A NUMERICAL REPRESENTATION FOR TAXONOMIC KEYS

Transcription

1 A NUMERCAL REPRESENTATON FOR TAXONOMC KEYS BY D. V. OSBORNE Physics Department, University of British Columbia * {Received 1 May 196) SUMMARY Some of the limitations of the usual type of dichotomous taxonomic key are pointed out. t is suggested that a particular method of representing the characteristic features of a species in numerical form can overcome some of these limitations. dentification by this method is rapid, can often be made correctly in spite of minor errois of observation, and requires no computation or computing equipment. The method may therefore be of particular use in the field. NTRODUCTON t is a common experience among amateur botanists, and not unknown among professionals, that keys for the assignment of plants to families, genera and species may on occasion yield incorrect results. For the present purpose we propose to ignore alike failures due to simple errors which it is the business of the amateur to avoid, and failures due to the difhculty of defining a species (e.g. Hieraciiim, Euphrasia) which it is the business of the professional to elucidate. Of the remaining sources of failure there are two which may be somewhat reduced by the method discussed below. The first arises when the observer is required by the key to arrive at a very precise description of a particular feature, e.g. leaf shape. The feature in question may be very uniform for all discoverable individuals and yet hard to describe to the standard of accuracy required for successful operation of the key. The second arises when the information needed for a particular decision early in the key is not available; e.g. it may be required to study the fruit which is not yet formed, or to know (of a fully developed plant) whether the leaves appeared before or after the flowers. The first difficulty demands a method of analysis which will allow the observer a small latitude of error without leading to wrong identification. The method described here takes a step in this direction for species which are not too variable, although it cannot replace statistical methods for taxonomic problems of real difficulty. The second difficulty demands the use of a key in which all the characters are studied simultaneously rather than step by step. This idea, by no means a new one (e.g. Balkovsky, i960), can best be realized in practice by using a computer, or at least by carrying out computation. The present method avoids such computation (for the user of the key) but still has some of the advantages of a simultaneous key. n what follows it will be assumed that strictly distinct species exist, that complete descriptions of them are available, and that the specimen in hand does actually answer one of these descriptions. t should be made clear that the information for the examples quoted below has been drawn from Clapham, Tutin and Warburg (195), and from Ross-Craig (198), and not from original observations of actual plants. * On leave from Department of Physics, St. Salvator's College, St. Andrews University, Scotland, until August

2 6 D. V. OSBORNE THE USE OF NUMBERS The use of numbers will best be made clear by a very simple example; it is possible (see later) to construct a key to those species of Geranium listed by Ross-Craig, in such a way that G. dissectum, G. molle and G. pusillum can be isolated as a group. Consider now the possibility of distinguishing between these three by using (a) the presence and size of the petal claw, and (b) the shape of the leaf lobes (See Table i). Table Ross-Craig Vol. V Plate No.,. 7 Geranium dissectum Short claw Leaf lobes linear G. molle Very short claw Leaf lobes rather broad 5 G. pusillum Fairly long claw Leaf lobes rather narrow, not divided to the base Two possible conventional keys are: fclaw long A \ [_Claw very short fleaf lobes narrow B \ [_Leaf lobes broad fleaf lobes broad X < lobes narrow fclaw long Y \ ^Claw very short or G. pusillum See B G. dissectum G. molle G. molle See Y G. pnsillwn G. dissectum n the first of these, the finder of a specimen of G. dissectum may be uncertain how to answer A, and therefore (in principle) uncertain how to proceed further. nverting the order of questions is no help, because, in the second key, question X will be hard to answer if the specimen in hand is G. pusillum. Yet it is evident that if the difficulty of question order could be overcome the three species are all clearly distinct from each other. A useful numerical representation will therefore (a) present all the relevant characters at once and (b) indicate to some extent degrees of difference between species. Table serves for the present example: Table A. Assigning of numerical values Claw: None C = i Leaf lobes: Linear L = i. Very short C = Narrow, divided to base L = Short C = Narrow, not divided to base L ^ Long C = Broad L = B. Numerical key for the three species Plate No. 7 Geranium dissectum C = L = i G. molle C = L = 5 G. pusillum C = L ^

3 Taxonomic keys 7 This table is to be used by assuming that the specimen in hand can be described with an accuracy of ± \ unit in any given feature. For example, the leaf lobes of a particular specimen of G. pusillum might be thought to be narrow, not divided to the base (L = ), or half-way between narrow and broad (L =!), or narrow and divided almost to the base (L = ^). But they could not reasonably be thought of as broad (L = ), or as narrow and divided to the base (L = ). f now observation of a given plant yields, for example, the description C = i\, L =, it is unambiguously clear from the table that it must be G. molle. The value C = i^ is compatible with G. dissectum, but L = is not. Likewise L = ^ is compatible with G. pusillum, but C = 1-- is not. The principle involved is that if the observer's description is only slightly wrong, the key can still lead to the correct identification. n order that this may be so, two conditions must be fulfilled: the key itself must contain a description accurate for all possible material, and the key must be constructed so that each species shows sufficient distinction from all others. The former point must be left to the botanist; it is an ideal which is sometimes hard to attain. So far as the latter is concerned, it is easily seen by inspection of Table B that for the tolerance specified (±i unit) the key descriptions of any two species are sufficiently distinct from one another if the descriptive numbers differ by at least units for at least one property (in this case C or L). t is assumed that the key descriptions make use only of integers. Using the symbols AL, AC, to refer to differences (without regard to sign) between values of L and of C for different species, a test table may be drawn up to show that Table B makes satisfactory distinctions between all possible pairs of species (see Table ). Table Plates No. 7 & Geranium dissectum G. molle AC = i AL = & 5 G. molle ^ G. pusillum AC = AL = i 5 & 7 G. pusillum G. dissectum AC = AL = n each row, at least one difference is greater than, or equal to,. Numerical treatment may be applied to almost any character; two further examples are the calyx shape of Vicia species (sepals equal i, upper two sepals shorter, upper two sepals very short ), and the styles of Saxifraga species in fiower (widely diverging i, short diverging, erect diverging, erect connivent ). n order to illustrate more fully this use of numerical grading, we now give a numerical key to the fourteen Geranium species illustrated by Ross-Craig. Two of the useful taxonomic characters have already been given (petal claw and leaf lobes) and numerical scales proposed for them in Table A. Other useful characters are shown in Table. Table Root or rhizome: Vertical R = Oblique or horizontal R = i Petal shape: Rounded, truncate or apiculate P = i Emarginate P = Hairs at base of petal: All over H = i Marginal only, or none H = All characters are expressed on a scale of four for reasons which will appear later. t would be perfectly possible in Table to make use of intermediate numbers, e.g. petals

4 8 D. V. OSBORNE rounded P = i, petals truncate P =^, petals emarginate P =, but unless and both figure in such a table, the presence of one of thenn is of no particular value. The fact that 'Rounded and truncate petals might be confused but both are distinct from emargmate petals' is just as well expressed by the use of P = i, P = as it is by P = i, P = and P = - The key is written down by listing the characteristics of each species m the order RHPCL, i.e. with the i, characters (RHP) first and the i,,, characters (CL) afterwards. The species themselves are placed in numerical order, treating RHPCL as an ordinary five digit number of which L is the least significant digit. The result is in Table 5. Ross-Craig Plate No \'ol. V Table 5. Geraniitm key Geiaiiiiiiii sylraticion G. pliaeinii CT. Vi'i'SJColor G. sangiiineuiu G. pratense G. columbinum G. rotundijoliini! G. pyreimicinn G. vobertianinii G. piirpiireiini G. hicidiun Cj. /nolle G. dissectiidi G. piisilliiiii (R, H. P from Table ; C from Table 6 below, L from Table A) One entry requires explanation; G. pltaeiun (0) has petals without claw, and would therefore have C = i if Table A were used. This, however, would render it nondistinct trom G. sylvaticum. Since the petals of G. phaeum are nearly black and carry a crimson blotch at the base, the confusion could be avoided by adding petal colour as a further character in the key, but this would complicate the key a good deal and would he useful only for this one purpose. We therefore modify the claw key (Table ) as in Table 6. Table 6 Claw: None, and no blotch at base of petal C = i None, and crimson blotch at base of dark petal Very short C = C ^ Short (^ _., Long C = This artificial procedure distinguishes G. phaeiim from G. sylvaticum, and it introduces no confusion between G. phaeum and the long-clawed species (5 and 9-1) because they are already fully distinguished by other characters (e.g. R). Many useful manoeuvres of this type are possible in this kind of key. Having compiled such a key, it is necessary to test it by constructing a table like Table to show that all possible pairs of species are distinguished. Unfortunately there are now fourteen species so that the total number of such pairs is 1C, which is 91. The work of testing can, however, be reduced very greatly when several columns (in this case RHP) contain only 's or 's. Any two species which differ at all in one of these three columns R H P C L

5 Taxonomic keys 0 are bound to differ by three units, i.e. significantly. The problem of testing is therefore reduced to that of taking each group of species having a given value of RHP, and testing each pair of species in it for distinctness as revealed by the remaining two digits (CL). n the present example it turns out that most groups can be tested at sight. The first group in Table 5, RHP = 111, contains species 1 and 0, which differ by a significant amount (>) in their value of C. Groups 11, 11 (one member only), 11 and 1 (one member only) are also plainly distinct. Group (species, 7 and 5) contains three members and is best tested by subtraction. t has already been shown (see Table ) that t makes satisfactory distinctions. n the remaining group 1, it is clear that species 9 is distinct from 0 and 1, but the latter are not distinct from one another. t would be possible to build an artificial distinction into the key, but it is perhaps easier in this case to treat them as a pair to be resolved when necessary by an auxiliary criterion. 0 (G. robertiamm), for example, has orange pollen, while 1 (G. piirpiireum) has yellow. The ehmination of the numbers and from the first three columns simplifies the ordering and testing of the key; it also simplifies its use. magine a plant has been found and its characters have been observed as: Root: vertical Hairs at base of petal: none R = H = Petal shape: emarginate Petal claw:?short or very short P = C = or, say '- Leaf lobes: not divided to the hase?broad or narrow L = or, say J- Description is therefore J- T,\ The last two digits are evidently not quite right; they might really be,, or. The first three digits, however, are correct, and the search is therefore reduced to the group which will be found in its proper place in numerical order. nspection of the last two digits shows that only G. molle () fits within the tolerance of ±5 unit. FURTHER DEVELOPMENT OF THE NOTATON The last stage of such an identification becomes much more tedious if the number of full columns (containing all four digits) is greater than two, but it will now be shown that a change of notation can be used to simplify the work very considerably. t is here that the importance emerges of arranging the characters so that they may be represented by one of four (or fewer) digits, rather than by say five or six. Consider a fictitious key referring to six different characters A-F, of which only A and B can be written unambiguously as i or. The group A = i, B = i of such a key might run as in Table 7A. f we now wish to identify a specimen described asi^i-j- -i (CDFF), there are eight possibilities for its true description, i.e. 111, 111, 11, 11,11,11, 1 and 1. n looking for one of these in Table 7A it is necessary to survey alniost half the table, the basic reason being that the numerical order of the table lays too much stress on the distinctions between i and and between and. We therefore change our notation, writing i for i, i for, for, and for. The result of this transcription is given in Table 7B. From this. Table 7C is formed by rewriting Table 7B in numerical order, treating i like i and like. We now rewrite the description of our specimen li i J- i i as 111, the rule being to transcribe i, i and as i, i as?, and, i- and as. On looking up 111 in its

6 0 D. V. OSBORNE proper numerical place in Table 7C, the specimen is identified as species a, and a reference to the original description (i i- i i ^ i) compared with the full description of a (111 in 7C, or 111 in 7A) verifies the identification as accurate within the prescribed tolerance of J^ i unit. Table 7A Table 7B Table 7C Species a b c d C D E F C 1 1 D E E ( Species,? b ] fl ] D 1 E F 1 e f gh c ] rf ] i i k 1 0 ' j n m n 0 P -^ -1- «' P Specimens which include a i in their description cost only slightly more trouble, as for example - i J- which is transcribed?i?. n seeking this number in key 7C, the?'s may be either i or (i.e. or in 7A) but not i or (i.e. i or in 7A). The four possibilities having the correct second and third digits are easily found; they are 111, 11, 11 and 1, and of these only 11, i.e. species /, has first and fourth digits which could be read as ^-. The theory of the use of such a key may be understood by remarking first that no two distinct species can occupy the same position in numerical order in Table 7C; e.g. 111, 111, 111, etc., are all indistinguishable. Once the table has been successfully constructed by a suitable choice of characters and of representation to make each species distinct, it follows that each species may be identified uniquely by its numerical order in Table 7C. f a specimen is described within an accuracy of zb? unit, its transcribed description (in terms of i,? and ) cannot therefore refer to a wrong species in the key, although the presence of one or more r may necessitate a moment's inspection to determine the right species. We conclude that, if any specimen is correctly described to an accuracy of +\ unit in each feature, if it is in fact referable to a species contained in the key, and if the species is correctly described by the key, then the rules given above will identify the specimen uniquely and correctly. f a given specimen lies outside the tolerance of ± \ unit, i.e. it differs from all the key descriptions by at least one unit in one character, its transcribed description may appear to be identifiable by Table 7C. Confirmation using the full description will, however, expose the error. For example, a description 11, transcribed for key 7C as 11 must be either species i or a mistaken description. Reference to the full description, 11, compared with that of / (11 in 7A, or 11 in 7C) shows that tbis identification cannot be correct. n such cases the species wrongly obtained (/ in this case) will usually resemble the description somewhat, and may offer a hint concerning the error. On the

7 Taxonomic keys 1 other hand, a description ^ will be transcribed?? and this does not even appear to be identifiable; it cannot be any of the species e, h, oxp having D =, E == in Table 7C, smce in each of these species either C or F (or both) is a definite' i or. Again, therefore, the description must be in some way erroneous. Rules may now be given for the testing of such a key when it has been constructed. n the first place, if A^ different characters are being used (in this case four, CDEF) '^ is the maximum number of distinct descriptions that can be written down, if the scale for each character runs from i to, whether or not the digits and are included. Table 7 therefore represents the maximum number of species (sixteen) that could possibly be distinguished using four characters of this type, and any real key based on four characters would usually contain considerably fewer species. t so happens therefore that Table 7C contains all the consecutive 'numbers' from 1111 to ; in practical cases some of these would usually be missing. n testing the key for distinctness, we take the first species from Table 7C and note those characters which are represented by 'definite' digits, i.e. i or rather than i or. This species is then compared with all others for which these particular characters are represented by the same digits, whether definite or indefinite. Remembering that 'distinction' means a difference of two units in at least one character (see Table ), each comparison must show that for at least one character the values in the two species are i and or and ; otherwise the key is non-distinct. The same test must be repeated for each species in turn. f no non-distinctness is revealed by this set of comparisons the key is satisfactory. For example species,«' (Table 7C) has only one definite digit, E = i, and has therefore to be compared with all other species having E = i or i, i.e. with b, c, d, 0, i, ni and k; species b has to be compared with all others having C = i or i, E = i or i, and F = or, i.e. with species d only; species a has to be compared with any others having C=iori,D= iori,e = or, and F = i or i, i.e. it requires no testing since there is no other such species. The total procedure involves, in the present example, thirty-nine separate comparisons; it is not laborious if carried out systematically. n constructing a key of this type, some characters will not at first lend themselves to classification in terms of four numbers (i,,, ) or two numbers (1,). t has already been remarked that a threefold classification of the type (i,, ) or (i,, ) may be replaced by a twofold one (i, ) without loss of precision. A classification represented by (1,, ), e.g. the calyx shape of Vicia referred to earlier, may be used as though it were (i,,, ). The fact that never appears will not affect the rules given earlier. f the character appears to require more than four different numbers, it is worth preserving the 1- system by splitting the character into sub-characters. As an example we take the petal colour and markings of Saxifraga which could be treated thus: J Petals white i; petals cream or pale yellow ; petals yellow ; petals mauve. K Petals white, cream or yellow i; petals mauve. L Petals spotted or having green veins i; petals unmarked. M Petals spotted i; petals with green veins or unmarked. This scheme makes all the distinctions required without resorting to means such as 'Petals unmarked i, green-veined, spotted 5'. At the same time each question (JKLM has been designed to have a definite answer for any relevant specimen. t is necessary to avoid 'K: Petals yellow i, mauve ' since this question has no answer for a whiteflowered plant.

8 D. V. OSBORNE SUMMARY OF THE METHOD We recapitulate here, free of explanatory material, tbe basic rules proposed above. Compiling and testing the key A set of different characters A,B,C... N is chosen, by means of wbicb tbe different species a, b...r in tbe selected group may be distinguished from one another. Species a is examined witb respect to character A and is assigned a numerical value Aa {Aa being an integer, i scaa ^^) for this character. n this way a set of key descriptions Aa Ba--- Na, Al, Bl,... Nb,..., Ar B,... Nr is built up for tbe r species (Table 7A). Tbe notation is now modified by writing everywhere i for i, i for, for, and for. Tbe modified key descriptions (Table 7B) are now placed in numerical order (Table 7C) gnoring at tbis point tbe distinctions between i and i, and between and. Tbe modified descriptions are tested for distinctness by the method described m the previous section, and tbe choice of characters or choice of numerical representation must be adjusted until tbe key is distinct. Using the key Tbe specimen x is examined witb respect to tbe characters A, B... N, and is assigned a numerical description Ax, Bx Nx using tbe same rules tbat were used in compiling tbe key descriptions. f, however, doubt exists about any character, it is permissible to assign a non-integral value to it, half-way between tbe two nearest probable values. Tbe specimen description is transcribed into modified form, writing for, i and, writing? for i, and writing for, ^ and. Tbis modified form is sougbt in its place in the numerically ordered key (Table 7C). f found, tbe original numerical specimen description is checked against the modified (or original) key description of the indicated species to see tbat tbey agree within ±i unit. f tbey do, the identification is unique and correct. f tbey do not, tben either tbe specimen description has errors greater than ±h unit, or tbe key was based on incomplete information. CONCLUSON f some of tbe botanical information used in tbe examples seems to botanists to be unsatisfactory or to have been roughly bandied, it is the author's hope tbat tbey will charitably overlook an amateur's shortcomings. The examples have been concerned with tbe identification of species within tbe genus. Tbe principles may be applicable at other levels, e.g. tbe identification of families and of genera, but tbis requires real botanical knowledge and will be humbly left to tbe professionals. Tbe main concern of tbe paper is to point out bow some of tbe limitations of the usual step-by-step taxonomic key may be overcome by tbe use of characters described by numbers, and the use of species arranged in numerical order. t is to be emphasized that tbe method described here, is, in its present form, applicable only to straightforward cases in which the species are in practice distinguishable. t would be likely, therefore, to be of most use to workers trying to identify in tbe field species already well known to science. t offers as much certainty of identification as a conventional key, it can be quicker in use, and it may allow identifications to be made even when the available material is in some respects incomplete. The actual identification process requires neither calculation, nor punched cards, nor a computer, and is therefore particularly

9 Taxonomic keys adapted for use in the field. The method is, however, no substitute for patient statistical work on those difficult genera in which distinct species are hard to define. ACKNOWLEDGMENT The author gratefully acknowledges the support afforded to this \\ ork by the grant from the President's Committee on Research at the University of British Columbia. REFERENCES BALKOVSKY, B. E. (i960). Methods of botanical research; a polytomic numerical coding system for the identification of plants. Bot. Zh., 5, 6. CLAPHAM, A. R., TUTN, T. G. & WARBURG, E. F. (195). Flora of the Britisli sles. Cambridge Uni\-ersity Press. ROSS-CRAG, S. (198). Draivings of Biitisli Plants. Bell, London.

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