Bull. Hiroshima Inst. Tech. Research Vol. 46 (2012) 173 – 179

ArticleNon-parametric Modelling and Its Application on Real-life Data

Prasenjit PAL*, Subhabaha PAL**, Satyabrata PAL*** and Sanpei KAGEYAMA****(Received Oct. 13, 2011)

Abstract
Some non-parametric models (kernel, loess and spline) have been compared with different parametric models in order to judge their precision by measuring the proximity between the observed and thepredicted values based on the data-sets on the pest incidence (measured in terms of the number ofpests) in respect of two important crops, namely, brinjal and chilli, with a view to describing the patternof the incidence of pests over the year based on weekly incidence data (28 weeks). The importance ofsuch studies stems from the fact that the farmers need the specialized scientific information in the formof advisory services in respect of the pattern of pest incidence over the year in advance and creationof pest-models helps us to obtain such important information. This paper is devoted to generate pestincidence models in cases of brinjal and chilli crops. Such pieces of generated information assist thefarmers in their planning and execution of the cultivation of the said crops over the year. It is revealed
that the precision levels (as are obtained by calculating the actual values of the well-known R2 criterion)
are much higher when the non-parametric models are applied on the data-sets considered under thepurview of the paper, in comparison to other methods exploited to obtaining the above-said information.
Key Words: Kernel, Spline, Loess, Parametric model, Non-parametric model, R2 criterion

exist problems in developing models on the dynamics

1. Introductionof pest incidence (pest population) on crops grown in

A model is a representation of the outcomes of a real–

field owing to the scarcity of information on the inci-life phenomenon presented in terms of analytical rules

dences of pest population over a long period of time (aexpressed by means of equations developed on theyear), hence, indeed, on their dynamics. Unless andbasis of an obser ved data-set. As presented in theother wise planned long-term experiments are con-Abstract that extensive data were collected on the inci-

ducted for the purpose, hardly, anyone is able to col-dence (measured by the number of pests) of different

lect the chronological data on pest population over apests on the crops, namely, Brinjal and Chilli, using

considerable duration of time to facilitate the studies

fixed plot technique, the outputs (predicted values ofon modeling of pest incidences in respect of the above

the incidences related to different pests) obtained after

crops of interest. So, from natural pest control point offitting of the different models generate a wealth ofview, modelling of incidences (Agarwal and Kumar,information related to the dynamics of the infestation-2007) of these predator-insects also has a very impor-pattern of insects over a year. The fact is that theretant role in their management. The dynamics of the* NCIPM, LBS Building, IARI Campus, New Delhi-12,India** Sikkim Manipal University and Consultant, Manipal Universal, Bangalore, Karnataka, India*** Regent Education and Research Foundation, Barrackpore, West Bengal, India**** Department of Environmental Design, Hiroshima Institute of Technology, Hiroshima 731-5193, Japan─ 173 ─

Prasenjit PAL, Subhabaha PAL, Satyabrata PAL and Sanpei KAGEYAMA
Table 1: Forms of different parametric models

populations of pests or insect vectors of disease arecomplex in nature, and consequently the models rep-Linear

b0 + b1t

resenting the prediction of their outbreaks are difficult

Quadraticb0 + b1t + b2t2

to achieve. To help unravel such complex systems andCubic

b0 + b1t + b2t2 + b3t3

to understand how pests and vectors interact with theCompoundb0 b1tenvironment and with other organisms as reflected inLogarithmic

b0 + b1log t

the dynamics, attempts have been made in this paper

Inverseb0 + b1 / t

to apply different models (parametric, non-parametric

S Typee b0 + b1 /tand semi-parametric) on the data on pest incidence

Exponential

b0 e b1t

obtained from the planned year-long field experiments

Powerb0 t b1

on two important crops, Brinjal and Chilli, in respect

Gompertz (GRO)

abz, where z = c t

of a specific ecological situation. Indeed, the non-parametric and semi-parametric methods have revealed

can be found in Thisted (1988), Simonoff (1996),

their worth in modelling real-life data on pest infesta-

Härdle et al. (2004) and Kageyama et al. (2006).

tions mentioned above owing to the degree of closeness of the predicted values (obtained from such mod-3. Results and discussions

els fitted on the data-sets) to the observed values they

The descriptive statistics in Tables 2 and 3 next con-can produce. Other related references are Markkula

cerning the incidences of the different pests in case of(2003), Matis et al. (2005), Kageyama et al. (2006), andthe crops, Brinjal and Chilli, for the year 2007-2008 are

Agrawal and Kumar (2007).

presented. The salient features, which are revealed,are:

2. Materials and methods(1) In case of Brinjal, there is maximum variability

For the purpose of formulating models on the datain the incidences in case of the pest, Whitefly and leastmentioned in the Abstract, the rainy season data werevariability (almost no variability) in the incidences in

not considered as there were almost no incidences ofcase of the pest, Borer. The incidence data are posi-pests at that time for both the crops. In this study wetively skewed in all cases. The incidence diagrams, ifhave used the data on the infestation of the pests

prepared from the observed data (on data on incidence(Thrips, Jassids, Whitefly, Borer) on Brinjal, and pestsof pests), will be seen to exhibit platykurtic (more flat-(Whitefly, Yellow Mite, Thrips) on Chilli for the period

ness over the entire range when the incidence data are(September, 2007 to March, 2008) obtained from anplotted on the Y axis against months on the X axis)

experimental station located in Southern part (Gayes-feature in case of all types of pests.

pur farm in Bidhan Chandra Krishi Viswavidyalaya) of(2) In case of Chilli, there is maximum variability

West Bengal, India. A reference to the methods

in the incidences in case of the pest, Yellow Mite andemployed in this paper is given below.

least variability in the incidences in case of the pest,Whitefly. The incidence data, however, exhibit positive

2.1 Parametric methods

skewness in all cases. The incidence diagrams, if pre-Parametric models can broadly be categorized into lin-pared from the observed data (on incidence of pests),

ear and non-linear. Different parametric models as inwill be seen to exhibit platykurtic (more flatness overTable 1 like, linear model, parabolic model, polynomial

the entire range) feature in case of all (excepting themodel, logarithmic model, Gompertz model, etc. are

Yellow Mite) pests.

tried.

Among the parametric models fitted, it is observedthat the best fitted models in Tables 4 and 5 next are

2.2 Non-parametric methods

found to be cubic (based on R2 criterion, values of R2

The non-parametric models, Kernel model and Loesslying between 0.6 to 0.8) in all the cases (for the twomodel are employed, also used is the semi-parametriccrops and over different pests). Though the obtainedmodel, Spline model. Such models are well known andR2 values are satisfactory, a search for models which─ 174 ─

Non-parametric Modelling and Its Application on Real-life Data
Table 2: Descriptive statistics with respect to the data on the incidence (number of pests) of different pests on Brinjal cropPestNMin

Max

Mean

Variance

Skewness

Kurtosis

Whitefly

28

0.63

15

6.59

22.64

0.34

– 1.19Thrips280.14

11.99

4.72

14.83

0.45

– 1.13

Jassid

28

0.10

5.12

1.95

2.03

0.86

– 0.44

Borer

28

0.01

0.83

0.22

0.07

0.97

– 0.21

Table 3: Descriptive statistics with respect to the data on incidence (number of pests) of different pests
on Chilli crop
Pest

N

Min

Max

Mean

Variance

Skewness

Kurtosis

Whitefly

28

0.20

3.93

1.49

1.62

1.01

– 0.49Yellow Mite

28

0.01

18.60

4.96

35.20

1.33

0.47

Thrips

28

0.01

14.00

4.48

20.71

0.75

– 0.88

Table 4: R2 values for different pests of Brinjal (2007–2008)
R2LinLog

Inv

Qua

CubCom

Pow

S

GRO

Expo

Whitefly

0.020

0.025

0.110

0.796

0.807

0.040

0.023

0.177

0.040

0.040

Thrips

0.014

0.033

0.117

0.796

0.830

0.010

0.062

0.220

0.010

0.010

Jassids

0.001

0.068

0.141

0.774

0.775

0.007

0.055

0.198

0.007

0.007

Borer

0.509

0.521

0.438

0.509

0.639

0.453

0.647

0.754

0.453

0.453

Table 5: R2 values for different pests of Chilli (2007–2008)
R2LinLog

Inv

Qua

CubCom

Pow

S

GRO

Expo

Whitefly

0.210

0.047

0.012

0.287

0.622

0.128

0.004

0.082

0.128

0.128

Yellow Mite

0.027

0.008

0.056

0.506

0.602

0.043

0.015

0.120

0.043

0.043

Thrips

0.032

0.137

0.138

0.622

0.637

0.246

0.550

0.588

0.246

0.246

can produce higher precision levels (in terms of higher
2values of R ) needs to be carried out.

major agricultural crops, models built up on the reallife data on pest infestations reveal the patterns ofIn this paper, along with the parametric models,intensity of pest incidence importantly useful to them.

two non-parametric and one semi-parametric models

The implication and importance of this study are thathave been tried. The R2 values as presented in thethe non-parametric model like Kernel and Loess andgraph plots (in the Annexure) next indubitably con-semi-parametric model like Spline are capable to pro-firm the superiority of the representative power (induce forecasts on the basis of magnitude of incidences

terms of higher precision) of the non-parametric (and(data acquired from field experiments) at any time-semi-parametric) modelling in comparison to paramet-point during the year most precisely so as to provide

ric modelling, when called for, in situations to depict

accurate advisor y ser vices against possible pest

the pattern of pest incidence based on weekly pest-

attacks to crop, an advance knowledge of which is ofincidence data on two important crops, namely, Brinjal

utmost help to the farmers. It is no denying that toand Chilli. As the reality demands that the farmers aremodel the fluctuations/characteristics/dynamics evi-keen to be provided with specialized scientific informa-denced in temporal real-life data situations, the non-tion in the form of advisory services in respect of theparametric and semi-parametric models offer better

pattern of pest incidence over the years in case of therepresentations almost often.

─ 175 ─

Prasenjit PAL, Subhabaha PAL, Satyabrata PAL and Sanpei KAGEYAMAANNEXURE
Crop – Brinjal
Y Axis - No. of incidences

X Axis – Week numbers

Graph 1: Spline Fit (Whitefly)

Graph 2: Kernel Fit (Whitefly)

R2 = 0.9879
Figure demonstrating Spline modelling of Whitefly incidence onBrinjal data
Red Line – Linear Fit
Violet line – Spline fit (good fit – nearing 1.0)

R2 = 0.9878
Figure demonstrating Kernel modelling of Whitefly incidence on
Brinjal data
Red Line – Linear Fit
Orange line – Kernel fit (good fit – nearing 1.0)

Graph 3: Loess Fit (Whitefly)

Graph 4: Spline Fit (Thrips)

R2 = 0.9855
Figure demonstrating Loess modelling of Whitefly incidence on
Brinjal data
Red Line – Linear Fit
Violet line – Loess fit (good fit – nearing 1.0)

R2 = 0.9999
Figure demonstrating Spline modelling of Thrips incidence on
Brinjal data
Red Line – Linear Fit
Violet line – Spline fit (good fit – nearing 1.0)

Graph 5: Loess Fit (Thrips)

Graph 6: Kernel Fit (Thrips)

R2 = 0.9881
Figure demonstrating Loess modelling of Thrips incidence on
Brinjal data
Red Line – Linear Fit
Violet line – Loess fit (good fit – nearing 1.0)

R2 = 0.9931
Figure demonstrating Kernel modelling of Thrips incidence on
Brinjal data
Red Line – Linear Fit
Violet line – Kernel fit (good fit – nearing 1.0)
─ 176 ─

Non-parametric Modelling and Its Application on Real-life Data
Graph 7: Spline Fit (Jassid)

Graph 8: Kernel Fit (Jassid)

R2 = 0.9841
Figure demonstrating Spline modelling of Jassid incidence on
Brinjal data
Red Line – Linear Fit
Violet line – Spline fit (good fit – nearing 1.0)

R2 = 0.984
Figure demonstrating Kernel modelling of Jassid incidence on
Brinjal data
Red line – Kernel fit (good fit – nearing 1.0)

Graph 9: Loess Fit (Jassid)

Graph 10: Spline Fit (Borer)

R2 = 0.9746
Figure demonstrating Loess modelling of Jassid incidence on
Brinjal data
Red line – Linear fit
Red line – Loess fit (good fit – nearing 1.0)

R2 = 0.9999
Figure demonstrating Spline modelling of Fruit and Shoot Borer
incidence on Brinjal data
Violet line – Spline fit (good fit – nearing 1.0)

Graph 11: Kernel Fit (Borer)

Graph 12: Loess Fit (Borer)

R2 = 0.9815
Figure demonstrating Kernel modelling of Fruit and Shoot Borer
incidence on Brinjal data
Violet line – Kernel fit (good fit – nearing 1.0)

R2 = 0.9649
Figure demonstrating Loess modelling of Fruit and Shoot Borer
incidence on Brinjal data
Red line – Linear fit
Violet line – Loess fit (good fit – nearing 1.0)
─ 177 ─

Prasenjit PAL, Subhabaha PAL, Satyabrata PAL and Sanpei KAGEYAMA

Crop – Chilli
Graph 13: Spline Fit (Whitefly)

Graph 14: Kernel Fit (Whitefly)

R2 = 0.9984
Figure demonstrating Spline modelling of Whitefly incidence on
chilli data
Red line – Linear fit
Violet line – Spline fit (good fit – nearing 1.0)

R2 = 0.9922
Figure demonstrating Kernel modelling of Whitefly incidence on
chilli data
Red line – Kernel fit (good fit – nearing 1.0)

Graph 15: Loess Fit (Whitefly)

Graph 16: Spline Fit (Yellow Mite)

R2 = 0.9828
Figure demonstrating Loess modelling of Whitefly incidence on
chilli data
Red line – Linear fit
Violet line – Loess fit (good fit – nearing 1.0)

R2 = 0.9973
Figure demonstrating Spline modelling of Yellow Mite incidence onchilli data
Violet line – Spline fit (good fit – nearing 1.0)

Graph 17: Kernel Fit (Yellow Mite)

Graph 18: Loess Fit (Yellow Mite)

R2 = 0.9978
Figure demonstrating Kernel modelling of Yellow Mite incidence
on chilli data
Red line – Linear fit
Violet line – Kernel fit (good fit – nearing 1.0)

R2 = 0.9965
Figure demonstrating Loess modelling of Yellow Mite incidence on
chilli data
Red line –Loess fit (good fit – nearing 1.0)

─ 178 ─

Non-parametric Modelling and Its Application on Real-life Data
Graph 19: Spline Fit (Thrips)

Graph 20: Kernel Fit (Thrips)

R2 = 0.9966
Figure demonstrating Spline modelling of Thrips incidence on
chilli data
Violet line – Spline fit (good fit – nearing 1.0)

R2 = 0.9959
Figure demonstrating Kernel modelling of Thrips incidence on
chilli data
Red line – Kernel fit (good fit – nearing 1.0)

Graph 21: Loess Fit (Thrips)

R2 = 0.9932
Figure demonstrating Loess modelling of Thrips incidence on
chilli data
Red line – Linear fit
Violet line – Loess fit (good fit – nearing 1.0)

Issue 1–4, 113–126.

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