Analysis of regression problem, modeling and prediction

What is regression analysis

Give you a series of related data , There is such an internal relationship between independent variables and dependent variables , Then by processing these data , Use mathematical methods to model linear equations , Then finally through the model you build , Predict independent variables that are not provided to

So what is prediction ？

Prediction , Is to use a trained model , According to the input arguments Get the corresponding output . In the prediction task , If the model Can accurately provide forecasts , that It doesn't matter what form it is , And if the The form of is very complex and difficult to explain , We can call it black box model (Black Box).
For example , hypothesis Xi It is the blood sample characteristics of a patient , The risk of serious adverse reactions after drug use was measured , So if there is a model that can pass well Xi To predict Yi , That's naturally the best thing . here , Form of model 、 The relationship between variables is not so important in the face of correct prediction .

What is inference ？

In many cases , We are right when X How does the change of affect Y More interested in , here , We estimate the model The purpose of is not to predict , But to understand the relationship between the two

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linear regression model

$$y=x+u$$
further

$$y=a_0+\sum _{i=1}^n{a}_i{x}_i$$

One dimensional linear model

$$y=a_0+a_1{x}_1+u$$
$$a_{0}Is\; the\; intercept\; term\; ,\; It\; can\; be\; understood\; asx=0WhenyThe\; expectations\; of\; the\; ,a_{1}It\; can\; be\; understood\; asxEvery\; additional\; unit\; ,yAverage\; increase{a}_{1}A\; unit\; of$$

Least square method

import statsmodels.api as sm

model = sm.OLS(endog, exog=None, missing='none', hasconst=None, **kwargs)

''' Parameters  endog: array_like  A one-dimensional dependent variable . exog: array_like  By default , Intercept is not included , It should be added by the user .  Use  statsmodels.tools.add_constant  add to . missing: str  The available options are  'none'、'drop'  and  'raise'. If  'none', Do not carry out  nan  Check . If  'drop', Then discard any use  nan  The observations . If  'raise', Throw an error . The default is  'none'. hasconst: None or bool  instructions RHS Whether to include constants provided by users . If True, Then the constant is not checked ,k_constant Set to 1, And calculate all the result statistics , Just like there is a constant . If False, Then the constant is not checked , And will k_constant Set to 0. **kwargs  Additional parameters used to set model properties when using the formula interface . '''

Normal distribution

from scipy.stats import norm
import numpy as np
#  Generate an average of 1, The variance of 2 Is a normal distribution 
distribution=norm(loc=1.0, scale=2.0)
#  Show mean and variance 
print(distribution.stats())
#  sampling 
samples=distribution.rvs(size=1000)
#  The mean value of the sample and the difference between the labels 
print(np.mean(samples), np.std(samples))
#  Fit the data 
print(norm.fit(samples))