LINEAR ALGEBRA AND LEGISLATIVE VOTING BEHAVIOR : RICE'S INDICES"

THOMAS w. CASSTEVENS

Oakland University

PREFACE

T egislative voting behavior has been extensively researched by L behavioral scientists using a variety of quantitative indices and mathematical techniques. Some of these indices and techniques have been used for decades, but their essentially mathematical properties have been subjects of debate or have not been thor­oughly explored. Stuart Rice's index of cohesion and index of like­ness, for example, have been used continuously since the 1920s but have rarely been investigated from a mathematical point of view.'

Despite continuous use since the 1920s, Rice's indices suffer from a well-known flaw, viz., legislators who neither vote aye nor vote nay are omitted. This shortcoming hardly needs elaboration, but two aspects deserve notice. ( i) Empirically, the number of omitted legislators may be rather large in relation to the total size of the party or other subset of legislators being studied. An extreme example is furnished by the 1910 British House of Commons. Dur­ing the session of July 18, on division number 106, the vote was 49 ayes, 1 nay, and approximately 620 MPs did not vote. (There were 670 seats in the House of Commons, but a few were vacant as a re­sult of deaths, resignations, etc.) The low turnout may be amus­ingly explained by the fact that the division occurred at 3 A.M. in the morning of July 19. During the 1910 Parliament, however,

0 With the exception of Figures 1 and 2 and related text, tl1is paper was written while I was on leave from Oakland University as a NSF Science Faculty Fellow and a Visiting Fellow, Mathematics Department, Dartmouth College. M. K. Bennett, D. Roeder, M. Vitale, and the students in Mathe­matics 36 (Mathematical Models in the Social Sciences) were helpful in diverse ways, but the paper would not have been written without K. Bo-gart's sustained encouragement and friendly criticism. ·

1Rice's seminal paper is reprinted in John C. Wahlke and Heinz Eulau, eds., Legislative Behavior (Glencoe, Illinois: The Free Press, 1959), 372-377. For a survey of the subsequent literature, see Lee F. Anderson, Meredith W. Watts, Jr., and Allen R. Wilcox, Legislative Roll-Call Analysis (Evanston, Illinois: Northwestern University Press, 1966), ch. 3.

[769]

770 THE JoURNAL oF PoLITics (VoL. 32

there were 159 recorded divisions, and in 80 of those divisions, fewer than 335 MPs voted. ( ii) Conceptually, few votes seem to fit the procrustean bed of a nvo-choice model. If a fixed number of votes (either aye or nay) is required for a legislative decision, perhaps the dichotomy benveen aye and non-aye or between nay and non-nay is plausible. For example, at any given time, a fixed number of aye votes is required for the U.S. House of Representa­tives to pass a constitutional amendment, and on occasion, a fixed number of objections is required for the British House of Commons to void an executive order. However, although the actual fre­quency has never been investigated (so far as I know), this type of vote appears to occur relatively infrequently.

This article embeds Rice's indices in a linear algebra structure, generalizes the indices so as to overcome the above shortcoming, and, finally, evaluates the indices from an algebraic point of view. Subsequent papers will apply linear algebra to other indices and techniques used to study legislative voting behavior.2

RICE's DEFINITIONS

Let S be the non-empty set of legislators who vote either aye or nay on a given motion. Let X and Y be non-empty subsets of S.

The index of cohesion, which measures the "cohesiveness" of a political party or other subset of legislators on a given vote, pro­vides a quantitative answer for such queries as: How cohesive is the farm bloc in the U.S. Congress?

Definition 1. (Rice) The index of cohesion for X is the abso­lute difference between the proportion of X's who vote aye and the proportion of X's who vote nay.

The index of cohesion obviously ranges from 0 to 1. The maximum ( 1) occurs if and only if the subset of legislators is either unani­mously voting aye or unanimously voting nay. The minimum (0) occurs if and only if the subset of legislators is evenly divided be­tween the ayes and nays.

The index of likeness, which measures the degree of "likeness"

2For an outstanding introduction to linear algebra for political scientists, see Hugh G. Campbell, Matrices With Applicatiot~.~· (New York: Appleton­Century-Crofts, 1968).

1970] fucE's INDICES 771

between two political parties or a brace of other subsets of legisla­tors on a given vote, furnishes a quantitative answer for such ques­tions as: How much alike are the Republicans and Southern Demo­crats in the U.S. Congress?

Definition 2. (Rice) The index af likeness between X and Y is the complement of the index of difference, viz., 1-index of difference. The index of difference between X and Y is the absolute difference between the proportion of X's who vote aye and the proportion of Y's who vote aye.

For computational ease, in applications, the index of likeness is frequently replaced by the index of difference. When convenient, that practice is followed in this paper. The index of difference ob­viously ranges from 0 to I. The maximum (I) occurs if and only if the subsets of legislators are each unanimous but in different direc­tions. The minimum ( 0) occurs if and only if the two subsets are identically (but not necessarily evenly) distributed between the ayes and nays.

For X and Y, the index of difference is related to the indices of cohesion as follows:

Theorem 1. (Porterfield) The index of difference equals either l/2 the sum of the indices of cohesion or l/2 the absolute differ­ence between the indices of cohesion.3

These definitions are nicely illustrated by a 1918 British exam­ple of some historical importance.• On April 9, Prime Minister David Lloyd George informed the House of Commons that "the Army in France was considerably stronger on January lst, 1918, than on January lst, 1917." On May 7, in a letter to the newspa­pers, General Frederick Maurice stated, 'That is not correct." On May 9, the Leader of the Liberal Party (H. H. Asquith) moved that a Select Committee of the House of Commons be appointed to investigate Maurice's allegations. Asquith's motion was defeated

3F or a detailed proof and extensions, see Thomas Casstevens and Owen Porterfield, "The Index of Likeness as a Mathematical Function of the Indices of Cohesion for Roll Call Voting," Behavioral Science, 13 (May 1968 ), 234-237.

•For source material, see The Liberal ll.fagazine, 1918 (London: The Liberal Publication Departn1ent, 1919).

772 THE JouRNAL OF PoLrrrcs [VoL. 32

by a vote of 108 ayes, 295 nays. (Approximately 2ff7 MPs did not vote. There were ff70 seats in the House of Commons, but a few seats were vacant as a result of deaths, resignations, etc.) The par­tisan breakdown of the ayes and nays was as follows:

Liberal= ( 98 ayes, 71 nays), Labour= ( 9 ayes, 15 nays), Conservative= ( 1 aye, 206 nays), and others= ( 0 ayes, 3 nays).

For the three political parties, routine calculations yield the fol­lowing values to three decimal places, for the indices of cohesion, difference, and likeness:

TABLE 1

PARTISAN INDICES FOR THE MAURICE VOTE

Index of cohesion

Liberal = 0.160 Labour = 0.250 Conservative = 0.990

Index of difference

Lib-Lab Lib-Con Lab-Con

=0.205 =0.575 = 0.370

VECIOR DEFINITIONS

Index of likeness

Lib-Lab Lib-Con Lab-Con

=0.795 =0.425 =0.630

The following properties of real vectors (i.e., vectors whose com­ponents are real-that is, ordinary decimal-numbers) are summa­rized for reference purposes.

Let X= ( x,xz, ... , x.) and Y = ( y,y2 , ••• , y.) be n-dimensional vectors, where n is a fixed integer > 2. Let c be a real number. The sum of two n-dimensional vectors X and Y is an n-dimensional vector, denoted by X+ Y, where

X + Y=(x,+ y.,x2+y2, . .. ,x.+y.).

The product of a real number c and an n-dimensional vector X is an n-dimensional vector, denoted by eX, where

eX= ( cx,,cx", . .. , ex.).

Subtraction is defined in terms of addition and multiplication. The

1970] RrCE's I NDICES 773

difference between two n-dimensional vectors X and Y is an n­dimensional vector, denoted by X-Y, where

X- Y =X+ ( -1 )Y = (X,- y, Xz - y~, ... , Xn- y.).

The inner product of two n-dimensional vectors X and Y is a real number, denoted by XoY, where

X0 Y=x,y1 + x"y" + ... + x.y •.

The norm of an n-dimensional vector X is the positive square root, denoted by I ! X I I· of the inner product of X with itself.

A probability vector is a real vector with non-negative com­ponents whose sum is 1.

Geometrically, a vector is a point or an arrow in space. For the Euclidean space of one, two, or three dimensions, the

notion of the distance between two points is quite intuitive. This notion is generalized and defined as follows:

Definition 3. Let X and Y be n-dimensional vectors. The distance between X and Y is I I X- Y I ], where

II X -Y I J = v' (x1 -y1 )2 + (x1 -y2 )

2 + ... + (x. - y.) 2•

If Y is the origin, represented by the n-dimensional zero vector (O,O, ... , O), then JJ X-Y J f = JfX f ]. Thus,thenormofthe vector X is the distance between the point ( x,, x1, •.• , x.) and the origin; this distance is the length of the vector, when the vector is

. represented as an arrow from the origin to the point. For the Euclidean space of two or three dimensions, if two

arrows are drawn between the origin and two points, then the idea of the angle between the arrows is quite intuitive. This idea is generalized and indirectly defined as follows:

Definition 4. Let X and Y be n-dimensional non-zero vectors. The cosine of the angle between X andY, denoted by cos (X,Y),

. XoY JS-------

1 !XJ J JJ Yif

The cosine of an angle obviously ranges from -1 to 1. In statistics, cos( X,Y) is known as the product-moment correlation coefficient.5

sFor a geometric interpretation of correlation, see Harry H. Harrnan, Modem Factor Analysis (2nd ed.; Chicago: The University of Chicago Press, 1967), 60-62.

774 THE JoURNAL oF PoLmcs [VoL. 32

The application of vector notation to legislative voting behavior is straightforward. Several instructive examples are readily con­structed from the previously cited data about the voting on Asquith's motion in the 1918 British House of Commons. Let the aves and

' nays constitute the universe of discourse, and let X be a two-dimen­sional vector whose components are x, =ayes, x2 =nays. The Liberal vote is conveniently written as the vector L = ( 98, 71) or, alter-

natively, as the probability vector L' = ( 1~9 , 1~t ). The lengths

of those vectors are I I L I 1= 121.017 and I I L' l 1=0.716. The Conservative vote is represented by the vector C=(1, 206) or, al-

ternatively, by the probability vector C' = ( 2~7 , ;J,~ ). The

lengths of those vectors are I I C I I= 206.002 and I I C' I I= 0.995. The distance between L and C is J J L- C I I= 166.235, but the distance between L' and C' is 1 ! L' - C' I I= 0.813. The cosine of

L•C the angle between L and C is --:-I-:-I __ L_,..I.;I:........:;::..I .,.....1

C:::-:-l ..,..1- = 0.591, and

between L' and C' is I L' • C' I I C'T I 0.591. I L' I I

For obvious reasons, in this paper, this type of applied vector is called a voting vector. A voting vector may have more than two dimensions. For example, if all the MPs constitute the universe of discourse, then the total vote on Asquith's motion is represented by a three-dimensional vector T = ( t1 =ayes, t, =nays, t3 =others) and, approximately, T= (108, 295, 267). Any finite number of dimen­sions is readily represented in vector notation.

N-oiMENSIONAL GENERALIZATIONs OF RicE's DEFINITIONS

Since Rice's indices are defined in terms of ayes, nays, and pro­portions, two-dimensional probability vectors are their algebraic model. The broad strategy for generalizing the indices is to con-

1970] RicE's INDICES 775

struct a suitable model from n-dimensional probability vectors. If any number of components is mathematically possible in a voting vector, then no relevant legislator need be omitted for mathematical reasons, since the voting vectors can have a residual "others" cate­gory as their n'h component.

Problem. Generalize the index of cohesion, subject to the fol­lowing four constraints. ( i ) The generalized (or general ) index should not omit legislators simply because they neither voted aye • nor voted nay. (ii ) For a two-choice u niverse of ayes and nays, the generalized index should be equivalent to the index of cohesion. (iii) The generalized index should range between 0 and 1. ( iv) The general index's maximum ( 1) should occur if and only if una­nimity exists, and its minimum ( 0 ) should occur if and only if the subset of legislators is evenly distributed . These four constraints are not wholly arbitrary: ( i) and ( ii ) ensure that the generalized index is more general than Rice's index, and (ii)-(iv) ensure that the general index is a generalization of Rice's rather than Whosit's index.

Definition 5. Let X=( x., x", ... , x.) be an n-dimensional prob­ability voting vector and let Y = ( 1/n, 1/n, . .. , 1/n) be an n­dimensional vector, where n > 2. The general index of cohesion

n

for X is~ I iX- Y! I· Theorem 2. The general index of cohesion has the four desired properties. Proof. ( i) The first constraint is obviously satisfied since a probability voting vector can have more than the two compo­nents x, = ayes, x" = nays.

{ ii) Let the general index be denoted by GC. For n = 2, we find that GC =(2 )'h ([x,- 1!2]' + [x" - 1h]')'h, where the frac­tional exponent denotes the positive square root. We note that x, and x" are equidistant from lh ( their mean ), so ( x, - 1h) • = {x2 -lh)' and GC = ( 2)'1.. (2[x, - %]")Y...

Thus GC = (2)'1.: {2)'1.: ([x, - lh]" )'h = 2 1 x, -lh 1, where the single pair of parallel bars denotes the absolute value. But 2 1 x, - 1h I is simply the value of the index of cohesion, so the second constraint is satisfied.

176 THE JouRNAL oF PoLITICS (VoL. 32

(iii) The general index

GC -=( n )Y.: ([x,- 1/n]'' + ... + [x.-1/n]");!. >O n- 1 -

because the first radical > 0 and the second radical > 0.

Expanding the second radical, we get

( [x,)2- 2x,/n + 1/n2 + ... + [x.F- 2x.fn + 1/n2) 'h,

and collecting like terms, we obtain

(~[x,F + ~[-2x,/n] + ~1/n2 ) ~~.

where the index of summation i runs from 1 to n. Since

~( -2x,/n) = ( -2/n )~x, = -2/n, and l:1/n2 = n/n2 = 1/n, the

second radical becomes

n . ..:..n~~.::..[ x._!.;, ]:....2 ~..:..1_ Thus GC=( )>6 (- )'·~ = ( n-1 n

n:S[~ ,F -1 n-1

But n~(x,) 2 :::; n since 0 < :S(x,)2 < 1, so GC < ( ~=~ )%= 1, as desired.

( iv) The general index is defined for probability vectors, i.e., vectors with non-negative components whose sum is 1.

f f (... h C n~[x,F -1 From the proo o m), we ave G = ( 1 n-

note that 0 < x, < 1 and that l:x, = 1, so ~(x,) 2 is (i.e., equals 1) if and only if some component x, = 1.

Thus GC = (-0 - 1 )% = 1 if and onlv if unanimitv n -1 ' '

)%. We

maximal

prevails.

Since :S(x,)2 is minimal (i.e., equals n/n2 = 1/n) if and only if

1970] RicE's INDICES 777

x, = 1/n for i = 1, 2, . . . , n, GC = ( 1-i ) 'h = 0 if and only n-

if ·the legislators are evenly distributed. This completes the proof.

Problem. Generalize the index of difference, subject to the following four constraints. ( i) The generalized (or general) in­dex should not omit legislators simply because they neither voted aye nor voted nay. ( ii) For a two-choice universe of ayes and nays, the generalized index should be equivalent to the index of differ­ence. (iii) The generalized index should range between 0 and 1. ( iv) The general index's maximum ( 1) should occur if and only if the two subsets of legislators are each unanimous but in different directions, and its minimum ( 0) should occur if and only if the subsets of legislators are identically (but not necessarily evenly) distributed. These four constraints are not entirely arbitrary: ( i) and ( ii) require that the generalized index be more general than Rice's index, and (ii)-(iv) require that the general index be a gen­eralization of Rice's rather than Whatshisname's index.

Definition 6. Let X = ( x,, x2, • • • , x.) and Y = ( y, y~, ... , y.) ben-dimensional probability voting vectors. The general index of difference between X and Y is v 1h I j X -

Y I l· Theorem 3. The general index of difference has the four de­sired properties.

Proof. ( i) The first constraint is obviously satisfied since nvo probability voting vectors, although haviug the same number of dimensions, can have more than the components x, = ayes, x" =

nays, and y1 = ayes, y2 = nays.

( ii) Let the general index be denoted by CD. For n = 2, CD = ( lh) 'h ( [x, - y,)2 + [x2 - Y2JZ) '!.:, We note that x, + x" = y, + Yz = 1, so x, -y1 = y"-x2 and (x1-Y1P..:. (y2 -x") 2

• Since (y2-x2F = (x2-Y2)2, we see that CD= (lh)'h (2[x,- y,J2 )'h = (lh)'h (2)'h ( [x,- y~P) 'h.

Thus CD = ( [x, - y,]') 'h = J x ~.- y 1 J, as desired.

778 THE JouRNAL OF Pouncs [VoL. 32

(iii) The general index GD =- (if2) 'h ([x1 - y,p + ... + [x"- y~P) 'h ? 0 because the first radical > 0 and the second radical ? 0. We recall that X and Y are probability vectors; each has non-negative components whose sum is 1. Thus, with­in the second radical, ~(x,-yY is maximal (i.e., equals 2) if and only if I x, - y, I =- 1 and I x. - y. I = 1, for some j f k. The desired result follows immediately since CD = ( 1f.z) 'h ( 2) 'h = 1 is maximal. ( iv) Since ( lh) 'h is always positive, we see that CD = 0 if and only if ( x,- y,) = 0, for i = 1, 2, ... , n. From the proof of (iii), we know that GD = 1 if and only if I x, - y 1 I = 1 and I x,- y .I = 1, for some j f k. Thus, for two subsets of legislators, GD is minimal ( 0) if and only if the subsets are identically distributed, and GD is maximal ( 1) if and only if the subsets are each unanimous but in different directions. (Geometrically, a component of a vector represents a direction). This completes the proof.

Problem. The problem of generalizing the index of likeness, subject to four appropriate constraints, is implicitly solved by the immediately preceding theorem and definition. Since the adequacy of the following definition is obvious, a proof is omitted.

Definition 7. The general index of likeness between X and Y is the complement of the general index of difference between X and Y, namely, 1 -general index of difference.

AN APPRAISAL

The general indices explicitly and Rice's indices implicitly are defined in terms of subsets of the set of voting vectors and the set of vector concepts. The indices may be illuminated by briefly considering other subsets of those sets.

The general indices are defined in terms of probability voting vectors, but these are not the only type of voting vectors. Some voting vectors have non-negative integers (viz., the number o~ ayes, etc.) as components; for convenience, these are dubbed data voting vecto1·s. These two types of voting vectors are obviously not equiv­alent. Loosely speaking, every data voting vector allows the construction of a single probability voting vector, but the converse is true only for special cases, e.g., if the components are written as

1970] RicE's I NotcES 779

non-reduced fractions. Some formalized comparative properties follow: Let X= (x,, ... , x.) andY= (y,, ... , y.) be data voting vectors. Let X' = (x; , . . . , x: ) and Y' = (y~ , . . . , y~ ) be the probability voting vectors <-'Onstructed from X and Y, respectively.

( i ) The distance between X and Y does not usually equal the distance between X' and Y' because 0 < I I X- Y I I whereas o < 1 1 X' - Y' 1 1 < v2 . ( ii ) Since the components of X, Y, X', and Y' are non-negative real numbers, the cosines between X and Y and between X' and Y' are always non-negative because 0 < cos(X,Y) < 1 and 0 < cos(X',Y' ) < 1.

( iii ) The cosines between X and Y and between X' and Y' are always equal because X' o Y' and I I X' I I I IY' I I contain a common factor which (upon cancelling ) reduces

X' o Y' X • Y I I X' I I r I Y' I I to I I X I I I I y I I

These three properties and both types of voting vectors are illus­trated above with the 1918 British example.

The general indices are defined in terms of the norm but this is not the only function of two vectors. The cosine function is also noteworthy for students of legislative voting behavior. These two types of functions are quite distinct. F rom a geometrical point of view, this is obvious since if X and Y are two non-zero vectors, rep­resented as arrows from the origin to points ( x,, ... , x.) and ( y., ... , y.) , the distance between the points tells nothing about the angle between the arrows. Thus, for a complete geometric charac­terization of voting vectors, norm-style indices must be supple­mented by cosines, and conversely. A corollary is the following:

Theorem 4. For two probability voting vectors, if at least one is non-unanimous, then the general index and Rice's index of like­ness cannot be derived from the cosine, and the converse is also true.

Geometrical representations are somewhat difficult to draw for three dimensions and impossible to graph for higher dimensions. A two-dimensional illustration is given in Figure 1. On November 21,

780 THE JouRNAL oF PoLITICS [VoL. 32

FIGURE 1

UNITED STATES SENATE VOTE ON }UDCE HAYNSWORTH, BY PARTY

Nay

00·

45

D (19,38)

15

d (33.3,66.7)

I I I I I I I I I

(26,17)

I I

r ( 00.5, 39.5)

I I I I I I

---4~------~------~~------~-------4----------AYc 3o 45 oo 15

Index of Difference In Pcr~"HDtaJ,!e Notation

1969, the United States Senate defeated President Nixon's nomina­tion of Judge Clement Haynsworth, for the Supreme Court, by a vote of 45 ayes to 55 nays. Every Senator voted; so, in this case, Rice's indices are equivalent to the general indices. The partisan breakdown of the vote was as follows: Republicans ( 26 aye, 17 nay) or, alternatively, (60.5% aye, 39.5% nay), Democrats (19 aye, 38 nay) or, alternatively, ( 33.3% aye, 66.7% nay). Probability vectors have been written in percentage notation so that Figure 1 has a reasonable size and all the entries are discernible. This mul-tiplication does not affect the angles in the graph. .

Two features of this graph deserve special emphasis. ( i) The two parties differ substantially in size. This variation is concealed by probability voting vectors ( r and d in the figure) but revealed by data voting vectors ( R and D in the figure). If sheer size is a

1970) RICE's INDICEs 781

FIGURE 2

UNITED STATES SENATE VOTE ON ]UDCE BLACKMUN, BY PARTY

Not Voting

]()()

80

60

40

20

G<~neral Index of Differenc·c· In Percentage Notation

theoretically significant variable, then data vectors are superior to probability vectors. For example, using data voting vectors, the voting behavior of two sets of legislators cannot be identical if the sets differ in size. ( ii) The cosine between two voting vectors does not facilitate locating a point in space. The cosine should be computed for the angle between each voting vector and the aye­axis; the latter serves as a standard basis vector for cosine calcula­tions in two dimensions.

782 THE J OURNAL OF POUTICS [VOL. 32

Following the Senate's rejection of Judge Haynsworth, President Nixon nominated Judge Harold Carswell for the Supreme Court vacancy. That nomination was also rejected, by a vote of 45 ayes to 51 nays, with four Senators not voting. Since the vote is three­dimensional, a graph is omitted . However, an additional two-di­mensional illustration is provided by the Senate vote on President Nixon's third nominee, Judge Harry Blackmun. On May 12, 1970i Judge Blackmun was confirmed by a vote of 94 ayes to 0 nays, with six Senators not voting. Although that vote is two-dimensional, every Senator did not vote; so, Rice's indices are not equivalent to the general indices in this case. The partisan breakdown of the vote was as follows: Republicans ( 40 aye, 3 not voting ) or, alterna­tively, (93.0% aye, 7.0% not voting ), Democrats (54 aye, 3 not voting) or, alternatively, ( 94.7% aye, 5.3% not voting). Figure 2 graphically represents this vote. The probability vectors, r and d in the graph, have been written in percentage notation; the data vec­tors, R and D. thus are easily squeezed into a figure of reasonable .

SIZe.

C oNCLusioNs

Rice's indices can be interpreted and generalized in terms of probability vectors, and thus the original indices' restriction to a two-option universe can be transcended. From a geometrical point of view, however, both the generalized indices and the original in­dices are incomplete, that is, they fail to specify or imply the loca­tion of a probability voting vector as a point in space. Further­more, from the perspective of probability vectors, both the general­ized and the original indices have an ad hoc or arbitrary appear­ance.

Mathematically similar but simpler expressions are obtained by using the norm of a probability voting vector as a measure of "co­hesion" and by using the distance between two probability voting vectors as a measure of "difference." If these measures are supple­mented by the relevant cosines, then the algebra of probability vec­tors can be fully exploited for the analysis of legislative voting be­havior.

Data vectors as well as probability vectors furnish a mathemati­cal notation or model for the study of legislative voting behavior. Probability voting vectors can be derived from data voting vectors, but the converse is not true in general. Thus probability voting

1970] RICE's I NDICES 783

vectors conceal or destroy information that is routinely obtained in the course of an analysis. For this reason, since no established the­ory for legislative voting exists, the model of data vectors seems preferable to the model of probability vectors. (Pace Rice.) If the norm (magnitude) and N-1 cosines (where N is the number of dimensions ) are calculated for each data voting vector, then linear algebra can be used without restraint.