A Novel Approach to Evaluation of Pest Insect Abundance in the Presence of Noise

Abstract

Evaluation of pest abundance is an important task of integrated pest management. It has recently been shown that evaluation of pest population size from discrete sampling data can be done by using the ideas of numerical integration. Numerical integration of the pest population density function is a computational technique that readily gives us an estimate of the pest population size, where the accuracy of the estimate depends on the number of traps installed in the agricultural field to collect the data. However, in a standard mathematical problem of numerical integration, it is assumed that the data are precise, so that the random error is zero when the data are collected. This assumption does not hold in ecological applications. An inherent random error is often present in field measurements, and therefore it may strongly affect the accuracy of evaluation. In our paper, we offer a novel approach to evaluate the pest insect population size under the assumption that the data about the pest population include a random error. The evaluation is not based on statistical methods but is done using a spatially discrete method of numerical integration where the data obtained by trapping as in pest insect monitoring are converted to values of the population density. It will be discussed in the paper how the accuracy of evaluation differs from the case where the same evaluation method is employed to handle precise data. We also consider how the accuracy of the pest insect abundance evaluation can be affected by noise when the data available from trapping are sparse. In particular, we show that, contrary to intuitive expectations, noise does not have any considerable impact on the accuracy of evaluation when the number of traps is small as is conventional in ecological applications.

Appendix: Finding a Credible Interval for the Relative Error in the Presence of Noise

We seek the upper and lower limit of the interval $[\tilde{E}_{\min}, \tilde{E}_{\max}]$ to which the quantity $\tilde{E}_{\mathrm{rel}}$ belongs with probability P(z) given by (11) as discussed in Sect. 2.3. We recall that the estimate of pest abundance $\tilde{I}$ calculated from measured data is a realisation of a normally distributed random variable with mean $\mu_{\tilde{I}}=I_{a}$ and standard deviation $\sigma_{\tilde{I}}$ as defined by (15). Thus any realisation $\tilde{I}$ lies within the interval $[I_{a}-z\sigma_{\tilde{I}}, I_{a}+z\sigma_{\tilde{I}}]$ with probability P(z). We use this credible interval for $\tilde{I}$ to construct a credible interval for $\tilde{E}_{\mathrm{rel}}$. We consider two cases based on the distance between the approximate integral formed from exact data I _a and the exact value of the integral I. Let us begin by finding the lower limit of the interval, $\tilde{E}_{\min}$.

Case 1. $|I-I_{a}|\le z \sigma_{\tilde{I}}$.

In this case, as can be seen from Fig. 7a, an estimate based on measured data $\tilde{I}$ which belongs to the range $[I_{a}-z\sigma_{\tilde{I}},I_{a}+z\sigma_{\tilde{I}}]$ can coincide with the exact value of the integral. Therefore, the lower limit of the range $[\tilde{E}_{\min}, \tilde{E}_{\max}]$ is

$$ \tilde{E}_{\min}=0. $$

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Fig. 7

Finding the interval $[\tilde{E}_{\mathrm{rel}}, \tilde{E}_{\max}]$ to which $\tilde{E}_{\mathrm{rel}}$ belongs with probability P(z). (a) Case 1. $|I-I_{a}|\le z \sigma_{\tilde{I}}$. In this case, the exact value of the integral I lies within the credible interval for $\tilde{I}$ thus the lower limit of the credible interval for $\tilde{E}_{\mathrm{rel}}$ is $\tilde{E}_{\min}=0$. (b) Case 2. $|I-I_{a}|> z \sigma_{\tilde{I}}$. The exact value of the integral I lies outside, thus the interval $[\tilde{E}_{\min},\tilde{E}_{\max}]$ does not include the zero value

Case 2. $|I-I_{a}|> z \sigma_{\tilde{I}}$.

In this instance, from Fig. 7b we can see that the range $[I_{a}-z\sigma_{\tilde{I}},I_{a}+z\sigma_{\tilde{I}}]$ does not include the exact value of the integral I. Either we have I _a ≤I in which case we can see that

$$\tilde{E}_{\min}=\frac{|I-I_a-z\sigma_{\tilde{I}}|}{|I|}, $$

or we have I _a >I; therefore,

$$\tilde{E}_{\min}=\frac{|I-I_a+z\sigma_{\tilde{I}}|}{|I|}. $$

In both cases,

$$ \tilde{E}_{\min}=E_\mathrm{rel}-\frac{z\sigma_{\tilde{I}}}{I}, $$

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which is a strictly positive quantity as the condition $|I-I_{a}|>z\sigma_{\tilde{I}}$, of course, means that $E_{\mathrm{rel}}>z\sigma_{\tilde{I}}/I$, where we recall that I>0.

It should be mentioned that a zero relative error is still possible in the second case, when the distance between the approximation based on exact data and the true value of the integral exceeds z multiples of the standard deviation $\sigma_{\tilde{I}}$ , however, we choose to fix $\tilde{E}_{\mathrm{rel}}$ as

$$\tilde{E}_{\min}=\left \{ \begin{array}{l@{\quad }l} \min \{E\ge 0: E\in[\mu_E-z\sigma_E, \mu_E+z\sigma_E] \}, & \mbox{for} \ \mu_E\ge0, \\ |\,{\max \{E\le 0: E\in[\mu_E-z\sigma_E, \mu_E+z\sigma_E] \}}|, & \mbox{for} \ \mu_E<0, \end{array} \right . $$

where E is defined by (16). In other words, we find the value of the quantity E closest to zero which lies within the range (18) and then take the absolute value as $\tilde{E}_{\min}$ (see Fig. 2).

Let us now consider the upper limit $\tilde{E}_{\max}$ of the credible interval of $\tilde{E}_{\mathrm{rel}}$. To find $\tilde{E}_{\max}$, we use the condition that any single value of $\tilde{E}$ lies within the range $[\tilde{E}_{\min}, \tilde{E}_{\max}]$ with fixed probability P(z) as defined by (11). As mentioned above, $\tilde{E}_{\mathrm{rel}}$ is a realisation of a random variable with a folded normal distribution. This distribution is formed by reflecting the negative quantities of the distribution (17) of the auxiliary error E in the y-axis. Unless the mean value of this underlying normal distribution is μ _E =0, if we take $\tilde{E}_{\max}=\mu_{E}+z\sigma_{E}$ then the probability $\hat{P}$ that $\tilde{E}_{\mathrm{rel}}$ lies within the above range will exceed P(z). We shall denote the additional contribution as P ^∗; therefore,

$$\hat{P}=P(z)+P^*. $$

We now seek the appropriate value of the upper limit $\tilde{E}_{\max}$ in order to satisfy the condition that $\hat{P}=P(z)$. Let us temporarily impose the restriction μ _E ≥0. As when constructing the lower limit $\tilde{E}_{\min}$, we consider the cases when the distance between the approximation based on exact data I _a and the true value of the integral I exceeds or is within z multiples of the standard deviation $\sigma_{\tilde{I}}$ separately.

Case 1. $|I-I_{a}|\le z \sigma_{\tilde{I}}$.

As shown in Fig. 2a, the probability P ^∗ is given by

$$ P^*=\int_{-\mu_E-z\sigma_E}^{\mu_E-z\sigma_E} p(E) \, dE. $$

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In order to satisfy the condition $\hat{P}=P(z)$, we must then find $\tilde{E}_{\max}$ such that

$$ \int_{\tilde{E}_{\max}}^{\mu_E+z\sigma_E}p(E) \, dE = P^*. $$

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Using the transformation

$$E\rightarrow \frac{E-\mu_E}{\sigma_E} $$

from (34) and (35), we obtain the following in terms of the standard normal distribution function Φ:

$$\varPhi(-z)-\varPhi \biggl(\frac{-2\mu_E}{\sigma_E}-z \biggr)=\varPhi(z)-\varPhi \biggl(\frac{\tilde{E}_{\max}-\mu_E}{\sigma_E} \biggr). $$

Rearranging gives

$$ \tilde{E}_{\max}=\mu_E + \sigma_E \varPhi^{-1} \biggl[2\varPhi(z)-\varPhi \biggl(z+2 \frac{\mu_E}{\sigma_E} \biggr) \biggr]. $$

(36)

Case 2. $|I-I_{a}|> z \sigma_{\tilde{I}}$.

Similar calculations for this case as illustrated in Fig. 2b yield

$$ \tilde{E}_{\max}=\mu_E+\sigma_E \varPhi^{-1} \biggl[\varPhi(z)-\varPhi \biggl(z-\frac{2\mu_E}{\sigma_E} \biggr)-\varPhi \biggl(z+\frac{2\mu_E}{\sigma_E} \biggr) +1 \biggr]. $$

(37)

Earlier we assumed μ _E ≥0. Since the probability density function (19) for the folded normal distribution is the same for mean μ _E as it is for −μ _E , we can replace the term μ _E for |μ _E | in Eqs. (36) and (37) so that they hold for arbitrary μ _E .