight='bold',alpha=.75)

2.二项式分布

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import matplotlib.style as style
from IPython.core.display import HTML# PLOTTING CONFIG
%matplotlib inline
style.use('fivethirtyeight')
plt.rcParams["figure.figsize"] = (14, 7)
plt.figure(dpi=100)# PDF (Probability Mass Function)
x_values = np.arange(21)  # X-axis values from 0 to 20
plt.bar(x=x_values, height=stats.binom.pmf(x_values, p=0.5, n=20), width=0.75, alpha=0.75)# CDF (Cumulative Distribution Function)
plt.plot(x_values, stats.binom.cdf(x_values, p=0.5, n=20), color="#fc4f30")# LEGEND
plt.text(x=4.5, y=0.7, s="pmf (normed)", alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=14.5, y=0.9, s="cdf", alpha=0.75, weight="bold", color="#fc4f30")# TICKS
plt.xticks(np.arange(21))  # Ensure the X-axis has ticks from 0 to 20
plt.tick_params(axis='both', which='major', labelsize=18)# Set X and Y limits to ensure the graph fits within 0-20 on X-axis
plt.xlim(0, 20)
plt.axhline(y=0.005, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-2.5, y=1.25, s="Binomial Distribution - Overview", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-2.5, y=1.1,s="Depicted below are the normed probability mass function (pmf) and the cumulative density\n""function (cdf) of a Binomial distributed random variable $y \sim Binom(N, p) $, given $ N= 20$ and $p=0.5 $.",fontsize=19, alpha=0.85)

plt.figure(dpi=100)
# PUF P = .2
plt.scatter(np.arange(21),(stats.binom.pmf(np.arange(21),p=.2,n=20)),alpha=0.75,s=100)
plt.plot(np.arange(21),(stats.binom.pmf(np.arange(21),p=.2,n=20)),alpha=0.75,)
#PDFP = .5
plt.scatter(np.arange(21),(stats.binom.pmf(np.arange(21),p=.5,n=20)),alpha=0.75,s=100)
plt.plot(np.arange(21),(stats.binom.pmf(np.arange(21),p=.5,n=20)),alpha=0.75,)
#PDFP = .9
plt.scatter(np.arange(21),(stats.binom.pmf(np.arange(21),p=.9,n=20)),alpha=0.75,s=100)
plt.plot(np.arange(21),(stats.binom.pmf(np.arange(21),p=.9,n=20)),alpha=0.75,)# LEGEND
plt.text(x=3.5, y=0.075, s="$p = 0.2$", alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=9.5, y=0.075, s="$p = 0.5$", alpha=0.75, weight="bold", color="#fc4f30")
plt.text(x=17.5, y=0.075, s="$p = 0.9$", alpha=0.75, weight="bold", color="#e5ae38")# TICKS
plt.xticks(range(21)[::2])
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0,color='black',linewidth=1.3, alpha =0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-2.5, y=0.37, s="Binomial Distribution - $p$", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-2.5, y=0.32,s="Depicted below are three Binomial distributed random variables with varying $p $. As one can see\nthe parameter $p$ shifts and skews the distribution",fontsize=19, alpha=0.85)

plt.figure(dpi=100)
# PUF N = 10
plt.scatter(np.arange(11),(stats.binom.pmf(np.arange(11),p=.5,n=10)),alpha=0.75,s=100)
plt.plot(np.arange(11),(stats.binom.pmf(np.arange(11),p=.5,n=10)),alpha=0.75,)
#PUF N = 15
plt.scatter(np.arange(16),(stats.binom.pmf(np.arange(16),p=.5,n=15)),alpha=0.75,s=100)
plt.plot(np.arange(16),(stats.binom.pmf(np.arange(16),p=.5,n=15)),alpha=0.75,)
#PUF N = 20
plt.scatter(np.arange(21),(stats.binom.pmf(np.arange(21),p=.5,n=20)),alpha=0.75,s=100)
plt.plot(np.arange(21),(stats.binom.pmf(np.arange(21),p=.5,n=20)),alpha=0.75,)# LEGEND
plt.text(x=6, y=0.225, s="$N = 10$", alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=8.5, y=0.2, s="$N = 15$", alpha=0.75, weight="bold", color="#fc4f30")
plt.text(x=11, y=0.175, s="$N = 20$", alpha=0.75, weight="bold", color="#e5ae38")# TICKS
plt.xticks(range(21)[::2])
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0,color='black',linewidth=1.3, alpha =0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-2.5, y=0.31, s="Binomial Distribution - $N$", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-2.5, y=0.27,s="Depicted below are three Binomial distributed random variables with varying $N$. As one can see\nthe parameter $N$ streches the distribution (the larger $N$ the flatter the distribution)",fontsize=19, alpha=0.85)

import numpy as np
from scipy.stats import binom#draw a single sample
np.random.seed(42)
print(binom.rvs(p=0.3, n=10),end="\n\n")#draw l0 samples
print(binom.rvs(p=0.3,n=10,size=10),end="\n\n")

from scipy.stats import binom#additional imports for plotting purpose
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.rcParams["figure.figsize"]=(14,7)#likelihood of x and y
x = 1
y = 7
print("pmf(X=1) = {}\npmf(X=7) = {}".format(binom.pmf(k=x,p=0.3, n=10), binom.pmf(k=y, p=0.3, n=10)))
#continuous pdf for the plot
x_s= np.arange(11)
y_s=binom.pmf(k=x_s,p=0.3, n=10)
plt.scatter(x_s,y_s, s=100)

from scipy.stats import binom#probability of x less or equal 0.3
print("P(X<3)={}".format(binom.cdf(k=3, p=0.3, n=10)))#probability of x in [-0.2,+0.2]
print("P(2<X<=8)= {}".format(binom.cdf(k=8, p=0.3, n=10) - binom.cdf(k=2, p=0.3, n=10)))

3.泊松分布

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import matplotlib.style as style
from IPython.core.display import HTML# PLOTTING CONFIG
%matplotlib inline
style.use('fivethirtyeight')
plt.rcParams["figure.figsize"] = (14, 7)
plt.figure(dpi=100)# PDF (Probability Mass Function) for Poisson Distribution
mu = 5  # Rate (lambda) for Poisson distribution
x_values = np.arange(20)  # X-axis values from 0 to 19 (since Poisson PMF is defined for non-negative integers)
plt.bar(x=x_values, height=stats.poisson.pmf(x_values, mu), width=0.75, alpha=0.75)# CDF (Cumulative Distribution Function) for Poisson Distribution
plt.plot(x_values, stats.poisson.cdf(x_values, mu), color="#fc4f30")# LEGEND
plt.text(x=8, y=0.45, s="pmf (normed)", alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=8.5, y=0.9, s="cdf", alpha=0.75, weight="bold", color="#fc4f30")# TICKS
plt.xticks(np.arange(21))  # X-axis ticks from 0 to 20
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0.005, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-2.5, y=1.25, s="Poisson Distribution - Overview", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-2.5, y=1.1,s="Depicted below are the normed probability mass function (pmf) and the cumulative density\n""function (cdf) of a Poisson distributed random variable $ y \sim Poi(\lambda) $, given $ \lambda=5 $.",fontsize=19, alpha=0.85)

plt.figure(dpi=100)# PUF LAM = 1
plt.scatter(np.arange(20),(stats.poisson.pmf(np.arange(20),mu=1)),alpha=0.75,s=100)
plt.plot(np.arange(20),(stats.poisson.pmf(np.arange(20),mu=1)),alpha=0.75,)
#PUF LAM = 5
plt.scatter(np.arange(20),(stats.poisson.pmf(np.arange(20),mu=5)),alpha=0.75,s=100)
plt.plot(np.arange(20),(stats.poisson.pmf(np.arange(20),mu=5)),alpha=0.75,)
#PUF N = 10
plt.scatter(np.arange(20),(stats.poisson.pmf(np.arange(20),mu=10)),alpha=0.75,s=100)
plt.plot(np.arange(20),(stats.poisson.pmf(np.arange(20),mu=10)),alpha=0.75,)# LEGEND
plt.text(x=3, y=0.1, s="$\lambda = 1$", alpha=0.75,rotation=-65, weight="bold", color="#008fd5")
plt.text(x=8.25, y=0.075, s="$\lambda = 5$", alpha=0.75,rotation=-35, weight="bold", color="#fc4f30")
plt.text(x=14.5, y=0.06, s="$\lambda = 10$", alpha=0.75,rotation=-20, weight="bold", color="#e5ae38")# TICKS
plt.xticks(range(21)[::2])
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0,color='black',linewidth=1.3, alpha =0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-2.5, y=0.475, s="Poisson Distribution - $\lambda$", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-2.5, y=0.425,s="Depicted below are three Poisson cistributed random variables with $\lambda $. As one can easily\nsee the parameter $\lambda$ shifts and flattens the distribution (the smaller $\lambda $ the sharper the function)",fontsize=19, alpha=0.85)

import numpy as np
from scipy.stats import poisson#draw a single sample
np.random.seed(42)
print(poisson.rvs(mu=10),end="\n\n")#draw 10 samples
print(poisson.rvs(mu=10,size=10),end="\n\n")

from scipy.stats import poisson
#additional imports for plotting pupose
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.rcParams["figure.figsize"]=(14,7)#continuous pdf for the plot
x_s= np.arange(15)
y_s=poisson.pmf(k=x_s,mu=5)
plt.scatter(x_s,y_s,s=100)

from scipy.stats import poisson#probabilityofxless or equal 0.3
print("P(X<=3) = {}".format(poisson.cdf(k=3,mu=5)))#probability ofxin [-0.2, +0.2]
print("P(2<X<=8) = {}".format(poisson.cdf(k=8,mu=5) - poisson.cdf(k=2,mu=5)))

from collections import Counterplt.figure(dpi=100)##:COMPLTATION
# DECLARING THE"TRUE"PARAVETERS LNDERLTING THE SAMPLE
lambda_real = 7# DRAT A SANPLE OF NEIOOO
np.random.seed(42)
sample=poisson.rvs(mu=lambda_real,size=1000)#ESTIMATE MU AND SIGMA
lambda_est = np.mean(sample)
print("Estimated LAMBDA: {}".format(lambda_est))## PLOTTTNG #
# SAMPLE DISTRIBUTION
cnt = Counter(sample)
_,values =zip(*sorted(cnt.items()))
plt.bar(range(len(values)),values/np.sum(values),alpha=0.25)plt.plot(range(18),poisson.pmf(k=range(18), mu=lambda_real), color="#fc4f30")plt.plot(range(18),poisson.pmf(k=range(18), mu=lambda_est), color="#e5ae38")#N LEGED
plt.text(x=6,y=.06,s="sample", alpha=.75, weight="bold", color="#008fd5")
plt.text(x=3.5,y=.14,s="true distrubtion", rotation=60, alpha=.75, weight="bold", color="#fc4f30")
plt. text (x=1, y=.08, s= "estimated distribution", rotation=60, alpha=.75, weight="bold", color="#e5ae38")# TICKS
plt.xticks(range(17)[::2])
plt.tick_params(axis='both',which='major', labelsize = 18)
plt.axhline(y=0.0009,color='black', linewidth = 1.3,alpha=.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-2.5, y=0.19, s="Poisson Distribution - Parameter Estimation", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-2.5, y=0.17,s="Depicted below is the distribution of a sample (blue) drawn from a Poisson distribution with $\lambda = 7$. \nAlso the estimated distrubution with $\lambda \sim 6.923$ is shown (yellow).",fontsize=19, alpha=0.85)

4.均匀分布

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import matplotlib.style as style
from IPython.core.display import HTML
from setuptools.command.rotate import rotate
from sympy import rotations# PLOTTING CONFIG
%matplotlib inline
style.use('fivethirtyeight')
plt.rcParams["figure.figsize"] = (14, 7)
plt.figure(dpi=100)# PDF
plt.plot(np.linspace(-4,4,100),stats.uniform.pdf(np.linspace(-4,4,100)))
plt.fill_between(np.linspace(-4,4,100),stats.uniform.pdf(np.linspace(-4,4,100)),alpha=.15)# CDF
plt.plot(np.linspace(-4,4,100), stats.uniform.cdf(np.linspace(-4,4,100)), )# LEGEND
plt.text(x=1.5, y=0.7, s="pdf (normed)",rotation=65, alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=0.4, y=0.5, s="cdf", rotation=55,alpha=0.75, weight="bold", color="#fc4f30")# TICKS
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-5, y=1.25, s="Nniform Distribution - Overview", fontsize=26, weight='bold', alpha=0.75)

plt.figure(dpi=100)# PDF  loc=0,scale=1
plt.plot(np.linspace(-8,8,100),stats.uniform.pdf(np.linspace(-8,8,100),loc=0,scale=1),)
plt.fill_between(np.linspace(-8,8,100),stats.uniform.pdf(np.linspace(-8,8,100),loc=0,scale=1),alpha=.15)# PDF  loc=0,scale=2
plt.plot(np.linspace(-8,8,100),stats.uniform.pdf(np.linspace(-8,8,100),loc=0,scale=2),)
plt.fill_between(np.linspace(-8,8,100),stats.uniform.pdf(np.linspace(-8,8,100),loc=0,scale=2),alpha=.15)# PDF  loc=-3,scale=3
plt.plot(np.linspace(-8,8,100),stats.uniform.pdf(np.linspace(-4,4,100),loc=-3,scale=3),)
plt.fill_between(np.linspace(-8,8,100),stats.uniform.pdf(np.linspace(-4,4,100),loc=-3,scale=3),alpha=.15)# LEGEND
plt.text(x=-1, y=0.65, s="loc=0,scale=1",rotation=65, alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=1, y=0.65, s="loc=0,scale=2", rotation=65,alpha=0.75, weight="bold", color="#fc4f30")
plt.text(x=-3, y=0.65, s="loc=-3,scale=3", rotation=65,alpha=0.75, weight="bold", color="#e5ae38")# TICKS
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-5, y=1.1, s="Nniform Distribution - loc nad scale", fontsize=26, weight='bold', alpha=0.75)

5.卡方分布

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import matplotlib.style as style
from IPython.core.display import HTML# PLOTTING CONFIG
%matplotlib inline
style.use('fivethirtyeight')
plt.rcParams["figure.figsize"] = (14, 7)
plt.figure(dpi=100)# PDF
plt.plot(np.linspace(0,20,100),stats.chi2.pdf(np.linspace(0,20,100),df=4),)
plt.fill_between(np.linspace(0,20,100),stats.chi2.pdf(np.linspace(0,20,100),df=4),alpha=.15)# CDF
plt.plot(np.linspace(0,20,100),stats.chi2.cdf(np.linspace(0,20,100),df=4),)# LEGEND
plt.xticks(np.arange(0,21,2))
plt.text(x=11, y=0.25, s="pdf (normed)",alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=11, y=0.85, s="cdf",alpha=0.75, weight="bold", color="#fc4f30")# TICKS
plt.xticks(np.arange(0,21,2))
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-2, y=1.25, s=r"Chi-Squared $(\chi^{2})$ Distribution - Overview", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-2, y=1.1,s="Depicted below are the normed probability density function (pdf) and the cumulative density\nfunction (cdf) of a Chi-Squared distributed random variable $ y \sim \chi^{2}(k) $, given $k$=4.",fontsize=19, alpha=0.85)

plt.figure(dpi=100)# PDF k=1
plt.plot(np.linspace(0,15,500),stats.chi2.pdf(np.linspace(0,15,500),df=1),)
plt.fill_between(np.linspace(0,15,500),stats.chi2.pdf(np.linspace(0,15,500),df=1),alpha=.15)# PDF k=3
plt.plot(np.linspace(0,15,100),stats.chi2.pdf(np.linspace(0,15,100),df=3),)
plt.fill_between(np.linspace(0,15,100),stats.chi2.pdf(np.linspace(0,15,100),df=3),alpha=.15)# PDF k=6
plt.plot(np.linspace(0,15,100),stats.chi2.pdf(np.linspace(0,15,100),df=6),)
plt.fill_between(np.linspace(0,15,100),stats.chi2.pdf(np.linspace(0,15,100),df=6),alpha=.15)plt.text(x=0.5, y=0.7, s="$ k = 1$",rotation=-65,alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=1.5, y=0.35, s="$ k = 3$",alpha=0.75, weight="bold", color="#fc4f30")
plt.text(x=5, y=0.2, s="$ k = 6$",alpha=0.75, weight="bold", color="#e5ae38")# TICKS
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-1.5, y=2.8, s="Chi-Squared Distribution - $ k $", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-1.5, y=2.5,s="Depicted below are three Chi-Squared distributed random variables with varying $ k $. As one can\nsee the parametek $k$ smoothens the distribution and softens the skewness.",fontsize=19, alpha=0.85)

import numpy as np
from scipy.stats import chi2#draw a single sample
np.random.seed(42)
print(chi2.rvs(df=4),end="\n\n")
#draw 10 samples
print(chi2.rvs(df=4,size=10),end="\n\n")

from scipy.stats import chi2# additional imports for plotting purpose
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.rcParams["figure.figsize"]=(14,7)# continuous pdf for the plot
x_s= np.arange(15)
y_s= chi2.pdf(x=x_s,df=4)
plt.scatter(x_s,y_s, s=100);

from scipy.stats import chi2# probability ofx less or equal 0.3
print("P(X<=3) = {}".format(chi2.cdf(x=3, df=4)))#probability ofxin [-0.2, +0.2]
print("P(2<X<=8) = {}".format(chi2.cdf(x=8, df=4)- chi2.cdf(x=2, df=4)))

6.beta分布

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import matplotlib.style as style
from IPython.core.display import HTML# PLOTTING CONFIG
%matplotlib inline
style.use('fivethirtyeight')
plt.rcParams["figure.figsize"] = (14, 7)plt.figure(dpi=100)# PDF
plt.plot(np.linspace(0,1,100),stats.beta.pdf(np.linspace(0,1,100),a=2,b=2),)
print(stats.beta.pdf(np.linspace(0,1,100),a=2,b=2))
plt.fill_between(np.linspace(0,1,100),stats.beta.pdf(np.linspace(0,1,100),a=2,b=2),alpha=.15)# CDF
plt.plot(np.linspace(0,1,100),stats.beta.cdf(np.linspace(0,1,100),a=2,b=2),)plt.text(x=.1, y=0.7, s="pdf (normed)",rotation=-52,alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=0.45, y=0.5, s="cdf",rotation=-40,alpha=0.75, weight="bold", color="#fc4f30")# TICKS
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-0.125, y=1.85, s=r"Beta Distribution-Overview", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-0.125, y=1.6,s="Depicted below are the normed probability density function (pdf) and the cumulative density\nfunction (cdf) of a beta distributed random variable $ y \sim Beta(\\alpha, \\beta)$, given $ \\alpha = 2 $ and $ \\beta = 2$.",fontsize=19, alpha=0.85)

plt.figure(dpi=100)# A=B=1
plt.plot(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=1,b=1),)
plt.fill_between(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=1,b=1),alpha=.15)# A=B=10
plt.plot(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=10,b=10),)
plt.fill_between(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=10,b=10),alpha=.15)# A=B=100
plt.plot(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=100,b=100),)
plt.fill_between(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=100,b=100),alpha=.15)plt.text(x=.1, y=1.45, s=r"$ \alpha=1,\beta=1 $",alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=0.325, y=3.5, s=r"$ \alpha=10,\beta=10 $",rotation=35,alpha=0.75, weight="bold", color="#fc4f30")
plt.text(x=0.4125, y=8, s=r"$ \alpha=100,\beta=100 $",rotation=80,alpha=0.75, weight="bold", color="#e5ae38")# TICKS
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-0.1, y=13.75, s=r"Beta Distribution -constant $\frac{\alpha} {\beta}$, varying $\alpha + \beta$", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-0.1, y=12,s='Depicted below are three beta distributed random variables with' + r'equal $\frac{\alpha} {\beta} $ and varying $\alpha+\beta$.' + '\n' +r'As one can see the sum of $\alpha + \beta$ (mainly) sharpens the distribution (the bigger the sharper).',fontsize=19, alpha=0.85)

plt.figure(dpi=100)# A/B=1/3
plt.plot(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=25,b=75),)
plt.fill_between(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=25,b=75),alpha=.15)# A/B=1
plt.plot(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=50,b=50),)
plt.fill_between(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=50,b=50),alpha=.15)# A/B=3
plt.plot(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=75,b=25),)
plt.fill_between(np.linspace(0,1,200),stats.beta.pdf(np.linspace(0,1,200),a=75,b=25),alpha=.15)plt.text(x=0.15, y=5, s=r"$ \alpha=25,\beta=75 $",rotation=80,alpha=0.75, weight="bold", color="#008fd5")
plt.text(x=0.39, y=5, s=r"$ \alpha=50,\beta=50 $",rotation=80,alpha=0.75, weight="bold", color="#fc4f30")
plt.text(x=0.65, y=5, s=r"$ \alpha=75,\beta=25 $",rotation=80,alpha=0.75, weight="bold", color="#e5ae38")# TICKS
plt.tick_params(axis='both', which='major', labelsize=18)
plt.axhline(y=0, color='black', linewidth=1.3, alpha=0.7)# TITLE, SUBTITLE & FOOTER
plt.text(x=-0.1, y=11.75, s=r"Beta Distribution -constant $\alpha + \beta$, varying $\frac{\alpha} {\beta}$", fontsize=26, weight='bold', alpha=0.75)
plt.text(x=-0.1, y=10,s='Depicted below are three beta distributed random variables with' + r'equal $\alpha + \beta$ and varying $\frac{\alpha} {\beta} $.'+'\n'+r'As one can see the fraction of $\frac{\alpha} {\beta} $ (mainly) shifts the distribution($\alpha$ towards 1, $\beta$ towards 0).',fontsize=19, alpha=0.85)

from scipy.stats import beta
# draw a single sample
print(beta.rvs(a=2,b=2),end="\n\n")
# draw 10 samples
print(beta.rvs(a=2,b=2,size=10))

from scipy.stats import beta
# additional import for plotting
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.rcParams["figure.figsize"]=(14,7)# continuous pdf for the plot
x_s=np.linspace(0,1,100)
y_s=beta.pdf(a=2,b=2,x=x_s)
plt.scatter(x_s, y_s)

from scipy.stats import beta#probability of x less or equal 0. 3
print("P<0.3)={:.3}".format(beta.cdf(a=2,b=2, x=0.3)))#probability of xin [-0.2, +0.2]
print("P(-0.2 <X< 0.2)= {:.3}".format(beta. cdf(a=2, b=2, x=0.2)- beta.cdf(a=2, b=2, x=-0.2)))

9.核函数变换

1.核函数的目的

2.线性核函数

3.多项式核函数

4.核函数实例

5.高斯核函数

6.参数的影响

10.熵与激活函数

1.熵的概念

2.熵的大小意味着什么

3.激活函数

4.激活函数的问题

11.回归分析

1.回归分析概述

2.回归方程定义

3.误差项的定义

4.最小二乘法推导与求解

5.回归方程求解小例子

6.回归直线拟合优度

7.多元与曲线回归问题

8.Python工具包介绍

9.statsmodels回归分析

import numpy as np
import matplotlib.pyplot as plt
import statsmodels.api as smnsample =20
x = np.linspace(0, 10,nsample)
x

X = sm.add_constant(x)
X

#β0,β1分别设置成2,5
beta = np.array([2,5])
beta

#误差项
e = np.random.normal(size=nsample)
e

#实际值y
y= np.dot(X, beta)+ e
y

#最小二乘法
model = sm.OLS(y,X)#拟合数据
res = model.fit()#回归系数
res.params

#全部结果
res.summary()

#拟合的估计值
y_= res.fittedvalues
y_

fig, ax=plt.subplots(figsize=(8,6))
ax.plot(x, y,'o',label='data')#原始数据
ax.plot(x, y_,'r--.',label='test')#拟合数据
ax.legend(loc='best')
plt.show()

10.高阶与分类变量实例

#Y=5+2*X+3*X^2
nsample=50
x= np.linspace(0,10,nsample)
X= np.column_stack((x,x**2))
X= sm.add_constant(X)beta = np.array([5,2,3])
e = np.random.normal(size=nsample)
y = np.dot(X, beta)+e
model = sm.OLS(y,X)
results = model.fit()
results.params

results.summary()

y_fitted =results.fittedvalues
fig, ax= plt.subplots(figsize=(8,6))
ax.plot(x,y,'o',label='data')
ax.plot(x, y_fitted,'r--.',label='OLS')
ax.legend(loc='best')
plt.show()

nsample=50
groups = np.zeros(nsample,int)
groups

import pandas as pd
import patsygroups[20:40]=1
groups[40:]=2# 创建一个 pandas Series
group_series = pd.Series(groups)# 使用 patsy.dmatrices 生成设计矩阵
# 公式 'group ~ 0 + C(group)' 表示 group 是因变量，C(group)表示将 group 作为分类变量，'0'表示不包含常数项
dummy = patsy.dmatrix('C(group) - 1', data={'group': group_series})
print(dummy)

x=np.linspace(0,20,nsample)
X=np.column_stack((x,dummy))
X= sm.add_constant(X)
beta =[5,2,3,6,9]
e= np.random.normal(size=nsample)
y=np.dot(X,beta)+e
result=sm.OLS(y,X).fit()
result.summary()

fig,ax=plt.subplots(figsize=(8,6))
ax.plot(x,y,'o',label="data")
ax.plot(x,result.fittedvalues,'r--.',label="OLS")
ax.legend(loc= 'best')
plt.show()

11.案例:汽车价格预测任务概述

#loading packages
import numpy as np
import pandas as pd
from pandas import datetime#data visualization and missing values
import matplotlib.pyplot as plt
import seaborn as sns # advanced yizs
import missingno as msno # missing values
%matplotlib inline
# stats
from statsmodels.distributions.empirical_distribution import ECDF
from sklearn.metrics import mean_squared_error, r2_score# machine learning
from sklearn.preprocessing import StandardScaler
from sklearn.linear_model import Lasso, LassoCV
from sklearn.model_selection import train_test_split, cross_val_score
from sklearn.ensemble import RandomForestRegressor
seed = 123#importing data(?=missing values)
data = pd.read_csv("Auto-Data.csv", na_values = '?')
data.columns

data.dtypes

# first glance at the data itself
print("In total:",data.shape)
data. head(5)

12.案例:缺失值填充

sns.set(style="ticks")
plt.figure(figsize=(12,5))
c ='#366DE8'#ECDF
plt.subplot(121)
cdf = ECDF(data['normalized-losses'])
plt.plot(cdf.x,cdf.y,label="statmodels",color = c)
plt.xlabel('normalized losses')
plt.ylabel('ECDF');# overall distribution
plt.subplot(122)
plt.hist(data['normalized-losses'].dropna(),bins = int(np.sqrt(len(data['normalized-losses']))),color =c)

13.案例:特征相关性

14.案例:预处理问题

15.案例:回归求解

12. 假设检验

1.假设检验基本思想

2.左右侧检验与双侧检验

3.Z检验基本原理

4.Z检验实例

5.T检验基本原理

6.T检验实例

7.T检验应用条件

8.卡方检验

9.假设检验中的两类错误

10.Python假设检验实例

11.Python卡方检验实例

13.相关分析

1.相关分析概述

2.皮尔森相关系数

3.计算与检验

4.斯皮尔曼等级相关

5.肯德尔系数

6.质量相关分析

7.偏相关与复相关

14.方差分析

1.方差分析概述

2.方差的比较

3.方差分析计算方法

4.方差分析中的多重比较

5.多因素方差分析

6.Python方差分析实例

15.聚类分析

1.层次聚类概述

2.层次聚类流程

3.层次聚类实例

4.Kmeans聚类算法

1.KMEANS算法概述

2.KMEANS工作流程

3.KMEANS迭代可视化展示

可视化 DBSCAN 聚类

5.DBSCAN聚类算法

1.DBSCAN聚类算法

2.DBSCAN工作流程

3.DBSCAN可视化展示

可视化 DBSCAN 聚类

6.聚类算法实践

1.多种聚类算法概述

2.聚类案例实战

16.贝叶斯分析

1.贝叶斯分析概述

2.概率的解释

3.贝叶斯学派与经典统计学派的争论

4.贝叶斯算法概述

5.贝叶斯推导实例

6.贝叶斯拼写纠错实例

7.贝叶斯解释

8.垃圾邮件过滤实例

8.经典求解思路

9.MCMC概述

10.PYMC3概述

11.模型诊断

12.模型决策