. . . . . .

Instar Determination Using Gaussian MixtureModels

Hao Wu

Department of Entomology and Plant Pathology

April 24, 2012

. . . . . .

Instar

I An instar is a developmental stage of insects, between eachmolt until sexual maturity is reached.(age)

I Insects have exoskeleton(different from us).I Discontinuous growth of exoskeleton (have to molt to increase

size)

I In many cases, the only obvious difference between successiveinstars is the increase in size that occurs after each molt (themolt increment).

. . . . . .

0

10

20

30

40

2 3 4 5

Histogram of Head With

Head Width of Instars

coun

t

. . . . . .

The GMMs Assumption

I There are finite numbers of componentsI majority of insects have a fixed number of instar.

I The data within each components can be modeled asGaussian distribution.

I The size of sclertized characters almost do not change in eachinstar.

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2 3 4 5

The Scatterplot of Ponotum Length and Head Width

Head Width

Pro

notu

m W

idth

2

3

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4 6 8 10

The Scatterplot of Ponotum Length and Head Width

Head WidthP

rono

tum

Len

gth

Figure: The Scatterplot of Three characters

. . . . . .

4

6

8

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4 6 8 10 12 14 16

The Scatterplot of Ponotum characters

Pronotum Width

Pro

notu

m L

engt

h

. . . . . .

Data Collection

I Period: Jan-Apr.2012

I Variables: Head Width, Pronotum Length and Width.

I 1925 dubia cockroaches.

. . . . . .

Outliers Detection

The function NNclean in the contributed R package prabclus is aimplementation of the nearest-neighbor method for cluster/noisedetection.This method is used in the instar determination with k = 2.

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Model Selection

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Figure: The BIC plot of each cluster in all models

I Selection:The unconstrained model(”VVV”),7 clusters

I mclustBIC in R package mclust

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Cluster Analysis

I GMMs based clustering are conducted based on the modelselected by BIC.

I Mclust in R package mclust

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Figure: The cluster of three characters

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Result Assessment

Instar 1 Instar 2 Instar 3 Instar 4 Instar 5 Instar 6 Instar 7Length 2.42 2.97 3.83 4.95 6.14 7.48 9.37Width 4.21 5.19 6.44 8.07 9.90 12.17 15.17Head 1.79 2.09 2.49 2.98 3.54 4.22 5.02

Table: The mean of the three characters in each instar

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More about Instar

According Dyar’s law (Dyar,1890)

InIn−1

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where In = postmolt size , In−1 = premolt size. It can be writtenas:

y = aebx

which is equal to:lny = lna+ bx

the constant growth ratio is eb.

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Result Assessment

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Figure: The linear regression of characters in each instar

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Result Assessment

Geometric sequence

Instar 1 Instar 2 Instar 3 Instar 4 Instar 5 Instar 6 Instar 7 RatioLength 2.42 2.97 3.83 4.95 6.14 7.48 9.37 1.26Width 4.21 5.19 6.44 8.07 9.90 12.17 15.17 1.24Head 1.79 2.09 2.49 2.98 3.54 4.22 5.02 1.19

Table: The mean of the three characters in each instar and growth ratio

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Conclusion

I The GMMs based clustering methods can be used todeterminate instars of the dubia cockroaches Blapitca dubia.

I The growth rates comply with Dyar’s law.I There are 7 instars in Dubia cockroaches Blapitca dubia.I The growth rates of pronotum length, pronotum width and

head width are 1.26, 1.24, 1.19 respectively.

I The GMMs based clustering methods should be applicable tobe used on other insects, since the dubia cockroaches complywith the same growth model with other insects