Private Outsourced Kriging Interpolation

Abstract

Kriging is a spatial interpolation algorithm which provides the best unbiased linear prediction of an observed phenomena by taking a weighted average of samples within a neighbourhood. It is widely used in areas such as geo-statistics where, for example, it may be used to predict the quality of mineral deposits in a location based on previous sample measurements. Kriging has been identified as a good candidate process to be outsourced to a cloud service provider, though outsourcing presents an issue since measurements and predictions may be highly sensitive. We present a method for the private outsourcing of Kriging interpolation using a tailored modification of the Kriging algorithm in combination with homomorphic encryption, allowing crucial information relating to measurement values to be hidden from the cloud service provider.

A Additional Details on Kriging

In order to apply the Kriging interpolation technique, the observed phenomena is viewed as a realization of a random field which satisfies certain properties related to the observed measurements. A random field generalizes the notion of stochastic process, by allowing the underlying parameter to take values other than real numbers. In the case of spatial interpolation, a random field Z is defined as a collection of real-valued random variables $\{Z(r)\}_{r\in R}$, all defined in the same probability space, and indexed by locations r in a fixed region $R\subseteq \mathbb {R}^{2}$.

Given a set of n samples S taken at positions P, every sample $z_{i}\in S$ can be viewed as a realization of the random variable $Z(r_{i})$, indexed by the position $r_{i}\in P$ in a random field Z. Given such realizations, a linear predictor $Z^{*}$ of the random field Z is defined as a random field of the form

$$\begin{aligned} Z^{*}(r)=\lambda _{0}+\sum _{i=1}^{n}\lambda _{i}Z(r_{i}), \quad \text { where } \lambda _{i}\in \mathbb {R}. \end{aligned}$$

We say a linear predictor $Z^{*}$ is unbiased if the expectation $\mathbb {E}(Z(r)-Z^{*}(r))=0$ for all $r\in R$. Moreover, we say that a linear predictor $Z^{*}$ is best or optimal if, for every location $r\in P$, it minimizes the prediction variance $\mathrm {Var}(Z(r)-Z^{*}(r))$ among all unbiased linear predictors.

The Kriging interpolation technique aims to find a best unbiased linear predictor for the random field Z derived from a Kriging dataset (P, S). In this sense, note that Kriging deals with the same problem as linear least squares in random fields. However, in order to derive such a predictor from sampled values, additional assumptions are usually made on the stationarity of the random field. The most widely applied Kriging process is Ordinary Kriging. This form of Kriging stems from two stationarity assumptions. The second-order stationarity assumption states that the first and second-order moments of the random variables in the random field are shift invariant:

Definition 4

A random field Z parametrized by elements of a region $R\subseteq \mathbb {R}^{2}$ is defined to be second-order stationary if the following conditions are satisfied:

• The mean $\mathrm {E}(Z(r))$ does not depend on $r\in R$, and

• The covariance $\mathrm {Cov}(Z(r),Z(r+h))$ is a function of only the separating vector h for every $r,r+h\in R$.

The intrinsic stationarity assumption considers variance of increments instead of covariance:

Definition 5

A random field Z parametrized by elements of a region $R\subseteq \mathbb {R}^{2}$ is defined to be intrinsic stationary if the following conditions are satisfied:

• The mean $\mathrm {E}(Z(r))$ does not depend on $r\in R$, and

• The variance of the increments $\mathrm {Var}(Z(r+h)-Z(r))$ is a function of only the separating vector h for every $r,r+h\in R$.

Second-order stationarity implies intrinsic stationarity [14] and thus we restrict our attention to the more general intrinsic stationarity assumption. Our techniques are, however, applicable to Ordinary Kriging in general.

The intrinsic stationarity assumption naturally leads to the notion of theoretical variogram [7, 10] which models the spatial dependency between the random variables Z(r). Given an intrinsic stationary random field Z, the theoretical variogram $\hat{\gamma }:R\rightarrow \mathbb {R}$ is defined as the function $\hat{\gamma }(h)=\mathrm {Var}(Z(r+h)-Z(r)).$ Under the intrinsic assumption, $\hat{\gamma }(h)$ depends only on the norm of h [14]. Hence, we may view $\hat{\gamma }$ as a function defined over positive real numbers.