Using LINEST for Least Squares Regression With More Than One Independent Variable

The use of least squares regression and curve fitting is well established in the applied sciences. It is discussed in detail in the monograph Least Squares Analysis and Curve Fitting. Most analyses of this type, however, are done with only one independent variable (the classic linear fit is a good example of this.)

For some problems it is necessary to consider two or more independent variables (a recent example is here.) A way to perform regression on such data is to use the LINEST function, which can be used for linear/planar types of correlations. It can be found in most of the current spreadsheet packages. It is tricky to use; about the only way to illustrate its use is through a video, and one from Dr. Todd Grande is featured here.

Fluid Mechanics Laboratory Video: Flow Meters, Gate Valve and Pipe Losses (Moody Chart)

Our main fluid mechanics laboratory page is here.  Other resources relating to this laboratory is here:

Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra

Theorem 1 (Eigenvector-eigenvalue identity) Let {A} be an {n \times n} Hermitian matrix, with eigenvalues {\lambda_1(A),\dots,\lambda_n(A)}. Let {v_i} be a unit eigenvector corresponding to the eigenvalue {\lambda_i(A)}, and let {v_{i,j}} be the {j^{th}} component of {v_i}. Then

\displaystyle |v_{i,j}|^2 \prod_{k=1; k \neq i}^n (\lambda_i(A) - \lambda_k(A)) = \prod_{k=1}^{n-1} (\lambda_i(A) - \lambda_k(M_j))

where {M_j} is the {n-1 \times n-1} Hermitian matrix formed by deleting the {j^{th}} row and column from {A}.

When we posted the first version of this paper, we were unaware of previous appearances of this identity in the literature; a related identity had been used by Erdos-Schlein-Yau and by myself and Van Vu for applications to random matrix theory, but to our knowledge this specific identity appeared to be new. Even two months after our preprint first appeared on the arXiv in August, we had only learned of one other place in the literature where the identity showed up (by Forrester and Zhang, who also cite an earlier paper of Baryshnikov).

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv a completely rewritten version of our previous paper, now titled “Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra“. This paper is now a survey of the various literature surrounding the following basic identity in linear algebra, which we propose to call the eigenvector-eigenvalue identity:

Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra

The Slow Suicide of American Science–ACSH

I’ve always been bullish about American scientific and technological supremacy, not in some starry-eyed, jingoistic way, but due to the simple reality that the United States remains the world’s research and development engine.

This is true for at least four reasons, which I detailed previously: (1) Superior higher education; (2) A cultural attitude that encourages innovation; (3) Substantial funding and financial incentives; and (4) A legal framework that protects intellectual property and tolerates failure through efficient bankruptcy laws. There’s a fifth, fuzzier reason, namely that smart and talented people have long gravitated toward the U.S.

The Slow Suicide of American Science–ACSH

President Grover Cleveland’s Secret Surgery on the Steam Yacht Oneida–Magic Masts and Sturdy Ships

The President stood at the rail of his friend’s yacht, the Oneida, watching the waves from Long Island Sound roll and tumble over each other. His fingers itched for his fishing rod. He had fished from this yacht many times in the past, but this time was different. This time, he faced something more serious than how many fish he caught. His tongue explored the contours of the tumor growing on the roof of his mouth. The economic panic threatened the country like his tumor threatened his mouth. He didn’t want to call it cancer. Cancer, the forbidden word that translated into a person just as forbidden. The operation to remove the growth from his mouth had to remain secret for the good of the country and for the good of his family.

From Magic Masts and Sturdy Ships

Too bad he didn’t choose one of George Warrington’s steam yachts, but alas the Warringtons (and many of Chicago’s grandees) were good Republicans…but that would pay off when Theodore Roosevelt appointed him as Engineering Commissioner of Lighthouses and Lightships a decade later.