Comparison of ordinary kriging and artificial neural network for spatial mapping of arsenic contamination of groundwater

Abstract

In this technical note, we investigate the hypothesis that ‘non-linearity matters in the spatial mapping of complex patterns of groundwater arsenic contamination’. The spatial mapping pertained to data-driven techniques of spatial interpolation based on sampling data at finite locations. Using the well known example of extensive groundwater contamination by arsenic in Bangladesh, we find that the use of a highly non-linear pattern learning technique in the form of an artificial neural network (ANN) can yield more accurate results under the same set of constraints when compared to the ordinary kriging method. One ANN and a variogram model were used to represent the spatial structure of arsenic contamination for the whole country. The probability for successful detection of a well as safe or unsafe was found to be atleast 15% larger than that by kriging under the country-wide scenario. The probability of false hopes, which is a serious issue in public health monitoring was found to be significantly lower (by more than 10%) than that by kriging.

Appendix

1.1 Training of artifical neural network

The algorithm for training of ANN employs damped Gauss–Newton method utilizing a damping parameter μ as follows,

$$ (J^{T} J + \mu I)h_{\text{lm}} = - g $$

(5)

$$ {\text{with}}\,g = J^{T} f\,{\text{and}}\,\mu \ge 0 $$

(6)

where, J ∈ R is a Jacobian Matrix, which contains the first partial derivatives of the function component $\left\| {f(x)} \right\|,$ that need to be minimized,

$$ \begin{aligned}& (J(x))_{i,j} = \frac{{\partial f_{i} }}{{\partial x_{j} }}(x) \hfill \\ &{\text{and}}\;F(x) = \frac{1}{2}\sum\limits_{i = 1}^{m} {(f_{i} (x))^{2} = \frac{1}{2}\left\| {f(x)} \right\|^{2} = \frac{1}{2}f(x)^{T} f(x)} \hfill \\ \end{aligned} $$

(7)

f has continuous second partial derivatives, that can be written from Taylor expression as follows,

$$ f(x + h) = f(x) + J(x)h + O(\left\| h \right\|^{2} ) $$

Here, h _lm is the step used in this LM method and g is a variable which depends on step size (h _lm) and damping parameter (μ).

Also, J = J(x) and f = f(x). The damping parameter μ has several effects as follows:

For all μ > 0 the coefficient matrix is positive definite. This ensures that h _lm is a direction downhill. For large values of μ we get, $h_{\text{lm}} \approx - \frac{1}{\mu }g = - \frac{1}{\mu }F^{\prime}(x)$ i.e. a short step in the steepest descent direction. This is good if the current iteration is far from the solution. If μ is very small, then h _lm ≈ h _gn; h _gn = Gauss–Newton step. This is beneficial in the final stages of the iteration, when x is close to x ^*. If F(x ^*) = 0 (or very small), then we can get (almost) quadratic final convergence.

The damping parameter μ influences both the direction and the size of the step. The choice of initial μ value should be related to the size of the elements in A ₀ = J(x ₀)^T J(x ₀), e.g. by letting μ ₀ = τ.max _i {a ⁽⁰⁾ _ii }, where τ is chosen by user.¹

Updating of the damping parameter μ is controlled by the gain ratio,

$$ \xi = \frac{{F(x) - F(x + h_{\text{lm}} )}}{{L(0) - L(h_{\text{lm}} )}} $$

(8)

Here,

$$ f(x + h) \approx \ell (h) \equiv f(x) + J(x)h $$

(9)

Inserting Eq. 9 in Eq. 7 we find that

$$ \begin{aligned} F(x + h) \approx L(h) & \equiv \frac{1}{2}\ell (h)^{T} \ell (h) \\ & = \frac{1}{2}f^{T} f + h^{T} J^{T} f + \frac{1}{2}h^{T} J^{T} Jh \\ & = F(x) + h^{T} J^{T} f + \frac{1}{2}h^{T} J^{T} Jh \\ \end{aligned} $$

(10)

The denominator of Eq. 8 is the gain predicted by the linear model of Eq. 10 as follows,

$$ \begin{aligned} L(0) - L(h_{\text{lm}} )\; & = - h_{\text{lm}}^{T} J^{T} f - \frac{1}{2}h_{\text{lm}}^{T} J^{T} Jh_{\text{lm}} \\ & = - \frac{1}{2}h_{\text{lm}}^{T} (2g + (J^{T} J + \mu I - \mu I)h_{\text{lm}} ) \\ & = \frac{1}{2}h_{\text{lm}}^{T} (\mu h_{\text{lm}} - g) \\ \end{aligned} $$

(11)

Both h ^T_lm h _lm and −h ^T_lm g are positive definite, so L(0) − L(h _lm) is guaranteed to be positive.

A large value of gain ratio, ξ indicates that L(h _lm) is a good approximation to F(x + h _lm), and we can decrease μ so that the next Levenberg–Marquardt step is closer to the Gauss–Newton step. If ξ is small (may be even negative), then L(h _lm) is a poor approximation, and we should increase μ with the twofold aim of getting closer to the steepest descent direction and reducing the step length.

The stopping criteria for the algorithm should reflect that at a global minimizer we have F′(x ^*) = g(x ^*) = 0 so we can use,

$$ \left\| g \right\|_{\infty } \le \varepsilon_{1} $$

Another relevant criterion is to stop if the change in x is small,

$$ \left\| {x_{\text{new}} - x} \right\| \le \varepsilon_{2} (\left\| x \right\| + \varepsilon_{2} ). $$

This expression gives a gradual change from relative step size ε ₂ when $\left\| x \right\|$ is large to absolute step size ε ²₂ if x is close to 0. Finally, as in all iterative processes we need a safeguard against an infinite loop,

$$ k \ge k_{\max } .$$

Also ε ₂ and k _max are chosen by the user.

The last two criteria come into effect e.g. if ε ₁ is chosen so small that effects of rounding errors have large influence. This will typically reveal itself in a poor accordance between the actual gain in F and the gain predicted by the linear model and will result in μ grows fast, resulting in small $\left\| {h_{\text{lm}} } \right\|$ and the process will be stopped by following the stopping criterion.