Using
Discriminant Analysis
& Principal Component
Analysis in Evaluating the
Impacts of Silvicultural
Treatments on Tree Growth
Impacts of Silvicultural Treatments
on Tree Growth
• Assuming that the relationship between diameter
at breast height (dbh, in) and tree height (ht, ft)
varies with treatment, a tree can be characterized
using a treatment class related criterion subject to
stipulated dbh and ht values. A higher rate of
correct judgement provides stronger evidence of
the significant difference among treatment groups
and vice versa. This method is so called
discriminant technique of multivariate data
analysis.
• Study data came from a research project at the
School of Forest Resources, UGA (CAPPS). The
discriminant technique was applied for evaluating
silvicultural treatment impacts on planted loblolly
pines from two installations in the lower coastal
plain of Georgia. The applied silvicultural
treatments were control (C), fertilization (F),
herbicide (H), and herbicide and fertilization (HF).
Figs. (1) and (2) show the distributions of dbh and
ht of study stems.
Figure 1. DBH distribution of study stems associated with treatments.
Figure 2. Height distribution of study stems associated with treatments.
• During the analysis, the prior probability was
assigned equally for each treatment (0.25) since
the number of stems in each class is approximately
the same. Cross validation was used to reduce bias
in which discriminant functions are obtained from
other n-1 observations and employed in
classifying the one left out. The overall means of
dbh and ht and the means of each treatment are
presented in Table 1.
• Table 1. Mean values of dbh (inch) and ht (feet)
for all and each treatment groups.
•
•
•
•
•
•
•
•
•
•
•
•
Group
All
C
F
H
HF
Variable N
DBH 1199
HT
1199
DBH
312
HT
312
DBH
272
HT
272
DBH
322
HT
322
DBH
293
HT
293
Mean
Standard
Deviation
6.78
1.66
57.74
10.07
5.48
1.49
48.31
8.47
7.71
1.45
64.58
5.44
6.32
1.19
53.40
6.66
7.78
1.25
66.22
5.65
• The multivariate analysis of variance showed
significant differences among treatment groups for
pooled dbh and ht, and a significant correlation
between dbh and ht (correlation coefficient 0.8019).
The derived linear discriminant functions are
presented below and classification result is listed in
Table 2:
• -29.9517-3.5473dbh+1.6420ht
• -51.7810-4.1344dbh+2.0973ht
• -35.5738-3.4972dbh+1.7468ht
• -54.9733-4.4319dbh+2.1814ht
for C
for F
for H
for HF
• Table 2. Number of observations and percentage classified
into a treatment (italic).
• TRT
C
F
H
HF
Total
• C
186
39
84
3
312
•
59.62
12.50
26.92
0.96
100.00
• F
5
92
33
142
272
•
1.84
33.82
12.13
52.21
100.00
• H
112
53
154
3
322
•
34.78
16.46
47.83
0.93
100.00
• HF
2
92
18
181
293
•
0.68
31.40
6.14
61.77
100.00
• Total 305
276
289
329
1199
•
25.44
23.02
24.10
27.44
100.00
• Error 0.40
0.66
0.52
0.38
0.49
• Large error rates of classification were between
treatments C and H, and F and HF, i.e., 27% of the
observations in C were misclassified into H and 35%
of the observations in H were misclassified into C.
Likewise, 52% of the observations in F were
misclassified into HF and 31% of the observations in
HF were misclassified into F.
• On the other hand, the classification for fertilized
stands versus unfertilized stands was more accurate,
i.e., only 15% of the observations from unfertilized
stands were misclassified into fertilized stands, and
10% of the observations from fertilized stands were
misclassified into unfertilized stands.
• The above facts implied that the growth responses
(for both dbh and ht) between treatment C and H
or F and HF are much less significant than those
between F involved treatments (F or HF) and F
excluded treatments (C or H). Figure 3 shows the
similar trends as the described above using the
group-specific density estimates obtained from
discriminant analysis.
Figure 3. Group-specific density estimates of each treatment versus dbh.
• The bell-shape distribution densities of treatments F
and HF are fully overlapped, implying that the
impacts of the two treatments on dbh growth are
hard to separate. Likewise, most of the density
estimates of treatments H and C are overlapped and
one may expect that more misclassifications occur
during separating the stems from H only and C
stands. However, distinct differences of the density
estimates exist between stems from fertilized stands
and unfertilized stands, implying a more accurate
classification of fertilized and unfertilized trees.
• In discriminant analysis, the impacts of silvicultural
treatments are evaluated using both dbh and ht
simultaneously. An observation is classified into a
group according to the nearest distance criterion. For
example, the calculated discriminant value for the
trees in F only stands is 52 and 36 for the trees in H
only stands. For a tree from F only stands with dbh
5.0 inches and ht 55 feet, the calculated discriminant
value is 43, which is closer to that of H only stands (7
versus 9 in absolute value). That is, this tree is too
small to be classified into the trees in F only stands
• Likewise, about 10% of such trees in fertilized
stands were called too small and classified into the
trees in unfertilized stands and about 15% of such
trees in unfertilized stands were called too large
and classified into the trees from fertilized stands.
On the other hand, tree sizes in F only and HF
stands were too close to call, which resulted in
large percentage misclassification, i.e., the
difference between impacts of the two treatments
is negligible.
Impacts of Silvicultural Treatments on Tree
Stem Development
• Table 1 shows that silvicultural treatments,
especially fertilization involved treatments,
increased dbh and ht growth, which may affect
stem profile. If the proportion of ht growth is
larger or smaller than that of dbh growth, the stem
could be slimmer or chummier, and vice versa. In
this study, we use principal component analysis
technique to address this problem
• Because only two continuous numerical variables
(dbh and ht) are involved, two principal
components can be constructed and the
coefficients equal to the eigenvectors of the
correlation matrix. Using study data, the two
eigenvalues obtained are 1.9 with proportion 0.93
and 0.13 with proportion 0.07 and the
eigenvectors obtained are presented below:
• P1 =0.4261(dbh-6.78)+0.0702(ht-57.74)
• P2 =0.4261(dbh-6.78)-0.0702(ht-57.74)
• where P1 and P2 are the first and second principal
components.
• Since the value of P1 increases with dbh and ht, it
reflects the variation of tree size, that is, larger
trees should have larger values of P1. Grouping all
1200 stems into four groups with 300 observations
each by ascending order of P1, the majorities from
the stipulated treatment stands in each group are C
and H (94%), C and H (74%), F and HF (63%),
and F and HF (94%), respectively (Table 3).
• Table 3. Numbers of stems in four even groups.
• Group Treatment Number DBH
HT
•
1
C
188
4.53
43.04
•
1
F
10
4.91
51.36
•
1
H
95
5.02
46.12
•
1
HF
7
4.70
52.03
•
2
C
78
6.44
54.13
•
2
F
41
5.90
59.11
•
2
H
145
6.40
54.20
•
2
HF
36
5.99
58.12
• Table 3. Numbers of stems in four even groups.
• Group Treatment Number DBH
HT
• 3
•
•
•
•
•
•
•
3
3
3
4
4
4
4
C
F
H
HF
C
F
H
HF
40
95
70
95
6
126
12
155
7.53
7.19
7.56
7.21
8.73
8.92
8.60
8.69
59.59
63.23
59.83
63.79
62.65
68.43
63.76
70.22
• On the other hand, the value of P2 decreases with
ht and increases with dbh, that is, chunky stems
have larger P2 value. Assuming that the stems in
the lower range are the slimmest and the stems in
the upper range are the chunkiest, study stems can
be grouped into three groups according to the
value of P2. The result showed that the
percentages of the stems from fertilized stands are
72% of the total 400 stems in the lower range with
a ratio of dbh to ht approximately 0.1 and 34% of
the total 400 stems in the upper range with a ratio
of dbh to ht approximately 0.13 (Table 4).
• Table 4. The stems in the first and third groups
obtained according to ascending order of the value
of the second principal component
• Group Treatment Number
DBH HT
•
1
C
52
4.80 49.99
•
1
F
122
6.74 64.34
•
1
H
59
5.48 52.96
•
1
HF
167
7.18 65.76
•
3
C
138
5.99 47.75
•
3
F
80
9.06 65.03
•
3
H
126
6.85 52.49
•
3
HF
55
8.95 66.51
• Note that more slim stems came from fertilized
stands and more chunky stems came from
unfertilized stands. The above statistics imply
that, in general, fertilization affects stem profile
development and the growth of these stems along
vertical direction is faster than horizontal
direction.