# Quasi-Least Squares Regression This specifies the Welsch function which can perform well in cases where the residuals have an exponential distribution. This function sets the tuning constant used to adjust the residuals at each iteration to tune. Decreasing the tuning constant increases the downweight assigned to large residuals, while increasing the tuning constant decreases the downweight assigned to large residuals.

This function sets the maximum number of iterations in the iteratively reweighted least squares algorithm to maxiter.

## Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression

This function assigns weights to the vector wts using the residual vector r and previously specified weighting function. This function computes the best-fit parameters c of the model for the observations y and the matrix of predictor variables X , attemping to reduce the influence of outliers using the algorithm outlined above. The -by- variance-covariance matrix of the model parameters cov is estimated as , where is an approximation of the residual standard deviation using the theory of robust regression.

In this case, the current estimates of the coefficients and covariance matrix are returned in c and cov and the internal fit statistics are computed with these estimates. This function computes the vector of studentized residuals for the observations y , coefficients c and matrix of predictor variables X.

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This function returns a structure containing relevant statistics from a robust regression. This contains the standard deviation of the residuals as computed from ordinary least squares OLS. This contains an estimate of the standard deviation of the final residuals using the Median-Absolute-Deviation statistic.

This contains an estimate of the standard deviation of the final residuals from the theory of robust regression see Street et al, This contains the coefficient of determination statistic using the estimate sigma. This contains the adjusted coefficient of determination statistic using the estimate sigma. This contains the number of degrees of freedom.

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This contains the final weight vector of length n. This contains the final residual vector of length n ,. This module is concerned with solving large dense least squares systems where the -by- matrix has ie: many more rows than columns. Therefore, the algorithms in this module are designed to allow the user to construct smaller blocks of the matrix and accumulate those blocks into the larger system one at a time. The algorithms in this module never need to store the entire matrix in memory. The large linear least squares routines support data weights and Tikhonov regularization, and are designed to minimize the residual.

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In the discussion which follows, we will assume that the system has been converted into Tikhonov standard form,. For a discussion of the transformation to standard form, see Regularized regression. The basic idea is to partition the matrix and observation vector as. The sections below describe the methods available for solving this partitioned system.

## SIAM Journal on Scientific Computing

The normal equations approach to the large linear least squares problem described above is popular due to its speed and simplicity. Since the normal equations solution to the problem is given by. Using the partition scheme described above, these are given by. Since the matrix is symmetric, only half of it needs to be calculated. Once all of the blocks have been accumulated into the final and , the system can be solved with a Cholesky factorization of the matrix.

The matrix is first transformed via a diagonal scaling transformation to attempt to reduce its condition number as much as possible to recover a more accurate solution vector. The normal equations approach is the fastest method for solving the large least squares problem, and is accurate for well-conditioned matrices. However, for ill-conditioned matrices, as is often the case for large systems, this method can suffer from numerical instabilities see Trefethen and Bau, The number of operations for this method is. This method is based on computing the thin QR decomposition of the least squares matrix , where is an -by- matrix with orthogonal columns, and is a -by- upper triangular matrix.

Once these factors are calculated, the residual becomes. The matrix on the left hand side is now a much smaller -by- matrix which can be solved with a standard SVD approach. The matrix is just as large as the original matrix , however it does not need to be explicitly constructed. The TSQR algorithm computes only the -by- matrix and the -by-1 vector , and updates these quantities as new blocks are added to the system.

Each time a new block of rows is added, the algorithm performs a QR decomposition of the matrix. This QR decomposition is done efficiently taking into account the sparse structure of. See Demmel et al, for more details on how this is accomplished. The typical steps required to solve large regularized linear least squares problems are as follows:. Choose the regularization matrix. Construct a block of rows of the least squares matrix, right hand side vector, and weight vector , ,. Transform the block to standard form ,. Accumulate the standard form block , into the system.

Determine an appropriate regularization parameter using for example L-curve analysis.

## Logistic regression models

Solve the standard form system using the chosen. This function allocates a workspace for solving large linear least squares systems. The least squares matrix has p columns, but may have any number of rows. The parameter T specifies the method to be used for solving the large least squares system and may be selected from the following choices.

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This specifies the normal equations approach for solving the least squares system. This method is suitable in cases where performance is critical and it is known that the least squares matrix is well conditioned. The size of this workspace is. This method is a good general purpose choice for large systems, but requires about twice as many operations as the normal equations method for. This function resets the workspace w so it can begin to accumulate a new least squares system. The block X , y is converted to standard form and the parameters , are stored in Xs and ys on output.

Optional data weights may be supplied in the vector w. This function calculates the QR decomposition of the -by- regularization matrix L. L must have. These functions convert a block of rows X , y , w to standard form , which are stored in Xs and ys respectively.

X , y , and w must all have the same number of rows. Optional data weights may be supplied in the vector w , where. This function accumulates the standard form block into the current least squares system. X and y have the same number of rows, which can be arbitrary. X must have columns. For the normal equations method, they are both unchanged. After all blocks have been accumulated into the large least squares system, this function will compute the solution vector which is stored in c on output. On output, rnorm contains the residual norm and snorm contains the solution norm.

After a regularized system has been solved with a regularization matrix , specified by LQR , Ltau , this function backtransforms the standard form solution cs to recover the solution vector of the original problem, which is stored in c , of length. This function computes the L-curve for a large least squares system after it has been fully accumulated into the workspace work.

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