Site reduction in redundant ecosystem sampling schemes

Abstract

Data collection for fresh-water regions of The Ecosystem Health Monitoring Program (EHMP), in southeast Queensland, Australia, involves the sampling of over 130 sites among 19 catchments twice per year and has been ongoing for over ten years. The sampling design was derived following an exhaustive process of indicator and site selection to develop a composite indicator that represented aquatic ecosystem health. After 13 years of implementation, there was an interest in identifying redundancies in sampling to reduce sampling costs without making a substantial impact on the integrity of the program and its capacity to report on ecosystem health. This paper focuses on identifying a subset of sites and times that could be removed from sampling with a minimal impact on the subsequent ecosystem health scores. Herein, Mixed models are employed to assess a variance structure from which optimality criteria are utilized to identify the scheme. Integer programs are then used to ensure specific practical constraints are observed.

Appendix

1.1 Annual/seasonal score components

Specific data collected at freshwater sites:

• 1. pH, Cond, Temp and DO are averaged to obtain the Water Quality Indicator,

• 2. DelC, R24 and GPP are averaged to form the Ecosystem Process indicator,

• 3. MacroRich, PET, and SIGNAL are averaged to form the Macroinvertebrate indicator,

• 4. PONSE, FishOE and PropAlien are averaged to form the Fish indicator, and

• 5. The index DelN by itself forms the Nutrient indicator.

1.2 Matrix model notation for mixed models

Section  2.3 introduces the matrix notation in Eq. 3 for Model  (2) described in Sect. 2.2. This section illustrates the explicit components of the $\mathbf{y }$, $\mathbf{X }$, ${\varvec{\beta }}$, $\mathbf{U }$, ${\varvec{\delta }}$, and ${\varvec{\varepsilon }}$, matrices and vectors in Eq.  3 in relation to the scalar notation and indices contained in both of Sects. 2.2 and  2.3. Based on Indices  (1) and Model  (2), let

$$\begin{aligned} \mathbf{y }_i= & {} [y_{i,t_i},\ldots ,y_{i,T_i}]',\\ {\varvec{\varepsilon }}_i= & {} [\varepsilon _{i,t_i},\ldots ,\varepsilon _{i,T_i}]', \end{aligned}$$

denote the $r_i \times 1$ vectors for the score y and error terms $\varepsilon$ for site i at times $t_i,\ldots ,T_i$ for $r_i$ adjacent and evenly spaced time points. Then the $N \times 1$ vectors $\mathbf{y }$ and ${\varvec{\varepsilon }}$ displayed in Eq. (3) for sites $i=1,\ldots ,n$ are defined as

$$\begin{aligned} \mathbf{y }= & {} [\mathbf{y }_1',\ldots ,\mathbf{y }_n']',\\ {\varvec{\varepsilon }}= & {} [{\varvec{\varepsilon }}_1',\ldots ,{\varvec{\varepsilon }}_n']'. \end{aligned}$$

The random effects and regressor coefficients, respectively, are denoted as

$$\begin{aligned} {\varvec{\delta }}_{n \times 1}= & {} [\delta _{i},\ldots ,\delta _{n}]',\\ {\varvec{\beta }}_{p \times 1}= & {} [\beta _{1},\ldots ,\beta _{p}]'. \end{aligned}$$

If we denote the $r_i \times p$ regressor matrix for site i as $\mathbf{x }_i$, then the full regressor matrix $\mathbf{X }$ in Eq. (3) is represented by the vertical concatenation of these matrices:

$$\begin{aligned} \mathbf{X }= & {} \left[ \begin{array}{c} \mathbf{x }_1\\ \vdots \\ \mathbf{x }_n \end{array} \right] _{N \times p} . \end{aligned}$$

Finally, the $\mathbf{U }_{N \times n}$ matrix consists of 0/1 entries based on whether or not observation $y_{ij}$ is observed in site i. If we denote an $r_i \times n$ matrix $\mathbf{u }_i$ for site i as a matrix with the ith column equal to unity and all other entries equal to zero, then the $\mathbf{U }$ matrix is represented by the vertical concatenation of these matrices:

$$\begin{aligned} \mathbf{U }= & {} \left[ \begin{array}{c} \mathbf{u }_1\\ \vdots \\ \mathbf{u }_n \end{array} \right] _{N \times n} . \end{aligned}$$

1.3 General covariance matrix for y

Consider $r_i \equiv T_i -t_i + 1$, and $N = \sum _{i=1}^{n} r_i$ for sites $i=1,\ldots ,n$ with the matrix ${\mathbf {J}}$ and matrices $R_i$ of dimension $r_i \times r_i$ defined as in Sect. 2.2. Then the resulting covariance matrix is expressed as

$$\begin{aligned} \mathbf{V }= & {} \mathbf{UGU} ' + \mathbf{R } \nonumber \\ \Leftrightarrow \left[ \begin{array}{ccccc} {\mathbf {V}}_{1} &{} &{} &{} &{} {\mathbf {0}}\\ &{} \ddots &{} &{} &{} \\ &{} &{} {\mathbf {V}}_{i} &{} &{} \\ &{} &{} &{} \ddots &{} \\ {\mathbf {0}} &{} &{} &{} &{} {\mathbf {V}}_{n} \end{array} \right] _{N \times N}= & {} \sigma _{\delta }^{2} \left[ \begin{array}{ccccc} {\mathbf {J}}_{r_1} &{} &{} &{} &{} {\mathbf {0}}\\ &{} \ddots &{} &{} &{} \\ &{} &{} {\mathbf {J}}_{r_i} &{} &{} \\ &{} &{} &{} \ddots &{} \\ {\mathbf {0}} &{} &{} &{} &{} {\mathbf {J}}_{r_n} \end{array} \right] \nonumber \\&+ \sigma _{\varepsilon }^{2} \left[ \begin{array}{ccccc} \ R_1 &{} &{} &{} &{} {\mathbf {0}}\\ &{} \ddots &{} &{} &{} \\ &{} &{} R_{i} &{} &{} \\ &{} &{} &{} \ddots &{} \\ {\mathbf {0}} &{} &{} &{} &{} \ R_{n} \end{array} \right] . \end{aligned}$$

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