A Semigraphical Method for the Analysis of Complex Problems

A Semigraphical Method for the Analysis of Complex ProblemsAuthor(s): Edgar AndersonSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 43, No. 10 (Oct. 15, 1957), pp. 923-927Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/89400 .

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ours in this preparation, many free spores were present, and intact asci broke after a minimum

nanipulation. 7 B. Ephrussi, H. Hottinguer, and H. Roman, these PROCEEDINGS, 41, 1065-1071, 1955. S The oil-chamber technique of P. DeFonbrune, La Technique de micromanipulation (Paris: usson & Cie, 1949), was used. 9 The synthetic medium of B. H. Olson and M. J. Johnson (J. Bacteriol., 69, 159-162, 1949) s used. Asparagine was omitted, and biotin and calcium pantothenate were the only vitamins led. This was supplemented as required with amino acids (10 mg/l) or 4-methyl-5-3-hydroxy- yl thiazole (0.1 mg/l). 0 M. Ogur, G. Lindegren, and C. C. Lindegren, J. Bacteriol., 68, 391-392, 1954. This medium s modified for replica plating by omitting the indicator, reducing the glucose to 0.025 per cent, i adding agar. 1 Kindly supplied by Dr. S. Spiegelman as 33-E-26. 2 B. Ephrussi, P. L'Heritier, and H. Hottinguer, Ann. Inst. Pasteur, 77, 64-83, 1949. 3 R. A. Fisher, Statistical Methods for Research Workers (Edinburgh: Oliver & Boyd, 1944). 4 R. E. Wright, Extranuclear Inheritance in Yeast Heterokaryons (doctoral dissertation, Uni-

sity of Wisconsin, 1957) (University Microfilms, 313 N. First St., Ann Arbor, Michigan).

A SEMIGRAPHICAL METHOD FOR THE ANALYSIS OF COMPLEX PROBLEMS

BY EDGAR ANDERSON*

NRY SHAW SCHOOL OF BOTANY, WASHINGTON UNIVERSITY, AND MISSOURI BOTANICAL GARDEN,

ST. LOUIS, MISSOURI

Communicated, August 26, 1957

Science and technology have many problems difficult to record, measure, or

alyze because many variables or complexes of variables have an important bear- r on the end result. It is difficult to measure and to get in mind the various inter-

ationships of several sets of facts for each of a considerable number of individuals.

lis is particularly true when some or all of the basic data are complex patterns, ficult or impossible to code in numbers. Such a problem is the relation (As/Ai) tween species differences and individual differences between plants or animals ,m which they were ultimately derived. After various attempts to estimate the relation (As/Ai) efficiently, a simple semi-

tphical method was gradually evolved. It is now being used on data from various

)ups of organisms in the analysis of variation in natural populations. With the

couragement of several mathematicians, particularly E. B. Wilson and J. W.

[key, I have recently been exploring its general applicability. On a trial basis it s given promising results with data from physiology, morphology, psychology, d linguistics. It is planned ultimately to illustrate the method with a series of joint papers, owing its applications to various fields. In the meantime (since only its special- d use in analyzing population variation has been published) a short, generalized -ount seems desirable.

The method had its inception in an attempt to measure variation in and between mize fields in Mexico.2 Two easily measured features of the maize ear-row mber and kernel width--were plotted on the X- and Y-axes of a Cartesian grid.

ours in this preparation, many free spores were present, and intact asci broke after a minimum

nanipulation. 7 B. Ephrussi, H. Hottinguer, and H. Roman, these PROCEEDINGS, 41, 1065-1071, 1955. S The oil-chamber technique of P. DeFonbrune, La Technique de micromanipulation (Paris: usson & Cie, 1949), was used. 9 The synthetic medium of B. H. Olson and M. J. Johnson (J. Bacteriol., 69, 159-162, 1949) s used. Asparagine was omitted, and biotin and calcium pantothenate were the only vitamins led. This was supplemented as required with amino acids (10 mg/l) or 4-methyl-5-3-hydroxy- yl thiazole (0.1 mg/l). 0 M. Ogur, G. Lindegren, and C. C. Lindegren, J. Bacteriol., 68, 391-392, 1954. This medium s modified for replica plating by omitting the indicator, reducing the glucose to 0.025 per cent, i adding agar. 1 Kindly supplied by Dr. S. Spiegelman as 33-E-26. 2 B. Ephrussi, P. L'Heritier, and H. Hottinguer, Ann. Inst. Pasteur, 77, 64-83, 1949. 3 R. A. Fisher, Statistical Methods for Research Workers (Edinburgh: Oliver & Boyd, 1944). 4 R. E. Wright, Extranuclear Inheritance in Yeast Heterokaryons (doctoral dissertation, Uni-

sity of Wisconsin, 1957) (University Microfilms, 313 N. First St., Ann Arbor, Michigan).

A SEMIGRAPHICAL METHOD FOR THE ANALYSIS OF COMPLEX PROBLEMS

BY EDGAR ANDERSON*

NRY SHAW SCHOOL OF BOTANY, WASHINGTON UNIVERSITY, AND MISSOURI BOTANICAL GARDEN,

ST. LOUIS, MISSOURI

Communicated, August 26, 1957

Science and technology have many problems difficult to record, measure, or

alyze because many variables or complexes of variables have an important bear- r on the end result. It is difficult to measure and to get in mind the various inter-

ationships of several sets of facts for each of a considerable number of individuals.

lis is particularly true when some or all of the basic data are complex patterns, ficult or impossible to code in numbers. Such a problem is the relation (As/Ai) tween species differences and individual differences between plants or animals ,m which they were ultimately derived. After various attempts to estimate the relation (As/Ai) efficiently, a simple semi-

tphical method was gradually evolved. It is now being used on data from various

)ups of organisms in the analysis of variation in natural populations. With the

couragement of several mathematicians, particularly E. B. Wilson and J. W.

[key, I have recently been exploring its general applicability. On a trial basis it s given promising results with data from physiology, morphology, psychology, d linguistics. It is planned ultimately to illustrate the method with a series of joint papers, owing its applications to various fields. In the meantime (since only its special- d use in analyzing population variation has been published) a short, generalized -ount seems desirable.

The method had its inception in an attempt to measure variation in and between mize fields in Mexico.2 Two easily measured features of the maize ear-row mber and kernel width--were plotted on the X- and Y-axes of a Cartesian grid.



Le individual dots of the resulting scatter diagram were replaced by precise but

nipictorial glyphs recording kernel shape and kernel texture. In analyzing riation in natural populations, these more or less pictorialized glyphs were re- bced by generalized glyphs,3 on each of which the variation of seven or more riables could be recorded. Subsequently1, 4these glyphs were modified to permit )re efficient scanning of the "pictorialized scatter diagrams" in which they were ad. It has since been realized that the glyphs (used quite apart from the scatter

Lgrams) can be arranged according to sequences in time, space, or development. .ey can be averaged or transferred from a scatter diagram to a map by making propriate modifications. It seems likely that, with their continued employment logical operations, various other uses for them may be found. It therefore be- nes convenient to have a generalized name for the glyphs apart from their use in ,torialized scatter diagrams, and they are accordingly named metroglyphs. rhe use of metroglyphs is illustrated in a simple generalized way in Figure 1. ven (upper left) four individuals-1, 2, 3, and 4-each of which has been measured

QUALITIES U

A ,B C D E . _ .

LOW W(D MEDIUM HIGH LOW LOW (2) 1 r, < P;

2HIGH HIGH HGH HIGH MEDIUM(4)

'3 MEDIUM?)MEDIUM HIGH MEDIUM LOW (I) j4 LOW () LOW MEDIUM MEDIUM LOW (I)

10

9

8

6

5 3 9 5 2 j 4

r3

21 7 7 FF1 1 1 31

2 3456789 10 0 2 3 4 5 6 7 8 9 10

QUALITY E INDEX FIG. 1.

r ranked in one of three grades) for each of five qualities-A, B, C, D, and E. ie key for coding this problem is shown in Figure 1, upper right, and the resulting atroglyphs for the four individuals are shown at right center. Each of the quali- s is diagramed by a ray, the rays for any one quality having the same position on



h glyph. For each quality a long ray indicates that the individual is high for t quality, a short ray that it has a medium value; no ray at that position indi- es that the individual in question has a low value. In other words, each glyph ;his problem, whatever the number of its rays, is diagraming five facts; a glyph h no rays indicates low values for all five variables. It is for this reason that

rays are placed at easily visualized and remembered points. There are three as from the upper pole of the glyph, two others at its equator to right and left. s then easy to train one's self to see the glyph as a whole and to take in all the ormation from it. The operator's eyes look at the glyph of number 1, and al- st immediately his mind interprets it as low A, medium B, high C, low D, low E. will also see, almost at a flash, that the total magnitude of the glyph, consider- all five qualities at once and giving them equal weight, is about one-third of the

y from the minimum total (low A, low B, low C, low D, low E) to the maximum al value (high A, high B, high C, high D, high E). 3y assigning values of 2 for each long ray and a value of 1 for each short ray, it is ssible to assign scores to each glyph and to arrange them as a frequency diagram in index (Fig. 1, lower right). The minimum total value per glyph (a glyph with

rays) is 0; the maximum total value (a glyph with five long rays) is 10. If re are logical or practical reasons for so doing, the index may be weighted, assign- different values to different rays.

in working with these glyphs, it has gradually been realized that they are a de- e for helping the eye to aid the mind. For maximum efficiency it is best to ke some concessions to the eye. The mind can be trained to adjust itself more

dily than can the eye. For instance, the medium values were originally scored

;h a ray which was made exactly half as long as the ray for the high values. It s learned from experience that the eye could do a better job, particularly in )blems involving large numbers of glyphs, if, in scanning the diagram, it never i to stop to decide whether a ray was medium or long. Therefore, it is better to ve the long rays at the very least almost three times the length of the short ones. r the same reason, in all problems involving much scanning, it is better to code

a data so simply that the eye can be trained to take in each glyph almost instan-

leously. Four examples of this principle will be discussed in order-quartiles, mber of rays, ray positions, accessory data, glyph variations. Quartiles.--Since the analysis of variability by medians and quartiles is a well- ablished technique, it was originally attempted to set the rays for such analysis, ling them as follows: no ray, values below lower quartile; very short ray, values ;ween the lower and the medium quartile; medium-short ray, values between the dium and the upper quartile; long ray, values above upper quartile. It was ind that, with any considerable number of glyphs, even a trained eye could not

ilyze the interaction of variables with this system as well as with the simpler one tlined above (but see below under "Glyph Variations"). Vumber of Rays.--Though, theoretically, there nlight be a very large number of rs, the eye works most efficiently with no more than three to seven, depending on a eye. If the problem has many important variables, it is better to take the four five which are most closely correlated and turn them into an index, as described >ve. This index can then be coded on- a ray in position C or it may be used on a X- or Y-axis of a pictorialized scatter diagram.



Ray Positions.-Originally, rays were used around all sides of each glyph. This ifuses the eye. By slanting them all in approximately the same direction, it is ssible to scan large complicated diagrams more efficiently. By using these ,ndardized positions in problem after problem, there is considerable carry-over m one problem to another. The data can be interpreted more efficiently by an )erienced analyzer if he is using the standard with which his eyes are already niliar.

4ccessory Data.-Many biologists, when first using this method, are apt to put various kinds of accessory data as little projections of one shape or another. ese all hinder the eye enough to reduce efficiency. If there are two kinds of in- [iduals which one wishes to distinguish, one kind may be given a white dot and

a other one a black dot. Other accessory information which one wishes to try t in one way and another is best managed by tiny pencil-point dots placed im-

idiately to the left or the right of the base of each glyph. They will scarcely in- fere with the scanning, and their association with the other variables can be rked out by the eye. Glyph Variations.-In so far as possible, it is better to vary the glyph from prob- i to problem by adapting the coding to the problem as much as possible and to

ange the glyph system as little as possible (see remarks under "Ray Positions"). r instance, if quartile data need to be worked with, one can diagram three vari- .es in quartiles-one variable at the left equatorial position, one at the right, d one at the upper pole. For each variable, one uses two rays. At the pole one as positions B and D, omitting C. At the equatorial positions one uses A and E d adds at each a twin horizontal ray originating at the equator. For each riable the four quarters of the data are diagramed as follows: no ray, one short

r, two short rays, two long rays. It has been found that to the eye the jump im no ray to one short ray is about as significant as that from two short rays to o long rays. If one needs to diagram certain variables in more than three or four categories, can use even more rays. A simple example is shown in F'igure 2, where two vari- les are coded in values from 1 to 10 each-one variable on rays slanting to the

ht, the other on rays slanting to the left.

A O 6 b tt t

B O 0 odd d 0 I 2 3 4 5 6 7 8 9 10

I 2345 A 9 2 4 8 0 B I 8 6 0 10 . . .

FIG. 2.-Top: variables A and B are coded in values from I to 10 each. Lower left: the scorinr five iindividuals with respect to A and B. Lower right: the corresponding metroglyphs fog Gh of the five individuals.



For problems in which the glyphs are merely employed in serial order, it is not

important to keep the rays down to two lengths. They may then be diagramed )portionate to the ranging of each variable or to the actual measured values. When going back and forth from glyphs used in serial order or on a map to

,phs used on a Cartesian grid, two rays may be qualities diagramed on X- and

axes, or measures on X- and Y-axes may be replaced by rays of appropriate gths. In Figure 1 at the lower left, individuals 1, 2, 3, and 4 are transferred to a rtesian grid. Qualities A and E (each measured on a scale of 1-10, using the

ginal values given in parentheses in the upper-left-hand corner of Fig. 1), are em-

)yed to locate each glyph on the scatter diagram. It will be seen that these

,phs are the same as those in the figure to the right, except that the ray at posi- n A has been replaced by the position of the entire glyph on the Y-axis, while the r at position E has been replaced by the position of the glyph on the X-axis. It i1 be noted that, as far as one can judge from four individuals, variables B and C a associated with each other, as are B and D; that B, C, and D, separately and in

nbination, are associated strongly with variable A and slightly with variable E. In attempting to work out complexes of related qualities, the analysis is facili- ;ed if the ray lengths are coded in such a way that all the extreme values charac- -istic of one complex are assigned long rays and tlhose characteristic of the other a assigned no rays. For example, in studying hybridization between two sub- acies of Campsis, one of the subspecies had a short tube, a wide limb, and much I in the flower; the other had a long tube, a small limb, and little red. Redness d limb width were coded with long rays for much red and for wide limbs, tube

gth was coded in reverse with a long ray for short tubes. This meant that those brids closely resembling one parent were coded as three long rays and those most

,sely resembling the other parent as a rayless dot.

I am indebted to the Guggenheim Foundation and to Princeton University for fellowships ich made it possible to explore the possibilities. E. Anderson, Biol. Revs., 28, 283-289, 1953. E. Anderson, Ann. Missouri Botan. Garden, 33, 175, 1946. E. Anderson, Introgressive Hybridization (New York: John Wiley & Sons, 1949), pp. 94-99,

E. Anderson, "Efficient and Inefficient Methods of Measuring Specific Differences," in Statis- and Mathematics in Biology, ed. O. Kempthorne (Ames, Iowa: Iowa State College Press

;4), Chap. VI, pp. 98-101.

'RMINA.L CONTROL, TIME LAGS, AND DYNAMIC PROGRAMMING

BY RICHARD BELLMAN

RAND CORPORATION, SANTA MONICA, CALIFORNIA

Communicated by Philip M. Morse, August 30, 1957

1. Introduction.-A type of control process that is common to economic and

5ineering fields is that of maximizing a functional of the form

J(v) = f0T h(x)dG(t) + f0T k(v)dt, (1.1)

ar all vector functions v(t) satisfying constraints of the form

For problems in which the glyphs are merely employed in serial order, it is not

important to keep the rays down to two lengths. They may then be diagramed )portionate to the ranging of each variable or to the actual measured values. When going back and forth from glyphs used in serial order or on a map to

,phs used on a Cartesian grid, two rays may be qualities diagramed on X- and

axes, or measures on X- and Y-axes may be replaced by rays of appropriate gths. In Figure 1 at the lower left, individuals 1, 2, 3, and 4 are transferred to a rtesian grid. Qualities A and E (each measured on a scale of 1-10, using the

ginal values given in parentheses in the upper-left-hand corner of Fig. 1), are em-

)yed to locate each glyph on the scatter diagram. It will be seen that these

,phs are the same as those in the figure to the right, except that the ray at posi- n A has been replaced by the position of the entire glyph on the Y-axis, while the r at position E has been replaced by the position of the glyph on the X-axis. It i1 be noted that, as far as one can judge from four individuals, variables B and C a associated with each other, as are B and D; that B, C, and D, separately and in

nbination, are associated strongly with variable A and slightly with variable E. In attempting to work out complexes of related qualities, the analysis is facili- ;ed if the ray lengths are coded in such a way that all the extreme values charac- -istic of one complex are assigned long rays and tlhose characteristic of the other a assigned no rays. For example, in studying hybridization between two sub- acies of Campsis, one of the subspecies had a short tube, a wide limb, and much I in the flower; the other had a long tube, a small limb, and little red. Redness d limb width were coded with long rays for much red and for wide limbs, tube

gth was coded in reverse with a long ray for short tubes. This meant that those brids closely resembling one parent were coded as three long rays and those most

,sely resembling the other parent as a rayless dot.

I am indebted to the Guggenheim Foundation and to Princeton University for fellowships ich made it possible to explore the possibilities. E. Anderson, Biol. Revs., 28, 283-289, 1953. E. Anderson, Ann. Missouri Botan. Garden, 33, 175, 1946. E. Anderson, Introgressive Hybridization (New York: John Wiley & Sons, 1949), pp. 94-99,

E. Anderson, "Efficient and Inefficient Methods of Measuring Specific Differences," in Statis- and Mathematics in Biology, ed. O. Kempthorne (Ames, Iowa: Iowa State College Press

;4), Chap. VI, pp. 98-101.

'RMINA.L CONTROL, TIME LAGS, AND DYNAMIC PROGRAMMING

BY RICHARD BELLMAN

RAND CORPORATION, SANTA MONICA, CALIFORNIA

Communicated by Philip M. Morse, August 30, 1957

1. Introduction.-A type of control process that is common to economic and

5ineering fields is that of maximizing a functional of the form

J(v) = f0T h(x)dG(t) + f0T k(v)dt, (1.1)

ar all vector functions v(t) satisfying constraints of the form



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A Semigraphical Method for the Analysis of Complex Problems