#########################
STAT 718A Howework 1: Solutions
### I have attempted to do the task
### without using knowledge of the actual co-ordinates
### I have used approximate co-ordinates which
### have some measurement error.
### This question was quite hard because of the task of
### obtaining the co-ordinates. I have been very flexible
### in interpreting the approximate answers.
DATA in R:
y
, , 1
[,1] [,2]
[1,] 0 0.00
[2,] 1 0.00
[3,] 0 0.46
, , 2
[,1] [,2]
[1,] 1.51 1.04
[2,] 2.81 1.79
[3,] 1.18 1.60
, , 3
[,1] [,2]
[1,] -0.93 0.63
[2,] -0.19 1.95
[3,] -1.51 0.96
, , 4
[,1] [,2]
[1,] 0 -1.98
[2,] 1 -2.00
[3,] 0 -1.19
, , 5
[,1] [,2]
[1,] -2.63 -0.11
[2,] -2.07 -0.67
[3,] -2.19 0.35
, , 6
[,1] [,2]
[1,] -2.07 1.32
[2,] -2.65 1.88
[3,] -2.19 2.33
, , 7
[,1] [,2]
[1,] -2.07 -1.32
[2,] -2.65 -1.88
[3,] -2.19 -2.33
ans<-procGPA( y)
ans$size
# Centroid size
A: 0.90
B: 1.34
C: 1.35
D: 1.05
E: 0.83
F: 0.84
G: 0.84
ans<-bookstein2d( y )
ans$bshpv
# Bookstein shape variables
U^B V^B
A: -0.50 0.46 }
B: -0.50 0.43 } approx equal shape
C: -0.50 0.44 }
D: -0.52 0.80 }
E: -0.52 0.80 } approx equal shape
F: 0.48 -0.80 }
G: 0.48 0.80 } reflections
#########################################
When using approximate calculations we need to decide
whether or not two quanities are the same, as we have
measurement error in locating the landmarks. If the centroid
sizes or Bookstein coordinates are within about 0.03 I'll
treat them to be approximately equal.
Q1.
B and C have the same size and shape (approx)
A, B, C have the same shape (approx) but A has a different size from B, C
D, E have the same shape (approx) but different sizes
F, G have the same reflection shapes (approx)
Q2.
Centroid size ranking
E , F, G < A < D < B, C
(although A is rather similar to E,F,G actually)
Baseline size ranking
E, F, G < A, D < B, C
Square root area: A < E,F,G < D < B,C
Q3. Distance to anti-clockwise labelled equilateral
D < E,G < A, B, C < F
Note F has clockwise labelling, whereas all the others are
anti-clockwise. Hence, F is the furthest away from the anit-clockwise
equilateral triangle shape.
Q4. I have been very flexible with your answers, due to
the approximate nature of the task. My approximate
Bookstein shape variables are:
U^B V^B
A: -0.50 0.46
B: -0.50 0.43
C: -0.50 0.44
D: -0.52 0.80
E: -0.52 0.80
F: 0.48 -0.80
G: 0.48 0.80
To obtain the Kendall shape variables, note
U^K = 2/sqrt(3) * U^B , V^K = 2/sqrt(3) * V^B
A: -0.58 0.53
B: -0.58 0.50
C: -0.58 0.51
D: -0.60 0.91
E: -0.60 0.93
F: 0.55 -0.92
G: 0.55 0.92
Q5. This is quite a hard task due to the imprecise descriptions of
the landmark classifications. There is some flexibility again but
reasonable classifications are:
(a) Type I (tissues join?) Anatomical
(b) Type II (extreme curvature) Anatomical or Mathematical
(c) Type I (at a junction of tissues) Anatomical
(d) Type III (constructed) Pseudo
(e) Type III (constructed) Pseudo or Mathematical
(f) Type II (extreme point) Mathematical
The easiest to locate are (a), (b), (c), (f)
The hardest to locate are (d), (e)