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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
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
Importance of an Accurate Estimate
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
Importance of an Accurate Estimate
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
Acquiring an Estimate of the Pest Abundance
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
Acquiring an Estimate of the Pest Abundance
• 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
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
• This is true when the pests are spread across the wholedomain.
(a) (b)
Figure: (a) Pest population distribution, (b) Accuracy of estimation.
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Introduction Midpoint Rule Study Statistical Method Study
• However, it is not the case when the pest density is localised.
(a) (b)
Figure: (a) Pest population distribution, (b) Accuracy of estimation.
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Introduction Midpoint Rule Study Statistical Method Study
• In such a situation, the error introduced by the method ofapproximation becomes important.
(a) (b)
Figure: (a) Pest population distribution, (b) Accuracy of estimation.
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Introduction Midpoint Rule Study Statistical Method Study
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
Accuracy for Very Sparse Data
Figure: Integration error for ten realisations of the random variable x∗.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
Figure: Range of γ where E ≤ 0.25 for h ≥ delta.
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Introduction Midpoint Rule Study Statistical Method Study
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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
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Introduction Midpoint Rule Study Statistical Method Study
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
Figure: Single peak distributions. Left: d = 10−4. Right: d = 10−5.
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Introduction Midpoint Rule Study Statistical Method Study
• 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.
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Introduction Midpoint Rule Study Statistical Method Study
• 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 .
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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).
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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)).
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Introduction Midpoint Rule Study Statistical Method Study
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∗∗.
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Introduction Midpoint Rule Study Statistical Method Study
Probability Theory
Figure: Probability p(N) of achieving an error of e ≤ τ = 0.25, whenthere is one trap within the peak subdomain.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
Ecological Test Cases
d = 10−4, δ ≈ 0.25
Figure: Left: Ecological distribution. Right: Probability curve.
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Introduction Midpoint Rule Study Statistical Method Study
d = 10−5, δ ≈ 0.0791
Figure: Left: Ecological distribution. Right: Probability curve.
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Introduction Midpoint Rule Study Statistical Method Study
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 .
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Introduction Midpoint Rule Study Statistical Method Study
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 .
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Introduction Midpoint Rule Study Statistical Method Study
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).
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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
.
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Introduction Midpoint Rule Study Statistical Method Study
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)
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
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.
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Introduction Midpoint Rule Study Statistical Method Study
Thank you for your attention.