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Deterministic relationship refers to the relationship that when the value of some variables is determined, the value of other variables is also completely determined .

1.2 Non deterministic relationship （ Correlation ）

Correlation refers to a certain dependency between variables , But when the value of some variables is determined , Although the value of other variables changes with it, it is not completely certain , At this time, the relationship between variables cannot be accurately expressed by functions .

1.3 regression analysis

regression analysis （regression analysis） It is a response variable in mathematical statistics （ The dependent variable ） With several prediction variables （ The independent variables ） An effective method of correlation between . You can use a definite functional relationship Roughly describe y And x The relationship between , be called Regression equation .

only one Regression analysis of predictive variables is called Univariate regression analysis ;

More than one Regression analysis of predictive variables is called Multiple regression analysis .

for example ：（1） The independent variables （ Forecast variable ）： Father's height ; The dependent variable （ Response variables ）： My son is tall

（2） The independent variables （ Forecast variable ）：IQ, Time T; The dependent variable （ Response variables ）： achievement

1.4 Least square method

And satisfied ：

Called random error , It is usually assumed that ~N（0,o²）

Least square method Namely , function m（x）=m（x,a1,a2,L,ak）, among a1 To ak Is an unknown parameter , We choose these appropriate parameters , Make the observations yi With the corresponding function value The sum of squares of deviations Minimum

Two 、 Univariate linear regression

3、 ... and 、 Multiple linear regression

According to the least square method , To make

Minimum , Yes bp Find partial derivatives and let them equal 0, secondly

When X'X Irreversible Or between variables Multicollinearity when , Least squares is not available , Principal component analysis and regression can be used

（X'： Transpose matrix ; For matrix A reversible , Existence matrix B bring AB=BA=E, Meet one ）

Four 、 Regression test

4.1 Univariate regression test

4.1.1 Analysis of variance （F Test method ）

Q total ： Total sum of squares , Reflect observations y1~yn Overall dispersion

Q Remnant ： The sum of the remaining squares , It reflects the degree to which the observed value deviates from the regression line , This deviation is caused by random factors such as observation errors

Q return ： Sum of regression squares , Reflect the dispersion of regression value , This dispersion is due to Y And X Caused by the linear correlation between

Q_ return And Q_ Remnant The ratio of reflects this linear correlation and random factor pair y Of influence , The ratio is about large , The stronger the linear correlation

When H0 To true ,

Given the level of significance α, if F≥F_α, Reject the assumption that H_0, That is, the linear relationship is significant . On the contrary, I think y Yes x There is no linear correlation , The linear regression equation is of no practical significance .

4.1.2 Correlation coefficient test （r Test method ）

r=S_xy/√(S_xx S_yy )

if r The absolute value of is very small , be y And x The linear correlation is not significant , Or there is no linear correlation ; if The absolute value is larger （ Close to the 1） when , It shows that the linear correlation is significant

（1） if |r|≤r_0.05 (n-2), Think y And x Linear correlation between No significant , Or there is no linear correlation ;

（2） if r_0.05 (n-2)<|r|≤r_0.01 (n-2), Think y And x Between The linear correlation is significant ;

（3） if |r|>r_0.01 (n-2), Think y And x Linear correlation between Particularly significant .

4.1.3 t Test method

4.2 Multiple linear regression test

4.2.1 Goodness of fit test （r）—— Model validity

Build an index to intuitively judge whether the fitting is good or bad ：R² = SSR / SST

R² The bigger it is , The more independent variables explain the dependent variables , The change caused by independent variables accounts for a high percentage of the total change . The denser the observation points are near the regression line . Value range ：0-1

4.2.2 Significance test of regression equation （F）—— Linear correlation

test Y And explanatory variables x1,x2,……xk Is the linear relationship significant .

The steps of testing ：

（1） suggest a hypothesis

（2） Calculation statistics

（3） Look up the table

（4） Make a test

4.2.3 Significance test of regression parameters （t）—— Influence of respective variables

Check whether each explanatory variable is correct Y remarkable

（1） suggest a hypothesis

（2） Construct and calculate statistics

（3） Look up the table

（4） test

|Ti|<t_a/2： Accept ; |Ti|>t_a/2： Refuse

Four 、 Prediction and control

4.1 forecast

For a given arbitrary x_0, Into the regression equation

y ̂_0=a ̂+b ̂x_0

Perform interval estimation , Given confidence 1-a, Find the confidence interval , That is, the prediction interval

(y ̂_0-σ ̑μ_(α/2),y ̂_0+σ ̑μ_(α/2) )

4.2 control

The inverse problem of prediction , That is to observe y To what extent , determine x The scope of the

{(&y_1^′=y ̂_1-σ ̑μ_(α/2)@&y_2^′=y ̂_2+σ ̑μ_(α/2) )

5、 ... and 、 Linearization of nonlinear regression

use Variable substitution Linearize the nonlinear model

（1）y = a + bsint ; Make x = sint

（2）y = a + bt + ct² ; Make x_1 = t ,x_2 = t² Multiple linear regression

（3） Make y = 1/y ;x = 1/x

（4） Make x = lnx

Sure excel、spss、lingo Realization

excel： Insert - Chart - Scatter plot

data - Data analysis - Return to ; Set the confidence level 95%,a>p Then refuse H_0,R² The closer the 1, The more significant the linear correlation ,F As big as possible , The better the model works

spss： Scatter plot

analysis - Return to - linear ;R and F The bigger it is ,Sig The smaller it is , The more significant the linear correlation

variance analysis , Such as ： if F>F_0.5(2,5)=5.2, The linear correlation is significant

if F>F_0.01(2,5)=10.2, The linear correlation is highly significant

Coefficient analysis ： Look at the coefficient table B

Draw a scatter diagram - Data analysis - Return to -...

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