1 Introduction
We consider the case where two types of sensors are deployed: the first consists of expensive, high quality sensors; and the second, of cheap low quality sensors, which are activated only if the intensity of the spatial field exceeds a predefined activation threshold (eg. wind sensors). This type of heterogeneous sensor networks approach has gained attention in the last few years due to the vision of the Internet of Things (IoT) where networks may share their data over the internet (gubbi2013internet, ; vermesan2011internet, ).
Two practical scenarios ^{1}^{1}1In particular, developing world countries are constrained by their budget when purchasing equipment. Meanwhile, these countries are heavily effected by climate change. This combination makes the robust and costeffective sensor selection in weather sensor networks a major concern for the developing world. that are of importance are: firstly, highquality sensors may be deployed by government agencies (eg. weather stations). These are sparsely deployed due to their high costs, limited space constraints, high power consumption etc. To improve the coverage of the WSN, lowquality cheap sensors can be deployed to augment the highquality sensor network (rajasegarar2014high, ); Secondly, Highquality sensors cannot be easily deployed in remote locations, for example in oceans, lakes, mountains and volcanoes. In these cases, battery operated lowquality cheap sensors can be deployed (werner2006deploying, ).
More specifically, the following two fundamental problems are the focus of this paper: Firstly, Spatial field reconstruction: the task is to accurately estimate and predict the intensity of a spatial random field, not only at the locations of the sensors, but at all locations (peters2015utilize, ; nevat2015estimation, ; nevat2013random, ), given heterogeneous observations from both sensor networks; Secondly, Query based sensor set selection with performance guarantee: the task is to perform online sensor set selection which meets the QoS criterion imposed by the user, as well as minimises the costs of activating the sensors of these networks (calvo2016sensor, ; joshi2009sensor, ; chepuri2015sparsity, ).
2 System model
We now present the system model for the physical phenomenon observed by two types of networks.

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Consider a random spatial phenomenon (eg. wind) to be monitored defined over a dimensional space . The mean response of the physical process is a smooth continuous spatial function , and is modelled as a Gaussian Process (GP) according to
(1) where the mean and covariance functions are assumed to be known.

Let be the total number of sensors that are deployed over a D region , with being the physical location of the th sensor, assumed known by the FC. The number of sensors deployed by Network and Network are and , respectively, so that .

Sensor network includes high quality sensors. The sensors have a threshold activation and each of the sensors collects a noisy observation of the spatial phenomenon . At the th sensor, located at , the observation is given by:
(2) where is i.i.d Gaussian noise . Sensor network includes low quality sensors. The sensors have a threshold activation and each of the sensors collects a noisy observation of the spatial phenomenon , only if the intensity of the field at that location exceeds the predefined threshold , (eg. anemometer sensors for wind monitoring (adafruit, ; Anemo4403, )). At the th sensor, located at , the observation is given by:
(3) where is i.i.d Gaussian noise .
3 Field Reconstruction via Spatial Best Linear Unbiased Estimator (SBLUE)
To perform inference in our Bayesian framework, one would typically be interested in computing the predictive posterior density at any location in space, , denoted . Based on this quantity, a point estimator, like the Minimum Mean Squared Error (MMSE) estimator can be derived:
We develop the spatial field reconstruction via Best Linear Unbiased Estimator (SBLUE), which enjoys a low computational complexity (kay:1998, )
. The SBLUE does not require calculating the predictive posterior density, but only the first two cross moments of the model. The SBLUE is the optimal (in terms of minimizing Mean Squared Error (MSE)) of all linear estimators and is given by the solution to the following optimization problem:
(4) 
where and .
The optimal linear estimator that solves (4) is given by
(5) 
and the Mean Squared Error (MSE) is given by
(6) 
4 Query Based Sensor Set Selection with Performance Guarantee
In this Section we develop an algorithm to perform online sensor set selection in order to meet the requirements of a query made by users of the system. In this scenario users can prompt the system and request the system to provide an estimated value of the spatial random field at a location of interest . We defined the activation sets of the sensors in both networks by . Then the sensor selection problem can be formulated as follows:
(7) 
where is the maximal allowed uncertainty at the query location , and and are the known costs of activating a sensor from Network and Network , respectively.
Suppose we wish to maximize a function over some set . Let us denote the maximum by ; thus,
(8) 
The Cross Entropy Method (CEM) solves this optimization problem by casting the original problem (8
) into an estimation problem of rareevent probabilities. By doing so, the CEM aims to locate an optimal parametric sampling distribution, that is, a probability distribution on
, rather than locating the optimal solution directly. To apply the CEM to solve our optimization problem in (7), we need to choose a parametric distribution. Since the activation of the sensors is a binary variable (eg.
), we choose an independent Bernoulli variable as our parametric distribution, with a single parameter (ie.). The Bernoulli distribution is a member of the NEF of distributions, hence, an analytical solution of the stochastic program is available in closed form as follows:
Since the optimization problem in Eq. (7) is a constrained optimization problem, we introduce an AcceptReject step which rejects samples which do not meet the QoS criterion , as follows
5 Experimental Results and Discussion
In order to test our algorithm on real data sets, we use fine grained datasets available from HansErtelCentre for Weather Research (HErZ) project^{2}^{2}2https://www.herztb4.unibonn.de/index.php/hansertelcentreforweatherresearch/funding. Particularly, we choose Feb 23, 2015 as testing data since it has the highest wind speed across the whole year. The left panel of Fig. 1 shows the region of interest on the map. Both high and low quality sensors are selected randomly within the region. In this figure we randomly deployed high quality and low quality sensors. The right panel of Fig. 1 shows daily wind speed intensity across the region. In left panel of Fig. 2 we present a quantitative comparison of the MSE for various values of high and low quality sensors. The result shows a clear trend of MSE with the increasing of high and low quality sensors. We also illustrate how our sensor selection algorithm performs. For comparison, we use an optimal selection method which only selects the sensor set collections that minimize the U values and ensures that the QoS criterion is being met. The simulation parameters we have are: . We fix the . The comparison is shown in right panel of Fig. 2. We also increase the number of iterations in CE method from 1 to 10. It shows CE method converges quickly to the optimal selection algorithm within 10 iterations for all the values.
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