For linear model $y=xbeta+e$, we can have a nice geometric interpretation of estimated model via OLS: $hat{y}=xhat{beta}+hat{e}$. $hat{y}$ is the projection of y onto the space spanned by x and residual $hat{e}$ is perpendicular to this space spanned by x.

Now, my question is: is there any geometric interpretation of generalized linear model (logistic regression, Poission, survival)? I am very curious about how to interpret the estimated binary logistic regression model $hat{p} =

textrm{logistic}(xhat{beta})$ geometrically, in a similar way as linear model. It even does not have an error term.

I found one talk about geometric Interpretation for Generalized Linear Models. http://statweb.stanford.edu/~lpekelis/talks/13_obs_studies.html#(7). Unfortunately, the figures are not available and it is quite hard to picture.

Any help, referencing, and suggestion will be greatly appreciated!!!

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#### Best Answer

I think that you best bet is the thesis of Dongwen Luo from Massey University, *On the geometry of generalized linear models*; it is available online here. In particular you want to focus on Chapt. 3 – *The Geometry of GLMs* (and more particular in section 3.4). He employs two different "geometrical domains"; one before and one after the canonical link transformation. Some of the basic theoretical machinery stems from Fienberg's work on The Geometry of an r × c Contingency Table. As advocated in Luo's thesis:

For a sample of size $n$, $R^n$ splits into an orthogonal direct sum of the sufficiency space $S$ and the auxiliary space $A$. The MLE of the mean $hat{mu}$ lies in the intersection of the sufficiency affine plane $T = s + A$ and the untransformed model space $M_R$. The link transformed mean vector $g(hat{mu})$ lies in the transformed mean space $g(M_R)$.

Clearly both $S$ and $A$ need to be at least 2-D and $R^n = S oplus A$. Under this theoretical framework $hat{mu}$ and the data vector $y$ have the same projection onto any direction in the sufficiency space.

Assuming you have differential geometry knowledge, the book of Kass and Vos Geometrical Foundations of Asymptotic Inference should provide a solid foundation on this matter. This paper on The Geometry of Asymptotic Inference is freely available from the author's website.

Finally, to answer your question whether there is "*any geometric interpretation of generalized linear model (logistic regression, Poisson, survival)*". Yes, there is one; and depends on the link function used. The observations themselves are viewed as a vector in that link transformed space. It goes without saying you will be looking at higher-dimensional manifolds as your sample size and/or the number of columns of your design matrix is increasing.

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