Hat Matrix Value Meaning

The hat values are the fitted values or the predictions made by the model for each observation. As you can see the two x values furthest away from the mean have the largest leverages 0176 and 0163 while the x value closest to the mean has a smaller leverage 0048. The invocation hatvalues vglmObject should return a n times M matrix of the diagonal elements of the hat projection matrix of a vglm object.

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Hat Matrix And Leverages In Classical Multiple Regression Cross Validated

Hat Matrix An Overview Sciencedirect Topics

It thus deserves a name.

Hat matrix value meaning. Properties and Interpretation Week 5 Lecture 1 1 Hat Matrix 11 From Observed to Fitted Values The OLS estimator was found to be given by the p 1 vector b XT X 1XT y. Thus H ij is the rate at which the ith tted value changes as we vary the jth observation the in uence that observation has on that tted value. The hat matrix provides a measure of leverage.

Check that Yb iY j H ij. Its easy to see that HT H. It is quite different from the Cooks distance.

Where H XXT X 1XT is an n nmatrix which puts the hat on y and is therefore. For a given model with independent variables and a dependent variable the hat matrix is the projection matrix to. Lets look at some of the properties of the hat matrix.

In fact if we look at a sorted list of the leverages obtained in. Where H the H matrix is known as the hat matrix because y ˆ Hy and X are for respective models in the V basis set and fi are filter values from Equation 13. Hence if you multiply H by y you project your observed values in y onto the space that is spanned by the variables in X.

The variables are vectors and span a space. I believe youre asking for the intuition behind those three properties of the hat matrix so Ill try to rely on intuition alone and use as little math and higher level linear algebra concepts as possible. It is defined as the matrix that converts values from the observed variable into estimations obtained with the least squares method.

It is useful for investigating whether one or more observations are outlying with regard to their X values and therefore might be excessively influencing the regression results. Therefore when performing linear regression in. Y X β ε1 where y Y Yi and X X Xij j and β1 is the vector of regression coefficients without the intercept the hat-value for the i th observation is 1 h hi i i i i i n h h x X X x.