I Vector visualization of multicollinearity

  • I
  • Thread starter Thread starter Trollfaz
  • Start date Start date
  • Tags Tags
    Regression analysis
AI Thread Summary
In regression analysis, multicollinearity occurs when there is significant correlation among the predictor variables, which can complicate the estimation of coefficients. A general linear model is represented as y = a_0 + Σ a_i x_i, where n observations of y are collected for k predictors. Perfect multicollinearity arises when at least one predictor is a linear combination of others, resulting in a rank deficiency in the matrix X, which is structured with observations as rows and predictors as columns. Strong multicollinearity indicates that some predictors make small angles with the subspace formed by others, impacting the model's reliability. Understanding and addressing multicollinearity is crucial for accurate regression analysis and interpretation.
Trollfaz
Messages
143
Reaction score
14
General linear model is
$$y=a_0+\sum_{i=1}^{i=k} a_i x_i$$
In regression analysis one always collects n observations of y at different inputs of ##x_i##s. n>>k or there will be many problems. For each regressor, and response y ,we tabulate all observations in a vector ##\textbf{x}_i## and ##\textbf{y}_i##, both is a vector of ##R^n##.So multicollinearity is the problem that there's significant correlation between the ##x_i##s. In practice some degree of multicollinearity exists. So perfectly no multicollinearity means all the ##\textbf{x}_i## are orthogonal to each other?ie.
$$\textbf{x}_i•\textbf{x}_j=0$$
For different i,j and strong multicollinearity means one of more of the vector makes a very small angle with the subspace form by the other vectors? As far as I know perfect multicollinearity means rank(X)<k. X is a n by k matrix with ith col as ##\textbf{x}_i##
 
Physics news on Phys.org
Perfect multicollinarity means that at least 1 predictor variable (columns) is a perfect linear combination of one or more of the other variables. Typically the variables are the columns of the matrix and observations are rows. In this situation, the matrix will not be full rank.
 
  • Like
Likes FactChecker

Similar threads

Replies
0
Views
449
Replies
8
Views
2K
Replies
30
Views
4K
Replies
5
Views
2K
Replies
4
Views
2K
Replies
27
Views
1K
Replies
2
Views
1K
Back
Top