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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Wireless Sensor Networks are presented as devices for signal sampling and reconstruction. Within this framework, the qualitative and quantitative influence of (i) signal granularity, (ii) spatial distribution of sensors, (iii) sensors clustering, and (iv) signal reconstruction procedure are assessed. This is done by defining an error metric and performing a Monte Carlo experiment. It is shown that all these factors have significant impact on the quality of the reconstructed signal. The extent of such impact is quantitatively assessed.

AWireless Sensor Network (WSN) consists of spatially distributed autonomous devices, which cooperatively monitor physical or environmental conditions, such as temperature, sound, vibration, pressure, motion or pollutants, at different locations [

The sensors scattered in a sensor field have the capability to collect, and aggregate data [

Most of current studies on WSNs focus on the sensors’ energy constraint as a key design feature. For this reason, techniques abound in the literature aiming at reducing energy consumption and, therefore, increasing the lifetime of the whole network. Since communication among nodes is the main cause of energy consumption, many techniques involving clustering and information fusion have been proposed to increase the network lifetime, some of them can be found in [

The aforementioned techniques have impact on the quality of the information delivered to users and, as consequence, have influence on the decisions they take. For instance, consider the case of a hierarchical WSN that uses information fusion to efficiently help the base station taking decisions about temperature management. Suppose the cluster head of each cluster sends the mean of the temperature measured by individual sensors of that cluster. This approach is prone to imprecisions and, among other issues, it is quite sensitive to outliers. In this case, for instance, information about data variability is lost in this process. For applications in which data dependability is critical, such issues are not acceptable.

Among the factors that impact the quality of the reconstructed signal, we emphasize the following:

The impact of such factors in the quality of the reconstructed information signal in WSN is seldom present in the literature. Even the studies of how wireless sensor networks are able to report data they collect by means of estimated errors are scarcely found. Some authors as [

The aforementioned proposals attempt to established theoretical limits for the reconstruction problem considering some aspects such as clustering and correlated random field data. The work presented below assesses the impact of those factors on the quality of the reconstructed signal by modeling WSNs as signal processing problems on ℝ^{2}, where the data might be irregularly sampled. We conclude that, using the error metric defined in this work, we observe smaller errors for (i) coarser processes, (ii) more regular sensor deployment, (iii) data-aware aggregation, and (iv) the reconstruction based on Kriging.

In particular, we quantitatively assess: data granularity by using a Gaussian field model for the data (disregarding temporal variation); sensors deployment by a new stochastic point process, which is able to describe from regularly spaced to tightly packed sensors distribution; the evaluation of two node clustering techniques (LEACH, a geographic clustering, and SKATER, which also incorporates data homogeneity; no clustering is considered as a benchmark); and two reconstruction strategies, namely, Voronoi cells and Kriging. A constant perception radius and the mean value as data aggregation are assumed. We show that all the aforementioned factors have significant impact on the quality of the reconstructed signal, for which we provide quantitative measures.

The paper unfolds as follows. Section 2. presents the main models we employ, namely the clustering strategy (Section 2.1.), WSNs as a whole (Section 2.2.), the data (Section 2.3.), the sensor deployment (Section 2.4.), and signal sampling and reconstruction (Section 2.5.). Section 3. describes the scenarios of interest and the methodology. Section 4. presents the results, and Section 5. concludes the paper.

This section presents the four central models for our work, namely clustering strategy (Section 2.1.), WSNs from the signal processing viewpoint (Section 2.2.), a model for the observed data (Section 2.3.) and a model for sensor deployment (Section 2.4.)

WSNs present several constraints such as battery capacity, and limited computing capabilities [

Clustering sensors into groups is a popular strategy to save energy [

In the following, two clustering approaches are detailed. The former creates clusters based on geographical information, while the later is based on a data-aware clustering technique. These approaches will be assessed in terms of the quality of reconstructed signal in Section 4.

LEACH (Low-Energy Adaptive Clustering Hierarchy) [

There are two different versions of LEACH proposed in [

Once the election finishes, CHs inform their role by an advertisement message. Thus, all other nodes receive this message and join only one group represented by the CH that requires the minimum communication energy. LEACH takes this decision based on the received signal strength of the advertisement message from each CH. Note that, typically, this will lead to choosing the closest CH, unless there is an obstacle impeding communication. After clusters are formed, each group member configures its power to reach its corresponding CH. The communication within the group uses a TDMA scheme and, outside the groups, CHs employ a direct-sequence spread spectrum. These schemes attempt to diminish intraand inter-group interferences.

The main goal of our assessment is to analyze the reconstruction error. Thus, questions related to energy consumption were not considered and CHs were chosen randomly in a similar manner of the distributed version of LEACH. The difference is that we forced the CHs to be far from at least

LEACH assumes that nearby nodes have correlated data, while SKATER (Spatial ‘K’luster Analysis by Tree Edge Removal) [

As spatially homogeneous clusters, SKATER looks for a partition with three properties: (i) nodes of the same group have to be similar to each other in some predefined attributes; (ii) the attributes are different among different groups, and (iii) nodes of the same group must belong to a predefined neighborhood structure.

SKATER works in two steps. First, it creates a minimal spanning tree (MST) from the graph representation for the neighborhood of the geographical structure of the nodes. The cost of the edges represents the similarity of the sensors’ collected data, defined as the euclidian square distance between them (data might be in ℝ^{p}

SKATER is a centralized clustering processing and presents high computational cost due to the exhaustive comparison of all possible values of the objective function. However, SKATER uses a polynomial-time heuristic for fast tree partitioning.

In our work, we used SKATER to build homogeneous clusters. The process is similar to that described in LEACH, but the cluster formation is performed in the same manner as in SKATER. CHs are chosen randomly among cluster members.

As presented in Aquino

A WSN collecting information can be represented by the diagram shown in _{1}, . . ., _{n}_{i}_{1}, . . ., _{n}

Most of the time, collecting all data from every sensor is a waste of resources since there is redundant information. In order to save resources, e.g., energy and, therefore, to extend the network lifetime, information fusion techniques are used [

The class of transformations Ψ we consider here is formed by two different steps: the first is the clustering of nodes, and the second is data aggregation. Aggregated data, with their corresponding locations, are used as input to a reconstruction process that runs in the sink, and then delivered to the user. The data sent to the user,

Besides the already defined clustering techniques, namely, LEACH and SKATER, Pointwise data processing that makes neither clustering nor aggregation is used in this work as a benchmark.

In our study, data aggregation will be done by taking the mean value of the data observed at each cluster; this reduction makes sense when these data can be safely summarized by a single value.

Signal reconstruction is performed with two strategies: Voronoi cells and Kriging. They require the same information, namely sensor position and value, being the latter more computationally intensive.

Sensors measure a continuously varying function

Random fields are collections of random variables indexed in a

We assume a stable covariance function exp(−^{s}

Sampling outcomes of

Point processes are stochastic models that describe the location of points in space. They are useful in a broad variety of scientific applications such as ecology, medicine, and engineering [

The isotropic stationary Poisson model, also known as fully random or uniformly distributed, is the basic point process. The number of points in the region of interest follows a Poisson law with mean proportional to the area. The location of each point does not have influence on the location of the other points. The other process we will use is a repulsive one, where points cannot lie at less than a specified distance. Using these two processes we build a composed point process able to describe many practical situations.

The Poisson point process over a finite region ^{2} is defined by the following properties:

The probability of observing _{0} points in any set _{A}^{−ημ(A)}[^{n}/n

Random variables used to describe the number of points in disjoint subsets are independent.

Without loss of generality, in order to draw a sample from a Poisson point process with intensity ^{2}. Assume _{1}, . . ., _{n}_{1}, . . ., _{n}_{i}_{i}_{1≤i≤n} are an outcome of the Poisson point process on _{i}_{i}_{1≤i≤n} are an outcome of the Binomial point process on

The Matérn’s Simple Sequential Inhibition process can be defined iteratively as the procedure that places at most _{max} is reached, a new location is chosen uniformly on

We build an attractive process by merging two Poisson processes with different intensities. A step point process in ^{2} with parameters

Without loss of generality, we define the compound point process ^{2}, ^{2} and η = 1, denoted by 𝒞(_{max} is the maximum exclusion distance, which we set to _{max} = ^{−1/2}. The 𝒞(

Repulsive processes are able to describe the intentional, but not completely controlled location of sensors as, for instance, when they are deployed by a helicopter at low altitude. Sensors located by a binomial process could have been deployed from high altitude, so their location is completely random and independent of each other. Attractive situations may arise in practice when sensors cannot be either deployed or function everywhere as, for instance, when they are spread in a swamp: those that fall in a dry spot survive, but if they land on water they may fail to function.

Without loss of generality, in the following we consider that the whole process takes place on ^{2} and ^{2} with intensity η = 1, and that there are _{1}, _{1}), . . ., (_{100}, _{100}), which, in turn, are the outcome of the compound point process 𝒞(100,

For each 1 ≤ _{i}_{i}_{i}_{i}_{pi} f

Once every node has its value _{i}

Two reconstruction methodologies were assessed in this work: Voronoi cells and Kriging. The former consists in first determining the Voronoi cell of each sensor,

Kriging is the second reconstruction procedure we employed. It is a geostatistical method, whose simplest version (“simple Kriging”) is equivalent to minimum mean square error prediction under a linear Gaussian model with known parameter values. No parameter was assumed known and, regardless the true covariance model imposed to the Gaussian field, we estimated a general and widely accepted covariance function: the Matérn model given by
_{ν}

Given the data and their location, the covariance function is estimated using maximum likelihood. Then, the means are estimated by generalized minimum squares using the covariance as weight: closer values have more influence than distant ones. Notice that such procedure requires the same information needed by Voronoi reconstruction, namely, the sampled data and their position; see

Ordinary Kriging was used by Yu

As a benchmark, the result of applying ordinary Kriging to the original _{1}, . . ., _{100} sampled values without clustering or aggregation is also presented. This approach, which provides the best possible input for any reconstruction procedure, is too costly from the energy consumption viewpoint, but provides a measure of the loss introduced by LEACH, SKATER or any other similar procedure.

It is noticeable that the coarse process is easier to reconstruct, regardless the deployment. Regardless the coarseness of the process and the reconstruction, the more repulsive the deployment the better the reconstruction. Regardless the coarseness and the deployment, ordinary Kriging provides better reconstruction than Voronoi; because of this, only results produced by Kriging are presented in the remainder of this work.

The performance of each procedure is assessed by the absolute value of the relative error between the true signal

The following scenarios are reported:

four levels of coarseness:

seven deployment situations:

three sensor clustering and data aggregation procedures: neither clustering nor aggregation (Pointwise data delivery), LEACH (geographic clustering), and SKATER (geographic data-aware clustering).

One hundred independent samples were generated for each of the 4 × 7 × 3 = 84 different situations, and the absolute value of the relative error, defined in

Simulations were performed using R [

The results are reported in next section.

Regarding the first factor,

Regarding the second factor, namely process granularity, it is clear that the coarser the observed phenomenon,

Regarding the third factor,

All the aforementioned dependencies of the reconstruction error with respect to granularity and deployment are augmented when no clustering/aggregation is performed, but since this situation was only presented as a theoretical reference, it is not further commented.

One can readily see that SKATER is consistently better than LEACH, the smaller the error the larger the difference (ranging from 72% in the best situation to 10% in the worst one).

The error, for each clustering procedure, increases with both attractivity, being the most sensitive situation SKATER on the coarse process (

The error decreases with granularity in both clustering procedures, being the most sensitive situation SKATER on the coarse process, where it doubles from the coarsest (

The first conclusion is that reconstruction by ordinary Kriging consistently produces smaller errors than those obtained by using Voronoi reconstruction: the values in

The study presented here leads us to the following conclusions. The reconstruction error reflects the performance of the WSN and provides an idea of the dependability of the data available to the user. This error is sensitive to process granularity, spatial distribution of sensors, clustering procedure, and reconstruction technique. Regarding the factors the user is able to control, for a given number of sensors; ordinary Kriging is consistently better than Voronoi reconstruction, the best strategy is regular deployment, otherwise the error may increase in up to 50%; the best clustering algorithm is SKATER, using LEACH may increase the error in up to 70%. Regarding the uncontrolled factor, namely granularity, the user should be aware that the finer the process the larger the error; for a fixed number of sensors, and when using SKATER, it may double in the best situation (regular deployment of sensors) and increase 74% when attractive deployment is used.

Representing WSNs as a sampling/reconstruction process guided the proposal and development of the simulation experiments. Each stage of the process can be modeled differently, leading to tailored results.

Future work includes further studies using non-isotropic sensing and communication, multivariate and non-Gaussian phenomena models, other clustering procedures and robust aggregation techniques. Direct estimation of granularity and other parameters using aggregated data will also be performed.

A WSN as a sampling/reconstruction process.

Gaussian random fields.

Samples of 100 spatial point processes.

General setup and alternatives for clustering, aggregation and reconstruction.

The influence of sensor deployment using SKATER, coarse data set.

The influence of sensor deployment using SKATER, fine data set.

Reconstruction errors: three clustering/aggregation procedures (LEACH, SKATER and Pointwise, top, middle and bottom lines), four granularities (scales 5, 10, 15, 20, first to fourth column) and seven deployment point processes (−1000, −30, −15, 0, 5, 15 and 30 in different colors)

Relative errors using reconstruction by Kriging.

Clustering | Deployment | Granularity
| |||
---|---|---|---|---|---|

SKATER | 1.08 | 1.31 | 2.01 | ||

1.05 | 1.17 | 1.45 | 2.01 | ||

1.04 | 1.18 | 1.45 | 2.03 | ||

1.11 | 1.25 | 1.57 | 2.06 | ||

1.14 | 1.30 | 1.62 | 2.09 | ||

1.29 | 1.51 | 1.82 | 2.17 | ||

1.50 | 1.66 | 1.96 | 2.24 | ||

| |||||

LEACH | 1.72 | 1.90 | 2.16 | 2.30 | |

1.74 | 1.93 | 2.16 | 2.30 | ||

1.73 | 1.91 | 2.17 | 2.30 | ||

1.75 | 1.96 | 2.18 | 2.31 | ||

1.79 | 1.96 | 2.19 | 2.32 | ||

1.81 | 2.00 | 2.23 | 2.33 | ||

1.81 | 2.02 | 2.22 | 2.34 |

Relative errors using reconstruction by Voronoi.

Clustering | Deployment | Granularity
| |||
---|---|---|---|---|---|

SKATER | 1.49 | 1.63 | 1.90 | 2.46 | |

1.49 | 1.63 | 1.90 | 2.46 | ||

1.57 | 1.74 | 2.03 | 2.56 | ||

1.63 | 1.81 | 2.15 | 2.64 | ||

1.68 | 1.86 | 2.20 | 2.69 | ||

1.82 | 2.04 | 2.40 | 2.83 | ||

1.99 | 2.24 | 2.62 | 2.94 | ||

| |||||

LEACH | 2.22 | 2.43 | 2.70 | 2.85 | |

2.24 | 2.46 | 2.71 | 2.86 | ||

2.24 | 2.44 | 2.72 | 2.86 | ||

2.24 | 2.48 | 2.73 | 2.87 | ||

2.29 | 2.49 | 2.75 | 2.88 | ||

2.32 | 2.54 | 2.79 | 2.90 | ||

2.33 | 2.56 | 2.80 | 2.92 |