Transcript Forecasting future technological needs for rice crop in

Pests and Diseases Forewarning System
Amrender Kumar
Scientist
Indian Agricultural Statistics Research Institute,
Library Avenue, New Delhi, INDIA
[email protected]
Crop – Pests - Weather
Relationship
Crop
Weather
Pests
• Diseases and pests are major causes of
reduction in crop yields.
• However, in case information about time and
severity of outbreak of diseases and pests is
available in advance, timely control measures
can be taken up so as to reduce the losses.
• Weather plays an important role in pest and
disease development.
• Therefore, weather based models can be an
effective scientific tool for forewarning diseases
and pests in advance.
Why pests and disease forewarning
• Forewarning / assessment of disease important
for crop production management
– for timely plant protection measures
• information whether the disease status is expected to
be below or above the threshold level is enough,
models based on qualitative data can be used –
qualitative models
– loss assessment
• forewarning actual intensity is required - quantitative
model
Variables of interest
–
Maximum pest population or disease severity.
–
Pests population/diseases severity at most damaging stage i.e. egg,
larva, pupa, adult.
–
Pests population or diseases severity at different stages of crop growth
or at various standard weeks.
–
Time of first appearance of pests and diseases.
–
Time of maximum population/severity of pests and diseases.
–
Weekly monitoring of pests and diseases progress.
–
Occurrence/non-occurrence of pests & diseases.
–
Extent of damage.
Data Structure
Historical data at periodical intervals for 10-15 years
Year
Observation
1
2
3
4
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1
y11
y12
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2
y21
y22
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10-15
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• Historical data for 10-15 years at one point
of time
– overall status
– disease intensity
– crop damage.
• Data for 5-6 years at periodic intervals
– For week-wise models, data points inadequate
– combined model for the whole data in two steps
• Data at one point of time for 5-6 years
– Model development not possible
• Qualitative data for 10-15 years
– Qualitative forewarning
• Occurrence / non-occurrence of disease
• Mixed data – conversion to qualitative categories
• Data collected at periodic intervals for one year
– Within year growth model
Choice of explanatory variables
•
Relevant weather variables
– appropriate lag periods depending on life cycle
•
Crop stage / age
•
Natural enemies
•
Starting /
pathogen
previous
year’s
last
population
of
Forecast Models
•
Between year models
–
–
•
These models are developed using previous years’ data.
The forecast for pests and diseases can be obtained by
substituting the current year data into a model developed upon
the previous years.
Within year models
–
–
–
Sometimes, past data are not available but the pests and diseases
status at different points of time during the current crop season
are available.
In such situations, within years growth model can be used,
provided there are 10-12 data points between time of first
appearance of pests and diseases and maximum or most
damaging stage.
The methodology consists of fitting appropriate growth pattern
to the pests and diseases data based on partial data.
Thumb rules
– Most common
– Extensively used
– Judgment based on past experience with no or
little mathematical background
Example
A day is potato late blight favorable if
- the last 5 - day temperature average is < 25.50 C
- the total rainfall for the last 10 days is > 3.0 cm
- the minimum temperature on that day is > 7.20 C
Trivedi et al. (1999)
Regression models
• Relationship between two or more quantitative
variables
• The model is of the form
Y = 0 + 1 X1+2 X2 ………. +p Xp + e ,
where
–
–
–
–
i’s are regression coefficients
Xi’s are independent variables
Y variable to forecast
e random error
• Variables could be taken as such or some
suitable transformations
Cotton
• % of incidence of Bacterial blight (Akola) –
Weekly models (42nd to 44th SMW)
• Data used: 1993-1999 on MAXTemp, MINTemp,
RH1 (morn), RH2 (aft) and RF – [X1 to X5)
lagged by 2 to 4 weeks
• Model for 44th SMW
Y= 133.18 - 3.09 RH2L4 + 1.68 RFL4 (R2=0.78)
Bacterial Blight (% )
Forecast of Bacterial blight in Akola (Cotton) - 2001 in different
SMW
60.0
50.0
40.0
30.0
20.0
10.0
0.0
2000
2001
42
2000
2001
43
2001
44
Year & Weeks
Observed
2000
Forecast
Potato
• Potato aphid is an abundant potato pest and vector of
potato leaf-roll virus, potato virus Y , PVA, etc.
• Potato aphid population – Pantnagar (weekly models)
• Data used: 1974-96 on MAXT, MINT and RH
– [X1 to X3) lagged by 2 weeks
• Model for December 3rd week
Y = 80.25 + 40.25 cos (2.70 X12 - 14.82)
+ 35.78 cos (6.81 X22 + 8.03)
Population
Aphid popn. in 3rd week of December at Pantnagar
600
500
400
Observed
Predicted
300
200
100
0
7475
7677
7879
8283
8485
8788
Year
8990
9192
9394
9596
GDD approach
GDD =  (mean temperature – base temperature)
• The decision of
• Base temperature
• Initial time
–
–
–
Not much work on base temperature for various diseases
Normally base temperature is taken as 50 C
Under Indian conditions, mean temperature is seldom below 50 C
 Use of GDD and simple accumulation of mean temperature will
provide similar results in statistical models
 Need for work on base temperature and initial time of calculation
• Under Indian conditions, other variables also important
• Model using simple accumulations not found appropriate
• Models based on weighted weather indices
p
p
Y  a 0   a i z i   b ii' z ii' e
'
i 1
ii
where
Zi 
n2
 riw X iw
w  n1
n2
Z ii'   rii' w X iw X i ' w
w  n1
Y
variable to forecast
xiw value of ith weather variable in wth period
riw
weight given to i-th weather variable in wth period
rii’w weight given to product of xi and xi’ in wth period
p
number of weather variables
n1 and n2 are the initial and final periods for which weather
variables are to be included in the model
e
error term
Experience based weights
– Subjective weights based on experience.
• Weather variable not favourable : weight = 0
• Weather variable favourable : weight = ½
• Weather variable very favourable : weight = 1
Example :
Favourable relative humidity  92%
Most favourable relative humidity  98%
Weather data
Year
Week No.
1
2
1993
88.7
90.1
1995
94.0
1996
90.3
3
4
5
6
94.4 98.3
98.0
95.0
93.3
94.9 93.3
92.0
88.1
91.9
90.4 87.9
86.4
89.7
-------------------------------------------------------------------------------------------------------------------------------
Weighted Index
1993 0x 88.7 + 0x90.1 + 0.5 x 94.4 + 1 x 98.3 +
1 x 98 + 0.5 x 95 = 271.0
1995
0.5 x 94 + 0.5 x 93.3 + 0.5 x 94.9 +
0.5 x 93.3 + 0.5 x 92 + 0 x 88.1 = 232.6
1996
0 x 90.3 + 0 x 91.9 + 0 x 90.4 + 0x 87.9 +
0 x 86.4 + 0 x 89.7 = 0.0
------------------------------------------------------------------------------------------------------------------------------
Interaction :
Both variables not favourable : weight = 0
One variable not favourable, one variable favourable : weight = 1/8
One variable not favourable, one variable highly favourable : weight = ¼
Both variables favourable : weight = ½
One variable favourable, one variable highly favourable : weight = ¾
Both variables highly favourable : weight = 1
Correlation based weights
riw
correlation coefficient between Y and i-th weather
variable in wth period
rii’w correlation coefficient between Y and product of xi and xi’
in wth period
Modified model
• Model using both weighted and unweighted indices
Y
p
p
1
1
a 0    a Z   b Z  e
i 1 j  0
ij
ij
i  i ' j 0
where
Zij 
Z
ii 'j

n2
 riw Xiw
j
w n1
n2
 rii 'w Xiw Xi 'w
w n1
j
ii ' j
ii ' j
• For each weather variable two types of indices have been
developed
• Simple total of values of weather variable in different periods
• Weighted total, weights being correlation coefficients between
variable to forecast and weather variable in respective periods
• The first index represents total amount of weather variable
received by the crop during the period under consideration
• The other one takes care of distribution of weather variable with
reference to its importance in different periods in relation to
variable to forecast
• On similar lines, composite indices were computed with products
of weather variables (taken two at a time) for joint effects.
Pigeon pea
Phytophthora blight (Kanpur)
• Average percent incidence of phytophthora blight at
one point of time
• Data used : 1985-86 to 1999-2000 on MAXT, MINT,
RH1, RH2 and RF (X1- X5) from 28th to 33rd SMW
Y = 330.77 + 0.12 Z121 ….. (R2 = 0.77)
Sterility Mosaic
• Average percent incidence of sterility mosaic
• Data used : 1983-84 to 1999-2000 for MAXT, MINT,
RH1, RH2 and RF (X1- X5) from 20th to 32nd SMW
Y = -180.41 + 0.09 Z121 …… (R2 = 0.84)
• Validation for subsequent years :
Forecast of Phythphthora-blight (Pigeonpea) - Kanpur
Average % incidence
100
80
60
40
20
0
2000-01
2001-02
Year
observed
forecast
Forecast of Sterility Mosaic (Pigeonpea) - Kanpur
Average % incidence
50
40
30
20
10
0
2000-01
2001-02
Year
observed
forecast
Groundnut
Late Leaf Spot & Rust – Tirupathi
• Disease indices at one point of time
• Data used : MAXT, MINT, RH1, RH2, RF and
WS from (X1- X6)
- 10th to 14th SMW (Rabi or post rainy)
- 41st to 46th SMW (Kharif or rainy)
Models for LSS and Rust Disease Index - groundnut
(Tirupati)
Model
R2
Disease
Data used
LLS
Kharif
1990 - 1998 Y = 39.40 - 0.00921 Z120 +0.00037 Z460 +
0.0022 Z141
0.84
LLS Rabi
1990 - 1999 Y = 15.95 + 0.12Z151 + 0.0057 Z350
0.83
Rust
Kharif
1990 - 1995 Y = 0.4213 + 0.0167Z231 - 0.147 Z10
0.94
Forecast of LLS and Rust (Groundnut) - Tirupati
10
9
Disease Index
8
7
6
5
4
3
2
1
0
LLS
Kharif
LLS
Kharif
LLS
Kharif
LLS
Kharif
LLS
Kharif
LLS
R abi
LLS
R abi
LLS
R abi
LLS
R abi
R ust
Kharif
R ust
Kharif
19 9 9
2000
2001
2002
2003
2000
2001
2002
2003
2002
2003
observed
f orecast
Principal component regression
• Independent variables large and correlated
• Independent variables transformed to principal
components
• First few principal components
desired variation selected
explaining
• Regression model using principal components
as regressors
Discriminant function analysis
• Based on disease status years grouped into different categories –
low, medium, high
• Linear / quadratic discriminant function using weather data in above
categories
• Discriminant score of weather for each year
• Regression model using disease data as dependent variable and
discriminant scores of weather as independent.
• Data requirement is more.
• Can also be used if disease data are qualitative
• Johnson et al. (1996) used discriminant analysis for forecasting
potato late blight.
Deviation method
• Useful when only 5-6 year data available for
different periods
• Week-wise data not adequate for modeling
• Combined model considering complete data.
• Not used for disease forewarning but in pest
forewarning
• Assumption : pest population / disease incidence
in particular year at a given point of time
composed of two components.
– Natural growth pattern
– Weather fluctuations
• Natural pattern to be identified using data in
different periods averaged over years.
• Deviation of individual years in different periods
from predicted natural pattern to be related with
deviations of weather.
Mango
• Mango fruitfly – Lucknow (weekly models)
• Data used: 1993-94 to 1998-99 on MAXT, MINT and RH
– [X1 to X3]
• Model for natural pattern
Yt 
33.64  1.79t
1  0.16t  0.0067t 2
t = Week no.
Yt = Fruitfly population count at week t
III
II
I
IV
III
II
I
Week
IV
III
II
IV
Au
gu
st
I
III
II
I
Observed
I
Ju V
ly
III
II
50
I
Ju V
ne
I
M
ay
Ap
ri l
Population
300
250
200
150
100
Expected
0
Forecast model
Y =  125.766 + 0.665 (Y2) + 0.115 (1/X222 ) + 10.658 (X212)
+ 0.0013 (Y23) + 31.788 (1/Y3)  21.317 (X12)
 2.149 (1/X233)  1.746 (1/X234)
Y = Deviation of fruitfly population from natural cycle
Yi = Fruitfly population in i-th lag week
Xij = Deviation from average of i-th weather variable (i =
1,2,3 corresponds to maximum temperature,
minimum temperature and relative humidity) in j-th lag
week.
Soft Computing Techniques
• With the development of computer hardware and software and
the rapid computerization of business, huge amount of data
have been collected and stored in centralized or distributed
databases
• Data is heterogeneous (mixture of text, symbolic, numeric,
texture, image), huge (both in dimension and size) and
scattered.
• The rate at which such data is stored is growing at a
phenomenal rate.
• As a result, traditional statistical techniques and data
management tools are no longer adequate for analyzing this
vast collection of data.
• One of the applications of Information Technology that has
drawn the attention of researchers is data mining, where
pattern recognition, image processing, machine intelligence i.e
concerned with the development of algorithms and techniques
that allow system to "learn“ are directly related
• Data Mining involves
― Statistics : Provides the background for the algorithms.
― Artificial Intelligence : Provides the required heuristics for
learning the system
― Data Management : Provides the platform for storage &
retrieval of raw and summary data.
• Pattern Recognition and Machine Learning principles applied to a
very large (both in size and dimension) heterogeneous database for
Knowledge Discovery
• Knowledge Discovery is the process of identifying valid, novel,
potentially useful and ultimately understandable patterns in data.
Patterns may embrace associations, correlations, trends, anomalies,
statistically significant structures etc.
• Without “Soft Computing” Machine Intelligence and Data Mining
may remains Incomplete
Soft Computing
• Soft Computing is a new multidisciplinary field that was
proposed by Dr. Lotfi Zadeh, whose goal was to construct new
generation Artificial Intelligence, known as Computational
Intelligence.
• The concept of Soft Computing has evolved. Dr. Zadeh defined
Soft Computing in its latest incarnation as the fusion of the fields
of fuzzy logic, neural network, neuro-computing, Evolutionary &
Genetic Computing and Probabilistic Computing into one
multidisciplinary system.
• Soft Computing is the fusion of methodologies that were
designed to model and enable solutions to real world problems,
which are not modeled, or too difficult to model. These problems
are typically associated with fuzzy, complex, and dynamical
systems, with uncertain parameters.
• These systems are the ones that model the real world and are of
most interest to the modern science.
• The main goal of Soft Computing is to develop intelligent
system and to solve nonlinear and mathematically unmodelled
system problems [Zadeh 1993, 1996, and 1999].
• The applications of Soft Computing have two main advantages.
– First, it made solving nonlinear problems, in which
mathematical models are not available, possible.
– Second, it introduced the human knowledge such as
cognition, recognition, understanding, learning, and others
into the fields of computing.
• This resulted in the possibility of constructing intelligent
systems such as autonomous self-tuning systems, and
automated designed systems.
soft computing tools
Soft computing tools include
• Fuzzy sets
– Fuzzy sets provide a natural frame work for the process in
dealing with uncertainty
• Artificial neural networks
• Neural networks are widely used for modelling complex
functions and provide learning and generalization capabilities
• Genetic algorithms
– Genetic algorithms are an efficient search and optimization
tool
• Rough set theory
– Rough sets help in granular computation and knowledge
discovery
• Why Neural Networks are desirable
–
–
–
–
Human brain can generalize from abstract
Recognize patterns in the presence of noise
Recall memories
Make decisions for current problems based on prior
experience
• Why Desirable in Statistics
–
–
–
–
Prediction of future events based on past experience
Able to classify patterns in memory
Predict latent variables that are not easily measured
Non-linear regression problems
Application of ANNs
• Classification:
– medical diagnosis
– signature verification
– character recognition
– voice recognition
– image recognition
– face recognition
– loan risk evaluation
– data mining
• Modelling and Control
– control systems
– system identification
– composing music
• Forecasting:
–
–
–
–
–
economic indicators
energy requirements
medical outcomes
crop forecasts
environmental risks
• Neural networks are being successfully applied across an
extraordinary range of problem domains, in areas as diverse as
finance, medicine, engineering, geology, biology, physics and
agriculture.
• From a statistical perspective neural networks are interesting
because of their potential use in prediction and classification
problems.
• A very important feature of these networks is their adaptive
nature, where “Learning by Example” replaces “Programming”
in solving problems.
• Basic capability of neural networks is to learn patterns from
examples
Type of neural network models
– Two types of neural network models
• Multilayer perceptron (MLP) with different hidden
layers and nodes
• Radial basis function (RBF)
Neural network based model
Steps in developing a neural network model
• Forming training, testing and validation sets
• Neural network model
– No. of input nodes
– No. of hidden layers
– No. of hidden nodes
– No. of output nodes
– Activation function
• Model building
• Sensitivity Analysis
Data sets
•
The data available is divided into three data sets
– Training set represents the input- output mapping,
which is used to modify the weights.
– Validation set is required only to decide when to stop
training the network, and not for weight update.
– Test set is the part of collected data that is set aside to
test how well a trained neural network generalizes.
– No. of input nodes
: more than one
– No. of hidden layers : one / two
– No. of hidden nodes : decided by various rules
– No. of output nodes : one
– Activation function
: hyperbolic
•
Activation function:
–
Activation functions determine the output of a processing node. Non
linear functions have been used as activation functions such as logistic,
tanh etc.
–
Activation functions such as sigmoid are commonly used because they
are nonlinear and continuously differentiable which are desirable for
network learning
–
Logistic activation functions are mainly used for classification problems
which involve learning about average behavior
–
Hyperbolic tangent functions are used for the problem involves learning
about deviations from the average such as the forecasting problem.
–
Therefore, in the present study, hyperbolic tangent (tanh) function has
been used as activation function for neural networks model based on
MLP architecture.
Output
Input
Learning of ANNs
• The most significant property of a neural network is that
it can learn from environment, and can improve its
performance through learning
• Learning is the process of modifying the weights in
networks
• The network becomes more knowledgeable about
environment after each iteration of learning process.
• There are mainly two types of learning paradigms
– Supervised learning
– Unsupervised learning
A learning cycle in the MLP
(Backpropagation Learning Algorithm)
=
Differences
Target vector
Adjust weights
Output vector
Input vector
ANN model
• Mustard
– Alternaria blight (Varuna, Rohini & Binoy)
Bharatpur (Raj)
Behrampur (WB)
Dholi (Bihar)
– Powdery mildew (Varuna and GM2)
S.K.Nagar
– Variable to forewarn
crop age at first appearance of disease
crop age at peak severity of disease
maximum severity of disease
• Cotton
– Bacterial blight (% of disease incidence) Akola
Pests / diseases forewarning-Mustard
• Data have been taken from Mission Mode Project under National Agricultural Technology
Project, entitled “Development of weather based forewarning system for crop pests and
diseases”, at CRIDA, Hyderabad.
• Models were developed for forecasting different aspects relating to diseases for Alternaria
Blight (AB) and Powdery Mildew (PM) in Mustard crop.
• The field trials were sown on 10 dates at weekly intervals (01, 08, 15, 22, 29 October, 05, 12, 19,
26 November and 03 December) at each of the locations viz., Bharatpur, Dholi and Berhampur
for Alternaria Blight and at S.K.Nagar for Powdery Mildew.
• Data for different dates of sowing were taken together for model development.
• Weekly data on weather variables starting from week of sowing up to six weeks of crop growth
were considered
• Forewarning models were developed for two varieties of mustard crop for
– Alternaria Blight on leaf and pod (Varuna and Rohini – Bharatpur, Varuna and Binoy – Behrampur
and Varuna and Pusabold – Dholi) and
– Powdery Mildew on leaf (Varuna and GM2 – S.K.Nagar)
• Models have been validated using data on subsequent years not included in developing the
models.
Mean Absolute Percentage Error of various models at Bharatpur in different
varieties in mustard crop for Alternaria blight (AB) - 2006-07
Character
Variety
MLP
RBF
WI
111.0
153.8
150.1
Age at First app
14.0
15.1
14.7
Age at Peak Severity
14.1
27.3
22.3
113.7
143.6
132.6
15.7
9.2
14.2
3.9
6.4
5.4
184.0
200.6
196.3
Age at First app
12.0
15.5
8.9
Age at Peak Severity
28.3
27.8
26.2
174.8
220.4
229.6
Age at First app
29.3
28.2
24.7
Age at Peak Severity
17.2
20.7
19.6
Maximum severity
Maximum severity
Varuna (on Leaf)
Varuna (on Pod)
Age at First app
Age at Peak Severity
Maximum severity
Maximum severity
Rohini (on Leaf)
Rohini (on Pod)
• Neural networks, with their remarkable ability to
derive meaning from complicated or imprecise
data, can be used to extract patterns and
classifications
 Neural networks do not perform miracles.
But if used sensibly they can produce
some amazing results
Model for qualitative data
– Data in categories
– Occurrence / non-occurrence, low / medium / high,
etc.
– Classified as 0 / 1 (2 categories);
categories)
0,1,2
(three
– Quantitative data / mixed data can be converted to
categories
Logistic Regression model
1
PY  1 
e
1  exp( L)
where, L= β0+ β1x1+ β2x2 ….βnxn
x1 , x2 , x3 ,…xn are weather variables/weather indices
e = random error
• Forecast / Prediction rule

If P < 0.5, then the probability of epidemic occurrence
will be minimal
 If P  0.5, then there is more chance of occurrence of
epidemic.
Rice
• Leaf blast severity (%) - Palampur at one point of time
• Data used: 1991-92 to 1998-99 on MAXT, MINT, RH1,
RH2, BSH & RF – [X1 to X6] from 23th to 31st SMW.
• Model :
L= 394.8 -0.0520 Z351-1.5414 Z10
• Validation for subsequent years :
Year
Observed
Forewarning
Probabilities
1999-00
1
1
0.88
2000-01
1
1
0.63
Mustard
Alternaria blight and White rust
• Data used: 1987-88 to 1998-99 on MAXT, MINT, RH1, RH2 and
BSH – (X1 to X5) from week of sowing (n1) to 50th smw (n2)
Model for Alternaria blight
L = - 8.8347 + 0.0163 Z120 - 0.00037 Z130 - 0.00472 Z450
Model for White rust
L = 5.8570 - 0.0293Z40 + 0.00264 Z230
• Forecasts of subsequent years are
Alternaria blight
White Rust
Year
Observed
Forewarning
Prob.
Observed Forewarning
Prob.
1999-00
1
1
0.51
1
1
0.96
2000-01
0
0
0.13
0
0
0.49
2001-02
1
1
0.62
0
0
0.37
Within year model
• Model using only one year’s data
–
Data availability for several dates of sowing
–
If adequate dates of sowing, models similar to between-year
models could be developed
• Use for forewarning subsequent years (?)
• Model for single date of sowing
–
Forewarning of maximum disease severity
–
Applicable when 10-12 data observations between first disease
appearance and maximum disease severity
–
Non-linear model for disease development pattern growth
using partial data
Mustard
• Alternaria blight cv. Varuna (% disease severity) - Kumarganj
• Data used: 1999-2000
Model :
Yt = A exp (B/t)
Yt = pds at time t, A and B are parameters,
t = week after sowing (1,2,…….)
Observed, predicted and forecasts of max. percent
disease severity (PDS)
Date
(std. week)
of sowing
Obs. max. Pred.
pds (std. max.pds.
week)
(Full
Model)
270999
(39)
041099
(40)
121099
(41)
73.88
(7)
75.60
(7)
70.62
(8)
Forecast at lag
1 week
2 week
3 week
75.15
69.69
69.07
65.02
75.60
75.66
76.68
79.28
66.83
63.98
73.47
79.57
• Reliable forecast of max. pds could be obtained for 2
weeks in advance
Models developed at IASRI
• Mustard
–
–
–
–
Alternaria Blight
White Rust
Powdery Mildew
Aphid
• Cotton
–
–
–
–
American boll worm
Pink boll worm
Spotted boll worm
Whitefly
• Groundnut
– Spodoptera litura
– Late leaf blast
– Rust
• Onion
– Thrips
• Sugarcane
– Pyrilla
– Early shoot borer &
– Top borer
• Pigeon pea
–
–
–
–
Pod fly
Pod borer
Sterility Mosaic
Phytophthora Blight
• Rice
– BPH
– Gall midge
• Mango
– Powdery Mildew
– hoppers
– fruit-fly
References
•
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variables on rice yields. Mausam, 34(2), 177-81.
•
Agrawal, Ranjana, Jain, R.C., Jha, M.P., (1986). Models for studying rice crop
weather relationship, Mausam, 37(1), 67-70.
•
Agrawal Ranjana, Mehta, S.C., Kumar, Amrender and Bhar, L.M. (2004).
Development of weather based forewarning system for crop pests and diseasesReport from IASRI, Mission mode project under NATP, PI, Dr. Y.S.
Ramakrishna, CRIDA, Hyderabad.
•
Denton, J.W., 1995. How good are neural networks for causal forecasting?
Journal of Business Forecasting, 14 (2), 17–20.
•
Desai, A.G., Chattopadhyay, C., Agrawal, Ranjana, Kumar, A., Meena, R.L.,
Meena, P.D., Sharma, K.C., Rao,
M. Srinivasa, Prasad,, Y.G. and
Ramakrishna, Y.S. (2004). Brassica juncea powdery mildew epidemiology and
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•
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Theory, Wiley, New York.
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statistical tools. American Statistician, 50, 284-93.
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Thank You