Python資料正態性檢驗實現過程
在做資料分析或者統計的時候，經常需要進行資料正態性的檢驗，因為很多假設都是基於正態分佈的基礎之上的，例如：T檢驗。
在Python中，主要有以下檢驗正態性的方法：
1.scipy.stats.shapiro ——Shapiro-Wilk test，屬於專門用來做正態性檢驗的模組，其原假設：樣本資料符合正態分佈。
注：適用於小樣本。
其函式定位為：
def shapiro(x):
"""
Perform the Shapiro-Wilk test for normality.
The Shapiro-Wilk test tests the null hypothesis that the
data was drawn from a normal distribution.
Parameters
----------
x : array_like
Array of sample data.
Returns
-------
W : float
The test statistic.
p-value : float
The p-value for the hypothesis test.x引數為樣本值序列，返回值中第一個為檢驗統計量，第二個為P值，當P值大於指定的顯著性水平，則接受原假設。
2.scipy.stats.kstest（K-S檢驗）：可以檢驗多種分佈，不止正態分佈，其原假設：資料符合正態分佈。
其函式定義為：
def kstest(rvs,cdf,args=(),N=20,alternative='two-sided',mode='approx'):
"""
Perform the Kolmogorov-Smirnov test for goodness of fit.
This performs a test of the distribution G(x) of an observed
random variable against a given distribution F(x). Under the null
hypothesis the two distributions are identical,G(x)=F(x). The
alternative hypothesis can be either 'two-sided' (default),'less'
or 'greater'. The KS test is only valid for continuous distributions.
Parameters
----------
rvs : str,array or callable
If a string,it should be the name of a distribution in `scipy.stats`.
If an array,it should be a 1-D array of observations of random
variables.
If a callable,it should be a function to generate random variables;
it is required to have a keyword argument `size`.
cdf : str or callable
If a string,it should be the name of a distribution in `scipy.stats`.
If `rvs` is a string then `cdf` can be False or the same as `rvs`.
If a callable,that callable is used to calculate the cdf.
args : tuple,sequence,optional
Distribution parameters,used if `rvs` or `cdf` are strings.
N : int,optional
Sample size if `rvs` is string or callable. Default is 20.
alternative : {'two-sided','less','greater'},optional
Defines the alternative hypothesis (see explanation above).
Default is 'two-sided'.
mode : 'approx' (default) or 'asymp',optional
Defines the distribution used for calculating the p-value.
- 'approx' : use approximation to exact distribution of test statistic
- 'asymp' : use asymptotic distribution of test statistic
Returns
-------
statistic : float
KS test statistic,either D,D+ or D-.
pvalue : float
One-tailed or two-tailed p-value.引數是：
rvs：待檢驗資料。
cdf：檢驗分佈，例如'norm'，'expon'，'rayleigh'，'gamma'等分佈，設定為'norm'時表示正態分佈。
alternative：預設為雙側檢驗，可以設定為'less'或'greater'作單側檢驗。
model:'approx'(預設值)，表示使用檢驗統計量的精確分佈的近視值；'asymp'：使用檢驗統計量的漸進分佈。
其返回值中第一個為統計量，第二個為P值。
3.scipy.stats.normaltest：正態性檢驗，其原假設：樣本來自正態分佈。
其函式定義為：
def normaltest(a,axis=0,nan_policy='propagate'):
"""
Test whether a sample differs from a normal distribution.
This function tests the null hypothesis that a sample comes
from a normal distribution. It is based on D'Agostino and
Pearson's [1]_,[2]_ test that combines skew and kurtosis to
produce an omnibus test of normality.
Parameters
----------
a : array_like
The array containing the sample to be tested.
axis : int or None,optional
Axis along which to compute test. Default is 0. If None,compute over the whole array `a`.
nan_policy : {'propagate','raise','omit'},optional
Defines how to handle when input contains nan. 'propagate' returns nan,'raise' throws an error,'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
``s^2 + k^2``,where ``s`` is the z-score returned by `skewtest` and
``k`` is the z-score returned by `kurtosistest`.
pvalue : float or array
A 2-sided chi squared probability for the hypothesis test.其引數：
axis=None 可以表示對整個資料做檢驗，預設值是0。
nan_policy：當輸入的資料中有nan時，'propagate'，返回空值；'raise' 時，丟擲錯誤；'omit' 時，忽略空值。
其返回值中，第一個是統計量，第二個是P值。
4.scipy.stats.anderson：由 scipy.stats.kstest 改進而來，用於檢驗樣本是否屬於某一分佈（正態分佈、指數分佈、logistic 或者 Gumbel等分佈）
其函式定義為：
def anderson(x,dist='norm'):
"""
Anderson-Darling test for data coming from a particular distribution
The Anderson-Darling tests the null hypothesis that a sample is
drawn from a population that follows a particular distribution.
For the Anderson-Darling test,the critical values depend on
which distribution is being tested against. This function works
for normal,exponential,logistic,or Gumbel (Extreme Value
Type I) distributions.
Parameters
----------
x : array_like
array of sample data
dist : {'norm','expon','logistic','gumbel','gumbel_l',gumbel_r','extreme1'},optional
the type of distribution to test against. The default is 'norm'
and 'extreme1','gumbel_l' and 'gumbel' are synonyms.
Returns
-------
statistic : float
The Anderson-Darling test statistic
critical_values : list
The critical values for this distribution
significance_level : list
The significance levels for the corresponding critical values
in percents. The function returns critical values for a
differing set of significance levels depending on the
distribution that is being tested against.
其引數：
x和dist分別表示樣本資料和分佈。
返回值有三個，第一個表示統計值，第二個表示評價值，第三個是顯著性水平；評價值和顯著性水平對應。
對於不同的分佈，顯著性水平不一樣。
Critical values provided are for the following significance levels:
normal/exponenential
15%,10%,5%,2.5%,1%
logistic
25%,1%,0.5%
Gumbel
25%,1%
關於統計值與評價值的對比：當統計值大於這些評價值時，表示在對應的顯著性水平下，原假設被拒絕，即不屬於某分佈。
If the returned statistic is larger than these critical values then for the corresponding significance level,the null hypothesis that the data come from the chosen distribution can be rejected.
5.skewtest 和kurtosistest 檢驗：用於檢驗樣本的skew（偏度）和kurtosis（峰度）是否與正態分佈一致，因為正態分佈的偏度=0，峰度=3。
偏度：偏度是樣本的標準三階中心矩。
峰度：峰度是樣本的標準四階中心矩。
6. 程式碼如下：
import numpy as np
from scipy import stats
a = np.random.normal(0,2,50)
b = np.linspace(0,10,100)
# Shapiro-Wilk test
S,p = stats.shapiro(a)
print('the shapiro test result is:',S,',p)
# kstest（K-S檢驗）
K,p = stats.kstest(a,'norm')
print(K,p)
# normaltest
N,p = stats.normaltest(b)
print(N,p)
# Anderson-Darling test
A,C,p = stats.anderson(b,dist='norm')
print(A,p)
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