bertin - matrix theory of graphics - idj - ocr

8/11/2019 Bertin - Matrix Theory of Graphics - IDJ - Ocr

http://slidepdf.com/reader/full/bertin-matrix-theory-of-graphics-idj-ocr 1/12

Matrix theory of graphics

The ‘matrix theory of graphics’ is based upon Sémiologie

g r a p h iq u e . It developed progressively after the publication of

L a G r a p h iq u e e l l e tr a i t e r n e n l g r a p h iq u e d e ¡ 'i n f o r m a t i o n in

1977. Since then, the theory has evolved. Some basic concepts

changed. Pertinent examples were reinterpreted more in

depth. Finally, it was necessary to restructure the whole to

underline its unity and its essential tenets. It had to be sim-

pler, probably more logical and more didactical. This article

summarizes this new structure.

Definitions

Graphics uses the properties of the visual image in order to

make relationships of difference/ similarity, order or propor-

tion appear among data. This language covers the universe of

diagrams, networks and topographies.

Graphics is applied to a set of data after it has been defined

'the table of data’, and thus it constructs the rational under-

pinning of the world of images within the logical classifica-

tion of fundamental sign systems (see Diagram 1).

Graphics pursues two goals:

to process data for understanding and extracting infor-

mation.

to communicate, i f necessary, this information or an

inventory of basic data.

The matrix theory, based on Sé m io lo g ie G r a p h iq u e , con-

structs a homogeneous and coherent system for the analysis

of the graphic language, its use and pedagogy.

It is essential to avoid any confusion between GRAPHICS

( l a g r a p h i q u e ) which processes only rigorously predefined

data sets (the data table) and GRAPHIC DESIGN whether

figurative or abstract which acts according to its own rules

within its own definition of the graphic world. Graphics is a

tool that obeys universal laws that are unavoidable and

undisputable but can be learned and taught. Graphic designas an art is free, but also subjective.

Diagram 1. The written transcriptions of music, words and

mathematics are techniques of memorizing fundamentally sonicsystems, thus keeping the linear and temporal character of these

systems. Through the telephone, the car can hear an equation,

but not a map.

Jacques Berlin



Natural properties of the graphical image

The three dimensions of the instantaneous image

In the plane, a mark can be on the top or the bottom, to the

right or to the left (1). Human perception constructs in theplane two independant dimensions, X and Y, distinguished

orthogonally. A variation of light energy (2) creates the third

4 dimension Z, which is independent of X and Y.

The image is a meaningful form that is perceived instan-

taneously and is created within the three dimensions X,

Y,Z (3). It can thus transmit the relationships between

three independent data sets.

The visual variables of the image are thus the X and Y dimen-

sions of the plane and for the Z dimension, the size or the

value of marks.

The properties of the plane: Points or lines.

Network or matrix

A datum' is a relationship between two entities. Correspon-

dingly, the plane offers us points and lines. Consequently,

one can represent entities by points and relationships by

lines (4): one then constructs a NETWORK . The X and Y

dimensions of the image arc not significant.

one can represent entities by lines and relationships by

points (5): one thus constructs a MATRIX. In that case,

the dimensions X and Y each have their own significance.

While any set of data can be constructed in these two ways,

each of the two construction types has its own properties.

The network describes the relationships between the ele-

ments. It is the best way to transcribe the topographic order,

but it is useless when transcribing reorderable data sets.

How, for example, could one discover in (6) the deviant

relationship which appears instantaneously in (7)? The

matrix construction is the basic construction of graphics. Its

three independent dimensions furnish the underpinning to

try and understand data, strengthened by the universality ofthe ‘double entry data table’ and the reordering applied to

data classification.



Fixed or transformable image?

Let us consider the data table (8), which shows the presence

of the products A, B, C ... in the countries 1.2.3 ... As given

in (9) or its one to one graphic translation it presents adaunting effort of analysis. But simply by displacing country

2 and product D we already discover groups of similar

elements (10) and we are able to reduce the 25 elements to

three groups that are characteristic of this data set.

This internal transformation of the image, obtained by

the permutation of rows and columns, based on the universal

principle of proximity resemblance, defines the ‘reorderable

matrix’ as a basis of the graphic data processing. The permu-

tations are symbolically represented by (11).

The properties of Z: Order, associativity, selection

Limitations of the image, and layering problems

Layering graphical images is like superimposing photographs:

the films mix and the images destroy each other. The image

has only three dimensions. How can we represent various

properties on a ‘map’, that is on a fixed XY plane, and still

separate their images? This is the problem of the selectivity of

visual variables.(12) The variables of the image are ordered (O) (A precedes

B). The size, as the plane, has the ability to show ratios (Q)

(A is n times B). In any combination of variables, size and

value are the variables which define order (by variation of

light energy) prior to the other variables. Size and value are

said to be dissociative.

(13) The other variables have a constant visibility and do not

disturb the action of the rest. They are said to be associative

(a) (A can be seen as similar to B). They are used to separate

elementary images.

(14) All variables are selective (*) (this is different from that),

but they arc so to various degrees (see page 17). The plane is

uniquely endowed with all perceptual properties.



The matrix theory of graphics

Why Graphics? Demonstration by example

The aim of Graphics is to understand the essence of data by

transforming them. Maps, diagrams: these are documents we

fire questions at. The data matrix (15) for example, which

shows the meat production of 5 countries, can be interrogat-

ed along three axes: X I am interested in one type of meat,

how does it fare in one or the other country? Y I am inter-

ested in a certain country, what is its meat production? Z

where do I find high percentages? But in every axis, the

questions range from the elementary to the global.

The elementary questions: The question: ‘In Italy, how much

pork?’ is answered by the content of the cell. At this level, it is

the only kind of answer we can memorize; the total content

of 25 cells is too much to absorb.

But to understand is to synthesize all the data. If we want

to come to a synthesis, we must condense the data into

groups of similar elements, and try to reduce the number of

groups as much as possible. This is the objective of the pro-

cessing of data, whether it be mathematical or graphical.

The global question: What patterns can we extract from the

data in X and Y? This is the essential question. The answer is

revealed by the construction (16), the reorderable matrix,

within which columns and rows have been pcrmutated to

create a meaningful pattern where the data (15), that is the 25

figures contained in the cells, are made visible as two groups

of countries A and B with contrasting structures. This is the

first ‘information’.

Country C is an exception. It does not fit in any group. But

this exception is important, because in this particular data

set, and in a situation where all partners are equal, it is the

answer per country that counts. The information extracted

from the patterns is not perceivable in the data table (15), nor

in any other construction (see (17)). And yet, this is the

second ‘information’. Graphical and algorithmic information

processing precedes interpretation and makes it valuable. On



THE REORDERABL E MATRIX (18) answers all types and

levels of questions. It is the basic construction of Graphics.

This construction embodies for the graphics operator the

optimal properties of the image. It allows a chain of logical

operations: data matrix reduction exceptions discus-

sion communication.

It structures reflection, gives a sense to computer manip-

ulation and by distinguishing differences, characterises

specific cases.

The choice of a graphic construction

The synopsis of useful graphic constructions

THE SY NOPSIS (19) classifies the useful constructions accord-

ing to the properties of the data table.

It indicates the construction best suited to each case and,

inversely, helps to define the data table corresponding to a

construction.

Given a data table within the X dimension items A,B,C

... and in the Y dimension, the variables 1,2,3, we canobserve:

1) the number of variables, 2) the ordered properties of the

item: i.e. ordered (o) or rcordcrable (*) of the items. These

are the two principles of the diagram classification. The

relationships between items define the networks.

More than 3 variables

Items * ( rcordcrable)

(1) Rcordcrable matrix. This is the basic construction.

Items O (ordered)

(2) Imagefile (data noted graphically on sets of records).

Permutation along Y only. Allows a maximum quantity of

data.

(3) Array of curves, when the slopes of the curves are mean-

ingful.

(4) Classifiable sets of ordered tables or 'maps’, for example

maps of sounds, of colours).

(5) Sets of geographical maps presenting one attribute.

3 variables and fewer

Each variables takes hold of one dimension of the image.

Patterns appear directly on the ordered table.

(9) (10) (11) scatter plots with 3 or 2 properties

(12) distribution of one variable.

Rcordcrable networks

The transformations of these networks are meant to simplify

the image, but they are limited by the number of items.

Ordered networks

(21) topography, basic maps(18) maps for one variable

(17) (16) exhaustive sets of chromatic superimpositions

(5) map sets

exhaustive sets of superimpositions (inventories)

simplified superimpositions (syntheses)

Transmitting information to ‘others’Graphic communication

Graphic communication is the best known function of

graphic representations. But should one communicate only

the elementary data as do classical constructions, or should

one rather communicate the means of ’understanding’?

Useful graphic representation, of course, enhances under-

standing. Its images are the simplest possible and there is noreason to emphasize groups unless the optimised image

remains complex.

But any optimisation is subject to discussion. Should one

favour discussion and retain the raw data or should one let

them disappear into better patterns one can easily capture,

which are more evident, but are then beyond discussion?

This is the real dilemma of scientific communication, of

dendrograms, multivariate clouds, cartographic models,

which make the raw data disappear and thus preclude critical

analysis.



Graphic representation fulfills also the / unction of reposi-

tory. This function is the characteristic of many topographic

representations and as such it maintains and rightly so

the questions at the elementary level. It also justifies fixed

orders alphabetic or chronologic which make searches

easier. It excludes unordered lists which compel the reader tobrowse everything until the search item is discovered.

Schematic representation of the graphic image

When graphics are used as a tool for information processing,

the ‘sender’ and ‘recipient’ of information arc either one and

the same person, or two ‘actors’ who formulate the same

basic questions. Because of this, they do not fit into the

diagram of polysemic communication, where we have

sender <» code «» recipient (diagram A)

Instead we have the monosemic diagram:

actor three relationships *, O, Q,

where *, O, Q, are relationships of similarity and of order

which allow the reduction of data. These relationships are

not submitted to conventional coding, as they are expressed

by visual variables which have corresponding properties.

Diagram A applies only when using language to answer the

first question.

The diagrams

Analogy and complementarity of algorithmic and graphic

information processing

We have a data set of 59 Merovingian artifacts, described

according to 26 characteristics (20). The data are at firstrearranged using three algorithmic techniques: automatic

classification (AC), multivariate analysis (MV) and hier-

archical analysis (HA). The images obtained differ. We have

to interpret the results. (Note: The Merovingian dynasty

reigned in France from the 5th to the 8th century.)

In order to interpret, we apply hierarchical analysis HA in

a first step. As part of the visual classification VC 1we insert

separation lines and isolate a subset (a). VC 2 simplifies (a)

by inverting the first three columns and reordering subset

(b). VC3 shifts (b) into (a). VC 4 simplifies VC 3 by creating

an evolutive pattern and by isolating exceptional artefacts

and characteristics, all of which can be clearly analysed.

A special graphical device: The imagefile

.An experimental tool. By placing in X a component with a

fixed order (in the present case, time), this device eliminates

one axis of permutation and thus simplifies the graphic infor-

mation processing.



The homogeneity of a collection of insects

An experiment takes place in three connected rooms: a light

one, a dimly lit one, and a dark one.

For each insect, the time (Tl) and (T2) spent in the two

first rooms before reaching the dark room is measured in

5 minutes chunks over one hour.

The experiment is repeated 12 times.

The problem is to discover whether:

the 12 experiments are comparable

the 8 insects produce distinguishable types of behaviour

there are diverging patterns.

(a) constructs the imagefile. It puts in X the time quantities

and in Y the 8 insects (*) X 12 experiments (at)(b) constructs one image for each experiment, based on the

classification of insects ABCDEFGH. The experiments appear

to form 2 groups. Experiments 5 and 11 differ from the ma-

jority. They are extracted and studied separately.

(c) constructs one image per insect based on the order of

experiments. Three types appear: slow, quick, chaotic.

(d) orders all the time measurements from the longest to the

shortest. Three thresholds appear: 10, 20 and 45 minutes.

Many other conclusions are possible: see also Graphics and

Graphic Information Processing, p. 78.



The reorderable networks

Nerwork graphs and flowcharts transcribe in the plane the

relationships (connectors) between objects (points). Process-

ing them implies simplifying the image (A) by suppressing

meaningless line crossings.

The following step (B) creates meaningful groups. Step

(C) attributes a meaning to the X and Y coordinates of the

plane. When the number of elements increases, these opera-

tions quickly become too complex. It is then necessary to

apply matrix processing (D) or matrix algebra computa-

tionally.

Map constructions (Ordered networks)

I n t h e c o n te x t o f g r a p h i c i n f o r m a t i o n p r o c es s in g , t h e a n c h o r i n g

i n t h e p la n e d e fi ne s t h e t o p o g r a p h i c im a g e s a n d t h e i r s p e c if ic

p r o b le m : t h e v i s u a l s e p a r a t i o n o f s u p e r im p o s e d v a r ia b l e s . T h e

so lu t ion var ies accord ing to the leve l o f the re leva n t ques t ions

a n d im p le m e n t s t h e l a ws o f s e l e c t i v it y a p p l i e d t o t h e v i s u a l

v a ri ab le s , t h e i r i m p l a n t a t io n a n d t h e p e r c e p ti o n o f e l e m e n t a r y

gestalt.



H e r e, t he a d e q u a te s e l e c t iv i ty o f th e o r i e n t a t io n o f p u n c t u a l

s igns a l lows the appearance o f reg ion a l g roup ing s, i .e . an an swe r

a t t he m e d i um l eve l.

Basic questions

Let us consider a cartographic problem with n characteristics,

and the corresponding data table containing in X the geo-graphic items and in Y the attributes. The first basic question

(what are X, Y, and Z) determines the contents of the map

legend. To answer to the other questions (what are the

groups, what are the exceptions...) we have to study the

attribute values assigned to the geographic items, discover

similarities and look at regionalisations, in order to answer to

the question ‘where is such and such an attribute to be

found?’ (A). Moreover, as an inventory, the map should

answer the question 'what is there at a given place?' (B). The

answers vary according to the constructions.

A o n e -a t tr ib u te m a p (1) answers the two types of questions.

The problem it creates is the representation of quantities in

Z. When the wrong choice is made as in (2) a representa-

tion lacking any order a type B question is the only one to

be answered.

A collection of oneattribute maps (3) will answer only a type

(A) question, but can be ordered in many ways.

A m a p w he re a l l a tt r ib u te s a re s u p e ri m p o se d as in (5) answers

only a type (B) question, as does a map like (4). Superimposi-

tion raises the problem of selectivity. If we want to answer all

questions in full, we have to construct both the collection (3)

and the superimposition (4).

A simplified map or synthetic map (7) (9) (10) is meant

to answer to all questions, but it does so by abandoning the

completeness of the collected data. It illustrates the dilemma

of choosing between different kinds of data processing: by

reordering a matrix (6) or by a purely cartographic represen



tation as in (8). It also evades the critical questioning of re-

gional groupings once the original data have been obliterated

by the synthesis (10).

The representation of quantities in the Z dimension(Semiology of graphics: Diagrams, Networks, Maps. p. 366)

Equalizing classes

Around the earth, the population of a country is related to its

area. Similarly, in statistics, the population of an age class

depends upon its bounds. In cartography as in statistics, it is

necessary to neutralize or to equalize classes in order to avoid

erroneous representations. This operation can be implement-

ed mathematically (ratios, percentages, indices) or graphical-

ly (grids).

Using size variation

In map (1) the higher land prices are immediately perceived.

It is a map for ‘seeing’. In map (2) the reader does not see

them. It is a map for ' reading', as is map (4).

Variations of levelling

When representing quantities in Z, one answers two ques-

tions: what are the characteristic thresholds of the distribu-

tion? At what level does the significant image appear (similar



to what other pattern, unifying ‘islands', covering a given

area)?

A large body of literature underscores the difficulty, if not

the impossibility, of answering both kinds of questions

within one map. The ability of replacing incremental varia-

tion by computed continuous variation of levell ing introduc-

es the possibility of an efficient solution.

Selectivity

(Semiology of graphics: Diagrams, Networks, Maps. p. 67;

Graphics and Graphic Information Processing, p. 213).

Selectivity is of importance in superimpositions and is in fact

defined by its opposite: it is what remains when ignoring the

rest.

Given an equal amount of reflected or transmitted light

for all shapes, the selection of squares in map (2) boils down

to ignoring all other shapes. This is an impossible task for the

human eye. Shapes do not induce selectivity.

When the perception of marks depends upon the varia-

tion of luminosity, the selection of dark marks amounts to

see light ones as a common background, which is immediate

in map (1).

Better selectivity is guaranteed by

difference of intensity: size and value, when they do not have

an ordering significance.

difference of implantation which allows the superimposition

of marks shaped as points, lines and zones.

colour, but only in as far it is not neutralized by the size of

marks. Minute red and green marks cannot be perceived as

having different colours when they arc seen at a distance,

whereas on the large surface of a wall, the eye can distinguish

up to a million of colours. The use of colour as a distinguish

ing variable is thus to be avoided for small sizes of implanta-

tion.

grain, combined with zonal implantation (3 levels)

orientation (see map 5) with marks spread as points (up to 4

levels) and with linear marks (2 levels).

Shape does not have any selectivity to speak of within any

kind of implantation, whether points, lines or zones.

The invention of the data table

What data table should one construct?

The matrix analysis of a problem helps answer this question,

and leads to three successive steps of reasoning.

I. The problem is translated into simple questions. The list of

all relevant attributes and elements should be constructed

freely and without technical constraints. One then notes the

span of the set of elements and of their relationships. This is

the 'apportionment table'.



2. Imagine the ideal homogeneous table containing the

maximum number of elements composing the list. In other

words, what is to be put in X (whatever its length) to get the

largest possible number of attributes in Y? Estimate the

extent of the work, the availability, the time and the means

involved. Trace the possible condensation of data by aggrega-

tion or by sampling and interpolation. This is the ‘homoge-

neity table’. The result is a graspable and usable table.

3. Verify the relevance of this table by noting in the margins

the correspondencesand the relationships defined in the initial

questions. This is the ‘pertinency table’, which specifics the

final data table (Graphics and Graphic Information Processing.

p. 233). This study of course precedes the data processing as

such, but cannot be conducted appropriately without knowl-

edge of mathematic and graphic data analyses and of their

methods.

Power and limitations of graphics

The three dimensions of the image impart a great power tohuman visual perception and could give graphics special

impact as an efficient pedagogical tool. This tool can, from

primary education on, translate information problems into

concrete instruments of reasoning and decision. Thanks to its

permutations, modern graphics materialises notions that

would otherwise stay abstract:

Graphics gives a visible shape to the steps and theoperations

of a research process, and in doing so organizes the work flow.

It gives data materiality and underscores the problems

raised by the design of the initial table, which is purely a matter

of creation, outside any computational setting. These prob-

lems are expressed by the question 'what is to be pul in X?’

It materialises the concept of 'data analysis', and renders it

more graspable in its graphical form than in its mathematical

form.

It underscores that work is only scientific i f its assumptions

are justi fied by the rigorous treatment of an explicit datatabic. Outside such a rigorous process, it is only a matter of

personal opinions.

Graphics renders visible the notions of discussion, reasoning

and understanding, notions which are determined by the level

of relevant questions.

But the image has only three dimensions. The consequencesof such a limitation are probably beyond our imagination, as

we are immersed in this natural situation.

While mathematical analysis introduces n dimensions, the

input listings are still expressed in one table with X, Y and Z

dimensions. And when we want to see the computational

results, we still do it through an image... which has still only

three dimensions, the fourth being time, which we sought to

downplay in the first place.

Thus interdisciplinary research will remain difficult, as the

geographers puts the space into the X dimension where the

historian puts time, the psychologist puts individuals, and

the sociologist puts social categories. Each of them is certain

of embodying 'scientific synthesis’, without being aware that

each discipline, each research centre is itself defined by its

own X and Y components which characterize its field of

information. It is the absence of a 4th dimension in the image

which in fact prohibits the birth of a scientific synthesis freeof disciplinary constraints.

Thus one can demonstrate the limitations of rationality. A

well justified information processing can only exist within the

frame of a finite set of data: the data table. But there is an

infinity of finite sets.

However powerful our rational efforts will be, they will

always be swept away in the infinity of irrationality.

(Translated by MynamDaru)

Note

SG refers to Sfmiologie graphtque (1967). The English transla-

tion is Semiology of graphics: Diagrams, Networks, Maps. Madi-

son: University of Wisconsin Press (1983).

GR refers to La graphique et le traitement graphtque de Vinfor-

mation (1977). The English translation is Graphics and Graphic Information Processing IGGIP) . Berlin: Walter de Gruyter

(1981).

bertin - matrix theory of graphics - idj - ocr

Documents