【Mathematical Modeling】Statistical Analysis Methods

1. Regression Analysis

• The amount of data needs to be large, and the larger the total sample size n, the better --> to ensure better fitting effect and better prediction effect
  • Generally n>40/45 is better
• method
  • Establish regression model yi=β0+β1i+……+βkxki+εi
    
  • Write down the estimated formula
  • Bring the data in to find the regression coefficient [Find out what β^ is by least squares estimation]
  • [Part] Test whether the regression coefficients β1, β2... βk are 0, and the coefficient is significant, indicating that the independent variable x is significant
    • If βi=0, it means that the regression equation is not affected by xi, and the regression equation is simplified
  • [Overall] Test the regression equation
    • 0<r^2<=1 [The closer R2 is to 1, the better the model, and the smaller R2 is definitely not good]
    • variance analysis
    • Sig [p value] the smaller the better Sig<0.01 is more significant
  • Fortune-telling

εi is generally iid, which means independent and identically distributed, ~N(σ, σ^2)

2. Logistic regression

• Dependent variables are attribute variables, categorical variables, at least one variable is continuous
• Model

3. Cluster Analysis

• Systematic clustering method [in the case of small samples]
  continuously reduces the number of classes, and the selection criteria are not unique
  • Cluster the samples
  • Cluster the variables

4. Discriminant analysis

• Selection criteria for unique, supervised learning
  
  

5. Principal Component Analysis

• Purpose: Dimensionality reduction! ! make variable reduction

• Take out a part of the principal components [example: y1, y2, y3]
• Regression on y with the extracted principal components
• Coefficient factors a1, a2, a3 before estimating y1, y2, y3
• Interpretation of the coefficients in the principal components of the independent variable x is a relatively large proportion [multiple use of factor analysis]

6. Factor analysis

• Most of the elements of the correlation matrix are >0.3, and the correlation coefficient is large
• Write out the factor model, then analyze

7. Correspondence Analysis

• Both horizontal and vertical coordinates are regarded as categorical variables