IEEE TRANSACTIONS ON COMPUTERS, VOL. C-18, NO. 5, MAY 1969

A Nonlinear Mapping for Data Structure Analysis

JOHN W. SAMMON, JR.

Abstract-An algorithm for the analysis of multivariate data ispresented along with some experimental results. The algorithm isbased upon a point mapping of N L-dimensional vectors from the L-space to a lower-dimensional space such that the inherent data"structure" is approximately preserved.

Index Terms-Clustering, dimensionality reduction, mappings,multidimensional scaling, multivariate data analysis, nonparametric,pattern recognition, statistics.

INTRODUCTIONHE purpose of this paper is to describe the non-linear mapping algorithm (NLM) which has beenfound to be highly effective in the analysis of mul-

tivariate data. The analysis problem is to detect andidentify "structure" which may be present in a list ofN L-dimensional vectors. Here the word structure refersto geometric relationships among subsets of the datavectors in the L-space. Some examples of structure arehyperspherical and hyperellipsoidal clusters, and linearand certain nonlinear relationships among the vectorsof some subset.The algorithm is based upon a point mapping of the

N L-dimensional vectors from the L-space to a lower-dimensional space such that the inherent structure ofthe data is approximately preserved under the mapping.The approximate structure preservation is maintainedby fitting N points in the lower-dimensional space suchthat their interpoint distances approximate the corre-sponding interpoint distances in the L-space. We shallbe primarily interested in mappings to 2- and 3-dimen-sional spaces since the resultant data configuration caneasily be evaluated by human observations in 3 or lessdimensions.

THE NONLINEAR MAPPINGSuppose that we have N vectors in an L-space desig-

nated Xi, i= 1, * * , N and corresponding to these wedefine N vectors in a d-space (d = 2 or 3) designated Yi,i=l, *, N. Let the distance' between the vectorsXi and Xj in the L-space be defined by dij*=dist [Xi,Xj] and the distance between the corresponding vectors-Y and Yj in the d-space be defined by dij= dist [ Yi, YP I .

Manuscript received August 26, 1968; revised February 2, 1969.The author was with Rome Air Development Center, Griffiss

AFB, Rome, N. Y. He is now with Computer Symbolic, Inc., Rome,N. Y.

1 Any distance measure could be used; however, if we have no apriori knowledge concerning the data, we would have no reason toprefer any metric over the Euclidean metric. Thus, this algorithmuses the Euclidean distance measure.

Let us now randomly2 choose an initial d-space con-figuration for the Y vectors and denote the configura-tion as follows:

Yii Y21 YN1

Y1= Y2 [Y...N

_yldJ__Y2d_L_yNdJ

Next we compute all the d-space interpoint distancesdij, which are then used to define an error E, whichrepresents how well the present configuration of Npoints in the d-space fits the N points in the L-space,i.e.,

1 N [dj* - dj]2E = vEE [dij*] i<j dij*

i<i

(1)

Note that the error is a function of the d XN variablesyp^ p= , ... , N and q= 1, * * , d. The next step inthe NLM algorithm is to adjust the y,, variables orequivalently change the d-space configuration so as todecrease the error. We use a steepest descent procedureto search for a minimum of the error (see Appendix I forfurther details).

SOME COMPUTER RESULTS

We have exercised the nonlinear mapping algorithmon several data sets in order to test and evaluate theutility of the program in detecting and identifyingstructure in data. Some of the results obtained for sev-eral different artificially generated data sets3 are re-ported for the case where d= 2. We have also run thealgorithm on many real data sets and have achievedhighly satisfactory results; however, for demonstrationpurposes it is useful to work with artificially generateddata in order that we can compare our results with theknown data structure. The test data sets were as follows.

1) Straight Line Data: These data consisted of ninepoints distributed along a line in a 9-dimensional space.The data points were spaced evenly along the line withan interpoint Euclidean distance of V,/9 units. The ini-tial 2-space configuration was chosen randomly.

2 For the purpose of this discussion it is convenient to think ofthe starting configuration as being selected randomly; however, inpractice the initial configuration for the vectors is found by project-ing the L-dimensional data orthogonally onto a d-space spannedby the d original coordinates with the largest variances.

I One exception is data set 3 which is a classical data set. Thisdata set was not artificially generated.

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IEEE TRANSACTIONS ON COMPUTERS, MAY 1969

2) Circular Data: The data consisted of nine points,eight of which were spaced evenly (450 apart) along a

circle of radius 2.5 units in a 2-dimensional space. Theninth point was placed at the center of the circle. Theinitial 2-space configuration was chosen randomly.

3) Iris Data: This data set is fairly well-known sinceit was used by Fisher [4] in several statistical experi-ments. The data were originally obtained by makingfour measurements on Iris flowers. These measurementswere then used to classify three different species of Irisflowers. Fifty sample vectors were obtained from eachof the three species. Thus, the data set consists of 150points distributed in a 4-dimensional space.

4) Gaussian Data Distributed at the Vertices of a Sim-plex: These data consist of 75 points distributed in a

4-dimensional space. There are five spherical Gaussiandistributions which have their respective mean vectorslocated at the vertices of a 4-dimensional simplex.4 Theintervertex distance is V5/4 units and each covariancematrix is diagonal with a standard deviation along every

coordinate of 0.2 unit. Fifteen points were generatedfrom each of the five Gaussian distributions, making a

total of 75 points.5) Helix Data: This data set consisted of 30 points dis-

tributed along a 3-dimensional helix. The parametricequations for this helix are

X = cos Z

Y = sin Z

.\/2z = t.

2

The points are distributed at one-unit intervals alongthe curve (i.e., t=0, 1, 2, , 29).

6) Nonlinear Data: This data set consisted of 29points distributed evenly along a 5-dimensional curve.

The parametric equations for this curve are

X = cos Z

Y = sin Z

U = 0.5 cos 2Z

V = 0.5 sin 2Z

Z = -/ t.2

The points were distributed at one-unit intervals alongthe curve (i.e., t = 0, 1, * , 28).

Figs. 1 through 9 display the results obtained usingthe nonlinear mapping algorithm. Convergence was es-

sentially obtained for each case in twenty or less itera-tions using the gradient searching technique described

4The vertices of a simplex are all equidistant from one anotheras well as from the origin.

in Appendix I. As was expected, the data structure in-herent in both data sets 1 and 2 was faithfully repro-duced under the mapping (see Figs. 1 and 2). This wasexpected since the data sets are 1- and 2-dimensional,respectively, and therefore a mapping to a 2-space canbe accomplished with zero error. In both cases the finalerror was 10-16.

Observing the result of the Iris data mapping (Fig. 3),we can essentially detect the three species of Iris. Thefinal error was 2 X 10-3 which is considered quite small.The results obtained on the simplex data (data set 4)

were quite interesting. The result of the mapping clearlyshowed the presence of five clusters (see Fig. 4). How-ever, when we compare this result with the projectionof the same data onto the 2-space defined by the twolargest eigenvectors of the estimated data covariancematrix, we can only detect four clusters. Two of theclusters overlap completely in the 2-space which fitsthe data in the least squares sense (see Fig. 5). The re-sulting NLM error was 0.05. The same experiment hasbeen conducted using Gaussian data distributed at thevertices of higher-dimensional simplexes. Figs. 6 and 7show the NLM and principal eigenvector plots, re-spectively, for a 19-dimensional Gaussian simplex dis-tribution. These experiments indicate that for some datasets, the NLM is superior to eigenvector projections fordata structure analysis.The results shown in Figs. 8 and 9 clearly indicate the

"istring structure" in data sets 5 and 6, respectively. Themapping error for the helix data was 6 X 10-4. The errorfor data set 6 was 1.6 X 10-3.The utility of any data analysis technique is somehow

more convincing when applied to "real" data as opposedto artificially generated data, presuming, of course, thatthe analysis results are correct. For this reason, theapplication of the NLM algorithm to an experiment indocument retrieval by content is reported here.The experiment, conducted jointly by Rome Air De-

velopment Center (RADC) and the University of Col-orado, involved the construction of a document classifi-cation space (referred to as the C-space) where everydocument in the library was represented as a 17-dimen-sional vector. The construction technique devised byOssorio [8], [9] describes a mapping of 1125 preselectedwords and phrases into the C-space. Documents, orequivalently retrieval requests, were located in the spaceby computing the vector average of the correspondingkey words or phrases which were contained in the docu-ment or the request. Retrieval was accomplished byrank-ordering the relevance of the library documents toa given request. The relevance measure was computedusing the Euclidean metric between the document vec-tors and the request vector, the concept being thatdocument vectors which are close in the C-space arerelated by content and therefore should be retrieved to-gether.

402

SAMMON: NONLINEAR MAPPING FOR DATA STRUCTURE ANALYSIS

NONLINEAR MAPPINGData: 9 pts. along a straight lineOriginal Dimension =9Starting Configuration: RandomMapping Error-10-1615

10 _ __/

-5t

-15 -10 -5 0 5 10 I5 20 25

Fig. 1.

NONLINEAR MAPPINGData: 9 pts. on o circleOriginal Dimension -2Starting Configurotion: RandomMapping Error = 10-16

12

10

8-.

-I -2 -3 -4 -5 -6 -7

Fig. 2.

NONLINEAR MAPPINGDota: ris dataOriginal Dimension =4 xStorting Configurotion Maximum Variance X X

Coordinate Plane X6 Mapping Error-*002 X- X X

xXx KX'xxx xx

4 0OoO @ :=: K~: x3 ° °° @ @ Q tw~~~ Xo0 0 0 6 o

000~~~~~

0 00

2 3 4 5 6 7 s9

Fig. 3.

NONLINEAR MAPPINGData:75pts. trom 5Goussion distributionsOriginal Dimension 4Starting Configuration Maximum variancecoord I note ploneMapping Error - 05

2.0

1.5

43~ ~ ~~tg 4.

001.0 0~~~~~~~~~~~

- -- -- E<EYCTRPO

0~~~~~000 00~0 0

0~~~~~~~

.5 1.0 IS5 2.0 2.5 S.C

Fig. 4.

EIGENVECTOR PLOTData: 75pt. from 5Gaussian distributionsOriginal Dimension -4Projection on the two principle eigenvectors

5 5513 3

55

5 555

5 55 5~ ~~ I35 35 tIll

3333 5 5533 3 5 5 .I

33 3 33

22 22

4 2 2

4 44421 22 22 2444 2222 2

44 22 2 2 24 22 2

4 44 2

444 4

4 4 2 4

Fig. 5.

403

NONLINEAR MAPPINGData: 100 pts. from 2OGaussiOn-spherical

distributions= 19Origina I d men sion a 19Starting Conf ig ura t ion s Random

4.850 .....ff~ ~~~~~~~~~MaDPing Error 144.850 H KK.K

4 66 _______ E~~~~~Fig 6

H E K A

4.484-- A A A

4.30: T0 T

4.11.

-0205 t ] | i | Ey L L L

3.934 - l-~ ~~~

~~~R NN

M M R

3.384 G.-1G G

G DD3.20:

B B G3.01 8 B B.S2.834

F S S

2.65: __50__24212__ 43_276 .4

.304 .608 .912 1.216 C1.0 84 2.2 243 276 300

Fig. 6.

EIGENVECTOR PLOTData- 100 pts from 2OGaussiian spherical

distributionsOriginal Dimension a 19Projection on the two principle esigenvectors

0. 630

0. 538

0.445 MN---M

0.352M

0.259R

0. 16( -R--N- -

R T

Q O7s F Nv N

Q -TJ ~~~~c S UGH N A A A

-0.205E 0 KK L8 LL L

-0.391

-0.4841 R__ __BI_

-0.654 0.531 - 0.407 -0.283 -0.159 -0.035 B O.O8

Fig. 7.

0.212 0.336 0.460

Briefly, the C-space construction proceeded as follows.First the subject content covered by the 188 documentsin the experimental library was subjectively partitionedinto 23 technical fields (see Appendix II for a listing ofthese fields). Several experts representing each fieldrated the relevance of each of the 1125 words or phrasesto his field, using a scale from 0 to 8. The rating by theexperts within each field were then averaged to obtain a

word-by-field relevance matrix designated X; the ijthelement of X represents the relevance of word or phrasei to field j. (It is convenient to think of the 1125 words

or phrases as being represented by vectors in a 23-dimensional space spanned by the 23 coordinate fields.)Next, a 23 X 23 field correlation matrix C was computed,where the ijth element represented the correlation be-tween the ith and jth fields. C was then factored usingthe minimum residual method and rotated to a Varimaxcriterion. Seventeen orthogonal factors were thenselected to define the 17-dimensional C-space.

All 1125 words and phrase vectors were mapped intothe 17-dimensional C-space using a simple nonlinearformula which tended to emphasize large coordinate

4i4 IEEE TRANSACTIONS ON COMPUTERS, MAY 1969

SAMMON: NONLINEAR MAPPING FOR DATA STRUCTURE ANALYSIS

NONLINEAR MAPPINGData1: 3Opts. along a helix

- - - -- - - - - - ~~~~~~~Original Dimension- 3Starting Configuration:' RandomMopping Error- 6xIOA4

IC5- NOUNA -.PiN

9.-

8.-~~~~~~~~~~~ 0I 2 3. . l4.15 6 7 8 l0 1 3 1 S 1 7 1 9 2

15 -H IOUNA MPIN

Daa79ps ln olna uv

IC

14 - -Ori-in- --im-ns-o-.

4 5 6 7 8 10 12 13 14 15 16 17 18 19 20 21

Fig. 9.

projections and minimize small coordinate projections.Finally, the 188 documents were located in the C-spaceby algebraically averaging the word or phrase vectorscorresponding to the word or phrases which appeared inthe documents.5

In order to evaluate the C-space as a potential methodfor document indexing, several individuals were asked togenerate English queries (see Appendix III for the per-tinent queries used here) which were then keypunched,automatically scanned for key word or phrase content,

5The entire document was never searched for key words orphrases. Rather, for one half of the documents only the abstractswere used, and for the remainder several paragraphs from each docu-ment were used.

and finally mapped into the C-space. Each requester wasthen asked to identify those documents of the entire 188which he felt were most relevant to his query. The C-space was then evaluated by examining the rank order-ing of the retrieved documents to compare them to thelist of relevant documents specified by the requester.The results of this evaluation can be found in Ossorio[8].The nonlinear mapping algorithm was used to evalu-

ate the "structure" of the documents in the C-space.Specifically, we were interested in how the documentsconsidered relevant to a particular request were clus-tered, and further, how these clusters were interrelatedto each other and to the entire library. To accomplish

405

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IEEE TRANSACTIONS ON COMPUTERS, MAY 1969

Fig. 10. Nonlinear mapping-photograph of CRT display. Data:1= eight Request 1 vectors; 2 =seven Request 2 vectors; 3 = six-teen Request 3 vectors; 4 = thirteen Request 4 vectors; 5 = sevenRequest 5 vectors. Starting configuration: maximum variancecoordinate plane. Mapping error = 0.062.

this analysis, all 188 17-dimensional vectors were usedas the input data to the NLM. The numerals 1 through5 were used in the resulting 2-dimensional mapping todesignate the documents labeled relevant to queries 1

through 5. In addition, the symbol D was used to desig-nate the remaining library documents. It is importantto note that the NLM algorithm did not utilize thenumeric query designations in computing the mapping.Only at the time of plotting the final 2-space configura-tion of the 188 points were the numeric and symbolicdesignators used to distinguish the data. The error inachieving the NLM shown in Fig. 10 was 0.062, whichwas considered to be acceptable for adequate 2-spacerepresentation.The following facts were obtained upon investigation

of the NLM result.

1) The documents considered relevant to a given re-

quest were clustered, lending evidence to support thehypothesis that related documents have C-space vectorswhich are close.

2) There does not appear to be any natural C-spacestructure relating subsets of documents. Instead, thedocuments tend to be uniformly distributed throughoutthe space.

3) Clusters 2 and 3 tend to overlap, yet they are well-separated from clusters 4 and 5. This can easily be ac-

counted for since requests 2 and 3 are both concernedwith the common subject of statistical data analysis,whereas 4 and 5 involve completely different subjects.In general, the intercluster relationships seem consistentwith their respective subject relationships.

In summary, we have found the NLM algorithm tobe of considerable value in aiding us in our understand-ing of the C-space as well as other document spaces.

Presently we are planning to incorporate a similar map-ping technique in an on-line document retrieval systemin order to improve the retrieval via geometric means.

The experimental system will operate as follows. Theon-line user would examine the 30 highest-ranked docu-ments by retrieving and reading their abstracts. Hewould then indicate those he considered relevant. Next,a scatter diagram similar to Fig. 10 would be presentedupon the CRT display where each of the 30 documentswould be indicated by an I or an R, depending upon itsrelevance. In addition, the original query vector will bedisplayed as a Q. After examining the relative positionsof the documents in the mapping, the user would select(using a light pen) one or more relevant documents tobe used to generate a new query vector(s). The conceptis that the query vector can be moved to highly relevantregions of the document space by interacting at a displayconsole with a geometric representation of the space.

RELATIONSIIIP OF NLM TO OTHERSTRUCTURE ANALYSIS ALGORITHMS

A mapping algorithm which bears a relationship tothe NLM algorithm is one developed by Shepard [11]and later improved by Kruskal [5], [6]. Briefly, theShepard-Kruskal algorithm seeks to find a configurationof points in a t-space such that the resultant interpointdistances preserve a monotonic relationship to a givenset of interelement similarities (or dissimilarities).Specifically, they wish to analyze a set of interelementsimilarities (or dissimilarities) given by Sij, i = 1, * * ,N, j = 1, * * *, N. Suppose these similarities are orderedin increasing magnitude, such that

SPJl1 < SP2q12 < <S...

The Kruskal-Shepard algorithm seeks to find a set ofN t-dimensional vectors yi, i = 1, . . . , N, such that theorder of the interpoint distances dij=dist[yi, y;] devi-ates as little as possible from the monotonic ordering ofthe corresponding similarities. Although the mathema-tical formulations are similar, the underlying mappingcriterions are quite different.

Ball [1 ] has compiled an excellent survey of cluster-ing and clumping algorithms which are useful in solvingthe "structure analysis" problem. However, it has beenour experience in using clustering techniques that thesealgorithms suffer to some extent from the following fourdeficiencies.

1) When using a particular algorithm, the resultingcluster configuration is highly dependent upon a set ofcontrol parameters which must be fixed by the user.Some examples of such parameters are:

a) the similarity measure;b) various similarity thresholds;c) number of iterations required;d) thresholds which control the increase or reduc-

tion of the number of clusters;e) the minimum number of vectors required to de-

fine a cluster.

406

SAMMON: NONLINEAR MAPPING FOR DATA STRUCTURE ANALYSIS

When choosing the control parameters for complexdata, the user must either have a good deal of a prioriinformation regarding the "structure" of his data, or hemust apply the algorithm many times for different val-ues of the control parameters. This second alternative is,at best, tedious.

2) Most of the existing clustering algorithms areparticularly sensitive to hyperspherical structure and areinefficient in detecting more complex relationships inthe data.

3) Perhaps the most serious deficiency involvingpresent-day clustering algorithms is that there do notexist really good ways for evaluating a resultant clusterconfiguration.

4) When two clusters are close, the vectors betweentend to form a bridge and cause spurious mergers [7].

We feel that the nonlinear mapping is a highly prom-ising structure analysis algorithm since it suffers littlefrom the listed clustering deficiencies. Consider thefollowing facts concerning the algorithm.

1) The routine does not depend upon any controlparameters that would require a priori knowledge aboutthe data. Specifically, the user must set the number ofiterations and the convergence constant (MF in Appen-dix I).

2) It is highly efficient in identifying hyperspherical,hyperellipsoidal, and other complex data structures.

3) The resulting mapping (scatter diagram) is easilyevaluated by the researcher, thereby taking advantageof the human ability to detect and identify data struc-ture.

4) The problem concerning extraneous data andspurious mergers is not present since humans easilyeliminate troublesome data points by making globalevaluations (machines have difficulty performing thisfunction).

5) The algorithm is simple and efficient.

LIMITATIONS AND EXTENSIONSThere are, of course, limitations to every algorithm

and the nonlinear mapping is no exception. There existtwo limitations which we are presently investigating.The first has to do with the reliability of the scatter dia-gram in displaying extremely complex high-dimensionalstructure. It is conceivable that the minimum mappingerror is too large (E>>0.1) and the 2-dimensional scatterplot fails to portray the true structure. However, we feelthat for data structures composed of superpositions ofhyperspherical and hyperellipsoidal clusters, the non-linear mapping algorithm will, in general, display ade-quate representations of the true data "structure."The second limitation of the nonlinear mapping al-

gorithm is related to the number of vectors that it canhandle. Since we must compute and store the interdis-

we are limited at present to N< 250 vectors.6 In thosecases where N> 250, we suggest using a data compres-sion technique to reduce the data set to less than 250vectors. Specifically, we propose to use the Isodata [2]clustering algorithm to perform data compression. Thisis actually a natural function of clustering since we re-

place several vectors with a typical representativevector (i.e., the cluster center). Our previous objectionsto present-day clustering algorithms do not apply heresince we are only concerned with fitting the data with250 cluster centers. We are specifically not using theclustering algorithm to detect structure.We have used the NLM to analyze multivariate data

from two or more classes for the purpose of determininghow well the classes can be discriminated from one an-

other. In these cases, it is recommended that the dimen-sionality be reduced to the smallest number of variableswhich still preserve discrimination.7 In many problemscertain measurements provide little discriminatory in-formation; yet if these measurements are included, theNLM will attempt to "fit" interpoint distances alongthese "noisy" directions as well as along discriminatingdirections. In truly high-dimensional problems, the re-

sulting mapping may show considerable overlap be-tween classes and still a high degree of discriminationmay be possible. This phenomena occurred whenanalyzing a 4-class, 24-dimensional data set. The result-ing NLM (the final error was 0.5, which was consideredhigh) showed considerable overlap among the datafrom three of the classes; yet, using a piecewise lineardiscrimination technique (based upon the use of a

Fisher's linear discriminant between all pairs of classes),94 percent correct classification was achieved. In thiscase, the NLM did not give an incorrect result since the

6 The nonlinear map is programmed in FORTRAN IV and runs on aGE-635 computer equipped with 128 K of core. The computationtime can be estimated by

T'c (1.1X 10-5)- (2 )

minutes, where

I= number of iterationsN= number of vectors.

7 A number of techniques may be used for this purpose. We oftenuse the following:

a) Discriminant measure

M(X) E -

i<Ki .i2+ j2

b) Interpoint measure

1 1 Ni NiM(X) =2 E E (Xp(fi) Xq(i))2

(TX i< NiNj p=1 q=.1

where

g2i= mean of class i along XO'Xi= variance of class i along X

2=variance of all data along XXp(i) the pth sample from the ith class along X

Ni= number of samples from the ith class.tance matrix, which consists of N(N-1)/2 elements,

407

c) Multilinear discriminant defined in Wilks [141.

IEEE TRANSACTIONS ON COMPUTERS, MAY 1969

classes greatly overlapped in approximately 20 dimen-sions and mildly overlapped in the remaining space. TheNLM weighted all coordinates equally in an attempt tofit the interpoint distances, and therefore the resultingmapping indicated the predominant overlap which actu-ally existed.The NLM algorithm described here is one of many al-

gorithms which are being programmed and incorporatedinto a large on-line graphics-oriented computer sys-tem, entitled the On-Line Pattern Analysis and Recog-nition System (OLPARS) [10].8 Once the NLM al-gorithm is incorporated into the OLPARS system, theon-line user will be able to designate a data set, and fromthe graphics console execute the NLM. The user shallspecify a mapping to a 2-space or a 3-space. For d = 2,the resultant scatter diagram will be displayed upon theCRT; for d = 3, a perspective scatter plot will be dis-played. If the 3-space option is selected, the user will beable to dynamically analyze the resultant perspectivescatter diagram by selecting various rotations of thethree space. When the user selects d = 2, he will be giventhe capability to designate subsets of data (via piecewiselinear boundaries drawn on the CRT) representing a col-lection of points in the scatter diagram which exhibitstructure, and thereby partition the initial data list intostructured subsets.

APPENDIX I

Let E(m) be defined as the mapping error after themth iteration, i.e.,

1 N

E(m) -- [dij*-dij(m) 2/dij*C i<j

where

and

Nc = E [dIj*]

iKi

V d

dij(m) = Z [yik(m) -yjk(m)]2 -

k=l

The new d-space configuration at time m +1 is given by

ypq(m + 1) = ypq(m) - (MF) *p.(m)where

Apq(m) = aE(m) / 92E(m),Oypq(m) Oypq(M)2

and MF is the "magic factor" which was determinedempirically to be MFm 0.3 or 0.4. The partial derivativesare given by

8 For other examples of interactive pattern analysis systems, seeBall and Hall [31, Stanley et al. [12], and Walters [13].

aE -2 N[d-ddpj-= Ld- d * j(ypq - yjq)(9yp,q c j=l , dpjdvj.* (Y- Yq

and

a2E -2 N

(9yV 2 C j=1 dpj*dpjj#p

[(d* - dp))- q+-Yi)2 *dp)]In our program we take precautions to prevent any

two points in the d-space from becoming identical. Thisprevents the partials from "blowing up."

APPENDIX IICLASSIFICATION SPACE FIELDS

1)2)3)4)5)6)7)8)9)

10)11)12)13)14)15)16)17)18)19)20)21)22)23)

Adaptive SystemsAnalog ComputersApplied MathematicsAutomata TheoryComputer Components and CircuitsComputer MemoriesComputer SoftwaveDisplay ConsolesHuman FactorsInformation RetrievalInformation TheoryInput-Output EquipmentLanguage TranslationLinear AlgebraMultivariate Statistical AnalysisNonnumeric Data ProcessingNumerical AnalysisPattern RecognitionProbability and StatisticsProgramming LanguagesStochastic ProcessesSystem Design and EvaluationTime-Sharing Systems.

APPENDIX I II

REQUESTS

Request 1: What is known about the statistical dis-tributions of words or concepts in English text? Whatimpact does this knowledge or lack of knowledge haveon the effectiveness of standard statistical methods toinformation retrieval problems? Are nonparametricmethods more applicable?

Request 2: I am interested in techniques for data anal-ysis. In particular, I wish information on "cluster-seek-ing" techniques as opposed to those of factorial analysisand discriminant analysis. "Cluster-seeking" techniquesmay be classified as follows: probabilistic techniques,

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IEEE TRANSACTIONS ON COMPUTERS, VOL. C-18, NO. 5, MAY 1969

signal detection, clustering techniques, clumping tech-niques, eigenvalue-type techniques, and minimal mode-seeking techniques.

Request 3: I would like any information concerningBayesian statistics. In particular, I would like to knowif one can define or devise multiple-decision proceduresfrom the Bayes approach. Also, how sensitive are Bayesprocedures to the prior distribution? Finally, I wouldlike a comparison of the Bayes approach to other classi-cal decision theoretic approaches.

Request 4: What is the structure and characteristicsof paging techniques?

Request 5: Are there survey documents (information)available which discuss or detail the relative practical-ity of memories; for example, capacity versus utiliza-tion, density, weight, environmental features, failurerates, economics, etc.?

ACKNOWLEDGMENTThe author expresses his appreciation to D. Elefante

for his efforts in developing an efficient FORTRAN iv pro-gram for the Nonlinear Mapping Algorithm.

REFERENCES[1] G. H. Ball, "A comparison of some cluster-seeking techniques,"

Rome Air Development Center, Rome, N. Y., Tech. Rept.RADC-TR-66-514, November 1966.

[21 G. H. Ball and D. Hall, "Isodata," portion of Stanford ResearchInstitute (SRI) Final Report to RADC Contract AF30(602)-4196, September, 1967.

[31 , "Promenade-an improved interactive graphics man/ma-chine system for pattern recognition," Stanford Research In-stitute, Menlo Park, Calif., Project 6737, October, 1968.

[4] R. A. Fisher, "The use of multiple measurements in taxonomicproblems," Ann. Eugenics, vol. 7, pp. 178-188, 1936.

[5] J. B. Kruskal, "Multidimensional scaling by optimizing good-ness of fit to a nonmetric hypothesis," Psychometrika, pp. 1-27,March 1964.

[6] -, "Nonmetric multidimensional scaling: a numerical meth-od," Psychometrika, vol. 29, pp. 115-129, June 1964.

[7] G. Nagy, "State of the art in pattern recognition," Proc. IEEE,vol. 56, pp. 836-861, May 1968.

[81 P. G. Ossorio, "Classification space analysis," RADC-TDR-64-287, October 1964.

[9] , "Attribute space development and evaluation," RADC-TDR-67-640, January 1968.

[101 J. W. Sammon, "On-line pattern analysis and recognition sys-tem (OLPARS)," RADC-TR-68-263, August 1968.

[11] R. N. Shepard, "The analysis of proximities: multidimensionalscaling with an unknown distance function," Psychometrika, vol.27, pp. 125-139, 219-246, 1962.

[12] G. L. Stanley, G. G. Lendaris, and W. C. Nienow, "Pattern rec-ognition program," AC Electronics Defense Research Labs.,Santa Barbara, Calif., TR-567-16, November 1967.

[13] C. M. Walters, "On line computer based aids for the investiga-tion of sensor data compression, transmission and delay prob-lems," 1966 Proc. Natl. Telemetry Conf., Boston, Mass.

[14] S. S. Wilks, Mathematical Statistics. New York: J. Wiley, 1962.

Mathematical Analysis of Ferrite Core

Memory Arrays

WILLIAM T. WEEKS

Abstract-A mathematical model for simulating pulse propaga-tion in ferrite core memory arrays is described. Although specificallydeveloped to analyze 3-dimensional arrays, the model is sufficientlygeneral to give a satisfactory analysis of pulse propagation, waveformdeterioration, and noise generation in a wide variety of memoryconfigurations. The model treats the memory as a generalized,mutually coupled, multiconductor transmission line system. Insofaras is possible, the transmission line parameters are calculated fromthe array geometry, thus leaving only a small number of parametersthat must be supplied empirically. Following a discussion of theequations which define the model and the methods by which theyare solved, a sample array calculation is given to illustrate the kindof information that can be obtained from the model.

Index Terms-Arrays, computers, ferrite cores, memories, pulsepropagation, transmission line system.

Manuscript received October 23, 1968; revised February 10, 1969.The auithor is with IBM Corporation, Components Division,

Poughkeepsie, N. Y. 12602.

INTRODUCTIONEU URING the past five years, considerable prog-

ress has been made in the development of mathe-matical models for simulating the electrical

properties of ferrite core memory arrays. The purpose ofthis paper is to describe the techniques available for theanalysis of 3-dimensional ferrite core memory arrays.The techniques presented here represent a substantialadvance in the state-of-the-art over earlier reportedwork [1], [2], which dealt mainly with the simulation of2-dimensional arrays.A precise mathematical description of a memory ar-

ray, backed up by a rigorous and practically realizablemethod for solving the resulting equations, would be ofinestimable value to a memory designer, for it would re-move much of the uncertainty from the design process

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