Using Discriminant Analysis

1
Using Discriminant Analysis Principal
Component Analysis in Evaluating the Impacts of
Silvicultural Treatments on Tree Growth
2
Impacts of Silvicultural Treatmentson 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.

3

• 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.

4
Figure 1. DBH distribution of study stems
associated with treatments.
5
Figure 2. Height distribution of study stems
associated with treatments.
6

• 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.

7

• Table 1. Mean values of dbh (inch) and ht (feet)
  for all and each treatment groups.
• Group Variable N Mean
  Standard
• 
  Deviation
• All DBH 1199 6.78
  1.66
• HT 1199 57.74
  10.07
• C DBH 312 5.48
  1.49
• HT 312
  48.31 8.47
• F DBH 272 7.71
  1.45
• HT 272
  64.58 5.44
• H DBH 322 6.32
  1.19
• HT 322
  53.40 6.66
• HF DBH 293 7.78
  1.25
• HT 293
  66.22 5.65

8

• 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
• -29.9517-3.5473dbh1.6420ht for C
• -51.7810-4.1344dbh2.0973ht for F
• -35.5738-3.4972dbh1.7468ht for H
• -54.9733-4.4319dbh2.1814ht for HF

9

• 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

10

• 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.

11

• 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.

12
Figure 3. Group-specific density estimates of
each treatment versus dbh.
13

• 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.

14

• 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

15

• 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.

16
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

17

• 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.

18

• 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).

19

• 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

20

• Table 3. Numbers of stems in four even groups.
• Group Treatment Number DBH HT
• 3 C 40
  7.53 59.59
• 3 F 95
  7.19 63.23
• 3 H 70
  7.56 59.83
• 3 HF 95 7.21
  63.79
• 4 C 6
  8.73 62.65
• 4 F 126 8.92
  68.43
• 4 H 12 8.60
  63.76
• 4 HF 155 8.69
  70.22

21

• 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).

22

• 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

23

• 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.