Estimating Pest Abundance for High Aggregation Density ... · Estimating Pest Abundance for High...

Introduction Midpoint Rule Study Statistical Method Study

Estimating Pest Abundance for High Aggregation

Density Distributions

Nina Embleton

School of Mathematics, University of Birmingham

Midlands MESS, 02 July 2012


Ecological Problem

• Pests pose a significant problem to farmers across the globe.

• The term pest can be used to describe any species whichcauses damage to mankind in some form.(1) We focus oninsect pests.

• Pest insects destroy approximately 14% of crops worldwide.(2)

• The monetary value of the crops lost per year has beenestimated at around $2,000 billion.(2)

(1)Dent, Insect Pest Management, 1991.(2)Pimentel, Pesticides and pest control, In: Integrated Pest Management:Innovation-Development Process, 2009.


Ecological Problem

• Knowledge of the pest population size in an agricultural fieldcan be used to determine if and when it is necessary tointervene.(1)

• An estimation of the pest abundance is made and comparedto some ‘critical density’.(2)

• The threshold value can take into consideration a variety ofcriteria; the most well known depend on economic factors.(3)

(1)Stern, Economic thresholds, 1973.(2)Binns, Nyrop & Werf, Sampling and Monitoring in Crop Protection, 2000.(3)Stern et al., The integration of chemical and biological control of thespotted alfalfa aphid, 1959.


Importance of an Accurate Estimate

• Too low an estimation of the pest abundance can lead to thefailure to take action when it is needed.

Figure: Coconut Hispine Beetlehttp://www.malaeng.com/blog/?s=coconut+hispine

• An outbreak of the CoconutHispine Beetle in South-eastAsia destroyed 16 % ofcoconut trees inCambodia.(1)

• Had no action been taken inVietnam, $1 billion ofdamage would have beencaused over a span of 30years. (1)

(1)Liebregts & Chapman, Impact and control of the coconut hispine beetle,Brontispa longissima Gestro (Coleoptera: Chrysomelidae), 2004.



• Too high an approximation can lead to a control action i.e.

pesticides, being used unnecessarily.

• Pesticides are known to contribute to air, soil, and waterpollution. (1)

• The dangers pesticides pose to human health are welldocumented. (2) (3)

• 3 million metric tonnes of pesticides are used per year acrossthe globe with a price tag of $40 billion.(4)

(1)Werf, Assessing the impact of pesticides on the environment, 1996.(2)Blair et al., Cancer among farmers, 1991(3)Harrison, Connecting air pollution and pesticide drift in California, 2004(4)Pimentel, Pesticides and pest control, In: Integrated Pest Management:Innovation-Development Process, 2009.



• It is important to avoid the unnecessary use of potentiallyhazardous pesticides, whilst protecting crops from pestoutbreaks.

• There is clear motivation from both economic andenvironmental viewpoints to obtain an accurate estimation ofpest population size.


Acquiring an Estimate of the Pest Abundance

Figure: Flatworm monitoring - trap installation.(1)

(1)Petrovskaya,Petrovskii, & Murchie, Challenges of ecological monitoring:estimating population abundance from sparse trap counts, 2011.



• Current approaches typically involve the implementation ofstatistical methods to the pest density function.

• A commonly used statistical method depends on thearithmetic average: (1)

I ≈ I = Au,

where

u =1

N

N∑

i=1

ui ≈ u.

(1)Davis, Statistics for describing populations. In: Handbook of SamplingMethods for Arthropods in Agriculture, 1994.



• In theory, the approximation tends to the true pest populationas the number of traps N becomes infinitely large.

• Practical limitations imposed by the ecological problem meanthat the number of traps is limited.

• Typically the number of traps installed is N ∼ 10.(1)(2)

(1) Mayor & Davies A survey of leatherjacket populations in South-westEngland, 1976

(2) Northing, Extensive field based aphid monitoring as an information toolfor the UK seed potato industry, 2009


Optimising Accuracy: The Approach of Ecologists

• Ecologists focus on minimising the error introduced by thesampling plan.

• Sampling the same population several times will producedifferent estimations of the pest abundance.

• Ecologist define the accuracy in terms of the bias and theprecision (e.g see (2)).

• The bias is the difference between the true value of theabundance and the expectation of the estimates.

• Precision is defined by the variance in the estimates.

(2)Binns, Nyrop & Werf, Sampling and Monitoring in Crop Protection, 2000.


Optimising Accuracy: The Approach of Ecologists

• From a theoretical perspective, random sampling is favoured.

• In practice it is rarely implemented as it is expensive and timeconsuming.

• If a sampling plan is considered to have acceptable accuracy(in terms of bias and precision), it is assumed that theresulting estimate will be close to the true abundance.


• This is true when the pests are spread across the wholedomain.

(a) (b)

Figure: (a) Pest population distribution, (b) Accuracy of estimation.


• However, it is not the case when the pest density is localised.

(a) (b)



• In such a situation, the error introduced by the method ofapproximation becomes important.

(a) (b)



• We first consider the case where the traps are placed at thenodes of a uniform grid.

• The problem can then be considered as one of numericalintegration.

• We then investigate the situation when the traps are placedrandomly.


Numerical Integration Problem

• The problem of obtaining an estimate of the pest abundancefrom the discrete density distribution is essentially one ofnumerical integration.

• Indeed the statistical method is a simple form of a numericalintegration method.

• Hence, we are faced with a common problem. We mustevaluate the integral I as I ,

I =

∫ 1

0

∫ 1

0u(x) dx dy ≈ I ,

where the population density function u(x) ≡ u(xi ) is onlyknown at a set of discrete points {xi}, i = 1, ...,N.


Restrictions Imposed

• Nature of the ecological problem means this is not a standardnumerical integration problem.

• The limitation on the number of traps that can be installedmeans we are forced to seek an accurate estimation fromsparse spatial data.

• Grid refinement, installing more traps and repeating theestimation process is not possible as the initial conditionscannot be recreated.

• Traps cannot be installed local to the patch of pestpopulation as its location is unknown.


• We have ui = u(xi ), i = 1, ...,N, the discrete density of thepest population where the number of traps N is small..

• We require that the relative error is

e =|I − I ||I | ≤ τ.

• The conventional way to conclude about the accuracy of amethod is to use the asymptotic error estimates.

• Ordinarily these estimates can be relied upon, however whenthe number of grid nodes is small we cannot use thisapproach.


The 1-D Problem

Figure: (a) Pest population density distribution u(x, y) at an early stageof patchy invasion. (b) A one-dimensional counterpart of the densitydistribution.


Consider the normal distribution

u(x) =1

σ√2π

exp(−1

2

(x − x∗)2

σ2)

Figure: (a) Normal distribution with the location of the maximumx∗ = 0.38. (b) Integration error.


Accuracy for Very Sparse Data

Figure: Integration error for ten realisations of the random variable x∗.


Grid Classification

• I Fine grids: the asymptotic error estimates hold.

• II Coarse grids: the asymptotic error estimates do not hold,however we can guarantee an accurate approximation.

• III Ultra-coarse grids: we can determine the probability ofachieving an accurate approximation.


Objectives

• To establish the probability of achieving e ≤ τ for a givennumber of grid nodes (traps) N.

• Find the threshold number of nodes (traps) Nt where theerror becomes deterministic.


Midpoint Rule

• The Symmetric Midpoint rule is given by

I =

∫ 1

0u(x) ≈ I =

1

N − 1

(

u1

2+

(

N−1∑

i=2

ui

)

+uN

2

)

.

• This method automatically introduces a length scale, thedistance between the traps must be h.

• We define the computational grid {xi} i = 1, ...N such thatx1 = 0, xN = 1 and xi = xi−1 + h, i = 2, ...N − 1 where h > 0.


Probability Theory

• We consider the high aggregation density distributionmodelled by the following peak function

u(x) =

{

f (x) > 0, x ∈ (xI , xII ),0, otherwise,

where f (x) has a single maximum point at x∗ = 0.5(xI + xII ).

• Expanding about x∗ up to second order terms we obtain aquadratic approximation in the vicinity of the peak

u(x) ≈ g(x) = B − A(x − x∗)2, x ∈ [xI , xII ],

u(x) = 0, otherwise,

where A = −12d2u(x∗)dx2

> 0 and B = u(x∗) > 0.


• The integral I is then given as

I =

∫ 1

0u(x) dx ≈

∫

xII

xI

g(x) dx =2

3Bδ,

where the peak width δ ≡ xII − xI = 2√

B/A

• Let the grid step size be h = αδ.

• The grid node location xi is defined as

xi = x∗ + γh γ ∈ [0, 1/2].

• We now calculate the approximation I by applying theSymmetric Midpoint rule.


a

0 1x

g(x)

xi-1/2 xi+1/2xI xIIx* xi

b

0 1x

g(x)

xi-1 xi+1xI xIIx* xi

Figure: Symmetric Midpoint rule approximation of the peak. (a) Onenode in the vicinity of the peak. (b) Two nodes in the vicinity of thepeak.


• We want to find when the condition e ≤ τ is satisfied. Weselect τ = 0.25. This condition is then equivalent to

3

4I ≤ I ≤ 5

4I .

• First we consider the case when h > δ, that is there is at mostone grid node in the vicinity of the peak. The approximation I

isI = (B − Aγ2h2)h.

• Solving for γ we obtain

1

2α

√

1− 5

6α≤ γ(α) ≤ 1

2α

√

1− 1

2α.


Figure: Range of γ where E ≤ 0.25 for h ≥ delta.


• We then consider the case when there is at most two gridnodes in the vicinity of the peak, δ/2 ≤ h ≤ δ. Theapproximated integral is then

I = −A((γ − 1)2 + γ2)h3 + 2Bh.

• Similar analysis is then performed to obtain the range of γwhich satisfies e ≤ 0.25 for all h ≥ δ/2.


0.2

0.4

0.6

0.8

1

b(α)p

αα t

theor

γ0IγγIII

γII

a(α)γ

1/2 2αα

t 1

D1D2

1/2

Figure: (a) The set of parametric curves defining the admissible range ofnode location γ. (b) Probability p(α) of achieving an error e ≤ 0.25.The error becomes deterministic for αt ≈ 0.8.


Numerical Test Cases

• We now check our theory for a series of test cases.

• For each test case, the peak width δ is fixed, and the locationof the peak maximum x∗ is a uniformly distributed randomvariable on the interval [δ, 1− δ].

• The peak location x∗ is generated r = 10, 000 times and ineach case the error e is calculated.

• The number of times we obtain e ≤ 0.25 is counted and fromthis we calculated the probability p(h) of achieving anaccurate integral evaluation.


u(x) = B − A(x − x∗)2, x ∈ [xI , xII ],

u(x) = 0 otherwise.

Figure: (a) Quadratic peak function: δ = 0.06, A = 1000, B = 0.9. (b)Probability curve.


u(x) = A((δ/2)4 − (x − x∗)4), x ∈ [x∗ − δ/2, x∗ + δ/2],

u(x) = 0 otherwise.

Figure: (a) Quartic peak function; A = 1200000; δ = 0.06. (b)Probability curve


u(x) = A(x−x∗+δ/3)(x−x∗−2δ/3)2 x ∈ [x∗−δ/3, x∗+2δ/3],

u(x) = 0 otherwise.

Figure: (a) Cubic peak function: δ = 0.06, A = 30000. (b) Probabilitycurve.


u(x) =1

σ√2π

exp(−1

2

(x − x∗)2

σ2)

Figure: (a) Normal distribution; σ = 0.01, δ = 6σ = 0.06. (b)Probability curve.


u(x) = sin(ω(x−x∗+π/(2ω))), x ∈ [x∗−π/(2ω), x∗+π/(2ω)],

u(x) = 0, otherwise.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1u(x)

x

a

0.1 0.2 0.3 0.4

0.1

0.2

0.3

0.4ht

theor

htnum

δ

ht

b

Figure: (a) Sine peak; (b) The threshold grid step size ht = αtδ forvarious peak widths δ.


Ecological Test Cases

• We simulate ecologically relevant data using a model.

• We use the spatially explicit form of theRosenzweig-MacArthur model (e.g. see (1)), which in itsdimensionless form is:

∂u(x , t)

∂t= d

∂2u

∂x2+ u(1− u)− uv

u + h

∂v(x , t)

∂t= d

∂2v

∂x2+ k

uv

u + h−mv .

• For any fixed time t, u(x) is a spatial distribution of the pestpopulation.

• We have d , h, k and m as the problem parameters, where d isthe diffusion coefficient.

(1)Murray, Mathematical biology, 1989


• For d << 1, single peak patterns can evolve. (1)

• The peak width is defined by the diffusion and is given by

δ = ω√d .

• The coefficient ω depends on the parameters of the model,but it is relatively robust and typically has a value of ω = 25.(1)

• The threshold grid stepsize can be approximated as

ht = αtδ ≈ αtω√d .

(1)Petrovskii et al. Quantification of the spatial aspect of chaotic dynamics inbiological and chemical systems, 2003.


Figure: Single peak distributions. Left: d = 10−4. Right: d = 10−5.


• For the wider peak, d = 10−4 so we have ht ≈ 0.2. HenceNt ≈ 6.

N 3 4 5 6 7 8

h 0.5 0.3333 0.25 0.20 0.1667 0.1429

e 0.6948 0.1119 0.5459 0.07983 0.1699 0.02305

Table: The integration error for the density distribution u1(x) on asequence of refined grids with grid step size h. N is the number of gridnodes on a uniform grid.

• The numerical results are in extremely good agreement withthe theoretical prediction.


• For the narrow peak d = 10−5, Nt ≈ 16.

N 17 18 19 20 21 22

h 0.0625 0.0588 0.0556 0.0526 0.05 0.0476

e 0.4127 0.5412 0.5101 0.4124 0.1960 0.0028

Table: The integration error for the density distribution u2(x) on asequence of refined grids with grid step size h. N is the number of gridnodes on a uniform grid.

• This number of traps exceeds the acceptable limit.

• For such a narrow peak, the value of the integral may be verysmall. Therefore we could consider a larger tolerance andrecompute Nt .


Conclusions

• We have introduced the concept of ultra-coarse grids.

• On ultra-coarse grids we cannot determine the accuracy ofintegral evaluation, but we can say what chance we have ofachieving an answer to within a chosen tolerance.

• Given the number of grid nodes (traps) N, we can estimatethe probability of achieving the error e ≤ 0.25.

• We have an estimate for the grid step size when the errorbecomes deterministic, namely

ht = αtδ,

where α ≈ 0.8.


Statistical Rule

• We now perform a similar study for the case where the trapsare located randomly.

• The statistical method is used to give an approximation of thepest abundance.

• We consider the approximation to the mean pest density,namely

E ≈ M(N) =1

N

N∑

i=1

ui .

• There is no spatial scale introduced by this method.

• This makes the probability analysis more complicated.


One Trap Case

x0

g(x)

x*x

b

0 10 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

au(x)

xx

Figure: (a) One peak distribution. (b) One trap is located in the peaksubdomain.


Probability Theory

• We paramaterise the trap location as

x0 = x∗ +γδ

2,

where γ ∈ [0, 1] is a uniformly distributed random variable.

• As with the Midpoint Rule analysis, we consider the peak as aquadratic function.

• The population density at x0 is approximated by

u(x0) = B − A(x0 − x∗)2 = B(1− γ2).


Probability Theory

• The sample mean density is then

M(N) =u0

N=

B(1− γ2)

N.

• We require the sample mean to be within a tolerance τ of thetrue mean E :

(1− τ)E ≤ M(N) ≤ (1 + τ)E ,

where E =∫ 10 u(x) dx = 2

3Bδ.


Probability Theory

• Solving for γ we find limiting values:

γ ≥ γI =

√

1− 2Nδ(1 + τ)

3,

which exists for N <= N∗ = 3/(2δ(1 + τ)), and

γ ≤ γII =

√

1− 2Nδ(1− τ)

3

which exists for N <= N∗∗ = 3/(2δ(1− τ)).


Probability Theory

• The admissible range of γ is then:

γI ≤ γ ≤ γII for N ≤ N∗

0 ≤ γ ≤ γII for N∗ < N ≤ N∗∗

φ for N > N∗∗.

• The probability of achieving an estimate within the toleranceτ is calculated as

pI = (γII − γI )/(γmax − γmin) for N ≤ N∗

pII = γII/(γmax − γmin) for N∗ < N ≤ N∗∗

pIII = 0 for N > N∗∗.


Probability Theory

Figure: Probability p(N) of achieving an error of e ≤ τ = 0.25, whenthere is one trap within the peak subdomain.


Numerical Test Cases

• We now compare the theoretical results for a series of testcases.

• For each test case, the peak width δ, the tolerance τ and thelocation of the peak maximum x∗ is fixed.

• We consider the trap location x0 as a uniformly distributedrandom variable on the peak subdomain.

• The location of the trap x0 is generated nr = 100, 000 timesand in each case the error e is calculated.

• The number of times the error satisfies the condition e ≤ τ iscounted and from this we calculate the probability p(N) ofachieving a sufficiently accurate estimate.


u(x) = B − A(x − x∗)2, x ∈ [xI , xII ],

(a) (b)

Figure: (a) Quadratic peak function: δ = 0.06, A = 1000, B = 0.9. (b)Probability curve.


u(x) = A(x−x∗+δ/3)(x−x∗−2δ/3)2 x ∈ [x∗−δ/3, x∗+2δ/3],

(a) (b)

Figure: (a) Cubic peak function: δ = 0.06, A = 30000. (b) Probabilitycurve.


u(x) = A((δ/2)4 − (x − x∗)4), x ∈ [x∗ − δ/2, x∗ + δ/2],

(a) (b)

Figure: (a) Quartic peak function; A = 1200000; δ = 0.06. (b)Probability curve


u(x) =1

σ√2π

exp(−1

2

(x − x∗)2

σ2)

(a) (b)

Figure: (a) Normal distribution with the location of the maximumx∗ = 0.38. (b) Integration error.


Ecological Test Cases

d = 10−4, δ ≈ 0.25

Figure: Left: Ecological distribution. Right: Probability curve.


d = 10−5, δ ≈ 0.0791

Figure: Left: Ecological distribution. Right: Probability curve.


2-D Analysis

• Similar analysis can be performed in 2-D.

• We consider the peak as a quadratic function in the peaksubdomain DI , and ignore the tail region:

u(x , y) ≈ g(x , y) = B − A((x − x∗)2 + (y − y∗)2), (x , y) ∈ DI

u(x , y) = 0, otherwise,

where A = −uxx/2 and B = u(x∗, y∗).

• The peak subdomain DI is the circular disc of radiusR =

√

B/A centred at (x∗, y∗).

• We define the peak width to be δ = 2R .


0

0.5

1

0

0.5

10

10

20

30

40

50

xy

u(x

,y)

(a)

xy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

5

10

15

20

25

30

35

R

(b)

Figure: (a) 2-D quadratic peak. (b) Peak subdomain DI .


2-D Analysis

• We assume there is one trap only in the peak subdomain.

• The location of the trap (x0, y0) is parameterised as:

x0 = r cos θ + x∗,

y0 = r sin θ + y∗,

where r ∈ [0,R] and θ ∈ [0, 2π] are uniformly distributedrandom variables.

• The population density at the trap location u(x0, y0) ≡ u0 isgiven as

u0 ≈ g(x0) = B − A((x0 − x∗)2 + (y0 − y∗)) = A(R2 − r2).


2-D Analysis

• The mean density M(N) is calculated as

M(N) =u0

N=

A(R2 − r2)

N.

• As before, by imposing the condition

(1− τ)E ≤ M(N) ≤ (1 + τ)E ,

we can obtain expressions for the probability p(N) ofachieving an accurate estimate.


2-D Analysis

pI =

√

1− N(1− τ)πR2

2−√

1− N(1 + τ)πR2

2, N ≤ N∗

pII =

√

1− N(1− τ)πR2

2, N∗ < N ≤ N∗∗

pIII = 0, N > N∗∗,

where

N∗ =2

(1 + τ)πR2

N∗∗ =2

(1− τ)πR2.


Linking the 1-D and 2-D problem

• The theoretical probability functions p(N) in both 1-D and2-D are of the form

pI =√

1− N(1− τ)∆−√

1− N(1 + τ)∆, N ≤ N∗

pII =√

1− N(1− τ)∆, N∗ < N ≤ N∗∗

pIII = 0, N > N∗∗.

• The definition of ∆ varies according to the dimension in whichwe are working:

∆1D = 2δ1D/3,

∆2D = πR2/2 = πδ22D/8.


Linking the 1-D and 2-D problem

• The theoretical curves for the 1-D and 2-D case will be thesame when ∆1D = ∆2D .

• The 2-D peak width can be written in terms of the 1-D peakwidth:

δ2D =

√

16δ1D3π

.


Numerical Results

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

N

p(N

)

(a)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

p(N

)

(b)

Figure: Probability curve for 2-D quadratic text case. (a) δ2D = 0.06.(b) δ2D =

√

16× 0.06/(3π)


Numerical Results

0

0.5

1

0

0.5

10

2

4

6

8

xy

u(x

,y)

(a)

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

N

p(N

)

(b)

Figure: (a) 2-D Normal distribution, δ =√

16× 0.06/(3π). (b)Probability curve.


Ecological Results

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a)

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Np

(N)

N*

(b)

Figure: (a) Ecological distribution; δ ≈ 0.848541. (b) Probability curve.


X

Y

0 2 4 6 80

1

2

3

4

5

6

7

8

9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a)

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

Np

(N)

(b)

Figure: (a) Ecological distribution; δ ≈ 0.0848541. (b) Probability curve.


Conclusions

• When the grid is uniform, increasing the number of trapsleads to the peak being resolved.

• For randomly distributed traps, however, this is not the case.

• Instead there is an optimum number of traps N∗ where theprobability of achieving and accurate estimate is at its highest.

• Beyond this point, the probability decays to zero.

• We recommend the traps be located at the nodes of a uniformgrid when the population distribution is highly aggregated.


Further Work

• Incorporate the probability that there is only one trap in thepeak subdomain.

• Consider the case where there are two traps within the peaksubdomain for random sampling plan.

• Investigate the effect of introducing noise to the functionvalue.


Thank you for your attention.

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