J. Zool., Lond. (1974) 172,267-288 A multivariate study of the external measurements of the Sperm whale (Physeter catodon) DAVID MACHIN* School of Plant Biology, University College of North Wales, Bangor, North Wales (Accepted 13 September 1973) (With 6 figures in the text) The methods of principal component analysis and of canonical analysis are applied to data of the external measurements of the Sperm whale (Physeter catodon). The principal com- ponent analysis summarizes the main features of the data in the first four components and suggests those characters of prime importance while the canonical analysis permits com- parisons between those whales caught at several locations. Contents Introduction. . . . . . . . . . . . The variables .......... The data ............ Principal Component Analysis . . .... Canonical Analysis ........ Discussion and conclusions .. References ............ .... Page .. ........ 267 .... ...... 267 ........ . . 269 .. ........ 274 .......... 282 ........ . . 286 .......... 288 Introduction Machin & Kitchenham (1971) have described a principal component analysis of the Humpback whale (Megaptera novaeanglia). We here report on a multivariate study of the Sperm whale (Physeter catodon). The data available are a series of measurements of the external variables of the Sperm whale recorded at several whaling stations over a 35 year period beginning in June 1926 and ending in May 1961. Full descriptions of the data collected are given by Matthews (1938), Fujino (1956) and Clarke & Paliza (1972). The method of analysis we have adopted, in this paper, falls into two parts. The first is a principal component analysis of the five data collections separately because of the disparity in the numbers of variables that can be considered and the second a canonical analysis of the data as a whole but on a restricted number of variables. The variables Information on 27 external variables of the Sperm whale were recorded. The characters measured and labellings followed are those described by Matthews (1938). Measurement (12), “Notch of flukes to posterior end of ventral grooves”, is omitted as this is applicable only to balaenopterid whales while measurement (9a), “span of flukes from tip to tip”, * Present address : Department of Mathematics, University of Stirling, Stirling, Scotland. 267
Transcript
J. Zool., Lond. (1974) 172,267-288
A multivariate study of the external measurements of the Sperm whale (Physeter catodon)
DAVID M A C H I N * School of Plant Biology, University College of North Wales, Bangor, North Wales
(Accepted 13 September 1973)
(With 6 figures in the text)
The methods of principal component analysis and of canonical analysis are applied to data of the external measurements of the Sperm whale (Physeter catodon). The principal com- ponent analysis summarizes the main features of the data in the first four components and suggests those characters of prime importance while the canonical analysis permits com- parisons between those whales caught at several locations.
Introduction Machin & Kitchenham (1971) have described a principal component analysis of the
Humpback whale (Megaptera novaeanglia). We here report on a multivariate study of the Sperm whale (Physeter catodon).
The data available are a series of measurements of the external variables of the Sperm whale recorded at several whaling stations over a 35 year period beginning in June 1926 and ending in May 1961. Full descriptions of the data collected are given by Matthews (1938), Fujino (1956) and Clarke & Paliza (1972).
The method of analysis we have adopted, in this paper, falls into two parts. The first is a principal component analysis of the five data collections separately because of the disparity in the numbers of variables that can be considered and the second a canonical analysis of the data as a whole but on a restricted number of variables.
The variables Information on 27 external variables of the Sperm whale were recorded. The characters
measured and labellings followed are those described by Matthews (1938). Measurement (12), “Notch of flukes to posterior end of ventral grooves”, is omitted as this is applicable only to balaenopterid whales while measurement (9a), “span of flukes from tip to tip”,
* Present address : Department of Mathematics, University of Stirling, Stirling, Scotland.
267
268 D . M A C H I N
reported by Clarke & Paliza (1972) is added as is variable (24a), “Tail flukes, tip to notch” reported by Fujino (1956).
The 27 variables are thus as follows:- (1) Total length, tip of snout to notch of flukes. (2) Projection of snout beyond tip of lower jaw. (3) Tip of snout to blow-hole. (4) Tip of snout to angle of gape. (5) Tip of snout to centre of eye. (6) Tip of snout to tip of flipper. (7) Centre of eye to centre of ear. (8) Notch of flukes to posterjor emargination of dorsal fin. (9) Width of flukes at insertion.
(9a) Span of flukes, tip to tip. (10) Notch of flukes to centre of anus. (1 1) Notch of flukes to umbilicus. (13) Centre of anus to centre of reproductive aperture. (14) Vertical height of dorsal fin. (15) Length of base of dorsal fin. (16) Axilla to tip of flipper. (17) Anterior end of lower border to tip of flipper. (1 8) Length of flipper along curve of lower border. (19) Greatest width of flipper. (20) Length of severed head from condyle to tip. (21) Greatest width of skull. (22) Skull length, condyle to tip of premaxilla. (23) Length of flipper from head of humerus to tip. (24) Depth of body at dorsal fin. (25) Height of head.
variables (23), (24) and (25) to:- (23a) Height of skull (misprinted “Length of skull” in Fujino (1956: 48)). (24a) Tail flukes, tip to notch. (25a) Tail flukes, total spread (measurement (9a) in the above list, see Clarke & Paliza (1972: 16)).
In addition sex, locality and date of capture were recorded for each whale. Variable (13) was omitted from the subsequent analyses since it is a variable whose main variation is between sexes and emphasizes a distinction between male and female not so emphasized by the other variables.
Fujino (1956), whilst retaining the numbering given by Matthews, has altered the
The variables may be grouped into measurements of: Variables
Total length (1) Head region Trunk Flipper Flukes (9) (94 (244 Dorsal fin (14) (15)
The data The available data may be summarized into five groups as follows:-
269
A . Information was recorded on a total of 67 male and 14 female Sperm whales caught
off whaling stations at Saldanha Bay, Durban and South Georgia between June 1926 and
South Africa and South Georgia (Matthews, 1938)
TABLE I Mean values and standard deviations of male and female whales caught off South Africa*
Male Female Number of Standard Number of Standard
Variable observations Mean deviation observations Mean deviation
23
22 2
18 17 9 2
13 2 1
16
21 22 21 20
-
13.33
1.01 0.55 3.38 3.77 5.80 0.55 4.88 2.56 4.80
2.08
4.91 4.15 6.73 1.65
-
-
0.96 1.21 1.58 0.64
2.26
0.3 1 0.14 0.89 0.95 1.55 0-00 0.89 0.37
14
I0
10.20
0.43
0.51
0.13
0.14 0.17 0.17 0-03 0.10
- 1.63 2.12 3.71 0.33 2.80
11 13 7 4 5
-
0.44
0.65 0.59 0.98 0.22
-
4 -
1.27
4.02 3.18 5.51 0.25
- 0.05
0.19 0.32 0.39 0.17 -
0.08 0.11
0.03 -
-
0.05 -
9 13 12 9
0.72 0.91
0.47 -
-
0.75 -
14 18 2
16
0.14 0.20 0.04 0.08
11 12
12 -
-
10 -
-
20 - -
2 3
1.01 - -
0.35 1.22
0.14 - -
0.07 0.13
3 1
0.23 1 .oo
0.04 -
~~
* Variables included in the multivariate analysis.
February 1931. The number of observations for each variable, its mean and standard deviation (in metres) are given in Table I for those caught off Durban and in Table I1 for those caught off South Georgia. The two whales recorded at Saldanha Bay are combined with those recorded at Durban.
270 D. MACHIN
Observations are missing from the data given by Matthews(1938: 166-167) for a variety of reasons as discussed in his report and together with the omission of variable (13) this leaves 12 variables for the subsequent analysis. One whale D24 was excluded as only its total length of 9-92 m was recorded. The 12 included variables are indicated by asterisks in Tables I and 11, the variables are also grouped into their respective body regions.
TABLE I1 Mean values and standard deviations of male whales caught off South Georgia*
Variable Number of
observations
Male
Mean Standard deviation
44
38 25 30 32 31 22 34 4 3
3
32 44 43 43
36 36 27 35 5
15.72
1.08 0.67 3.85 4.57 6.99 0.49 5.62 2.1 1 4.78 -
2.59
5.10 4,52 7.36 1.80 -
0.99 1.37 1.42 0.71 1.48
0.92
0.21 0.13 0.50 0.52 0.62 0.06 0.54 0.08 0.13
0.14
0.3 1 0.25 0.43 0.17
-
-
0.18 0.10 0.10 0.06 0.08
* Variables included in the multivariate analysis.
B. Bonin Islands (Fujino, 1956) Information was recorded on a total of 34 male and 2 female Sperm whales caught off
the Bonin Islands between April and June 1950. The data matrix given by Fujino (1 956 : 74-76) is almost complete except for variables (9a) and (18) which are thus excluded from the analysis. Table I11 gives the number of observations for each variable, its mean and standard deviation.
EXTERNAL MEASUREMENTS OF T H E SPERM WHALE 271
C. Antarctic males (Fujino, 1956) Information was recorded on a total of 50 male Sperm whales caught in the Antarctic
in November and December 1950. There are no corresponding records of females as these are not taken in the Antarctic (Fujino, 1956: 47). The number of observations for each variable, its mean and standard deviation are given in Table IV.
TABLE I11 Mean values and standard deviations of male and female whales caught off the Bonin Islands*
Male
Mean
Female
Mean Number of observations
Standard deviation
Number of observations
Standard deviation
0.13
0.05 0.09 0.12 0.12 0.07 0.01 0.03 0.04 0.08 0.04
Variable
34
32 31 33 34 34 28 33 34 34 31 -
33 34 34 33 -
31 31 23 30 -
32 21 32
32 33
13.39
1.02 0.55 3.25 3.64 5.88 0.42 4.53 1.78 3.77 1.43
1.58
028 0.14 0.70 0.75 0.91 0.06 0.88 0.26 0.69 0.22
2
2 2 2 2 2 2 2 2 2 2
10.85
0.49 0.49 1.67 2.17 3.70 0.30 2.77 1.25 2.49 1.01
4.33 3.84 6.19 1.31
0.44 0.35 0.58 0.23
3.85 3.64 5.69 0.20
0.14 0.20 0.20
0.87 1.19 1.23 0.60
0.12 0.14 0.15 0.08
0.74 0.97 1 .00 0.50
0.06 0.10 0.08 0.04
0.98 3.60 1.82
0.27 1.12
0.15 0.39 0.27
0.07 0.22
0.95 2.95 1.50
0.23 0.94
0.07 0.07 0.00
0.01 0.45
* Variables included in the multivariate analysis.
Observations are missing from the data given by Fujino, (1956: 70-73), although no explanation is offered in his report, and this leaves the 15 variables indicated in Table I11 for the subsequent analysis.
D. Japanese coastal waters (Fujino, 1956) Information was recorded on a total of 68 male and 30 female Sperm whales caught in
waters adjacent to the Japanese coasts between July 1950 and September 1951. There are
272 D. M A C H I N
many missing values in the data matrix (Fujino, 1956: 77-81) in particular variables (9a) and (18) are totally absent and only 16 observations of variable (7) are available. For pur- poses of analysis the data were reduced to the 14 variables indicated in Table V.
TABLE IV Mean values and standard deviations of measurements of mak sperm whales caught in the Antarctic*
* Variables included in the multivariate analysis.
E. Information was recorded on a total of 30 male and 20 female Sperm whales caught in
the South East Pacific off the four whaling stations at Paita, Pisco, Iquique and Talcahuano between August 1959 and May 1961. Only one whale, a male, was recorded at Talcahuano (Clarke & Paliza, 1972: 74-75).
There was some confusion, reported by Clarke & Paliza, over the definition of variable (24). Missing observations reduced the number of variables to the 18 indicated in Tables
South East PaciJc (Clarke & Paliza, 1972)
EXTERNAL MEASUREMENTS O F T H E SPERM WHALE 273
VI, VII, and VIII. The immature whale, Pisco 890 of total length 5-75 m, is omitted from Table VII.
Table IX gives the correlation coefficients, averaged over the five groups, between the variables for males and females separately. All variables are not available from each locality so that the correlation coefficients presented in Table IX, pooled by the method
TABLE V Mean values and standard deviations of male and female whales caught off the Japanese coast*
Male Female Number of Standard Number of Standard
deviation Variable observations Mean deviation observations Mean
t Whale number HN68 is recorded as 2.25 m for this variable rather than, we presume, 025 m. * Variables included in the multivariate analysis.
described by Fisher (1967), are based on differing numbers of observations and thus are not readily comparable. It can be seen however that all the correlations for the male whale are positive and quite large while those for the female are correspondingly smaller and often of differing sign. This sexual dimorphism of the two correlation matrices suggests that the sexes should be treated separately in subsequent analysis.
274 D. MACHIN
Principal Component Analysis The method of Principal Component Analysis seeks to economize on the number of
variates by means of linear transformations of the original variables chosen in such a way that the new variates are mutually uncorrelated arld have importance of successively decreasing magnitude so that it may be possible to ignore all but the first few. In addition
TABLE VI Mean values and standard deviations of male and female whales recorded at Paira*
Number of observations
Male
Mean Standard deviation
Number of observations
Female
Mean Standard deviation
12
12 12 12 12 12 11 12 4
11
10
12 12 12 12 10
11 12 11 11 10
3
-
-
-
12 12
1343
0.92 0.59 3.38 3.79 6.1 1 0.45 4.77 1.70 4.1 1
2.20
4.54 4.04 6.53 1.63 1.41
0.87 1.27 1.36 0.66 1.36
0.99
-
0.31 1.21
2.04
0.25 0.12 0.82 0.87 1.24 0.10 1.01 0.37 0.91
0.44
0.54 0.46 0-72 0.23 0.20
0.15 0.21 0.21 0.10 0.22
0.10
-
0.07 0.32
9
9 7 9 9 9 7 1 1 1
7
9 7 9 3 5
7 7 7 7 1
7
-
-
7 7
9.46
0.37 0.36 1.67 1.94 3.61 0.29 2.85 1.30 2.60
1.69
3.37 2.95 4.95 0.25 1.79
0.70 0.92 1 .oo 0.49 1.05
0.71
-
0.25 0.85
0.43
0.05 0.04 0.20 0.18 0.24 0.03
- 0.21
0.23 0.18 0.36 0.05 0.27
0.13 0.08 0.08 0.04
0.1 1 -
0.07 0.12
* Variables included in the multivariate analysis.
it would suggest those original variables of prime importance. The necessary details and calculations are described by Seal (1 966).
Now, since different variables have been looked at in the five different groups just described, separate Principal Component calculations have been done on each group, male and female separately where appropriate, making a series of eight calculations in all.
EXTERNAL MEASUREMENTS OF T H E SPERM WHALE 275
The calculations were performed on an Elliott 4130 computer. Tables X, XI and XI1 give the percentage of variation accounted for by the first two of the new variates calculated from the original variables. Also included in these tables are the eigen values for these two components and the weightings for the original (standardized) variables in each component.
TABLE VII Mean values and standard deviations of male and female whales recorded at Pisco*
Malet Female Number of Standard Number of Standard
Variable observations Mean deviation observations Mean deviation
t The immature male whale Pi 890, of length 5.75 m, is omitted from these calculations. * Variables included in the multivariate analysis.
For the males it is clear that components I and I1 are of major significance indeed every other component accounts for less variation than an individual standardized original variable except for the Antarctic males (Fujino, 1956) for which the first four components have associated eigen values greater than unity. The first component is obviously a general measure of whale size with approximately equal weightings given to all variables except
276 D. MACHIN
perhaps variable (1) with a larger weighting and variables (3), (7), (14), (15) and (16) with smaller weightings.
The second component for the South Georgia and South African males is mainly a contrast between the head region and the flipper. For the Bonin Islands whale it is a complex contrast between variables (3) and (7) in the head region and (14) and (15) in the
TABLE V I I I Mean values and standard deviations of male and female whales recorded at Iquique*
Number of observations
Male
Mean Standard deviation
Number of observations
5
5 5 5 5 5 5 4 4 4
4
5 5 5 5 5
5 5 5 5 4
5 1
-
-
5 5
12.36
0.78 0.58 2.66 3.1 I 5.26 0.40 4.25 1.61 3.44
1.68
4.28 3.66 5.96 I .46 1.32
0.68 1.19 1.27 0.59 1.22
0.93 3.30
-
-
0.30 1.36
1.81
0.20 0.15 0.55 065 0.89 0.06 0.79 0.27 0.67
0.3 I
0.56 0.37 0.62 0.18 0.24
0.05 0.14 0.17 0.05 0.14
0.13
-
-
-
0.08 0.35
3
3 3 3 3 3 3 2 3 3
1
3 3 3 3 2
3 3 3 3 2
3 1
-
-
3 3
Female
Mean
9.20
0.46 0.28 1 -68 1.94 3.58 0.28 2.69 1.20 2.33
1.23
3.29 2.95 4.88 0.20 1.19
0.54 0.88 0.94 0.44 0.98
0.74 2.60
-
-
0.26 1.03
Standard deviation
0.36
0.01 0.03 0.07 0.07 0.14 0.04 0.02 0.06 0.20 - -
0.12 0.19 0.40 0.00 0.23
0.01 0.05 0.08 0.02 0.00
0.08 - -
0.01 0.26
* Variables included in the multivariate analysis.
dorsal fin together with high weightings for variable (8) and, of the opposite sign, variables (16) and (17). Variable (3) has very high weighting for the Antarctic whales as has the contrast between variables (16) and (17). The Japanese coastal whales show a strong contrast between variables (14) and (1 5) together with high weights for variables (3) and (9). The whales caught off South America however have high weights for variables (14)
TA
BL
E IX
A
vera
ged
corr
elat
ions
betw
een
mea
sure
d va
riabl
es (m
ale
valu
es a
bove
the p
rinc
ipal
dia
gona
l and
fem
ale
belo
w)
100
09
40
51
65
77
51
54
49 -
-
56
37
61
39
43
17
49 -
21
50
19
07
(2) 78
100
- 20
28
20
11
01
15
17
-
-
- 22
08
04
- 32
- 09
- 23
- 04
-
-17
- 38
14
03
(3)
(4)
(5)
70
91
91
51
84
77
100
62
64
50
100
93
44
76
100
26
68
73
19
57
39
41
48
59
_-
-
41
39 -
--
-
28
35
28
09
15
24
21
43
52
42
08
05
32
34
39
39
42
43
50
29
35
36
13
06
13
21
33
31
11 -
01
-02
-10
13
00
--
(6) 95
84
64
94
94
100 56
49
45
-
-
38
35
57
19
72
10
42
-
23
29
23
13
(7)
(20)
(2
1)
(22)
(2
3a)
63
92
75
92
90
66
75
59
12
19
46
77
46
64
64
66
89
66
91
90
64
88
74
91
89
63
96
17
90
90
100
48
61
55
56
-
100
94
95
89
- - 1
00
70
84
- 1
00
93
- - -
-10
0
30
13 - -1
6 -
19 -
26 - -3
3 -
08
65 -
58 -
22
60
--
-
41
42 - - -
39
--
--
29
72
- - -
--
-_
--
-
-12
19 -
IS -
28 -
10
15 -
25 -
26 -
21 - -2
0 -
-_
-
(8)
(10)
(1
1)
78
86
90
65
61
64
52
56
59
68
72
78
66
73
80
75
71
84
41
50
57
69
71
78
51
53
58
71
82
83
52
79
83
100
68
73
48
100
89
35
14
100
22
02
08
43
39
40
67
48
49
30
29
25
--
-
27
09
10
17 -
04
24
15
19
04
06
10
01
43
25
51
52
64
49
33
53
61
73
45
68
59
31
55
67
51
71
46
31
58
70
60
74
68
55
62
80
53
75
68
65
65
80
56
78
80
90
69
80
60
57
60 -
37
47
50 -
53 -
60
81
38 -
57
35
72
69
50 -
64
65
71
80
69 -
73 -
65
75
41
50
68
57
40
57
68
40
55
73
66
51
67
79
44
67
68
70
66
65
82
100
39
58
51
35
42
68
47
100
94
63
13
45
59
65
86
100
75 -
37 -
39
54
43
100
53
52
60
-10
0
--
46
16
44
19 -
100
7s
37
100
-05
04 -
03
02 -
08
18
-16
-09
06 -
36 -
04
02
--
-
--
--
-
(14)
59
57
48
60
55
58
46
59
55
62
71
41
54
53
41
39
36
55
-
43
50
100 08
(1 5)
41
41
35
39
41
40
46
37
56
36
37
33
41
44
33
38
59
55
-
33
37
27
100
x > r m
278 D. MACHIN
and (15), but now of the same sign contrasting with high values for variables (8) and (10) of the trunk region and (16) and (17) of the flipper. In addition variable (7) in the head region is prominent.
TABLE X Component I weightings of the original variables for the males
Figure 1 gives the half monthly variation in component I for the males reported by Matthews and indicates an average decline in whale size caught from October to August; together with Fig. 2 (the projection of components I and 11) it shows that the South Georgian males caught were mostly large whilst those caught off South Africa varied over the whole size range.
EXTERNAL MEASUREMENTS OF THE SPERM WHALE 279
The single male whale recorded at Talcahuano (South America) had a high component I1 coordinate but is thought to be not untypical of the whales caught off South America.
It is clear from Table XI that, for the females, only 50 to 60% of the variation is ex- plained in the first two components: indeed in all cases the first four components (five for
TABLE XI Component I1 weightings of the original variables for the males
~
Variable A B
-
D
~
E Location
C
0.09
0.37
0.15 -0-01
-
0.02
- 0.03 0.23
- 0.09 -0.07 -0.01 -0.29
0.05 0.09
-0.02 -0.17
-0-01
- 0-20 0.70
- 0.01 0-05 0.02
005
001 0.21 0.03
-001 005
006 -
0.10
0-06 0.12 0.04
-0.03 - 0.01
0.33 -
-0.15 0.20 -
0.07 -
- 0.09 -0.09 - 0.06
0.46 -0.07 - 0.00
-0.15 -0.11
0.03
0.14 - 009 -006
- 0.27 - 0.22 -0.10
-0.66 - 0.07
- 0.22 -
- 0.26 - 0.23 -
0.15
-0.35 0.47 -
-0.07 0.17
-0.21 - 0.1 5
0.03 0.05
0.09
-0.01
- 0.27 0.60
1.03
- -0.20
-0.09
0.23
-
- 0.90
0.99
- 0.28 -
-0.16 - -
0.58 0.50
1.32 1.05 1 40
8.75
65.87
4.89
77m-l
9.35
62.38
6.60
83.08
7.78
78.26
the South American females) have eigen values in excess of unity. This one might expect from the form of the female correlation matrix of Table IX. The first component seems to be a general measure of whale size as for the males with variable (1) given high, but not the highest weighting. Low weightings are given consistently to variables (9), (14) and (15) and
280 D. MACHIN
variously to (2), (8) and (10) whereas variables (4), (5), (11) and (20) each have high weightings.
For the South Georgia and South African females components I1 appear to be a contrast between variables (4) and (5), of the head region, (8) and (lo), of the trunk, against (16),
TABLE XI1 Component I and II weightings of the original variables for the females
I Location
Variable A D
Component I1
Location E A D E
0.40
0.39
0.30 0.35
-
-
-
0.42 -
-
-
-
0.33 0.13 0.34 -
0.26 0.17
0.28 -
-
0.21 -
-
-
-
4.80
40.02
40.02
0.35
0.02 0.29 0.34 0.37 0.35
0.33
0.31
-
-
- -
0-05 0.09 0.34 -
-
- -
- -
0.17
0.21
0.14 - 0.02
5.27
-
35.16
35.16
0.36 0.00
0.00 0.02 0.17 -
0.27 0.32 0.28 0.30 0.34 -
0.21 -
- -0.14 - -
0.29 036 0.27 032 0.23 0-03
0.14 - 0.39 035 - 0.46 0.33 -
0.23 -0.43 - -
0.06 - 0.02 - -
000 -
0.02 -
6.58 2.28
38.73 19.03
38.73 59.05
0.22
-0.34 0.02
-0.10 - 0.04
0.19 -
-0.24
- 0.20 -
- -
0.48 0.42
-0.17 -
-
-
-
- -
0.26
0.21
026 0.28
2.52
-
16.79
51.96
0.04
0.49 - 0.27
0.27 0.28 0.1 5 0.11 - -
- - -
-0.13 0.04 0.09 -
- 0.46 - 0.04 - 0.20 - 0.09 -
-0.35 - -
0.05 0.29
2.78
16.37
55.1 1
(1 7) and (19) of the flipper. This is not the case for Japanese coastal females whose second component has highest weightings for variables (8) and (10) of the trunk, moderately high weighting for variables (9) and (24a) of the flukes and (14) and (15) of the dorsal fin
EXTERNAL MEASUREMENTS OF THE SPERM WHALE
I I A A
I I I U A A
-8 -6 -4 . - 2 A -10
28 1
A A 0 A A A O
I . A D 2 D A 4 A O A A A A
I
Souih Georgia ‘f 2
Numberof whales
Jan. Fc -8
5 1 3 6 6
Mar. A p .
oulh Africa South Georgia
ec.
FIG. 1. Seasonal variation of Component I for the South Georgia and South Africa males.
contrasted with variables (2), (20) and (22) of the head region. For the South American females high weightings to variables (2), (3), (4) and (5) of the head region are contrasted with (16) and (17), of the flipper. Variables (9) and (15) also have high weights.
Jolliffe (1972, 1973) has described several alternative methods for the selection of those original variables of prime importance in a Principal Component Analysis. One suggestion
3 c 0
m
-3
0
A
A
A
FIG. 2. Projection of the first two components of the Principal Component Analysis for the South Georgia and South African males.
282 D. MACHIN
chooses those variables, one from each successive component starting with component I, with highest weightings in the components from those components with an eigen value greater than 0.7 together with a restriction that a minimum of four original variables should be taken. These selected variables are listed in Table XIII.
TABLE XI11 Important variables suggested by the Principal Component Analysis
Locality Important variables in their component order
Male Female
* The weighting for variable (1) differed from that of variable (6) by only 0.003.
Canonical Analysis The statistical technique of Canonical Analysis is described by Seal (1966) and is well
summarized by Ashton, Healy & Lipton (1957) on which the following description is based.
Suppose we have observations from two p-variate populations with a common within population dispersion matrix Wand with sample mean vectors x,, x,. Then the generalized distance between the two samples is given by
02=(x,-x,) w-1 (x,-x,)
If the populations are multivariate Normal then it can be shown that the amount of overlap between them is a function of P.
Suppose next that we have observations that h( >2) p-variate populations. These observations may be depicted as points in a p-dimensional space for which we seek a repre- sentation in a space of fewer dimensions (say) k. To do this, we replace the original p variates on each individual by a set of k linear functions of these variates in such a way as to maintain the greatest possible separation between the samples by maximizing the sum of all the D2 between the different pairs of population samples. It has been shown that the appropriate functions (canonical variates) are the solutions of the equation
(B-kW)x=O where B is the between populations dispersion matrix, the values of h being proportional to the between-population variances of the corresponding functions. The resulting functions are uncorrelated within samples.
The technique described assumes that the populations have a common dispersion matrix. We have shown earlier however that this is not the case for the two sexes so that separate analyses are reported on for the males and females.
It is clear from Tables I and VIII that although we have in effect observations from eight male populations and six female populations different numbers of variables are
EXTERNAL MEASUREMENTS OF T H E SPERM WHALE 283
available for study in each population. In fact only the p = 7 variables indicated in Table XIV are recorded in sufficient detail in all populations to be included in the canonical analysis. It would have been preferable rather if we could have used the most informative variables suggested by the Principal Component Analysis described earlier.
TABLE XIV Mean values of those variables (after transformation by logarithms to the base 10) included in the Canonical Analysis
(males in upper-row, at each locality, females in lower)
The logarithm to the base 10 of the individual observations was taken before analysis to improve the multivariate normality of the data. The means (after transformation) and number of observations available from each population are summarized for males and females separately in Table XIV. It is clear that the females are, for each variable in each population, smaller than the males. The correlation matrices for the males and females separately are given in Table XV for all the populations combined. It is clear from Table XV
TABLE XV Correlation matrices for male and female whales of those variables included in the Canonical
Analysis (male above and female below the diagonal)
Variable Variable
(4) (5)
100 - 07
54 72 75 67 74
80 100 07 10
- 05 07 08
91 93 85 79
100 90 73 100 25 50 35 45 49 69
(8)
85 67 76 78
100 55 59
(10)
89 67 79 82 79
100 41
(1 1)
92 67 82 84 82 92
100
284 D. MACHIN
that a correlation for the female is smaller than the corresponding correlation for the males, the correlation matrices cannot therefore be regarded as homogenous.
A major contributor to the heterogeneity is variable (2), (Projection of snout beyond tip of lower jaw). The results of the separate canonical analyses on the males and females are given in Table XVI and Figs 3,4 and 5. The latter show the group means and approximate 90 % contours.
TABLE XVI Weighrings of the original variables for the first rhree canonical axes
Canonical Variate Variable 1 I1 111
13.65
0.93 0.98
- 10.45
- 16.52 - 15.78
6.94
0.68
58.98
2.19 - 5.61 - 5.98
7.22 -1.64 - 0.27
-4.19
-8.50 0.8 1
- 7.80
3.77 - 3.77 36.99
0-35
-40.19
6.17
32.02
38.70 -13.00
4.57
-27.13
- 59.47
4.70 -0.04
5.08
30.83 12.42 4.35
0-25
- 6.70
7.26
6.25 - 10.92
- 14.65 21.88 5.55
Eigen value 3.51 2.21 0.18
For the males the first three canonical variates are significant. The first canonical variate distinguishes those whales caught off South Georgia, South Africa and in the Antarctic from the remainder. Whereas the second variate makes a clear distinction between the whales caught in Japanese coastal waters and those caught off the Bonin Islands. The third variate distinguishes those whales caught off South Africa from those caught by the Japanese in the Antarctic and those caught off South Georgia.
Clarke (1957 : 244) gives data for one adult male Sperm whale caught in the Azores in 1949 and suggests it provides no evidence of significant differences between the proportions of the individual and the range of those in southern whales. Figures 3 and 4 show the relative position of this whale, coordinates (-0.47, 1-49, 0.83). One can see the Azores whale is smaller than those caught off South Georgia and smaller than the average of those males caught off South Africa and the Antarctic. Its values for the second and third axes are larger than the corresponding values for the whales caught off the remaining localities.
EXTERNAL MEASUREMENTS O F THE SPERM WHALE
Males
Azores
I I
\ lquique \ / ' 7''"
285
FIG. 3. Projection of the first two canonical axes for the male whales.
For the females discrimination between the populations is effectively obtained by the first two canonical variates. There are three clear groupings indicated by the first canonical variate whereas the second canonical variate makes no distinction between the Western South American (Paito, Pisco and Iquique) and Eastern Asian (Japanese Coastal and Bonin Islands) groupings but isolates the South African whales.
FIG. 4. Projection of the first and third canonical axes for the male whales.
286 D. MACHIN
The size of the coefficients in the canonical axes suggest the prime importance of variables (l), (8), (10) and (11).
Discussion and conclusions Although the interpretation of the Principal Component Analysis has been somewhat
complicated by the differing numbers of variables available at each locality it has brought out those variables of major interest. It is clear from Table XI11 for instance that for the males the total length of the whale is of prime importance and that, where the data are available, variables associated with the dorsal fin are important for both sexes. If full data
Females
\ / I *Pisco I \
-d Bontn Islands I
South Africa
FIG. 5. Projection of the first two canonical axes for the female whales.
could be collected on whales the principal component analyses suggest concentration on the following ten variables, i.e. (I), (2), (3), (8), (9), (lo), (14), (15), (16) and (17), on a greater number of whales.
Applying Jolliffe’s rule to the Humpback study of Machin & Kitchenham (1971) variables (l), (lo), (14) and (15) are suggested each of which is considered important in the present study.
We have pointed out earlier the lack of homogeneity of the respective correlation matrices which is typically illustrated in Fig. 6 where the scatter diagram for the variables (2) and (3) for the Japanese coastal whales is given. The contrast in observed correlation coefficients may well be due to the restricted size range of the females caught rather than a true sexual dimorphism, a point which needs further investigation.
EXTERNAL MEASUREMENTS OF THE SPERM WHALE 287
0.6
- m
- n a : 0'4-
Y
u
0.2
0
The canonical analysis, albeit on a restricted number of variables, indicates several clear distinctions between the various localities. In the females there is clear separation into three groups, (i) the Japanese and Bonin Islands whales, (ii) the South African whales and (iii) the South American whales. Although this may be thought of as an overall difference in size this is not entirely so as the second axis separates the South African whales from the others. For the males the picture is not so clear but the whales from South Georgia and the Bonin Islands seem distinct from the remainder. The South African males seem relatively larger than their female counterparts.
a 0 0 *a a
- a a a
0 0 a * a
a 0
0 0 0 t a a a a 'a a a
o o ~ a 0:- a a .
a a a a a 0 00
o a coo
Length ronge of moles 1 0 ~ 6 8 t o 1 6 ~ 9 5 m Correlation coefficient I = 0.64 -
1 I I I 1 I 0.2 0.4 0.6 0.8 1.0 1.2
Length ronge of femoles 10-60 to l l .60m Correlation coefficient r =-040
a
a a a
0
a
am a a
FIG. 6. The correlation between variable (Z), the projection of snout beyond tip of lower jaw, and (3), tip of snout to blow hole, for the Japanese coastal whales.
We cannot agree, certainly with respect to the males, that the Sperm whales from the Antarctic, Bonin Islands and adjacent waters of Japan can be regarded as having no differences in body proportions as is reported by Fujino (1956). Figures 3, 4, and 5 show clear distinctions between these groups especially with respect to the second canonical axis and are contrary to Clarke & Paliza (1972) who group the Japanese and South African whales together.
Our studies agree with Clarke & Paliza (1972) however when they indicate that the males from South Georgia and Durban differ. There is some tentative support also to Tomilin's recognition (1957) of northern and southern subspecies of the Sperm whale although the groups do not divide quite so clearly as that would suggest.
The canonical analysis emphasizes too the importance of variables (l), (8) and (10). In conclusion there are several cautionary points to note:
(1) The whales caught and measured cannot be regarded as random samples from the whales present in their locality of capture.
288 D . M A C H I N
(2) The lapse of years between samples from the differing localities confounds their
(3) Measurements have been taken by several different observers at the different stations. (4) Lack of homogeneity of the correlation matrices.
Each point emphasizes the necessarily tentative conclusions to be drawn from these data.
possible differences with time.
I am very grateful to Mr B. Sanderson of the Computing Laboratory, U.C.N.W., Bangor, for much help in processing these data. I would also like to thank Dr Ray Gambell, Whale Research Unit, National Institute of Oceanography, who saw an earlier version of this paper and pointed out the wealth of additional data available that had escaped my notice.
R E F E R E N C E S Ashton, E. H., Healy, M. J. R. & Lipton, S. (1957). The descriptive use of discriminant functions in physical
Clarke, R. (1957). Sperm whales of the Azores. ‘Discovery’ Rep. 28: 237-298. Clarke, R. & Paliza, 0. (1972). Sperm whales of the Southeast Pacific. Part 111. Morphometry. Hvalrad. Skr. 53:
Fisher, R. A. (1967). Statistical methods for research workers. 13th Edition Reprinted. Edinburgh: Oliver and Boyd. Fujino, K. (1956). On the body proportions of the Sperm whales (Physeter catodon). Scient. Rep. Whales Res. Znst.
Jolliffe, I. T. (1972). Discarding variables in a principal component analysis. I. Artificial data. Jl R. statist. SOC.
Jolliffe, I. T. (1973). Discarding variables in a principal component analysis. 11. Real data. Jl R. statist. SOC. (C) 22:
Machin, D. & Kitchenham, B. L. (1971). A multivariate study of the external measurement of the Humpback
Matthews, L. H. (1938). The Sperm whales, Physeter carodon. ‘Discovery’ Rep. 17: 93-167. Seal, H. L. (1966). Multivariate statistical analysis for biologists. London : Methuen. Tomilin, A. G. (1957). [Mammalsof U.S.S.R. andadiacent regions.] 9. Cetacea. Moscow: Akad. Nauk. [In Russian]
anthropology. Proc. R. SOC. (B) 146: 552-572.
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Tokyo 11: 47-83,
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