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  • 1. Instar Determination Using Gaussian Mixture Models Hao Wu Department of Entomology and Plant Pathology April 24, 2012 . . . . . .
  • 2. Instar An instar is a developmental stage of insects, between each molt until sexual maturity is reached.(age) Insects have exoskeleton(different from us). Discontinuous growth of exoskeleton (have to molt to increase size) In many cases, the only obvious difference between successive instars is the increase in size that occurs after each molt (the molt increment). . . . . . .
  • 3. Histogram of Head With 40 30count 20 10 0 2 3 4 5 Head Width of Instars . . . . . .
  • 4. The GMMs Assumption There are finite numbers of components majority of insects have a fixed number of instar. The data within each components can be modeled as Gaussian distribution. The size of sclertized characters almost do not change in each instar. The Scatterplot of Ponotum characters The Scatterplot of Ponotum Length and Head Width The Scatterplot of Ponotum Length and Head Width q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q qq qqq q qq qq qqqqq q q qq qq q q qq q qq qq q qq q 10 qqqq q qq qq q q q q q q qqq qqqq q qq qq qqq q q q qqqqq q qq q q qq q qqq q q 16 q q qq qqqqqq q qq qq q q qq q q qq q q qq qq q q q qqq qqqqq q q q qq qq q q q q q qqq q qqq qq q q qqq q qqq qqq qq qqqq qq q q q q q q q q q q qq q q q q qqqqq q q q q q qq q qqq qq qqqq qq q q q q q q qq q q q qqqq q qqq q qq qq qqqqq qqqqq qqqq q q q q qq q qq q q q qqqq qqqq q q q qqq q q q q q qq qq q q qq q q qqqq q q q qq q q q qq q qq q q q qqqqqqqqqqq q q qq q qqqq qq qqq q qq qq qqq qq qq qq q qq q q qq qq q q qqq q q q q qqq q q qqqqqq q qqq q q q qqq qq qq q q qqqqq q qqqqqqqqqq q 5 q qqqqqq qq q qq qq q q q qq q q qq q qqqqqq q q qq q qqqqq q q qq q q q qqqqqqqqq qq q qqq q q q qq q qq qq q q q qqqq q qq qq qqq qqqqqqqq q q q qqqq qqq qqqq qq qq q qq q q q q q qq qq q qqq q q q q qqqqqqqqq qq q qq qq q q q qqqq qq q qq q q qq q q q q q qqq q q q q q q qq q q q qq q qq q q q q qq q q q q qq q q qq q qq q q q qq q q qq q qqqqq q q q qqq qq q qq q qq q q qq q qq qqqqq qq q qq qq q q q q qq q q 14 q qq q q q qq q q qqqq q qq q qqqq qq q qq q q q q q q qq qq qq q q qqq q qq q q qq q qq q q q qq q q q qq qq qqq q q qq q q q qq q q q q q qq q qq q q q q q q q qq q q q q q q qq qqq qqqq q q qq q q q q q q qqqq qq q q qqq q qq q qq q q qqq q qq q q qq q q q q q q qq qqq qqq q q qq q q qq q q qq q qqqqq q q q q qq q q q q q q q qq q qqqqqqqq q q q qqqqqqq q q q qqqq q q q q q qq qq q q q q qq qq q q q q q qq q q q q qqqqqq q q q q qqq q q 8 q qqqqqq qqq q q q qq q q q qq q q qqqqqqq qq q q qq qq q q q q q q q q qqq qqqq q q qq q qq q qqq q qqq q q qq qq q q q q qqq qq q q q q q qqq q q q q q q q qq qq q qq q qqqq qqqqq q q qq q q qq q q qq q q qq q qq q qqq q q q qqqqq qqq q q q q q q q qq qqqqqqq qqqqqqqqq q qqqq qqq qq q qq qqq q qq q q q qqqqqq q q q q qq q q q q qqqqqqqq q qqqq qqq q q q q q q q Pronotum Length qqqqq qqqq Pronotum Length q q q q q qqqqqqq q q q qq q q q qqqq q q q qq q q q qqq q q q q qqqqq q qqq qq qqqq q q q qq q q q qqq q q Pronotum Width q q qqqqq qq q q q qqq q q q q q q qqqqq q q q qq q qq q qq qq q qq qq qq q q q q qqqq qq qq qq q 12 q qqqq q q qq q q qqqq qqqqqq qq q q q qqqqqqqq q q qqq q q q qqqq q q q qq q qq qq qqqq q q q q q q qq q q qqqqq qq q q q qq q qq q qqqqq q q q q 4 q qqqqqqq q q q q q qq q q q q qqq qqqqq q q q q q qq q qq qq q q qqqqqq q q qqqqqqq q qqqqqq q qq q q q qq qqq q q q q q qq q q q qqqqqq q qq q q q q q q q qqq q q qqqq qq q qq q qqq qq qq q qqq q qq q q qqqq qqq q qqqq q q q q qq qq q qq q q qqqqqq q q qq q q qq q qq q q qq q q qq q q q qqqq q qq q q qqq qq q q q qqqqqqqq q q q q qqq q q q q q q qqq qqq q qqqqq qq q q q q qqqq qqqq q q qqqqqqqq q q q qq q q q qqqqq q q qqq q q q q q qqqq q q qqqq q q qq q qqqqqqqq q q q qq q q q qq q q q qq q qq q q qq q q qqq qq q q q q qqqqq qq qqqqqq q q qqqqqqqq q q q qq qq q q q q qq q q q qqqqq q q qqq q q q qqqqq q q q q qqq q qqqqqqqqq q qqqqqq q qq qqqqqq q qqq qq q qqqqqq q q qqqq q qq q q q qq qqqqq q qqq qqqqq q qq q qq q q q qqqqqq qq q qqqq qqqqqqqqq q q qq q qq q q qqqq qqqq q q q qq q qq q q q qqqqq q q q q qq qqqqqq q qqq q qqq q q qqq q q qq q qqq q q qqqqqqqq q qqqq qqqq qq q qq q q q q q q qq q qq q qqqqq q q q q q qq q q qqqqqqqqqqq qqqqqqqq qq qq q q q q q qq q q q qqqq qq qq qqq q qqqqq q qqq q q qqqqqq q qq q qq q qq qq q qqq q q q q qq q q q qq qqq q qqqqqq qq q q qqq qqqq q qqq qqqqqqq qqqq qqqqqqqq 10 q qq q q q q qqqq qq q q q qqq q q qqq q q q q q qqqqqqqq qq q q q q q qq qqqqqq q q 6 q qq q q q q qq q q qqqq q q qqq q q q qq q q q q qqqq qq q qqqqqq qqqqqq q q qqqq qq qqqqq q q qq qqqqqq q qq qq q q qqqq q qq q qq q qqqq qq q q qqqq q q q qqqqqqq q q qq q q q q q q q qqqqqq qqq qq q q qqqqqq q qq q q qq q qq qqq q q qq q q q qq q q qqq q q q q qqq q q q qqqq q qq qqqqqqqq qq qq q qq q q qqqqqqqq q qq q q qqq qq q qq qqqq q q q qq q qq qqq q q qq q q qq q q q qq qq q q q qqq qqq q qq q qq q q qq q q q q qqqqq q q q qq q q q q qqq q q q q q qq q q q q qqqqqq qq qq q q qq qqqqqq q q q q qq q q q qq qq q q q q qq qq qqqqq q q q q q qqqq q q qq q q q q q qqqqq q qqq q qq q q q q q q qq q q q qq qqq q q qqqqq q q q qqq q qq q q qqqqq q q qq q q qq qq q q qqq qq q qqqqq q qq qq q q q qq qqqq q qqqqq qq qqq qq q q qqqq qqqq q q qq qqq q qqq qqq qq qq q qq qq q q q qq q qq qq q q q qqq qqqqq q q q q q q qqqqq q q q q qq qqq q q q qqq q q qqqq q qq q q qq q qq q qqqq q q 3 q qq q qqqqqq q q qqqq q qqqqq q qq q q q q q qqqqqqqq q q q q qqqqq q q q qq q q q q qqq q qq qq q q q qq q qq q q q q qq qq q q qqqq qq q q q qq q q q q qqq q qq qq qq q q qq q q qq qq q q q qqqq qq q qq q q q q q qqq qq q 8 q q qq q q q qqq q qq q q qqq q q q q q qqq q q q qq qq qq q qqq q q q q qqqq q q qqqqq qq qq qqq q qqqq q q q q q q qq q q qqq qq qqq q q qq q qq q qqqqq q q qq q q q qq q q q qq q qq qq q q q q qqqq qq q qqqq qq q qq q q q q q q qq q qq q q qq q q q q qq q q q q qqq qq q q q q q q qqq q q q q qqq q q q q q q q qqq qq q q q qq q qq q qq q q qq q q qq q qq qqq q q qq q q qq q q q q q qq q q qq q q q qqq qqq q q qq q qq q qq q q q q qq q q q q q q q qq q q qq q q qqq qqq q qqq q q q q qq q q qqqq q q q q qqq q q q q q q qq q qq q qq q q q qq qqqq q qqqq q q qqq qqq q q q qq qqq q q 4 q qqqq qq q qqqq qq qq q q qq qq qq q q qq qqqqqqqq q q qqq qq qqq q q qqqq qq qqqq qqqqqq qqq q q qqqq qqqq q qqqqqq q q q qqq qq q qq q q q q q qq qq q qqq qqq q qq q qq q q qqq q qq q q q qqqq q q qq q qq qq q qq qqqqqqq qq qqqq qqqq qq q qq q q q q qq q q q q qq q q q q q q qqq q q qq qq qqqq q qq qq q q q qq q qqq qqqqqqqq q q qq q q q q q q qqqq 6 q qqqq qq q q q q qqqq q q qqq q q q q q q qq q q q qq qq q q qqqqqq q qq q q q qqqq q qq q q qqqq qq q qq q qq q q q qq q qq qq q q qqq qq q q qq q qq qqqqqq qqqqqq qqq qq qq q q q qq qq qqq q qq qqqqq qq qq qqqqq qqqq q q q qqqq q q q q q qq q qq q q q q q qqqqq q qq qqqqq qqqq q qqq qqq q qqq qq 2 qq q qq q q qq q q qqq q q qqqqq qqq qqqq q qq q q qq q q q qqqqq qqqqqq qq qqq qq qq qq q qqqqqqqq qq qqq qq q qq q q qqq q qq q qqq q q qq qq qqqq q qqq q q qqqq qq qq q qq qq q qqq qq qq q q qq q qq qqqqq q q qqqq qqqq q qqq qqq q q qq q q qq q q qqqqqq q qqq qqqq qqq q qqq q q q qq q qqqq qqq qq qq qqq qqq qqq qqqqq qqqq qqq q q qq q q qq q qq qqqq q qq q qqqqq qqqq qqq q q q qq qqqq qqq q q 4 qq qqqqq qqqqq q qq qq qq q q q 4 6 8 10 12 14 16 2 3 4 5 4 6 8 10 Pronotum Width Head Width Head Width Figure: The Scatterplot of Three characters . . . . . .
  • 5. The Scatterplot of Ponotum characters q q q q q q q q q q q q qqq q q q q qq qq qq 10 qqqq qq qqqqq q qq q q qq q q q q qq q q qq q q qq qq q q q q q q qqq q q qq q q q q qqqqq q q q q q qq q qq q qq qqqqq qqqqq qqqq q q q q qq q qq qqqq q q q qq q q q qq q qq q qq qq q q qqq q q q q qqq q q qqqqqq qqq q qq qq q q q qq qqqqq qq q qq q q qq q q q q qq q q qq q q q q q qqqq qq qq q q q qq q q qq q qq q q q q qq q q q q qq q q qq q q qq qqqqq qq q q q qq q q qq q qq qqq q qqqqq q qq q q qq q q q q q qqqq qq q q q qqq q qq qq q q qq q q q q qqqqq q q q qq qq q q q q qq qq q q 8 q qqqqqq qqq q q q qq q q q qqq q qqqqqqq qq q q qqqqq q q qq qqq q qqqq qqqqq q q q qq q q qq q qq qqqq qq q q qqq q qq qPronotum Length q q q q q qq q q qqqqq qq q q q qq qqq q q qq q q qqqq q q qqq q qq q q qqqq q q qq qqq qqq qq qq q qq q q qq qq qq q qq q q qqqqq qq q qq q q q qqqq qqq q qqqqqq q q qq q q qqq q qqq q qqqqqq q q q q q qqqqqq q q qqq q q qq qq q q qqq q q q qq q qq q q qqqq qqq q q q q q q qqqqqq qq q q qqqqqqqq q q q qqq q q q qq qq qqqqqq q qqqq q q qqqqqqqq q q qqqqqq qq qqqqqqq q q q q qqqqq q q qqq q q qqq q q qqq q qq qq qqqqqqqqq q q qq q q q q qqqqqqqq q qq q qq qq q qqqq q q q qqq qqq qqq q q q q qqq qq qq qq q q q qqq qqqq q q qqq q q 6 q qqq qqqqqqq qqqq qqqqqqqq q q q q qq qqqqqq qq qq qqqqq q q qq qq q q qqqq q qqqqqq qqq qq q q qqqqqq q q qqqq q q q qqqqqqqq qq qq q qq qqq qq q q q q q qq q q q q q qqqqq q q qq qqqqqqq qq q qqqq q qq q q qq q q qqqqq q q q q q q q q qq q qq q q qqqq qqqq q q qq q q qqq qqq qq q q q qq q qq q q q q qq q qqqqqqq q qq q q q qqq qq q qq q q qqqqqqqq q qqq q qqqqqq q q qq q q q qq q q q q q qqq qq q q q q q qq qq qq q qqq q q qq q q qqq q qqq qq q q q q qq q q q q qq q qq q q q q q q qqq qq q q q qq q q q q q qq q qq q q q q q qq q q qq q q q qqq qqq q q q qq q q qqq q q q q q q qq q q 4 q qqqq qq q qqqq qq qqqq q qqqq qqqqqq qq q q qqq q q qqq qqq q qq q q q qqqq q q qq q q qqq q q qq q qqq qqqqqqqq q q qq q q q q q q qqqq q q qqqq qq qq q q qqq qq qq q q q q qqqqq q q qq q qqqqqq qq q q qq q qqqq q qqqqq q q qqq q q q qqqqq qqq q qqq qqqq qqq qqq qq qqq qqqq q q q qq qq q qqq qq q q qq q qqqqqq q qq qq qqqq qqq qq qqq qqq qqq q qq q qqqq qqq q q qq qq 4 6 8 10 12 14 16 Pronotum Width . . . . . .
  • 6. Data Collection Period: Jan-Apr.2012 Variables: Head Width, Pronotum Length and Width. 1925 dubia cockroaches. . . . . . .
  • 7. Outliers Detection The function NNclean in the contributed R package prabclus is a implementation of the nearest-neighbor method for cluster/noise detection. This method is used in the instar determination with k = 2. q q q qqq q q q q q q qq q q qqqqq qqq q qq qqqqq q qq qq qqq 16 q qq q q qq qq q q q qqqq qq qqq q q qqqq q qqq q q q q qqqqq q qqqqq q q qq qqq qqqqq q q qqqqq qqq qqq qqqq qqqqq q q qq q qqq q qqqq q q q qqqqq q q qq qq q qq qqqq qqqqqq q qqqq qqqq qq qqqqq q qqq qq qq q qqqqq qq qqq q 14 qqq q q qq q qq q q q cluster q 1 12 2 Width 3 4 10 5 6 7 8 6 4 4 6 8 10 Length . . . . . .
  • 8. Model Selection BIC for all models up to 12 components q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q −5000 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Models q q q EII q q q q q VII −10000 q q q EEI q q q q VEI BIC q q q q EVI q q VVI q EEE q q −15000 q EEV q q q VEV q VVV q q q −20000 q 2 4 6 8 10 12 Number of components Figure: The BIC plot of each cluster in all models Selection:The unconstrained model(”VVV”),7 clusters mclustBIC in R package mclust . . . . . .
  • 9. Cluster Analysis GMMs based clustering are conducted based on the model selected by BIC. Mclust in R package mclust The cluster of Ponotum characters The cluster of Ponotum Width and Head Length The cluster of Ponotum Length and Head Width q q qq q q q q qq q q qq q q q qq q qq q qq q q q q qq q qq q qq q q qqq q qqq q q q qq q qq q q qqq q q q q q 16 qqq q q q q q q qq q qq q qq q q q qq qqqq q qqq qqq q qqqq q qq qqq qq q qq qq qq qq qq q qq q q q q qq q qq q q qqqq q q qq qq qq q qq q q qq q q q qq qq qqqqq qqqqq q q qq qqqqqqq q qq q qqqqq q 5 q qq q qq qq qq q 5 qqqq q qq qq qqq q qqqqq qqqq q qqq q q q qqqq q qq q q q qqqqqq qqqq qq q qq q qq qqq qqq q q qq q q q qqq q qqq q qqq qq q qqq qq q q qq q qq q qq qqqq q qqq qqqqq q qq qq q q qqqq q qq q qq qqq q qq q qqq q q qq q qq q qq qq q qq q q q q qq qq qqq qq qq q qq qq q q 14 q q cluster cluster cluster q 1 q 1 q 1 Pronotum Length Pronotum Length 12 4 4 Head Length 2 2 2 3 3 3 10 4 4 4 5 5 5 6 6 6 3 3 8 7 7 7 6 2 2 4 4 6 8 10 4 6 8 10 12 14 16 4 6 8 10 Pronotum Width Pronotum Width Head Width Figure: The cluster of three characters . . . . . .
  • 10. Result Assessment Instar 1 Instar 2 Instar 3 Instar 4 Instar 5 Instar 6 Instar 7 Length 2.42 2.97 3.83 4.95 6.14 7.48 9.37 Width 4.21 5.19 6.44 8.07 9.90 12.17 15.17 Head 1.79 2.09 2.49 2.98 3.54 4.22 5.02 Table: The mean of the three characters in each instar . . . . . .
  • 11. More about Instar According Dyar’s law (Dyar,1890) In = constant In−1 where In = postmolt size , In−1 = premolt size. It can be written as: y = ae bx which is equal to: lny = lna + bx the constant growth ratio is e b . . . . . . .
  • 12. Result Assessment Linear regression for the mean of three characters 2.5 Natural Logarithm of measurements q q 2.0 q Characters q Length q Width 1.5 Head q q 1.0 q 1 2 3 4 5 6 7 Instar Number Figure: The linear regression of characters in each instar . . . . . .
  • 13. Result Assessment Geometric sequence Instar 1 Instar 2 Instar 3 Instar 4 Instar 5 Instar 6 Instar 7 Ratio Length 2.42 2.97 3.83 4.95 6.14 7.48 9.37 1.26 Width 4.21 5.19 6.44 8.07 9.90 12.17 15.17 1.24 Head 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 . . . . . .
  • 14. Conclusion The GMMs based clustering methods can be used to determinate instars of the dubia cockroaches Blapitca dubia. The growth rates comply with Dyar’s law. There are 7 instars in Dubia cockroaches Blapitca dubia. The growth rates of pronotum length, pronotum width and head width are 1.26, 1.24, 1.19 respectively. The GMMs based clustering methods should be applicable to be used on other insects, since the dubia cockroaches comply with the same growth model with other insects . . . . . .

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