Analysis and Design of Simulation Experiments with Linear Approximation Models

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

ANALYSIS

AND DESIGN OF SIMULATION

BCPEKDEIWS WITH

LINEAR APPROXIMATION M O D E S

V. Federov A Korostelev*

S. Leonov*

October.

1984

W - 8 4 - 7 4

All-Union Institute

of

Systems Studies, Moscow. USSR

Working %pers a r e interim reports on work of t h e International

Institute for Applied Systems Analysis and have received only limited review. Views o r opinions expressed herein do not necessarily represent those of the Institute or of i t s National Member Organizations.

II+I''ITRNATIONAL I N S I ? m FOR APPLIED SYSTEMS ANALYSIS

2361 Laxenburg. 'Austria

Understanding t h e n a t u r e and dimensions of t h e world food problem and t h e policies available to alleviate it has been the focal point of the IIASA Food and Agriculture Program since it began in 1977.

National food systems a r e highly interdependent, and yet t h e major policy options exist a t t h e national level. Therefore, t o explore these options, i t is necessary both to develop policy models for national economies and to link t h e m together by trade a n d capital transfers. For greater realism t h e models in this scheme a r e kept descriptive, r a t h e r than normative.

Over t h e y e a r s models of some twenty countries, which together account for nearly 80 percent of important agricultural attributes such a s area, production, population, exports, imports and so on, have been linked together to constitute what we call t h e basic linked system

(BLS)

of national models.

These models represent large and complex systems. Understanding t h e key interrelationships among t h e variables in such systems is not always easy. Communication of results also becomes difficult. To over- come t h i s problem, one may consider approximating these "primary models" by more transparent "secondary models".

In t h i s paper Valeri Federov, k Korostelev a n d S. Leonov describe the package of programs for t h e design and analysis of simulation experi- ments with such secondary models. The package was prepared in the All- Union Institute of Systems Studies in Moscow. It is one of the first attempts in t h i s field, a n d we hope t h a t more experience, comments and critiques will help t o improve and extend t h e package in a useful and practical way.

Kirit S. Parikh Program Leader Food and Agriculture Program.

I

am very grateful t o Lucy Tomsits for editing a n d typing t h e paper a n d t o Valerie Khaborov for his help in installing t h e software

at

IIASA.

There is a necessity i n a n u m b e r of IIASA's r e s e a r c h e s t o deal with analyzing t h e properties of t h e computerized versions of complex models. The use of simulation experiments is one of t h e m o s t successful tools in solving this problem. In this paper, t h e package of programs for t h e t h e design a n d analysis of simulation experiments is described. The package was prepared in t h e All-Union Institute of Systems Studies in Moscow.

I t

is one of t h e first a t t e m p t s in this field, and t h e a u t h o r s did not expect t o have c o n s t r u c t e d a very comprehensive variant, but hope t h a t m o r e experience, r e m a r k s a n d critiques will help t o improve and extend t h e package i n a m o s t useful and practical way.

-

^vii

-

CONTENTS

1. Introduction

2. Structure of the Interactive System 3. Construction of Experimental Design 4. Experimental Analysis

5. Stepwise Regression with Permutations 6. Example of System Utilization

References

ANAL= AND DESIGN OF SIMULATION EXPERIMENTS WlTH LINEAR -TION MOD=

by

V.

Fedorov,

A.

Korostelev* a n d S. Leonov*

The construction and computer realization of mathematical models of the natural a n d social phenomena is nowadays one of the stable ten- dencies of systems analysis. Sometimes those models a r e so compli- cated t h a t they look like "black boxes" even for their authors. That is why methods for the investigation of s u c h models a r e extremely interesting. The ideas a n d methods of t h e simulation experiment a r e rather old (Naylor, 1971). Some aspects of design and analysis of simula- tion experiment were described by Fedorov (1983). and we shall follow the concepts of this paper. The main object of o u r study is a computer realization of

a

model calledprimary model which is described below.

The aim of this paper is to describe t h e general s t r u c t u r e of the interactive system for design and analysis of simulation experiment, and

to show its potentialities.

2.

SI'RUCTURE OF THE

INTEXACTlV'E SYSlTM

The current version of the system contains two main programs intended for construction of experimental design and data analysis.

These programs are independent of each other and a r e linked only through input-output files of data.

I t

is necessary to point out that the treatment of any specific model requires an exchange module. This module makes i t possible to repeatedly call the primary model varying input data. The principle scheme of interactive system may be illus- trated by Figure 1.

The comparatively simple approximation function, methods of optimal design, construction and statistical methods of data analysis.

were deliberatedly used in the system. The choice of these simple mathematical tools can be explained as an attempt to balance between the reliability of a s e c o n d m y m o d e l ; its simplicity and lucidity taking into account t h e reasonability of the calculation volume. The following sections show the potentialities of programs and are illustrated by test examples.

I t

is necessary to underline that some potentialities not fore- seen in the system may be assigned to t h e exchange module.

3. CONSl'RUCTION OF

m A L

DESIGN

While investigating the primary model it is assumed t h a t input vari- ables z (factors, independent variables) are separated into groups a t the heuristic level according to: Firstly. prior information on their nature;

and Secondly. the expected degree of their influence on dependent

KEY:

d o t t e d l i n e s s h o w p o s s i b l e " f e e d b a c k s "

-

FORMULATION OF

SIMULATION

- - - - -

EXPERIMENT

HEURISTIC ANALYSIS OF RESULTS

TARGETS AND STRUCTURE OF SECONDARY MODELS

A

INPUT

T

OUTPUT

PROGRAMS OF

DESIGN CONSTRUCTION ^t

- - - - -

CONSTRUCTION OF

D e s i g n

C h a r a c t e r i s t i c s

)"

Y S

X

EXCHANGE MODULE

X

A

Y

v

PRIMARY MODEL

variables

Y.

The factors a r e usually separated into t h e following groups:

a) Scenario and exogenous variables;

b) Parameters of t h e model of which values a r e obtained on t h e stage of identification (usually t h e y have r a t h e r large intervals of uncertainty);

c ) Variables h o w n with "small" errors, which can often be con- sidered a s random ones.

The program for t h e construction of experimental design can gen- e r a t e designs of different types for variables from a f f e r e n t groups. In t h e c u r r e n t version, the following types of designs a r e available t o gen- e r a t e

-

Orthogonal design

-

(i) two-level design

X = 1% 1;

where

T j =

*l.i

=

1.N, j=rm.i is a number of an observation, j is a number of a variable,

X

is Hadarnard matrix, i.e.,

XTx =

NIN, where

IN

is identity matrix,

N =

4k ,k i s a integer number;

(ii) three-level design

X = [&-j.Xlj =

-1,0.+1;

X

is conference matrix, i.e.

XrX =

(N-l)IN; N

=

4k +2.

It is recommended to use orthogonal design for group (a), if a detail investigation for t h e factors from (a) is required;

-

Random design with two- and multilevel independent variation of factors. Usually i t is used for t h e factors from group ( b ).

-

Random design for simultaneous variation of all factors of t h e group.

I t

may be applied for block analysis.

-

Random design with continuous law of distribution: uniform a n d normal.

I t

may be applied for those factors which a r e known u p t o small random error.

The criterion for design construction is t h e correlation coefficients of column vectors of X-matrix: t h e columns m u s t have correlation which is a s small a s possible. The design may also be generated (for some groups) in a purely random mannner, without examination of correlations.

There exists a vast literature on the methods of constructing orthogonal design. One of t h e simplest approach based on t h e Paley con- cept (see Hall, 1967) is used here.

Conference matrices (C-matrices)

Let G F ( ~ ) be a k i t e Galois field of cardinality q , q = p r , where

p

is a prime odd n u m b e r . Let ~ ( z ) be a c h a r a c t e r defined on G F ( ~ ):

R ( z )

= rl

1

¹

if t h e r e exists ~ E G F ^{hu:: does not} s u c h t h a t f = z ,

If a,

=

O , a l

...,

aq-I are t h e e l e m e n t s of GF(q) t h e n a m a t r i x Q

=

IR(ai-a,)] i s called Jacobsthal m a t r i x a n d satisfies t h e equation

QJ

=

JQ

=

0 where J f i

=

1 for all i , k

= -

1.q. Let

and

Then Cq is C-matrix of order q + l .

Hadamard matrices (H-matrices)

Hadarnard matrices are constructed on the basis of the concept of Kronecker matrix product.

If

q

=

1 (mod 4), then

is H-matrix of order 272, n

=

q + l . Further, if

4

and Hm a r e H-matrices of orders n and m . t h e n %@Hm is H-matrix of order n m .

It m u s t also be noted t h a t if q = p T

=

-1 (mod 4), t h e n Hq+l

=

c q + ~

+ Iq+l -

H-matrix of order q + l . With the use of the methods described, the program

PLAN

constructs:

-

Hadamard H-matrices for all n s 1 1 2 , n is 4-tuple, except n =92;

-

Conference C-matrices of the following orders m ( m = 2 (mod 4)): 6, 14, 18, 26, 30, 38, 42, 54, 62, 74, 90, 98, 102

I t

is clear t h a t t h e above methods allow the construction of saturated (number of observation equals to number of factors plus one) orthogonal designs, which a r e optimal for the majority of statistical cri- teria, only for t h e above enumerated dimensions of the factor space.

Therefore, for intermediate dimensions, the orthogonal design for t h e nearest larger dimension has to be used.

I t

will also be orthogonal (but not saturated) in these cases.

There a r e two variants of the application of generated design

(i) X-matrix is written (row by row) into t h e auxiliary file (HELP.DA7') for application in the exchange module and further analysis of simulation experiment.

(ii) The levels of factors may be s e t in the real scale: in t h a t case, mean values and scale of variation a r e chosen by t h e user. The design in t h e real scale a r e obtained with the help of t h e evi- dent formula

h e r e the j-th factors belongs to the chosen group k ; vk is t h e scale of variation for group k ;

5

stands for their value of t h e j-th factors.

Matrix FN

=

tFN=j] is stored (row by row) into the file

HELP. DAT.

4. EXPWIMENTAL ANALYSIS

The aim of t h e simulation of analysis is t h e construction of secon- dary model of t h e following form:

where

y

is a response (dependent variable); 60,

.

.

.

, T ? ~ a r e pararne- t e r s to be estimated (regression coefficients); f f

*...,

^{f k} a r e lmown functions depending on z-vector of input variables.

Since k is usually r a t h e r large, one of the main problems of experi- mental analysis is t h e screening of significant factors. Following is t h e

statement of t h e problem: input data is set

XI, X,,. _.... ..XI, Xzl XZz..

....

.X2, ...

x , , , , x

...

, x

where

N

is number of an observation, rn is number of variables. One variable is taken as a response a n d is denoted by

y

(sometimes

y

is not a variable itself, it could be some transformation

--

the set of the most usable transformations a r e provided by the program). Then k functions

1 12,...,

fk. depending on the r e s t of variables, a r e chosen (mainly heu- ristically) and can be constructed with the help of t h e above-mentioned transformations of zl,z2,

. .

^, .z,. That is the final step in the formulation of model (1); screening experiments can be carried out now.

Here we shall e n u m e r a t e t h e possibilities of t h e program for the analysis of results provided by simulation experiments.

1) Input variables can be separated into groups with the help of identification vector; variables from one group only are analyzed simultaneously, but identification vector may be changed, a n d t h e groups can be rearranged easily.

2)

It

is possible t o make transformations of factors, include their interaction and take any variable as a response.

3) The program provides the stepwise regression procedure; fac- tors may be included into regression or deleted from t h e equa- tion (Efroymson, 1962). Technically. this program for screen- ing significant factors is based on the subroutines from

SSP

package. 1970; some modifications of these subroutines a r e being carried out for t h e implementation of interactive regime and Efroymson procedure. Interpretation of input a n d output information in this module will cause no difficulties for a user familiar with t h e SSP package.

4) A u s e r may obtain both statistics analogous to SSP subroutines and some additional information, for instance, correlation matrix of regression coefficients, a n d detailed analysis of resi- duals.

5) If a secondary model is used for interpolation or extrapolation, values of input variables (predictors) a r e being chosen by a user. The standard errors of t h e prognoses a r e calculated.

6) A heuristic method of random permutations for testing significance of entered variables is provided in t h e program. Its description is given in section 5.

Program for experimental analysis utilizes 3 files: SYSIN.DAT and SYSOUT.DAT for input a n d output information respectively, and an auxili- a r y file SYSS7,DAT for intermediate information.

5.

SllPWEE

REGRESSION

WlTH PEFMJTATIONS

I t is well-known (see for instance, Pope & Webster, 1972; Draper, Guttman, and Lapczak. 1979) t h a t the application of standard statistical criteria (F-test, for example) for testing significance of entered variables in t h e stepwise regression procedure is not correct by its nature. That is why a heuristic method of random permutations is used in t h e interac-

tive system for testing significance of entered variables. Such an approach enables one to avoid complicated analytical methods t h a t are necessary for calculating statistic of criterion. It must also be under- lined t h a t this method does not require the assumptions concerning the distribution of variables. Therefore it may be rather useful in practice (Devyatkina e t al., 1981).

Method of random permutations is based on the following concept:

two models a r e compared based model y

= g

( z ) and a model

5 =

i ( z )

where response function G(z) is constructed according to permuted values of response:

yil,y

..., y . , here i l , i 2 ,..., iN is a random permutation

22' k

of indexes 1.2,

..., N. If

the first (basic) model gives a n adequate approxi- mation of the primary model, then for example, residual sum of squares for the 1st model will be significantly less t h a n for the 2nd model. Such a comparison of statistics usually applied in stepwise procedure for test- ing adequacy of secondary model, underlies t h e method of random per- mutations.

Now we give a short description of the screening algorithm with per- mutations.

(1) 1st Step. The most significant variable is entered into regression

--

Xw. Student's T-statistic ( T o ) F-statistic (Fo) and

SS-

statistic (percentage of variance explained on this step, SSO) a r e com- puted.

Random p e r m u t a t i o n is carried out for all rows of X-matrix except t h e e l e m e n t s f r o m c o l u m n

M ,

corresponding t o t h e response function

y :

l e t i l ,..., iN be a r a n d o m p e r m u t a t i o n of indexes 1 ,..., N. For every 1-th permutation, (1

=

T L ) stepwise procedure is c a r r i e d out, t h e most significant variable is e n t e r e d into regression a n d corresponding values of TL

-,&

- a n d S q - s t a t i s t i c s a r e computed.

( 2 ) j t h Step, j

>

1. j t h variable, X

N.5'

i s e n t e r e d i n t o regression;

To,FO a n d SSo-statistics a r e computed for t h e e n t e r e d variable.

Random p e r m u t a t i o n is carried out for all rows of X-matrix except t h e e l e m e n t s f r o m c o l u m n s NVl.NVZ,...,Wj-l,M (Totally L- permutations should be done). After every permutation stepwise procedure is being carried out, variables XNV,.XNV, ,..., Xw,-, a r e being forced into regression.

q , F L

,Sq -

s t a t i s t i c s a r e c o m p u t e d a t every 1 t h p e r m u t a t i o n for t h e vari- able e n t e r e d i n t o regression on t h e j t h step.

ARer j t h s t e p G 1 1 ) t h e following information is given:

-

index of e n t e r e d variable, N 5 ;

-

value of To;

-

m e a n a n d s t a n d a r d deviation of q - s t a t i s t i c s , 1

=

- 1.L, minimal a n d m a x i m a l values of T-statistic a f t e r permutations; a histo- g r a m f o r T-statistics (after permutations); p e r c e n t a g e of those

q ,

forwhich1

qI IT,^.

Analogous information is given for

F-

a n d SS-statistics.

If

t h e n u l l hypothesis

H,:

"response function y (x) is independent of Xq" i s not satisfied, i t s e e m s n a t u r a l t o expect t h a t To-value (T-statistic

for basic model) is g r e a t e r (in absolute value) t h a n t h e "significant"

majority of Ti-values (analogously for

F-

a n d SS-statistics). A rule for testing null hypothesis can be formulated a s follows: null hypothesis H ' is rejected with significance level

if Fo-statistic is g r e a t e r t h a n ( ~ i + l ) - values of 4 - s t a t i s t i c s after per- mutations ( t h e s a m e for

T-

and S - s t a t i s t i c s ) .

Let's assume t h a t t h e r e is a model with 30 input variables, and we s u s p e c t t h a t only t h e first 7 variables hzve g r e a t influence on t h e output variable

y;

t h e next 8 variables may be significant. It is also known t h a t variables 16-23 can take values on t h r e e levels -1,0,+1; t h e remaining 7 variables may be continuous and will be t r e a t e d as a r a n d o m n o i s e in t h e model.

A priori information concerning input variables in t h e primary m o d e l often looks like t h e one above. Experimental design will be chosen on t h e basis of this information.

The aim of t h e experiment is to construct t h e s e c o n d a y m o d e l with a few significant variables. In t h e model under consideration we will t r y t o approximate t h e primary m o d e l by t h e model with 5-6 variables.

Now let us assume t h a t t h e t r u e model in t h e "black box" has the

following form:

y

=

5X1

+

6X2

+

7x3

+

BX.4

+

9X5

+

+ lox6 + x8 +

2xQ

+

3x1,

+

4x1,

-

- x1x, -

XlX1,/ 2

-

X1Xl3/ 3 +

+ ^RANDOM

NOISE variables.

The system's potentialities will be demonstrated with t h e help of some simple examples using this model.

I t

should be pointed out t h a t these illustrative examples cannot comprehend all features of t h e system.

More detailed information on t h e m a r e contained in SYS INSTRUCTION which a r e available from t h e IlASA computer c e n t e r .

REFERENCES

Devyatkina, G.N., and A.T. Tereokhin (1981) Statistical Hypothesis Tests in Stepwise Regression Analysis Based on t h e Permutation Method, in Linear and N o n l i n e a r A r . m m e t e r i z a t i o n in k k p e r i m e n f a l Design R o b l e m s . P r o b l e m in Q b s r n e t i c s ,

V.

Fedorov a n d V. Nalimov (eds.). pp. 111-121. Moscow (in Russian).

Draper, N.R.,

I.

Guttrnan, and

L

Lapczak (1979) Actual Rejection Levels in

a

Certain Stepwise Test. C o m m u n . S t a t i s t .

- mar.

Metho., AB(2):99-

105.

Efroyrnson.

M.A.

(1962) Multiple Regression Analysis, in Mathematical Methods f o r Dqital C o m p u t e r s . A Ralston,

H .

Wilf. (eds.). New York:

Wiley.

Fedorov, V. (1983) Analysis and Design of Simulation Experiments for the Approximation of Models. WP-83-71. Laxenburg, Austria: Interna- tional Institute for Applied Systems Analysis.

Hall,

M.

(1967) Combinatorics. New York.

Naylor,

T.H.

(1971) Computer Simulation Experiments with Models of Economic Systems. New York.

Pope,

P.T.,

and

J.T.

Webster (1972) The Use of an F-Statistics in Stepwise Regression Procedures. Technornetrics 14:326-340.

System/360 Scientific Subroutine Package, Version 3, Programmer's Manual. New York: IBM, Technical Publications Department. 1970.

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...

A NUMIER Or OBSERVATIONS 12 TAULI OF VAHIAILES

Y O W L VARIABLE 1 HEAL VAHlAILE POHMAL VAHIABLE 2 REAL VARIABLE P O W VAHIABLE 3 HEAL VAHIAILB FORMAL VAHIABLE 4 HEAL VAHIAILE P O W L VAHIABLE 1 HEAL VAHIAULE Y O H M L VAHIAILE 8 HEAL VARIAILE Y O H U L VAHIAILE 7 HEAL VAHIAILE P O W L VAHlAILE 8 HEAL VARIABLE

www... STEpwISE PHOCEUUHE

...*...

VARIABLE MEAN STANDARD

NO. DEVlATlON

1 0.18667 1.02888

2 0 . 1.04447

3 0.68687 0.77850

4 0 . 1.04447

5 0 . 1.04447

8 -0.33333 0.88473

7 -0.33333 0.88473

8 0 . 1.04447

9 2.16887 22.90681

NUMIER OP SELECTION 1 COOES

U O O O O O O O

--wwwwwww-... STEP 1

...

VAHIAILE ENTEHEU . . . 4

SUM OF SQUARES REDUCED IN THIS STEP..

..

3050.704 PRUPOHTlON REDUCED IN THIS STEP

...

0 . 6 2 8 CUMULATIVE SUM OF SQUARES REDUCED

...

3050.704

CUMULATIVE PPOPOHTION REDUCED..

...

0 . 6 2 9 OP 5771.334 MULTIPLE CORRELATION COEPYICIENT... 0 . 7 2 7

P-VALUE Y O H A N A L Y S I S O F V A R I A N C E . . . 11.213 STANUAHD ERROH OY ESTIMATE...

..

18.484

V ~ ~ I W L E REG. COEPP. EHHOH T-VALUE

4 15.84444 4.78150 3.346

INTEHCEPT 2.16687

---

...

^{STEP 2}

...

VAHIAILE ENTERED . . . 5

SUM OF SQUAHES HEDUCED IN THIS STEP.... 715.593 PROPOHTION REDUCED IN THIS STEP

...

0 . 1 2 4 C U M U L A T I V E sum or s a u u r s ~auuceo... 3 1 8 6 . 2 ~ 1

CUMULATIVE PPOPOHTION HEUUCED

...

0 . 8 6 3 OF 1771.334 MULT1Pl.E COHHELATION COEFFICIENT... 0 . 8 0 8

Y-VALUE POR ANALYSIS OP VARIANCE... 8 . 4 5 3 STANDAHD EHROH OF ESTIMATE... 1 4 . 8 2 8

VAH I ABLE REG. COEPP. ERROR T-VALUE

4 15.94444 4.30U73 3 . 7UO

5 7.72222 4.30873 1 . 7 8 2

INTEHCEW 2.18867

...

* * * * * FNII ***** ^STFPYISF wnrFnlluC: ***** ^PHI) *****

Forpal and rmal numberr of varlablmr of tits 2nd block dlffer from each other.

Formal varlabler 4 and 1. 1.0.. real varlablmr 11 and 12, mxplaln 85.3% of varlance. Tliat'r why varlablmr 1 - 12 have to be analyzed jolntly.

IDENTIPICATION VECTOR :

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 0

BLOCK 7 1

DEPENDENT VARIABLE 7 3 1

WITH INTERACTION 3 ( YES

-

^{1, NO}

-

^{0 )}

0

* * * m * * * * * * * * * * * * m * * * * * * m m * * * * * * m m * * * * * e * * * * * * * * * * * * * a * * * * * * a A NUMBER OY OBSERVATIONS 1 2

T W L E OP VAR Y O W L VARIABLE C O W L VAHIABLE YORWAL VAHIABLE POWWlsL VARIABLE F O W L VARIABLE P o n u L VARIABLE YORWAL VARIABLE POHWAL VARlAULE Y O W L VAHIABLE FORMAL VAHIABLE YORMAL VAltlABLE P O W L V A H l W L l !

LES

HEAL VARIABLE HEAL VARlAULE REAL VARIABLE HEAL VAHlAtlLE REAL VARIABLE HEAL VAHIABLE HEAL VARIABLE HEAL VAH IABLE HEAL VARIABLE HEAL VAHIAB1.E REAL VAHlABLE HEAL VAHlABLE

* * * * * * a * * * S T E P Y ~ S E PROCEUUHE **********

VAHlABLE ^)(UN STANUIUAHU

NO. DEV 1 AT l ON

1 0 . 1.04447

2 0 . 1 . 0 4 4 4 7

3 0 . 1.04447

4 0 . 1.04447

5 0 . 1 . 0 4 4 4 7

8 0 . 1.04447

7 0 . 1 . 0 4 4 4 7 8 0.16687 1 . 0 2 8 8 8 9 0 . 1 . 0 4 4 4 7 1 0 0 . 8 6 6 8 7 0 . 7 7 8 5 0

1 1 0 . 1.04447

I 2 0 . 1 . 0 4 4 4 7 1 3 2 . 1 6 6 6 7 2 2 . 9 0 6 6 1 NUMUtR OP SELECTION I COULS

U 0 0 0 0 0 1 a a a

0 0

A f t e r n a r l d a n t l f l c a t l o n t h e y f o r m b l o c k 1, t h a r e v t a r a I n b l o c k 2 .

V o r l o b l a r may b a c l a l a l n d dur111y ill* r c r e a ~ ~ l n y o ~ ~ o l y r l o . V o r l a b l a r 7

-

1 0 a r e d o l n t e d h a r a .

...

...

VARIABLE SQUANES

SUMO. ..UUC....THISSTE... 58.004

PH.PO...O....UC...P... 0 . 0 1 0

...

CUMULATIVE SUNOY SQUARES

CUMULATIVIPPOPORTIONREDUCEO... 0.888 OF 0771.334

Y U L T I P L ~ C 0 H W E l . A T I 0 N C 0 E P P I C I I I T . . . 0 . 8 8 1 Elght v a r l o b l a a a x p l a l n 88.8% o f vnrlanca, but they ANALYSIS VARIANCE... armantlally d l t f a r I n u l g n l f l c a r ~ c a (c f . t - v a l u a u ) .

I'-VALUEYOR 0. 31.330

STANUAHO BHWOHOP ESTIUTE... . . . U S T h s r r f o r r I t l a n a t u r a l l y t o r u y g a r t t h a t d e l a t l n g VANIABLE REG. COEPP. ERROR t-VALUE uoaa o f Cham wl11 n o t d a t r r l o r o t e our approxlmatlon

11 5.03508 1.89681 2.521 r r u u n t l n l l y .

8 8.73100 1.42766 6 . 1 1 6

4 7.38587 1.83050 4 . 6 3 0

5 8.80818 1.83050 4 . 0 5 3

3 8.47853 1.51782 5 . 5 8 8

2 8.65487 1.45837 4 . 5 6 3

1 3.58942 1.45837 2 . 4 6 8

12 2.87368 1 . I 8 8 1 3 1.685

INTERCEPT 2.18867

...

...

SUMO. SQUARES ...UC.O..THISST.P.... 142.081 P.O.O.T.O....UC... 0 . 0 2 1 CUMULAT.VESUNOPSQUAH.SREDUCED... 1814.7 18

CUMUUT~VE I ~ . O P O H T I ~ W H I U U C E U . . . 0.980 OP 8171 .a14 MULTIPLE COHHElATlON C O B Y . I C I . N T . . . 0 . 8 8 0

Y-VALUEPOH ANALYSIS 0. VANIANCE... 27.70. two v a r l a b l a n r r a d a l a t a d by the backward procadura.

STANDARD ERROROP E S T I M A T E . . . 5 . 1 0 0

VAN1 ABLE UEG. COBPP. ERROR T-VALUE

1 I 4.28167 2.33808 1 . 8 3 8

8 8.72222 1.55871 8.237

4 8.62500 1.74268 4 . 8 4 8

1 7.84722 1.74268 4 . 5 0 3

3 7.73611 1.74288 4 . 4 3 8

2 8.80278 1 . I 4 2 6 8 3 . 961

1 3.84722 1.74268 2.208

INTBNCEPT 2.18887

...

***mm.mmmmmmmmmmm STEP 8 *****************

VAHLAULE ENTERED

...

¹

SUM OY SQUAHES REDUCED IN THIS STEP

....

^334.259

PROPORTION REDUCED IN THIS STEP

...

^0.058

CUMULATIVE SUM OP SQUAHBS REUUCED..

....

^1558.4U3

CUMULATIVE PPOPORTION REDUCED.

...

^{0.983 OF 6111.334}

MULTIPLE COHRELATION COEPPICIENT... 0.981 Y-VALUE POR ANALYSIS OP VARIANCE... 21.552 STANUAHD ElUIOR OY ESTIMATE..

...

^8.555

VAHIABLE REO. COBYY. ERROR T-VALUE 4 10. OLLLL58 I. UU231 1.114

U Y.12122 1 .UY232 1. 13s

5 Y .27718 1. (19292 4.Y03

3 0. IUUBI 1 .ue232 4 . ~ 4 4

2 0.33333 1.110232 4.404

1 5.21118 1.89232 2.189

INTERCEPT 2.16887

---

Varlnblae 11 ~ a d 12 have been deleted wlthout esventlnl Incraarl~~y ot the sum ot rusldual sequals. The most slynltlcaot vnrlablav In the aquatlun are varlablas 1

-

^8.

Hara Ie tha smcondery model:

*** MODEL ANALYSlS ANU PORECASTING ***

REGRESSION EQUATION

...

NO. SlONlPlCANT VARIABLES RESPONSE S. D. CORRELATION MATRIX OF REGRESSION COEPPICIENTS (PERCENTAOE)

...

DEP. VAR. X( 31 ) - 2.181

Such correlntlon matrlx or rayreuvlon coettlclentm 1s due to or thogo~tal 1 ty ot the deulgn.

8 VARIABLE 2 0.333 l X ( 2 ) ( 4.404)

8 VARIABLE 1 t 8.278 X( 1 ) 1.892 0 0 0 0 0 1 0 0

( 2.189)

...

TEST STATISTICS

CUMULATIVE PHOYORTION REDUCED... O.UB3 MULTIPLE CORRELATION COEPPICIENT.... 0.981 YTANUARU ERROR ESTIMATE (SIGMA)

...

^tl.355

I-VALUK IOW ANALYIllU OF VARIANCI.... Pl.18O

...

..

.

L. cu a F 2 m

a .

C w

+ I 0 J= c- Y c - m w "

4 .

N L. I

rs 0

a-ol 0 I- s-ol

> - a

Z 0 - - a

ULOCK 7 1

DEPENDENT VARIABLE 7 3 1

WITH I N T E M C T l O N 7 ( YES

-

^{1. NO}

-

^{0 )}

0

...

A NUMBER OF OBSERVATIONS 12 TABLE OF VARIABLES

FORMAL VAHlABLe 1 REAL VARIABLE 1 P O W L VAHIAULE 2 HEAL VAHlAYLE 2 P O W L VAHIABLE 3 REAL VARIABLE 3 P O W L VAHlABLE 4 HEAL VAHIAULB 4 Y O W L VAHIABLE 8 REAL VAHIAULE 5 POHlUL VAHIABLE B WEAL VARIABLE B Y O W L V A H I A B L I 7 REAL VARIABLE 7 FOWL VAHIABLY. 8 n e A L VAH~ABLE 8 Y O W L VAHIABLE O WEAL VARIAULE O F O W L VAHIABLE 10 H U L VAHIAULY 10 Y O W L VAHIABLP 11 HEAL VAHlABLE 11 P O W L VAHIABLB 12 REAL VAHlAULE 12

******** STEPWISY. REGRESSION WITH PEMUTATIONS ********

... ...

NUMBEY UP OBSEWVATIONS 12 NUNUEW OY VARIABLES 13

...

SCALE POW A H I S T O G M I : A POINT CORRESPONDS TO 1 VALUE/S/

NUMBER OY INTERVALS FOR A H I S T O G W 5 MX.NUMYEW OP STEPS 12

NUNBPH UP PERMUTATIONS 30

. . . ...

TOTAL NUMUEW OY DELETED VARIABLES-?

... 1 THEIR INDEXES-7

*..**....*.*.*********** 6 ...

... ...

ENTERED VARIABLES AND THEIR T-STATISTICS.STEP- 1 1 I

3.35

...

STEP 1

V A H I W L E ENTEWED 11

A regrenslon model ulth 13 varlsbles Is s~~slyzecl.

a numbar of obnervatlonr

-

^12.

A scale and a numbrr of Intervals for s hlutogram.

Haxlmal number of rteps - ^12.

a number of prrmutatlonr

-

^30.

Verlabla B Ir deleted from the reyresslonal equallon.

Vsrlsblr 11 lu entered Into reyreuulor~ on the 1st step, Itr T-rtstlstlcr Is T

-

3 . 3 8 .

0

8 8 8 8 * * * 8 8 * * * * 8 8 * * * * 8 * * * * I * 8 * * * * * 8 I * *

U S l C T-STATISTIC 3.35

T - N U N AND ST.DEVIATION 0.5845 2.2838 HIM. ANU U.T-STATISTICS APTYR PEWI(UTATl0N

-2.84 4.40

Y l I T O G W POH T-STATISTIC

-2.U448 -1.3048

...*

-1.3848 0.0548 0.0548 1.5048 l

1.5048 2.8644

...*

2.8544 4.4042

.*

b OF T.FOH WHICH AUYtT) IS LESS THAN W S ( T 0 ) 88.87

B M I C F-STATISTIC 11.213

F - N U N AND ST.DEVIATION 5.395 3.107 YIY. AND M.F-STATISTICS AFTER PBRNUTATlOW

2.688 18.387 YIITOGRAN FOR P-STATISTIC

2.6884 6.8501

...

²

5.8501 8.3117

...*

8.3117 12.8733 12.8733 18.0350 l

18.0350 18.3088

.*

1 09 F.LESS THAN YO 98.87

***8888****88*8***88**** . . .

M S l C SY-PROPORTION HLUUCEU 0.6288

IS-MEAN AND ST.DBVIATION 0.3330 0.0834 YIY. AND )WI.SS-YHOYOHTlON8 AVTYH PIMUTATION

0.2058 0.8888

YISTOGIW FOB SY-STATIElTlC

0.2058 0.2965

...

0.2886 0.3873 . . . *

0.3873 0.4781

.... .*

0.4781 0.5800 0.6880 0.8588

.*

b OP SS.L6SS T U N SSO 88.87

... ...

PMTEHED VARIABLES AND THEIR T-STATISTIC6,STEP- 2 11 11

3.10 1.7)

b a n m d rtrndrrd devirtlon of T-rtrtirtic after prrmutrtlon: 0.6841 and 2.2838.

Ilnlrrl rnd maxlmrl vrluor of T-rtrtlrtlc after porautrtlon: -2.84 and 4.40.

Ulotograa for T-rtrtlrtic: one point corrurpol~dr to one valur (according to rrrgned acute).

88.67& of T-rtatirticr alter purmutatlon ratlrfy tllu lorqqurllty:

IT 1 < 3.31

-

^T

1 0 .

hlapour inlormrtlon Ir given lor P - rsd 89

-

rtetlrtlcr.

,Lrlrblrr that arm rntsred Into reyrerrlon sler 2 rtepr Cerrrrpoadlny T-rtatlrt lcr.

Vulrblr 11 Ir enterrd llrto rryrrrrlun on tLo 211d rtrp.