Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> In an epidemiological study the regression slope between a response and predictor variable is underestimated when the predictor variable is measured imprecisely. Repeat measurements of the predictor in individuals in a subset of the study or in a separate study can be used to estimate a multiplicative factor to correct for this ‘regression dilution bias’. In applied statistics publications various methods have been used to estimate this correction factor. Here we compare six different estimation methods and explain how they fall into two categories, namely regression and correlation‐based methods. We provide new asymptotic variance formulae for the optimal correction factors in each category, when these are estimated from the repeat measurements subset alone, and show analytically and by simulation that the correlation method of choice gives uniformly lower variance. The simulations also show that, when the correction factor is not much greater than 1, this correlation method gives a correction factor which is closer to the true value than that from the best regression method on up to 80% of occasions. We also provide a variance formula for a modified correlation method which uses the standard deviation of the predictor variable in the main study; this shows further improved performance provided that the correction factor is not too extreme. A confidence interval for a corrected regression slope in an epidemiological study should reflect the imprecision of both the uncorrected slope and the estimated correction factor. We provide formulae for this and show that, particularly when the correction factor is large and the size of the subset of repeat measures is small, the effect of allowing for imprecision in the estimated correction factor can be substantial. <s> BIB001 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> We consider the problem of locating and orienting a network of ::: unattended sensors that has been deployed in a scene with unknown ::: sensor locations and orientation angles, when no 'anchor' nodes ::: are present. Many localization problems assume that some nodes ::: have known locations and propagate location information about ::: other nodes using triangulation procedures. In our formulation, ::: we do not require such anchor nodes, but instead assume prior ::: probability density function for the nominal locations of a subset ::: of the nodes. These nominal locations typically have high ::: uncertainty, on the order of tens of meters. The self-calibration ::: solution is obtained in two steps. Relative sensor locations are ::: estimated using noisy time-of-arrival and direction-of-arrival measurements of calibration source signals in the scene, ::: and absolute location calibration is obtained by incorporating ::: prior nominal location knowledge. We consider a Bayes approach to the calibration problem and compute accuracy bounds on the calibration procedure. A maximum a posteriori estimation algorithm is shown to ::: achieve the accuracy bound. Experiments using both synthetic ::: data and field measurements are presented. <s> BIB002 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> We describe an ad-hoc localization system for sensor networks and explain why traditional calibration methods are inadequate for this system. Building upon previous work, we frame calibration as a parameter estimation problem; we parameterize each device and choose the values of those parameters that optimize the overall system performance. This method reduces our average error from 74.6% without calibration to 10.1%. We propose ways to expand this technique to a method of autocalibration for localization as well as to other sensor network applications. <s> BIB003 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Numerous factors contribute to errors in sensor measurements. In order to be useful, any sensor device must be calibrated to adjust its accuracy against the expected measurement scale. In large-scale sensor networks, calibration will be an exceptionally difficult task since sensor nodes are often not easily accessible and manual device-by-device calibration is intractable. In this paper, we present a two-phase post-deployment calibration technique for large-scale, dense sensor deployments. In its first phase, the algorithm derives relative calibration relationships between pairs of co-located sensors, while in the second phase, it maximizes the consistency of the pair-wise calibration functions among groups of sensor nodes. The key idea in the first phase is to use temporal correlation of signals received at neighboring sensors when the signals are highly correlated (i.e. sensors are observing the same phenomenon) to derive the function relating their bias in amplitude. We formulate the second phase as an optimization problem and present an algorithm suitable for localized implementation. We evaluate the performance of the first phase of the algorithm using empirical and simulated data. <s> BIB004 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Calibration is the process of mapping raw sensor readings into corrected values by identifying and correcting systematic bias. Calibration is important from both off-line and on-line perspectives. Major objectives of calibration procedure include accuracy, resiliency against random errors, ability to be applied in various scenarios, and to address a variety of error models. In addition, a compact mapping function is attractive in terms of both storage and robustness. We start by introducing the nonparametric statistical approach for conducting off-line calibration. After that, we present the non-parametric statistical percentile method for establishing the confidence interval for a particular mapping function. Furthermore, we propose the first model-based on-line procedure for calibration. The calibration problem is formulated as an instance of nonlinear function minimization and solved using the standard conjugate gradient approach. A number of trade-offs between the effectiveness of calibration and noise level, latency, size of network and the complexity of phenomena are analyzed in a quantitative way. As a demonstration example, we use a system consisting of photovoltaic optical sensors. <s> BIB005 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Sensornet systems research is being conducted with various applications and deployment scenarios in mind. In many of these scenarios, the presumption is that the sensornet will be deployed and managed by users who do not have a background in computer science. In this paper we describe the "tiny application sensor kit" (TASK), a system we have designed for use by end-users with minimal sensornet sophistication. We describe the requirements that guided our design, the architecture of the system and results from initial deployments. Based on our experience to date we present preliminary design principles and research challenges that arise in delivering sensornet research to end users. <s> BIB006 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Rapidly deployable sensor networks are portable, reusable, and can take advantage of a human user in the field attending to the deployment. Unfortunately, even small disruptions or problems in collected data must be addressed quickly, as the overall quantity of data gathered is small relative to longterm deployments. In this paper we describe a procedure for calibration and a system for online fault remediation. Care in the calibration process for ion selective electrodes used for water quality assists interpretation of the data. Scientists will have more confidence in the data obtained from a rapid deployment if in-field users can detect and compensate for problems as they occur. We have designed and implemented a tool for use in the field to detect potential faults and provide actions to remedy or validate the faulty data. In January of 2006 we deployed 48 sensors over a period of 12 days in Bangladesh in order to aid in validating a hypothesis on the mass presence of arsenic in the groundwater. Our system is based on the the approximately 25,000 measurements we collected. <s> BIB007 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> This paper proposes a system for activity recognition using multi-sensor fusion. In this system, four sensors are attached to the waist, chest, thigh, and side of the body. In the study we present two solutions for factors that affect the activity recognition accuracy: the calibration drift and the sensor orientation changing. The datasets used to evaluate this system were collected from 8 subjects who were asked to perform 8 scripted normal activities of daily living (ADL), three times each. The Naive Bayes classifier using multi-sensor fusion is adopted and achieves 70.88%-97.66% recognition accuracies for 1–4 sensors. <s> BIB008 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> We present a long-term and cross-sectional study of a vibration-based water flow rate monitoring system in practical environments and scenarios. In our earlier research, we proved that a water flow monitoring system with vibration sensors is feasible by deploying and evaluating it in a small-scale laboratory setting. To validate the proposed system, the system was deployed in existing environments—two houses and a public restroom—and in two different laboratory test settings. With the collected data, we first demonstrate various aspects of the system's performance, including sensing stability, sensor node lifetime, the stability of autonomous sensor calibration, time to adaptation, and deployment complexity. We then discuss the practical challenges and lessons from the full-scale deployments. The evaluation results show that our water monitoring solution is a practical, quick-to-deploy system with a less than 5p average flow estimation error. <s> BIB009 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> This paper presents a novel low-cost instrument that uses a sensor based on conductivity changes of heated tungstic oxide, which is capable of accurately measuring ambient concentrations of ozone. A combination of temperature steps and air flow-rate steps is used to continually reset and re-zero the sensor. A two-stage calibration procedure is presented, in which a nonlinear transformation converts sensor resistance to a signal linear in ozone concentration, then a linear correlation is used to align the calibration with a reference instrument. The required calibration functions specific for the sensor, and control system for air flow rate and sensor temperature, are housed with the sensor in a compact, simple-to-exchange assembly. The instrument can be operated on solar power and uses cell phone technology to enable monitoring in remote locations. Data from field trials are presented here to demonstrate that both the accuracy and the stability of the instrument over periods of months are within a few parts-per-billion by volume. We show that common failure modes can be detected through measurement of signals available from the instrument. The combination of long-term stability, self-diagnosis, and simple, inexpensive repair means that the cost of operation and calibration of the instruments is significantly reduced in comparison with traditional reference instrumentation. These instruments enable the economical construction and operation of ozone monitoring networks of accuracy, time resolution and spatial density sufficient to resolve the local gradients that are characteristic of urban air pollution. <s> BIB010 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Technological advances in nanotechnology enabled the use of microelectromechanical systems (MEMS) in various application areas. With the integration of various sensor devices into MEMS, autonomously calibrating these sensors become a major research problem. When performing calibration on real-world embedded sensor network deployments, random errors due to internal and external factors alter the calibration parameters and eventually effect the calibration quality in a negative way. Therefore, during autonomous calibration, calibration paths which has low cost and low error values are preferable. To tackle the calibration problem on embedded wireless sensor networks, we present an energy efficient and minimum error calibration model, and also prove that due to random errors the problem turns into an NP-complete problem. To the best of our knowledge this is the first time a formal proof is presented on the complexity of an iterative calibration based problem when random errors are present in the measurements. We also conducted heuristic tests using genetic algorithm to solve the optimization version of the problem, on various graphs. The NP-completeness result also reveals that more research is needed to examine the complexity of calibration in a more general framework in real-world sensor network deployments. <s> BIB011 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> We introduce a control algorithm for automated calibration of sensor networks and fault compensation that reduces overall system error. Our approach uses a novel combination of models (for the spatial phenomena and sensor faults) and data-driven methods, and it works under novel conditions of dynamic sensor noise and bias in which the phenomenon of interest is both time- and space-variant. Our algorithm combines noise compensation using the EM (expectation-maximization) algorithm and a gossip-based protocol for sharing calibration information. We demonstrate its effectiveness in simulations that model real-world sensor fault scenarios. <s> BIB012 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Internet-of-Things (IoT) is a concept referring to interconnected people and objects and smart city is one of the many applications of IoT. Wireless Sensor Network (WSN) is a specific technology that helps to create “Smart Cities”. It aims at creating a distributed network of intelligent sensor nodes which can measure various parameters for efficient management of the city. The data thus collected through a range of sensors is processed and is delivered wirelessly in real-time to the citizens or the appropriate authorities. Since the application framework for smart city application is huge, it would require a large number of different types of sensors for its implementation and the project could be viable only if we use low resolution, low precision but inexpensive sensors. The sensors in sensor network can suffer from random or systematic errors. Most common problem with inexpensive sensors used in WSNs for smart city applications is of drift and bias. They can be calibrated at the time of deployment, but they develop drift, which is the slow change in the reading of sensor from actual value as time progresses. In this paper we have proposed a framework to automatically detect and correct the drift of the sensor nodes to keep the WSN usable. Kriging based interpolation of the sensor readings of neighboring sensors is used to predict actual value at the sensor node and the measured drift is then kalman filtered to get correct drift estimates. We have demonstrated the results of this algorithm on real sensor data obtained from Intel Research Berkeley Laboratory deployment and shown that our system is able to detect and correct smooth drift and bias generated in the sensors. We have also shown that our system is robust with respect to the number of sensor nodes drifting and significantly outperforms the traditional averaging based interpolation methods. <s> BIB013 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Frequent sensor calibration is essential in sensor networks with low-cost sensors. We exploit the fact that temporally and spatially close measurements of different sensors measuring the same phenomenon are similar. Hence, when calibrating a sensor, we adjust its calibration parameters to minimize the differences between co-located measurements of previously calibrated sensors. In turn, freshly calibrated sensors can now be used to calibrate other sensors in the network, referred to as multi-hop calibration. We are the first to study multi-hop calibration with respect to a reference signal (micro-calibration) in detail. We show that ordinary least squares regression---commonly used to calibrate noisy sensors---suffers from significant error accumulation over multiple hops. In this paper, we propose a novel multi-hop calibration algorithm using geometric mean regression, which (i) highly reduces error propagation in the network, (ii) distinctly outperforms ordinary least squares in the multi-hop scenario, and (iii) requires considerably fewer ground truth measurements compared to existing network calibration algorithms. The proposed algorithm is especially valuable when calibrating large networks of heterogeneous sensors with different noise characteristics. We provide theoretical justifications for our claims. Then, we conduct a detailed analysis with artificial data to study calibration accuracy under various settings and to identify different error sources. Finally, we use our algorithm to accurately calibrate 13 million temperature, ground ozone (O3), and carbon monoxide (CO) measurements gathered by our mobile air pollution monitoring network. <s> BIB014 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Abstract The performances of several field calibration methods for low-cost sensors, including linear/multi linear regression and supervised learning techniques are compared. A cluster of ozone, nitrogen dioxide, nitrogen monoxide, carbon monoxide and carbon dioxide sensors was operated. The sensors were either of metal oxide or electrochemical type or based on miniaturized infra-red cell. For each method, a two-week calibration was carried out at a semi-rural site against reference measurements. Subsequently, the accuracy of the predicted values was evaluated for about five months using a few indicators and techniques: orthogonal regression, target diagram, measurement uncertainty and drifts over time of sensor predictions. The study assessed if the sensors were could reach the Data Quality Objective (DQOs) of the European Air Quality Directive for indicative methods (between 25 and 30% of uncertainty for O 3 and NO 2 ). In this study it appears that O 3 may be calibrated using simple regression techniques while for NO 2 a better agreement between sensors and reference measurements was reached using supervised learning techniques. The hourly O 3 DQO was met while it was unlikely that NO 2 hourly one could be met. This was likely caused by the low NO 2 levels correlated with high O 3 levels that are typical of semi-rural site where the measurements of this study took place. <s> BIB015 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Abstract Sensors play an important role in guiding building systems to achieve desired operation and efficiency. However, sensors are subject to continuous degradation and failures over time. Although a periodical calibration is needed, it is exceptionally difficult and/or impractical to many sensors with a conventional manual approach. Uncalibrated problematic sensors could significantly compromise the systems' performance and lead to unintended loss of energy efficiency in buildings. We propose a methodology, termed virtual in-situ calibration, to solve this critical issue. It is developed by mathematically extracting the characteristics of essential aspects involved in a calibration, including the environment assessment, benchmark establishment, and uncertainty quantification. A case study of a supply air temperature sensor in rooftop units illustrates the implementation process; the erratic uncertainty is reduced from ± 19.2 °C to ± 0.7 °C after the virtual in-situ calibration. The calibration method can be implemented online to significantly improve the reliability of sensor networks in buildings. <s> BIB016 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> A novel distributed algorithm for blind macro-calibration of large sensor networks is introduced. The algorithm is in the form of a system of gradient-type recursions for estimating parameters of local sensor calibration functions. The method does not require any fusion center. The convergence analysis is based on diagonal dominance of the dynamical systems with block matrices. It is proved that the asymptotic consensus is achieved for all the equivalent sensor gains and offsets (in the mean square sense and with probability one) in lossy sensor networks with possible communication outages and additive communication noise. An illustrative simulation example is provided. <s> BIB017 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Over the past few years, many low-cost pollution sensors have been integrated into measurement platforms for air quality monitoring. However, using these sensors is challenging: concentrations of toxic substances in ambient air often lie at sensors' sensitivity boundaries, environmental conditions affect the sensor measurements, and the sensors often suffer from poor selectivity, i.e. are cross-sensitive to multiple pollutants. Datasheet information on these effects is scarce or may not cover deployment conditions. Consequently the sensors need to undergo extensive pre-deployment testing to examine their feasibility for a given application and to find the optimal measurement setup that allows accurate data collection and calibration. In this work, we propose a novel method to conduct in-field testing of low-cost sensors. The algorithm proposed is based on multiple least-squares and leverages the physical variation of urban air pollution to quantify the amount of explained and unexplained sensor signal. We verify (i) whether a sensor is feasible for air quality monitoring in a given environment and underpin our analysis with positive and negative examples of sensors available on the market, (ii) model sensor cross-sensitivities to interfering gases and environmental effects and (iii) compute the optimal sensor array and its calibration parameters for stable and accurate sensor measurements over long time periods in a given environment. Finally, we provide an experimental evaluation of our approach using over 9 million measurements of various low-cost sensors collected in an urban area. Based on the results from our testing methodology we propose an optimized sensor array setup. Further, we show---compared to a state-of-the-art calibration technique---a significantly lower calibration error with better long-time stability of the calibration parameters. <s> BIB018 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Abstract In this work the performances of several field calibration methods for low-cost sensors, including linear/multi linear regression and supervised learning techniques, are compared. A cluster of either metal oxide or electrochemical sensors for nitrogen monoxide and carbon monoxide together with miniaturized infra-red carbon dioxide sensors was operated. Calibration was carried out during the two first weeks of evaluation against reference measurements. The accuracy of each regression method was evaluated on a five months field experiment at a semi-rural site using different indicators and techniques: orthogonal regression, target diagram, measurement uncertainty and drifts over time of sensor predictions. In addition to the analyses for ozone and nitrogen oxide already published in Part A [1], this work assessed if carbon monoxide sensors can reach the Data Quality Objective (DQOs) of 25% of uncertainty set in the European Air Quality Directive for indicative methods. As for ozone and nitrogen oxide, it was found for NO, CO and CO 2 that the best agreement between sensors and reference measurements was observed for supervised learning techniques compared to linear and multilinear regression. <s> BIB019 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Position from Ground-Truth Nodes <s> Urban air pollution monitoring with mobile, portable, low-cost sensors has attracted increasing research interest for their wide spatial coverage and affordable expenses to the general public. However, low-cost air quality sensors not only drift over time but also suffer from cross-sensitivities and dependency on meteorological effects. Therefore calibration of measurements from low-cost sensors is indispensable to guarantee data accuracy and consistency to be fit for quantitative studies on air pollution. In this work we propose sensor array network calibration (SCAN), a multi-hop calibration technique for dependent low-cost sensors. SCAN is applicable to sets of co-located, heterogeneous sensors, known as sensor arrays, to compensate for cross-sensitivities and dependencies on meteorological influences. SCAN minimizes error accumulation over multiple hops of sensor arrays, which is unattainable with existing multi-hop calibration techniques. We formulate SCAN as a novel constrained least-squares regression and provide a closed-form expression of its regression parameters. We theoretically prove that SCAN is free from regression dilution even in presence of measurement noise. In-depth simulations demonstrate that SCAN outperforms various calibration techniques. Evaluations on two real-world low-cost air pollution sensor datasets comprising 66 million samples collected over three years show that SCAN yields 16% to 60% lower error than state-of-the-art calibration techniques. <s> BIB020
Depending on the relative position of the uncalibrated node with respect to the reference model, calibration can be performed between collocated nodes, iteratively hop-by-hop (multi-hop) from the ground-truth, or with the local presence of a reference model (model-based), as illustrated in Figure 6 . Following the definition of Saukh et al. BIB014 , a process of interest P is a continuous measurable signal η:T × L → D with domain T ⊆ R + and L ⊆ R × R, where T represents time and L location. Let us assume two sensors s 1 and s 2 that take samples at two times and locations such as (t,l) and (t',l'); then, the process P is bounded in time and space: with γ being a monotonically non-decreasing function. As the authors note, the slower the function γ grows, the more correlated will be the data of the two sensors. Saukh et al. BIB014 defined a rendezvous as the set of spatially and temporally close pairs of measurements Φ s1,s2 between sensors s 1 and s 2 within a time interval T as where {x s 1 }∈ D s 1 , {x s 2 }∈ D s 2 , and t and d are temporal and spatial constraints on the required closeness of measurements, respectively. As it can be observed, the rendezvous defines how close two sensors have to be to calibrate the uncalibrated sensor with respect to the reference sensor and the temporal synchronization level of the samples from both sensors. Collocated calibration refers to the situation where a node is calibrated using interaction with a close neighbor node (collocated) that can sense the same phenomenon and provide reference measurements. In terms of the process P, collocated calibration involves a sensor at place (t,l) that gives ground-truth values, a sensor at place (t',l') that is going to be calibrated, and equation BIB010 to be fulfilled with closed temporal and spatial constraints. This approach, then, is associated with non-blind calibration where a node is exposed to the true values available from a reference ground-truth node BIB006 BIB015 BIB016 BIB018 BIB019 BIB009 . In iterative, or multi-hop, calibration, a node is calibrated using a set of nodes already calibrated, which are not necessarily ground-truth nodes. The name of multi-hop comes from the case in which the uncalibrated sensor is h-hops from the ground-truth node, where each hop has iteratively been calibrated. For illustrating the process, let us define the calibration graph ( Figure 6 ) as a directed graph G=(V, E), where V={1, 2, ..., N} is the set of nodes, and E⊆V×V is the set of edges. The edge e i j between sensor node s i and sensor node s j means that s j is calibrated using node s i as reference node. In this case, an uncalibrated sensor can be calibrated using an h-hop path. An example ( Figure 6 ) is a path s 1 → s 2 → s 3 → s 4 → s 6 , with s 1 being a ground-truth node. Then, s 2 would be calibrated using node s 1 , node s 3 using node s 2 , s 4 using node s 3 , and s 6 using node s 4 . Collocated calibration would be the case of having a 0-hop calibration. Another special case is when h=1, called singlehop. In this situation, a node is calibrated using already existing neighboring calibrated sensor nodes BIB007 . In a multi-hop calibration approach, there can be many paths to calibrate an uncalibrated sensor. Thus, the objective is to minimize the errors committed as a result of the calibration process BIB004 BIB014 BIB020 . Different paths contribute differently to the post-calibration skew of the sensors, where skew is defined as the difference between the calibrated value of sensor j and the actual value BIB011 . The goal in multi-hop, or iterative, calibration is to study how the errors propagate along the network path and find calibration paths with minimum skews. The skew depends on the length of the paths and on the individual calibration errors of the sensors that the path traverses. As an example, if sensor s 1 is calibrated against a reference sensor and produces measured data in the range [x 1 -1 ,x 1 + 1 ] and then a sensor s 2 is calibrated against s 1 , reporting measured data in the range [x 2 -2 ,x 2 + 2 ], the postcalibration skew of sensor s 2 will be at most [ 1 + 2 ]. Akan BIB011 proved that the skew is at most the sum of the absolute errors along the calibrating path. In a more recent work, Saukh et al. BIB014 investigated whether linear fitting models suppress multi-hop error propagation in a large-scale mobile sensor network. Saukh et al. BIB014 showed that ordinary least squares regression suffers from regression 12 A pair-wise calibration is also used in this context. Pair-wise implies a rendezvous between two sensor nodes in the calibration. If one of them is a ground-truth node, we call the pair-wise a 0-hop, or collocated, calibration. If one of them is an already calibrated node, we call the pair-wise a 1-hop, or single-hop, calibration. It was common in the early literature on calibration in which multi-hop was not well defined to call pair-wise any rendezvous, regardless of whether it was an already calibrated node or a ground-truth node. attenuation or regression dilution BIB001 . Regression dilution is the biasing of the regression slope towards zero caused by errors in the independent variable. This effect increases as the number of hops increases. The reason is that at every hop, the independent variable x of the linear fitting adds an error to the model. For example, let us have a path s 1 →s 2 →s 3 , where s 1 is a ground-truth node; then, assuming a simple linear model, with 2 and 3 normal distributed errors, x 1 is the ground-truth data, and x 2 and x 3 are respectively the measured sensor data and are defined according to equation (14) in subsection 5.4. In this case, at each hop, the calibration coefficient is decreased proportionally by the variance of the error given by that hop (regression dilution) BIB001 . The solution is compensating the bias in the slope estimation if the variance of the error of the sensor node is known. The authors of BIB014 mentioned the difficulty of this estimation due to drift in real sensors and changing environmental conditions. As an alternative solution, BIB014 proposed to use other fitting techniques that do not suffer from regression dilution, such as geometric mean regression (GMR). They tested the multi-hop calibration in a real network of O 3 and CO sensors, showing that ordinary least regression (OLR) produces regression dilution, that GMR has very low hop-by-hop error propagation, and that the calibration quality is independent of the calibration error of its calibration parent. Maag et al. BIB020 proposed sensor array network calibration (SCAN), a multi-hop micro-calibration scheme for mobile sensor arrays. SCAN minimizes the accumulated error over multiple hops of calibrated sensors. SCAN formulates a constrained least squares regression optimization problem reducing, and even eliminating under certain conditions, the regression dilution effect. In model-based calibration, a node is calibrated using a set of ground-truth nodes that are not in the vicinity of the noncalibrated node. In this case, the ground-truth values allow by means of a mathematical model to produce a reference value at the position of the non-calibrated node. The values produced by this model now act as ground-truth reference data for the estimation of the calibration parameters BIB005 BIB008 BIB002 BIB012 BIB013 . The accuracy of model-based calibration is less than that of the collocated calibration, since the values estimated in the noncalibrated node are produced by a model that introduces an error, even if this model uses ground-truth data. An example is a source of light. The intensity of the light at source location d can be obtained by an exponential decaying function BIB005 that depends on the intensity of the light source and distance (see equation BIB017 . In another example, Moses et al. BIB002 considered the problem of location and orientation of sensors by measuring the time of arrival (TOA) and direction of arrival (DOA) of the signal emitted by that source. A third example is given by Whitehouse et al. BIB003 , who used received signal strength information (RSSI) in fusion with acoustic time of flight (TOF) as a method of auto-calibration for localization services. Some distances between sensors are known a priori, and this allows taking advantage of anchor nodes or pre-calibrated nodes.