NBSIR 79-1920

A Collection of Test Problems

for Discrete Linear Li Data

Fitting

Operations Research Division

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Washington, DC 20234

July 1, 1979

Issued November 1979

U.S. DEPARTMENT OF COMMERCE

D. R. Shier

S. J. Neupauer

P. B. Saunders

.U56

79-1920
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NBSIR 79-1920

A COLLECTION OF TEST PROBLEMS

FOR DISCRETE LINEAR L, DATA

FITTING

07 atAtrDAsam UBMAXT

JUL 2 1 1980

Q-UOO

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D. R. Shier

S. J. Neupauer

P. B. Saunders

Operations Research Division

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

U S. Department of Commerce

Washington, DC 20234

July 1, 1979

Issued November 1979

U.S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Jr., Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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ABSTRACT

This document assembles 27 test problems representing a variety of examples in

which least absolute deviation (or Lj^) data fitting has been used. The problems were collected from the literature, from the authors of several Lj

approximation codes, and from examples encountered in practice. Optimal L]^ solutions to these problems (objective function value and solution vector)

have been obtained using a double-precision computer code designed for checking the Kuhn-Tucker conditions and for performing an accurate reinversion of

the optimal basis. . Special problem characteristics such as alternative optima, degeneracy, and rank loss are also noted. This set of test problems has

proven useful in evaluating and Improving the performance of L| codes as well

as in suggesting types of problem structures that might be mimicked by problem

generators.

Key Words : Algorithm testing; approximation; curve fitting; least absolute

deviation; polynomial approximation; spline fitting; test data; test problems
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ACKNOWLEDGMENT

The authors gratefully acknowledge the support provided by the Computer Services Division, National Bureau of Standards, Washington, DC In allowing use

of the UNIVAC 1108 for conducting this research project*
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TABLE OF CONTENTS

1 • Introduction 1

2. Lj^ Data Fitting Problems 3

3* The 27 Hand*-’Plcked Problems 4

4* Verification of Optimal Solutions 8

5. Listing of Input Problem Data 11

6* Summary of Problem Solutions ....108

7. References 135
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1. INTRODUCTION

The fitting of various models to existing data and the discrete approximation

of given functional relations are tasks frequently entountered in applied

mathematics. An extensively studied and used technique in this regard is that

of least squares estimation [9, 19, 21], in which the sum of squared residuals

between actual and predicted values is minimized. This approach enjoys a num- ber of useful advantages: the least squares estimates can be found by solving

(with appropriate care) a system of linear equations; the statistical properties of such estimators are well established; and, under the assumption of

normally distributed errors, a number of statistical tests of hypotheses can

be carried out.

In recent years interest has been rekindled in alternative estimation methods,

which can be used either in place of or in conjunction with the technique of

least squares [3, 10, 22, 26]. Of particular Interest is the method of least

absolute deviation (or L]^) data fitting, where the sum of absolute values of

the residuals is minimized. This approach also has certain desirable properties: the L]^ estimators are maximum likelihood estimators when the errors derive from a Laplace distribution [22]; they can be more efficient than least

squares estimators [22, 23]; they are "robust” with respect to long-tailed

distributions [13, 14] and do not give undue weight to observations which re- present outliers. Moreover, with the advent of reasonably efficient procedures for constrained optimization (descent methods, linear programming), obtaining such estimates is computationally feasible.

A number of algorithms for Lj data fitting problems have been recently advanced [1, 2, 5, 6, 24, 25], and preliminary comparative testing has been performed at the National Bureau of Standards using four generally available Lj

computer codes [12]. Controlled experiments with these codes are facilitated

by the availability of well-defined classes of test problems so that results

obtained by sampling a finite number of test problems from a class will possess some measure of validity. For this reason, generators for problem

classes with controllable characteristics have been developed [8, 16] and used

in performing comparisons of Lj codes [12]. In addition, a set of 27 handpicked problems was assembled and employed in an evaluation of the codes.

It is significant that although all four codes "correctly" solved each generated problem within certain classes, this was decidedly not the case for the

27 hand-picked problems.

It is believed, therefore, that specifying these 27 problems in some detail

would be a useful service both to code developers and to those involved in

testing Lj codes. That is the purpose of this document. In addition to presenting the data values themselves, we also report relevant problem characteristics (e.g. degeneracy, rank deficiency, uniqueness of solution) and verified

solution values. This material should allow code developers to perform an

initial benchmarking of their procedures and to identify possible areas for

algorithmic enhancement. In addition, the problems described here may suggest

types of problem structures to be mimicked by problem generators [8, 16, 17,

19].
Section 2 defines the general Lj data fitting problem and reviews certain

relevant features of such problems. Section 3 discusses the selection of the

27 hand-picked problems and documents the format of these problems as well as their solutions. The procedure used for verifying optimal solutions to the

test problems is described in Section 4. Individual data values are tabulated

in Section 5, and verified solutions are given in Section 6.

2
2. Lj DATA FITTING PROBLEMS

Given n sets of observations on one dependent variable y and m independent

variables , the discrete linear data fitting problem is that of

determining to minimize

= 2 \y^- ^ ^i^lj*

i=l j=l

Here, refers to the i-th observation on variable y, and x^. refers to the

i-th observation on variable x.. In other words, given n observations we wish

to estimate m < n parameters b^ minimizing the sum of the absolute values of

the residuals

e

.

1

m

A set of for which

1 m

(j)* = (j>(3*,.. . ,3*) = min{())( 3, , • .

. ,3 ) : 3jeR; j=l,...,m>

1 m

i m

is termed an optimal solution vector to the data fitting problem, and (p* is

termed the optimal objective function value*

Clearly, the problem data can be represented using an n x 1 column vector ^

and an n X m design matrix X. The rank of matrix X is denoted by r _< m; a given Lj problem has full rank if r = m and has rank loss m - r otherwise.

Unlike the case for least squares (L

2

) approximation, the optimal solution

vector may not be unique, even when the problem has full rank. In the presence of rank loss, the optimal solution vector will definitely not be unique.

It is known [4,11] that an optimal solution vector can always be found that

interpolates at least r of the data points: that is, at least r residuals are Identically zero at the solution. For our purposes, the problem is said to

exhibit degeneracy if z > r residuals are zero at an optimal solution, with

degree of degeneracy z - r.

A basic solution for the data fitting problem is obtained by selecting r data points to be interpolated exactly—data points corresponding to r independent rows (observations) of the design matrix are chosen to have zero re- siduals. A number of algorithms for solving discrete Lj data fitting problems

examine only these basic solutions and proceed in a systematic fashion from

one basic solution to the next. A collection of r rows from X that defines a basic solution is said to constitute a set of r active contraints . In the

presence of degeneracy, the same optimal solution vector will correspond to

several selections of active constraints, or basic solutions. If the problem

has full rank, then a unique set {3j,...,3^} is determined by selecting any

r = m constraints to be active.

3
3. THE 27 HAND-PICKED PROBLEMS

The 27 problems compiled and studied in this report were obtained in the following ways.

1. By selecting problems described or solved in journal articles or technical reports

2. By soliciting test problems from the authors of various data

fitting codes^

3. By collecting problems from actual applications

Table 1 shows various sources for the hand-picked problems categorized by

these three headings. In addition, the problems can be roughly classified

into the following structural types.

1. General data fitting problems

2. Spline-fitting problems

3. Polynomial approximation problems

4. Others

The disposition of the 27 hand-picked problems by these problem types is shown

in Table 2.

We have provided listings of the input data for all 27 problems in Section 5. The first line is a header which gives a brief Identification of the problem.

The next line gives the values for n (the number of observations) and m (the

number of parammeters) in FORTRAN format (215). The succeeding lines contain

the input data for successive observation sets (yj^jX^j, . .

. ,X£j^) in FORTRAN

format (5E15. 7).

Input values are given to seven significant figures when necessary. Because

conversion of real numbers to the internal representation of any computer generally Involves a round-off or truncation error, the given (single-precision)

input values may define different internally represented problems for different machines. The choice of seven significant decimal digits was made to

enhance the "portability" of such problems to other machines. Our current

study, it should be noted, was performed on the National Bureau of Standards*

UNIVAC 1108, Exec 8 Operating System. On this computer, single-precision

floating-point numbers are represented using a 36-bit word, with one sign bit,

eight exponent bits and twenty-seven mantissa bits (approximately 8.1 decimal

digits).

^We are especially Indebted to N. Abdelmalek and R. Bartels for a number of

these test problems.

4
Table 1

Sources of the 27 Hand-Picked Problems

Articles Authors Applications

' HPl HP3 HP4

HP2 HP8-HP26 HP5

HP6 HP27

HP7

HP8-HP26

5
Table 2

Structural Types for the 27 Hand-Picked Problems

General Spline Polynomial Other

HP4

HP15

HP25

HP16

HP27

HP1-HP3

HP6-HP14

HP17-HP20

HP26

HP5*

HP21-HP24**

* Matrix Inversion

** Trigonometric Approximation

6
Section 6 contains a summary, for each problem of its source, its characteristics, and its verified solution(s). Each of the items in this summary is discussed in turn.

1. Identification. Each problem is assigned a unique identifying code number (HP1-HP27).

2. Source . A brief reference is provided to an article discussing the problem, or to the type of application giving rise to the problem.

3. Characteristics . Various characteristics of the input problem are given,

including the number of parameters (m), the number of observations (n),

and the rank of the design matrix.

4. Solution. An optimal set of active constraints and associated numerical

solution values are provided as follows.

(a) the (unique) optimal objective function value 'i** is given,

accurate to the number of significant digits shown.

(b) A set of active constraints defining an optimal solution vector

is given. In the case of degeneracy, alternative sets of active

constraints defining the same solution vector are indicated.

(c) Numerical values for the solution vector 3*= (3*,..., 3*) are

given, accurate to the number of significant digits shown.

(d) An estimate of the 1-condition number [7, 20] of the final basis

matrix B defined by the set of active constraints. Since the

solution vector is obtained by solving a linear system with

coefficient matrix B, the 1-condition number provides Information

on the accuracy obtainable in from the input data (accurate to

seven decimal digits). Namely, if this quantity is approximately

10^, then the components of ^ can usually be expected to have d

fewer significant figures of accuracy than the elements of B [7j.

(e) An indicator is given of whether or not the problem possesses a unique optimal solution vector ^*. If there exist alternative

optimal solution vectors to the probvlem (in which case there are infinitely many, since the set of solution vectors forms a convex

set), then another optimal basic solution vector is given,

together with its defining set of active constraints and the

associated condition number.

Solutions are given, when possible, as simple fractions. Otherwise, solution

values are provided to seven significant decimal digits. Optimal solutions

were verified by means of a computer routine designed to check the Kuhn-Tucker

conditions of an associated linear program. (Details of the procedure used to

verify optimal solutions are provided in Section 4. ) Essentially, for a given

set of active contraints, the routine calculates (in double-precision) both

the solution vector _3 and the objective function value It is then determined whether or not this set of active contstraints, and the corresponding

solution vector, do indeed represent an optimal solution to the given

problem.

7
4. VERIFICATION OF OPTIMAL Lj SOLUTIONS

In this section, we describe a method for verifying whether a purported solution to a discrete linear Lj data fitting problem is in fact an optimal solution. Recall that the problem of interest is that of determining to

n m

(1) minimize Z - Z • i=l j=l

Without loss of generality, we can assume that the matrix X = (x^j) has rank

r ^ m and that the first r columns of X are linearly independent. Since an optimal solution can be obtained with 3j “ 0 for j > r, the original L^

approximation problem can be formulated [16] as the following linear program:

minimize

subject to

n

Z ttj 1=1

+ Z 3 iXi j y^

j=l

ai - Z 3iXi:j > -y^ , j=l

j.*"” l}***^tl ^

where represents the absolute value of the i-th residual

Necessary and sufficient conditions for a*,..., a* , 3*>»»»>3* to be an optimal

1 n 1 r solution to the above problem are provided by the Kuhn-Tucker conditions [15]:

there must exist uj,...,Uj^ , vj^,...,Vj^ such that

( 2 . 1 ) a* + Z 3* X4 4 > yj , i=l,...,n , i j=i J

(2.2) a* - Z 3* Xj^4 > -y^ , 1=1,..., n , 1 j=i j

’

(2.3) Uj^ + Vi = 1 , 1=1,... ,n ,

8
(2.4)

n

E i=l

(Ui“Vi)xij = 0 > j-l , . .

.

, r ,

(2.5) ^i

r a* + E

L 3*Xij

J yi "“0 y y

( 2 . 6 ) ^i

r a* - E _i j=l

3*Xij

J yi "“0 y y

(2.7) Vi >_ 0 , i- 1 , . .

.

,

n •

These conditions can be considerably simplified and tlien applied to judge

whether a given 3 = represents an optimal solution to (1). In

order to achieve this simplification, let us partition, for a given 3 , the

index set N = {l,...,n} into three disjoint subsets N"*", N”, ~

N"*” = {ieiN: e^^O) ,

N" = {ieN: ej^<0} ,

= {ieW: ex=0 > .

When inequalities (2.1) and (2.2) are summed for fixed i, we obtain a* > 0.

-{i

Thus, if ieN (so that > 0), then a* + e^ ^ 0. From (2.6) it follows that

i vx = 0 for ieN'*’, and from (2.3) that u^ = !• In a similar fashion, if ieN

i

then e^ < 0 and a* - e^ > 0, whence using (2.5) and (2.3) we obtain u^ = 0 and

i

Vi = 1.

Equation (2.4) can be rewritten as

I ie N'*’(Ui-Vx)Kij + >: (ux-Vi)xij + E (ui-vx)xxj = 0

ieN" ieN^

. or, using the above values for u^ and v^ >

E X

iCN+

ij E xxj + E (ux-Vi)xxj = 0 . i£N" ieN'J

9
This latter relation can be recast, via (2.3), as

(3) E = 1/2 { E Xjij + E - E x^j) , j=l r . IenO ieN" ieN^ ieN"*" r Recall that for given values have 04 = |y4 - E 3iXj^4| , i=l

i=l,...,n . Then conditions (2.1) and (2.2) are automatically satisfied, and

the remaining conditions obtain if and only if there exist values n =

(ui,...,Uj^) that satisfy equation (3) and

(4) 0 < UjL jC 1 , i=l,...,n .

The relations of (3) simply represent a linear system ^B=c of r equations in z = |N^| unknowns. In the absence of degeneracy (r=z) and if matrix B is

nonsingular, then equation (3) can be solved explicitly as ti = cB“^ and

condition (4) can be readily checked. In the cases when either B is singular or z>r (degeneracy), we can determine whether there is a feasible solution to

(3) and (4) by using linear programming. A "Phase I" calculation can be

readily carried out using artificial variables [15] to discover whether the

following system does indeed admit a feasible solution:

^B = c^ ,

I ^

(5) ^ H<i.

u > 0 .

In the computer program used to implement these id(feas, the data values ^ and X

are input, together with a candidate set of r active constraints (i^, . .

. ,ij.},

associated with Independent rows of the design matrix. The associated

solution vector ^is then obtained by solving in double-precision the

(nonsingular) system A=(x^i^,j) is the r x r submatrix

corresponding to the active constraint rows and the first r columns of X. From these 3j values, the residuals e^^ (i=l,...,n) are determined and the

index sets N"^, N“, N® are obtained. At this point the corresponding objective

function value

n

<l>(3i, .. . ,Bj-) “ ^ l®il

i=l

is also calculated in double-precision. Finally, the existence of values u^

that satisfy (3) and (4) is checked either by direct (double-precision)

solution of or by applying a revised simplex computer code to system (5).

If such values exist (to within pre-specif led tolerances), then {i^,...,lj.} is

deemed an optimal set of active constraints, with optimal solution vector ^

and optimal objective function value <|). Otherwise, the candidate solution is

reported as nonoptimal, to working precision.

10
5. LISTING OF INPUT PROBLEM DATA

HP I : PCLYNT'V. T 4L

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• 10 000 00 #-0 I

. lOOOCOO+0

1

. 2000000+0

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• JO 0 0 00 0+0

1

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1

APPROX IW AT ICN PR

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• 1000000 + 0 1 . 1 000 000 f 01

. 1 OOOCOO + 0 I

• 1 OOuOOO + 0

I

I I 0000 00+0 1

, 2000000+01

,3000000+0

1

,4000000+01

.5COOO 00+0 1

11
HP2 ; POLYNOMIAL

I 7 3

• 2F^00000^-C2

- • 3900000+02

-.4000000+02

- • 1000000+0 I .2300000+02

.34 00000+02

.2500000+02

.4000000+02

.A 300000 + 02

.2900000+02

. 140C000 + 02

.1200000+02

- . 1600000 + 02

-.1400000+02

- . 460 000 0 + 02

- .6-3 000 00+02

- . 24 0 0 00 0+02

APPPGX IMAT I CN

. 1 0 OOCOO + 0

I

. 1000000+01

. 1000000+01

. 1000000+01

• 1 000000 + 01

. 1000000+01

. 1000000+01

. 1000000+01

. 1000000 t-01

. 1000000 + 01

. 1000000+01

. 1 OOCOOO +0

1

. 1000000 + 01

. I 000000 + 0

1

. I 0 00000 + 0

1

. I 0 00 00 0 + 0 I

. 1 0 00000 + 01

PRCiiLE M

- . 3000000+0 1 -. 7000000+01

- .6000000+ 0 I -.5000000+01

- .4000000+0

1

-.3000000+01

-. 2000000+01

-. 1 000000+0

1

. OOOOO 30

. I 00 0 0 U 0 + 0 1

. 2000000+0

1

.3000000+0 I .4 000 000 + 0 I .5000000+01

.6000000+0

1

. 7000000+0

1

. 6oe(i jOo + 0 1

.6403000+32

.49 Ou 0 6 0 + 0 2

.3600000+02

.2500000+02

.1600000+02

.9000000+01

.4000000+01

. 1 000000 + 0

1

.0003003

.10000 0 0 1-0 1 .4000000+31

.9000000+01

.1630 003+02

.2500000+02

.3600000+32

.4 900 00 0+02

.6403000+02

12
MP3 : polynomial

2

I 1 4 JC 30+0 3

1 240000 +03

14300C0+C3

I 5 60 0C.) + 03

1660000+CJ

At-’PRCX I iAT I CN

• 1 000 00 0 fOl

• 10 0000 0 + 0 I

• lOOOCOO+0

1

• 1 OOCOOO+Ol

• 1 OOOCOO+Ol

P PC -OLE A •350OCO0+O2

*4bOCf 00+02

•5500000+02

*6500000+02

• 7500000 + 02

13
HP4 : GENERAL OATA FITTING RRODL E'A 13 5 6

. 1 600000 + OP . 1400CO0 1-02 . 1 OOCOOO+0

1

.1 000000+C

1

. 000 09 0 0

1 <.1960000+03 . 1 000000+0

1

.2600000+0? . 19 00000+02. . 1 OOOC 00 + 0

1

• 1 000000 + 01 .0000000

•36 1 0000+03 . 1 000000+01

. 1700000+02 • 1400000 + 02 . 1000000+01 . 1 000000+01 .0000000

• 1960000+03 . 1 000000+01

.2300000+02 • 1600000 + 02 . 1 000000+0

1

. 1 COOOOO+01 .0000000

.2560000+03 .’1000000+01

.2300000+02 .9 000000 + 0

1

. 1 oooooo+bi .1000000+01 .0000000

•6100000+02 • 1 000000 + 0

1

.2900000+02 . 1900000+02 • 1 0000 00+9 1 .1006000+01 . 9000000

.36 1 0000+03 . 1 000000+01

.2400000+0? . 1 700000+02 • lCOfiCOO + 0 1 • 1 000000 + 0 1 .OOOOOOO

.26^0000+03 . 1000000+01

. 27 ) 0000 + 02 . 1400000 +02 • 1 0900 00 + 0

1

• 1 0000(» 0+0 1 • OOC 0000

. 136 0 00 0+0 ji . 1 000 000 + 0

1

* 2300000 + 02 . 1 000000+02 . 1 00<'009 + 01 .1000000+01 .0000000

• 10 00000+05 . 1 000000 + 01

. 5500000 + 02 .19 CO 00 0 +02 . 1 0 006 06+0 1 .1000000+01 • 00 9 coo 0

.3610000+03 . 1000000+01

. 2500000+0? . I 700000+0? . 1 000 0 00 + 0

1

• 1 000090 + 01 • 0 0 0 0 0 C 0

.2890000+03 . 1000000+01

. 23000CO + 0? . 1200000+02 • 1 00060 0 + 0

1

.1000000+01 .9090000

. 1440000 + 03 . 1 0 00 00 0 + 0

1

.2300000+0? . 1 1 00000+02 . 1 0 jr 0 0 0 + 0

1

• 1 OOOCOO + 01 • 000090 0

.1210000+03 . iOOuOOO+01

.38 00000 + 0? . 19 CO 00 0+0? • 1 06 00 00 + 0 1 .1000000+01 .0000000

.3610000+03 . 1 000000+01

.2500000+02 .2100000+02 •1000000+01 . 1000000+0

1

. 9 0 0 0 0

• 4 4 10 000+03 . 1 0 00 UC 0 +0 1

. 3700000+0? . 23 00 000 +02 • 1 OOOCOO + 0 1 • 1 oooogc + 0

1

.009 0 0 (. 9

.529 0000+03 . 10 00000 + 0

1

. 3000000+02 . 1 2 0000 0 + 02 • 1 OOCOOO + 0

1

. 1 000000+0

1

. 9 9 0 0000

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apprgximati

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•5074080+03

•4O20000+05

• 1624890+04

.7098000+04

.3701 372+03

.2443520+04

• 36 1 3 0 79 + 03

.1334624+05

.3823035+04

.3447360+04

• 1 704708+04

.3256220+04

.6961598+03

.561 7630+04

.11 79324+04

.2913130+04

.4803 1 79+03

. 7444980 + 04

.1062291+04

.4397870+04

•6436979+03

.1207272+05

.1458258+04

.4924660+04

.9413230+02

-.3221 600+01

-.2227050+02

-.3598900+01

- .3392570+02

-.6356700+01

-.4364920+02

-.9384300+01

-. 1 168837+03

-•2859780+02

-. 1857210 + 02

-.5779 200+01

. 1 195900+01

. 1346000+01

.6166770+02

• 1890877+03

.2827160+02

• 9167970 + 02

.1396590+02

. 7954630+02

.2416450+02

• 15 15549+03

• 1 105230 + 02

• 1 755064+03

.2476150+02

.4892 1 38+03

. 1512600+02

. 3562247+03

. 3442890+02

.9183255+03

.2275300+01

.5749803+03

.3340444+03

.5083639+03

.6547047+03

.1753156+04

• 2785(^ 14 + 03

-. 1 793330+02

-.9244161+03

-.4237870+03

-.2092694+03

-.3620637+03

-.1654627+03

-.3706247+03

-.2 26 3651 + 03

-.5151772+03

-.33999 10+02

-.83 51 I 16 + 03

-. 12721f.l +04

-.1 636763+04

-.4382894+04

-.6964045+03

.4483340+02

.231 1069+04

• 1059481+04

.5231 8^38+ 03

.9051 785+03

. 4136720 + 0.3 .9265996+ 03

.5659378+03

. 1288003+04

. 8500390+02

.?36 1600 +04

.2825630+04

.715031 6+04

. 1047034+04

- .5874920 +02

-.3028363+04

136831 4+04

-.6355575 + 0

-.1186104+04

-.5420427+03

-.12141 34+04

-.741 5507+03

-. 1 68 7669+ 04

-.111 3756 + 03

107
6. SUMMARY OF PROBLEM SOLUTIONS

IDcNTIFICATIOM HPl a

SOURCE: I. Barrodale and F.D.K. Roberts,

Discrete Linear Approximation,

(1973), ppT 843-844.

"An Improved Algorithm for

" SIAIl J. Numer. Anal., 10

CHARACTERISTICS

# parameters 2 # observations 5 rank full

other ^ .

SOLUTION

objective function value 2.0

active constraints 1,3

solution vector 0.5, 0.5

condition number 7_

alternate solution vector yes

active constraints

solution vector .75, .2 5

condition number ^

108
IDENTIFICATION HP2

SOURCE ; R. R. Singleton, "A Method for Minimizing the Sum of Absolute

Values of Deviations," Ann. Math. Stat. , 11 (1940), p. 308.

CHARACTERISTICS

# parameters 3 # observations 17 rank full

other

SOLUTION

objective function value 10939/42

active constraints 2,10,16

solution vector 197/6, -29/14, -37/21

condition number 1 X 10^

alternate solution vector no

active constraints

solution vector

condition number

109
IDENTIF [CATION HP3

SOURCE i R. H. Bartels, "CLl: A FORTRAN Subroutine for Unconstrained and

Constrained (LAR) Regression,” Report to the National Bureau

of Standards, November 1976, p. 16.

CHARACTERISTICS

// parameters 2 // observations 5 rank full

other

SOLUTION

objective function value

active constraints

solution vector 68.5, 1.3

2

condition number 2 x 10

11.0

1,5

alternate solution vector

active constraints

solution vector

condition munber

yes

1,3

63.25, 1.45

2 X 10^

no
IDENTIr’ICATIQN HP4

SOURCE : P. W. Holland, Educational Testing Service; see "Robust

Test Problems for Robust Regression Problems," Proceedings

of the Ninth Interface Symposium on Computer Science and

Statistics, April 1976, pp. 99-110.

CHARACTERISTICS

# parameters 6 // observations 135 rank full

other Repeated rows in X matrix; Some variables qualitative (-1,0,1)

SOLUTION

objective function value 284269/504

active constraints 5,20,37 ,39,68,110

-643 -647 2573 -1459 43 601

solution vector inoR* S04* SOi » inns*

3

condition number 6 x 10

alternate solution vector yes

active constraints 5 .20.37.39.68.102

-643 -647 869 -509 43 1348

solution vector 1008’ 504’ 168’ 168’ 1008’ 63

condition number 7 3

X 10
IDENTIFICATION HP5

SOURCE ; U.S. Department of Labor, workload scheduling problem, 1967.

CHARACTERISTICS

# parameters 8 // observations 8

other Does not contain a column of I's

rank full

SOLUTION

objective function value

active constraints 1,2, 3, 4, 5, 6, 7,

8

126.3588, -3.378714, 1.022894, 16.35045,

solution vector -45.39628. 33.13378. -34.60532. .41740 25

3 10^

condition number ^

,

alternate solution vector

active constraints

solution vector

condition number

I

112
IDENTIFICATION HP6

SOURCE: Modification of problem from I. Barrodale and A. Young,

"Algorithms for Best L- and Linear Approximations on a Discrete Set," Numer . Hath

.

, 8 (1966), p. 298.

CHARACTERISTICS

# parameters J? observations 6 rank ^“11

other

SOLUTION

objective function value 11/15

active constraints 1,4

solution vector 76/5, -151/30

condition number 2 .;r.

alternate solution vector no

active constraints

solution vector

condition number

113
IDKNTIFICATION HP7

SOURCt. ; K. Spyropoulos, E. Kiountouzis and A. Young, "Discrete Approximation in the Norm," Computer J. , 16 (1973), p. 184.

CHARACTERISTICS

# parameters ^ /i* observations 7 rank

other degeneracy

SOLUTION

objective function value 16.0

active constraints Any two distinct indices from {1,3, 4, 5, 7}

solution vector 0,1

1 2

conditior i|umber 7 x. 10 to 5 x 10 ^ .

•

alternate solution vector

active constraints

solution vector ; condition number

114
IDENTIFICATION HP8

SOURCE : I. Barrodale and F. D. K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

#1172, January 1972.

CHARACTERISTICS

# parameters i # observations 201 rank full

other

SOLUTION

objective fimction value 21. 8l^$8

active constraints 107

solution vector . 1023800

condition number 1

alternate solution vector

active constraints

solution vector

condition number

no

.

115
IDENTIFICATION HP9

SOURCE ; I. Barrodale and F. D. K. Roberts, "An Improved Algorithm for

Discrete Linear i^proximation, " University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

ji^llT2, January 1972.

CHARACTERISTICS

# parameters 2 # observations 201

other

SOLUTION

objective function value 9.271^9^

active constraints 9^ >1^9

solution vector . 3^39^7^ > -.104^933

condition number 13

rank full

alternate solution vector no

active constraints

solution vector

condition nTimber

116
IDENTIFICATION HPIO

SOURCE ; I. Barrodale and F. D. K. Roberts, "An In5>roved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Snmmary Report

#1112 , January 1972.

CHARACTERISTICS

# parameters 3 # observations 201 rank full

other

SOLUTION

objective function value 9.078925

'

active constraints 30,101,172

solution vector .39678^18, -.1625170, .01282731

condition number 5 X 10^

alternate solution vector no

active constraints

solution vector

condition number

117
IDENTIFICATION HPll

SOURCE ; I. Barrodale and F. D. K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Svimmary Report

#1172, January 1972.

CHARACTERISTICS

# parameters 4 # observations 201 rank full

other

SOLUTION

objective fvinction value 4. 6407l4

active constraints 20,70,132,182

solution vector .l64o699, .3230841, -,2kk692k, .039221^3

2

condition number 4 x 10

alternate solution vector no

active constraints [ solution vector

condition number

118
IDENTIFICxVTION HP12

SOURCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete X.- Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

y/1172, January 1972.

CHARACTERISTICS

# parameters ^ f observations rank

other

SOLUTION

objective function value 1.055658

active constraints 51, 101, 151, 188 ^

solution vector ^03041441, .8248866, -.7469379, .2235657,

condition number 4 X 10'

alternate solution vector

active constraints

solution vector

condition number

.02236884

119
IDENTIFICATION HP13

SOURCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report #1172,

January 1972.

CHARACTERISTICS

# parameters 6 # observations 201 rank full

other ;

SOLUTION

objective function value .06050032

active constraints 4,18,84,127 ,165,192

-.0003814120, 1.008495, -1.041157“

solution vector .4120115, -.07444122, .005177852

condition number 2 x 10^

alternate solution vector no

active constraints

solution vector

condition number
IDENTIF.TCATIOM HP14

SOURCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

//1172, January 1972.

CHARACTERISTICS

# parameters 7 # observations 201 rank full

other

SOLUTION

objective function value

active constraints

-.002115490, 1.024030, -1.078237, .4485926,

solution vector -.09159510. .008979690, -.0003203885

condition number 4 x 10^

.03471234

8,30,63,101,139,172,194

alternate solution vector no

active constraints

solution vector

condition number

<

121
IDENTIFICATION HP15

SOURCE : I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report #1172,

January 1972.

CHARACTERISTICS

# parameters 6 // observations IQQ rank full

other

SOLUTION

objective function value 3.877737

active constraints 19,35,46,70,82,97

1.129434, -.7975623, -1.628313

solution vector 1.316736. -.1082232. .1593164

2 condition number 4 x 10

alternate solution vector no

active constraints

solution vector

condition number

122
IDENTIFICATION HP16

SOURCE: I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

#1172, January 1972.

CHARACTERISTICS

# parameters ^ # observations rank

other

SOLUTION

objective function value >05340562

active constraints 1,4,8,20,28,35,43,47

0., 6.262394, -44.95925, 141.1803,

solution vector - 136.6328. -3.195617. -1.100961, -.1572059

condition number 1 X 10'

alternate solution vector no

active constraints

solution vector

condition number

123
IDENTIFICATION HP17

SOURCE: I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

y/1172, January 1972.

CHARACTERISTICS

y/ parameters ^ // observations rank

other degeneracy

SOLUTION

objective function value .001867521

Any six distinct indices from

active constraints f2. 7. 11. 15. 20. 22. 23}

solution vector 0.. .9995966. 0., -.1652754, 0., .00 7352267

condition n’omber 1 X 10^ to 4 X 10^

alternate solution vector

active constraints

solution vector

condition nvimber
IDENTIFICATION HP18

SOURCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of VJisconsin,

Mathematics Research Center, MRC Technical Summary Report

#1172, January 1972.

CHARACTERISTICS

# parameters 6

other degeneracy

# observations 53 rank full

SOLUTION

objective function value .004344250 ' Any six distinct indices from

active constraints {5.15.26.37.47.52.53}

solution vector 0. , .9995389, 0., -.1652942, 0., .007358741

condition number 1 x 10^ to 3 x 10^

alternate solution vector no

active constraints

solution vector

condition number

125
IDENTIFICATION HP19

SOURCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

/if 1172, January 1972.

CHARACTERISTICS

// parameters 6 # observations 103 rank full

other degeneracy

SOLUTION

objective function value

active constraints

solution vector 0.

condition number

.008568850

Any six distinct indices from

flQ, 29, 51, 73, 92,102,103}

. .9995529, 0., -.1653271, 0.,

1 X 10^ to 4 X 10^

.007366283

alternate solution vector no

active constraints

solution vector

condition number

I 126
IDENTIFICATION HP20

|,

i| SOUPvCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

i; Discrete A. Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

y/1172, January 1972.

CHARACTERISTICS

// parameters 6 // observations 203 rank

other degeneracy

SOLUTION

objective function value .01694253

active constraints

Any six distinct indices from

{18.57,101.145.184,202,203}

solution vector 0., .9995427, 0., -.1653031, 0., .007360762

condition number 1 X 10^ to 4 X 10^

alternate solution vector no

active constraints

solution vector

condition number

i

127
IDENTIFICATION HP21

SOURCE ; I. Barrodale and F.D.K. Roberts, ”An Improved Algorithm for

Discrete Linear Approximation,” University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

//1172, January 1972.

CHARACTERISTICS

// parameters 11 # observations 21 rank full

other

SOLUTION

objective function value .1057780

active constraints 1,3,4,6,7,12,15,16,18,20,21

602.3115, -781.8188, 94.48467, -463.9671, -1326.421

solution vector 1248.389, 605.0711, -612.3761, 98.38093, 82.99064,

-72:W6'E^

condition number 4 y I_

alternate solution vector

active constraints

solution vector

condition number

128
IDENTIFICATION HP22

SOURCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

//1172, January 1972.

CHARACTERISTICS

// parameters 11 observations 51 rank full

other .

SOLUTION

objective function value .2667665

active constraints 1,2,5,12,13,17,23,30,39,45,50

49.79747, -117.3993, 328.0159, -425.9049, -627.8460

solution vector 642.0325. 213.0777. -283.7580, 76.16677 , 35.84595,

-38.21172

g condition number 5 x 10 "

alternate solution vector no

active constraints

solution vector

condition number

129
IDENTIFICATION HP23

SOURCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

//1172, January 1972.

CHARACTERISTICS

P parameters H // observations 101 rank full

other

SOLUTION

objective function value .5450632

active constraints 2,11.16.28,33,44,60,71,77,89,101

178.6954, -242.1546, 114.2572, -254.2264, -525.6558

solution vector 527.5312. 207.5384. -247.8918, 59.526 77, 32.13306,

-33.36465 . condition number 2 x 10°

alternate solution vector no

active constraints

solution vector

condition number

130
il)H:NTIFTCATIO>f HP24

SOURCE ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

y/1172, January 1972.

CHARACTERISTICS

// parameters 11 # observations 201 rank full

other

SOLUTION

objective function value 1.129209

active constraints 3,9,17,24,30,53,117,155,160,181,198

403.3163, -277.9515, -269.0950, -210.9572,

solution vector 497.1644. 102.2011. -234.4954. 88.3365 3, 29

-36.77963 ^ condition number 4 x 10

alternate solution vector no

active constraints

solution vector

condition number

286.9815

47814,

131
TDENTIFICATION HP25

SOCRCS ; I. Barrodale and F.D.K. Roberts, "An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

y/1172, January 1972.

CHARACTERISTICS

// parameters 5 // observations 15 rank 3

other Does not contain a column of I's

SOLUTION

objective function value 1677/49

active constraints

solution vector 25/49, 10/49, 4/49, 0, 0

condition number ^

alternate solution vector

active constraints 4,7,11

solutior* vector 29/49, 14/49, 0, 0, -4/49

condition number 3

132
,T iJKNTI FICATION HP26

I* Barrodale and F.D.K. Roberts, *'An Improved Algorithm for

Discrete Linear Approximation," University of Wisconsin,

Mathematics Research Center, MRC Technical Summary Report

//1172, January 1972.

CHARACTERISTICS

parameters 5 observations 51 rank full

other

SOLUTION

objective function value 16.73596

active constraints 4,13,26,38,51

.2884646, -5.961055, 41.81349, -82.04005,

solution vector 5 4.89915 ^ 3 condition number 1 x 10 -

alternate solution vector no

active constraints

solution vector

condition number

133
HP 2 7

^OLIRO^: Treasury Department, spline-fit ting problem, 1976.

CH/.EACTERISTICS

# parameters ^ observations 72 rank full

other

SOLUTION

objective function value 11948.75

active constraints 11,34,52,61,66,71

5.399219, 6.926597, 7.180670, 7.532025,

solution vector . .7.-n.aZQ3.,^a..3^aZQ3

condition number

altertiate solution vector jiu

active constraints

solution vector

condition number

134
7. REFERENCES

[1] Abdelmalek, N. N. , "An Efficient Method for the Discrete Linear Lj

Approximation Problem," Mathematics of Computation, Vol. 29 (1975), 844-

850.

[2] Armstrong, R. D. and Frome, E. L. , "A Comparison of Two Algorithms for

Absolute Deviation Curve Fitting," Journal of the American Statistical

Association, Vol. 71, No. 354 (1976), 328-330.

[3] Barrodale, I., "Lj^ Approximation and the Analysis of Data," Applied Statistics, Vol. 17 (1968), 51-57.

[4] Barrodale, I. and Roberts, F. D. K. , "An Improved Algorithm for Discrete

il Linear Approximation," SIAM Journal on Numerical Analysis, Vol. 10,

No. 5 (1973), 839-848.

[5] Barrodale, I. and Roberts, F. D. K. , "Solution of an Overdetermined System of Equations in the Aj Norm," Communications of the Association for

Computing Machinery, Vol. 17 (1974), 319-320.

[6] Bartels, R. H. , Conn, A. R. , and Sinclair, J. W. , "Minimization Techniques for Piecewise Differentiable Functions: The Solution to an Overdetermined Linear Systen," SIAM Journal on Numerical Analysis, Vol.

15, No. 2 (1978), 224-241.

[7] Dongarra, J. J. , Moler, C. B. , Bunch, J. R. , and Stewart, G. W.

,

"LINPACK Users' Guide," Society for Industrial and Applied Mathematics,

Philadelphia, 1979.

[8] Domlch, P. , Lawrence, J. , and Shier, D. , "Generators for Discrete Polynomial L]^ Approximation Problems," Journal of Research, National . Bureau

of Standards, Vol. 84, No. 6 (1979), to appear.

[9] Draper, N. R. and Smith, H. , Applied Regression Analysis , John Wiley & Sons, New York, 1966.

[10] Gentle, J. E. , "Least Absolute Values Estimation: An Introduction,"

Communications in Statistics, Vol. B6, No. 4 (1977), 313-328.

[11] Gentle, J. E. . , Kennedy, W. J. , and Sposito, V. A., "On Least Absolute

Values Estimation," Communications in Statistics, Vol. A6, No. 9 (1977),

839-845.

[12] Gilsinn, J. , Hoffman, K. , Jackson, R. H. F. , Leyendecker, E. , Saunders,

P. , and Shier, D. , "Methodology and Analysis for Comparing Discrete

Linear Lj^ Approximation Codes," Communications in Statistics, Vol. B6,

No. 4 (1977), 399-413.

[13] Hampel, F. R. , "A General Qualitative Definition of Robustness," Annals

of Mathematical Statistics, Vol. 42 (1971), 1887-1896.

135
/

[14] Harter, H. L. , ''Nonuniqueness of Least Absolute Values Regression,"

Communications in Statstics, Vol. A6, No. 9 (1977), 829-383.

[15] Hillier, F. S. and Lieberman, G. J. , Introduction to Operations

Research , Second Edition, Holde-Day, San Francisco, 1974.

[16] Hoffman, K. and Shier, D. , "A Test Problem Generator for Discrete Linear

Lj Approximation Problems," Transactions on Mathematical Software, to

appear.

[17] Hollard, P. W. , "Robust Test Problems for Robust Regression Programs,"

Proceedings of the Ninth Interface Symposium on Computer Science and

Statistics, D. C. Hoaglln and R. E. Welsch (eds.), April 1976, 99-^105.

[18] Johnston, J. , Econometric Methods, Second Edition, McGraw-Hill, New

York, 1972.

[19] Kennedy, W. J., Gentle, J. E. , and Sposito, V. A., "A Computer Oriented

Method for Generating Test Problems for Lj Regression," Communications

in Statistics, Vol. B6, No. 1 (1977), 21-27.

J [20] Ortega, J. M. , Numerical Analysis, A Second Course, Academic Press, New

York, 1972.

[21] Rao, C. R. , Linear Statistical Inference and its Applications , John

Wiley & Sons, New York, 1965.

[22] Rice, J. R. and White, J. S. , "Norms for Smoothing and Estimation," SIAM

Review, Vol. 6, No. 3 (1964), 243-256.

[23] Rosenberg, B. and Carlson, D. , "A Simple Approximation of the Sampling

Distribution of Least Absolute Residuals Regression Estimates," Communications in Statistics, Vol. B6, No. 4 (1977), 421-437.

[24] Schlossmacher, E. J. , "An Iterative Technique for Absolute Deviations

Curve Fitting," Journal of .the American Statistical Association, Vol.

68, No. 344 (1973), 857-859.

[25] Spyropoulos, K. , Kiountouzis, E. , and Young, A., "Discrete Approximation

in the L^ Norm," Computer Journal, Vol. 16, No. 2 (1973), 180-186.

[26] Tukey, J. W. , "The Future of Data Analysis," Annals of Mathematical Statistics, Vol. 33 (1962), 1-67.

136
NBS>114A (REV. »-7S)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

4. TITLE AND SUBTITLE

A Collection of Test Problems for Discrete Linear

Data Fitting

7. AUTHOR(S)

D. R. Shier, S. J. Neupauer, P. B. Saunders

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

5. Publication Date

A . 8. Performing Organ. Report No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS fSfreef, City, state, ziP)

Operations Research Division

Center for Applied Mathematics

National Bureau of Standards

Washington, DC 20234

13. Type of Report & Period Covered NBSIR

15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200^word or Ic33 factual summary of mo3t aignificant information. If document indudes a significant bibliography or

literature eurvey, mention it here.) This document assembles 27 test problems representing a variety of examples in

which least absolute deviation (or L ) data fitting has been used. The problems

were collected from the literature, from the authors of several approximation

‘ codes, and from examples encountered in practice. Optimal solutions to

these problems (objective function value and solution vector; have been obtained

using a double-precision computer code designed for checking the Kuhn-Tucker

conditions and for performing an accurate reinversion of the optimal basis.

Special problem characteristics such as alternative optima, degeneracy, and

rank loss are also noted. -This set of test problems has proven useful in

evaluating and improving the performance of codes as well as in suggesting

types of problem structures that might be mimicked by problem generators.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first tetter of the first key word unless a proper name;

separated by aemicolona)

Algorithm testing; approximation; curve fitting; least absolute deviation;

polynomial approximation; spline fitting; test data; test problems.

18. AVAILABILITY ®Unlimited

I I For Official Distribution. Do Not Release to NTIS

I I Order From Sup. of Doc., U.S. Government Printing Office, Washington, DC

20402, SD Stock No. SN003-003-

[ I Order From National Technical Information Service (NTIS), Springfield, VA. 22161

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

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