Fits a multiple linear regression model using either the Least Absolute Value (L1), Least Lp norm (Lp ), or Least Maximum Value (Minimax or L∞ ) method of multiple linear regression.

The type double function is imsls_d_Lnorm_regression.

imsls_f_Lnorm_regression returns a pointer to an array of length n_independent + 1 containing a least absolute value solution for the regression coefficients. The estimated intercept is the initial component of the array, where the i‑th component contains the regression coefficients for the i‑th dependent variable. If the optional argument IMSLS_NO_INTERCEPT is used then the (i-1)-st component contains the regression coefficients for the i‑th dependent variable. imsls_f_Lnorm_regression returns the Lp norm or least maximum value solution for the regression coefficients when appropriately specified in the optional argument list.

IMSLS_RETURN_USER, float coefficients[] (Output)
Storage for array coefficients is provided by the user. See Return Value.

Function imsls_f_Lnorm_regression computes estimates of the regression coefficients in a multiple linear regression model. For optional argument IMSLS_LAV (default), the criterion satisfied is the minimization of the sum of the absolute values of the deviations of the observed response yi from the fitted response

for a set on n observations. Under this criterion, known as the L1 or LAV (least absolute value) criterion, the regression coefficient estimates minimize

The estimation problem can be posed as a linear programming problem. The special nature of the problem, however, allows for considerable gains in efficiency by the modification of the usual simplex algorithm for linear programming. These modifications are described in detail by Barrodale and Roberts (1973, 1974).

In many cases, the algorithm can be made faster by computing a least-squares solution prior to the invocation of IMSLS_LAV. This is particularly useful when a least-squares solution has already been computed. The procedure is as follows:

When multiple solutions exist for a given problem, option IMSLS_LAV may yield different estimates of the regression coefficients on different computers, however, the sum of the absolute values of the residuals should be the same (within rounding differences). The informational error indicating nonunique solutions may result from rounding accumulation. Conversely, because of rounding the error may fail to result even when the problem does have multiple solutions.

Optional argument IMSLS_LLP computes estimates of the regression coefficients in a multiple linear regression model y = Xβ + ɛ under the criterion of minimizing the Lp norm of the deviations for i = 0, ..., n-1 of the observed response yi from the fitted response

for a set on n observations and for p ≥ 1. For the case when IMSLS_WEIGHTS and IMSLS_FREQUENCIES are not supplied, the estimated regression coefficient vector,

(output in coefficients[]) minimizes the Lp norm

The choice p = 1 yields the maximum likelihood estimate for β when the errors have a Laplace distribution. The choice p = 2 is best for errors that are normally distributed. Sposito (1989, pages 36−40) discusses other reasonable alternatives for p based on the sample kurtosis of the errors.

Weights are useful if the errors in the model have known unequal variances

In this case, the weights should be taken as

Frequencies are useful if there are repetitions of some observations in the data set. If a single row of data corresponds to ni observations, set the frequency fi = ni. In general, IMSLS_LLP minimizes the Lp norm

The asymptotic variance-covariance matrix of the estimated regression coefficients is given by

where R is from the QR decomposition of the matrix of regressors (output in R-Matrix). An estimate of λ2 is output in square_of_scale.

In the discussion that follows, we will first present the algorithm with frequencies and weights all taken to be one. Later, we will present the modifications to handle frequencies and weights different from one.

Option call IMSLS_LLP uses Newton’s method with a line search for p > 1.25 and, for p ≤ 1.25, uses a modification due to Ekblom (1973, 1987) in which a series of perturbed problems are solved in order to guarantee convergence and increase the convergence rate. The cutoff value of 1.25 as well as some of the other implementation details given in the remaining discussion were investigated by Sallas (1990) for their effect on CPU times.

In each case, for the first iteration a least-squares solution for the regression coefficients is computed using routine imsls_f_regression. If p = 2, the computations are finished. Otherwise, the residuals from the k-th iteration,

are used to compute the gradient and Hessian for the Newton step for the (k + 1)-st iteration for minimizing the p-th power of the Lp norm. (The exponent 1/p in the Lp norm can be omitted during the iterations.)

For subsequent iterations, we first discuss the p > 1.25 case. For p > 1.25, the gradient and Hessian at the (k + 1)-st iteration depend upon

and

In the case 1.25 < p < 2 and

and the Hessian are undefined; and we follow the recommendation of Merle and Spath (1974). Specifically, we modify the definition of

to the following:

In order to compute the step on the (k + 1)-st iteration, the R from the QR decomposition of

is computed using fast Givens transformations. Let

denote the upper triangular matrix from the QR decomposition. The linear system

is solved for

are nonzero, this is exactly the Newton step. See Kennedy and Gentle (1980, pages 528−529) for further discussion.

If the first attempted step does not lead to a decrease of at least one-tenth of the predicted decrease in the p-th power of the Lp norm of the residuals, a backtracking linesearch procedure is used. The backtracking procedure uses a one-dimensional quadratic model to estimate the backtrack constant p. The value of p is constrained to be no less that 0.1.

An approximate upper bound for p is 0.5. If after 10 successive backtrack attempts, α(k) = p1p2...p10 does not produce a step with a sufficient decrease, then imsls_f_Lnorm_regression issues a message with error code 5. For further details on the backtrack line-search procedure, see Dennis and Schnabel (1983, pages 126−127).

Convergence is declared when the maximum relative change in the residuals from one iteration to the next is less than or equal to epsilon. The relative change

in the i‑th residual from iteration k to iteration k + 1 is computed as follows:

where s is the square root of the residual mean square from the least-squares fit on the first iteration.

For the case 1 ≤ p ≤ 1.25, we describe the modifications to the previous procedure that incorporate Ekblom’s (1973) results. A sequence of perturbed problems are solved with a successively smaller perturbation constant c. On the first iteration, the least-squares problem is solved. This corresponds to an infinite c. For the second problem, c is taken equal to s, the square root of the residual mean square from the least-squares fit. Then, for the (j + 1)-st problem, the value of c is computed from the previous value of c according to

Each problem is stated as

For each problem, the gradient and Hessian on the (k + 1)-st iteration depend upon

and

where

The linear system [R(k+1)]TR(k+1)d(k+1)= XTz(k+1) is solved for d(k+1) where R(k+1) is from the QR decomposition of [V (k+1)]1∕2X. The step taken on the (k + 1)-st iteration is

where the first attempted step is with α(k+1) = 1. If necessary, the backtracking line-search procedure discussed earlier is used.

Convergence for each problem is relaxed somewhat by using a convergence epsilon equal to max(epsilon, 10−j) where j = 1, 2, 3, ... indexes the problems (j = 0 corresponds to the least-squares problem).

After the convergence of a problem for a particular c, Ekblom’s (1987) extrapolation technique is used to compute the initial estimate of β for the new problem. Let R(k),

and c be from the last iteration of the last problem. Let

and let t be the vector with elements ti. The initial estimate of β for the new problem with perturbation constant 0.01c is

where Δc = (0.01c − c) = −0.99c, and where d is the solution of the linear system

Convergence of the sequence of problems is declared when the maximum relative difference in residuals from the solution of successive problems is less than epsilon.

The preceding discussion was limited to the case for which weights[i] = 1 and frequencies[i] = 1, i.e., the weights and frequencies are all taken equal to one. The necessary modifications to the preceding algorithm to handle weights and frequencies not all equal to one are as follows:

These replacements have the same effect as multiplying the i‑th row of X and y by

and repeating the row fi times except for the fact that the residuals returned by imsls_f_Lnorm_regression are in terms of the original y and X.

Finally, R and an estimate of λ2 are computed. Actually, R is recomputed because on output it corresponds to the R from the initial QR decomposition for least squares. The formula for the estimate of λ2 depends on p.

For p = 1, the estimator for λ2 is given by (McKean and Schrader 1987)

with

where z0.975 is the 97.5 percentile of the standard normal distribution, and where

are the ordered residuals where rank zero residuals are excluded. Note that

For p = 2, the estimator of λ2 is the customary least-squares estimator given by

For 1 < p < 2 and for p > 2, the estimator for λ2 is given by (Gonin and Money 1989)

with

The estimation problem can be posed as a linear programming problem. A dual simplex algorithm is appropriate, however, the special nature of the problem allows for considerable gains in efficiency by modification of the dual simplex iterations so as to move more rapidly toward the optimal solution. The modifications are described in detail by Barrodale and Phillips (1975).

When multiple solutions exist for a given problem, IMSLS_LMV may yield different estimates of the regression coefficients on different computers, however, the largest residual in absolute value should have the same absolute value (within rounding differences). The informational error indicating nonunique solutions may result from rounding accumulation. Conversely, because of rounding, the error may fail to result even when the problem does have multiple solutions.

A straight line fit to a data set is computed under the LAV criterion.

Different straight line fits to a data set are computed under the criterion of minimizing the Lp norm by using p equal to 1, 1.5, 2.0 and 2.5.

A straight line fit to a data set is computed under the LMV criterion.