Keywords

Estimation is the process of inferring the value of an unknown given quantity of interest from noisy, direct or indirect, observations of such a quantity. Due to its great practical relevance, estimation has a long history and an enormous variety of applications in all fields of engineering and science. A certainly incomplete list of possible application domains of estimation includes the following: statistics (Bard 1974; Ghosh et al. 1997; Koch 1999; Lehmann and Casella 1998; Tsybakov 2009; Wertz 1978), telecommunication systems (Sage and Melsa 1971; Schonhoff and Giordano 2006; Snyder 1968; Van Trees 1971), signal and image processing (Barkat 2005; Biemond et al. 1983; Elliott et al. 2008; Itakura 1971; Kay 1993; Kim and Woods 1998; Levy 2008; Najim 2008; Poor 1994; Tuncer and Friedlander 2009; Wakita 1973; Woods and Radewan 1977), aerospace engineering (McGee and Schmidt 1985), tracking (Bar-Shalom and Fortmann 1988; Bar-Shalom et al. 2001, 2013; Blackman and Popoli 1999; Farina and Studer 1985, 1986), navigation (Dissanayake et al. 2001; Durrant-Whyte and Bailey 2006a,b; Farrell and Barth 1999; Grewal et al. 2001; Mullane et al. 2011; Schmidt 1966; Smith et al. 1986; Thrun et al. 2006), control systems (Anderson and Moore 1979; Athans 1971; Goodwin et al. 2005; Joseph and Tou 1961; Kalman 1960a; Maybeck 1979, 1982; Söderström 1994; Stengel 1994), econometrics (Aoki 1987; Pindyck and Roberts 1974; Zellner 1971), geophysics (e.g., seismic deconvolution) (Bayless and Brigham 1970; Flinn et al. 1967; Mendel 1977, 1983, 1990), oceanography (Evensen 1994a; Ghil and Malanotte-Rizzoli 1991), weather forecasting (Evensen 1994b, 2007; McGarty 1971), environmental engineering (Dochain and Vanrolleghem 2001; Heemink and Segers 2002; Nachazel 1993), demographic systems (Leibungudt et al. 1983), automotive systems (Barbarisi et al. 2006; Stephant et al. 2004), failure detection (Chen and Patton 1999; Mangoubi 1998; Willsky 1976), power systems (Abur and Gómez Espósito 2004; Debs and Larson 1970; Miller and Lewis 1971; Monticelli 1999; Toyoda et al. 1970), nuclear engineering (Robinson 1963; Roman et al. 1971; Sage and Masters 1967; Venerus and Bullock 1970), biomedical engineering (Bekey 1973; Snyder 1970; Stark 1968), pattern recognition (Andrews 1972; Ho and Agrawala 1968; Lainiotis 1972), social networks (Snijders et al. 2012), etc.

Chapter Organization

The rest of the chapter is organized as follows. Section “Historical Overview on Estimation” will provide a historical overview on estimation. The next section will discuss applications of estimation. Connections between estimation and information theories will be explored in the subsequent section. Finally, the section “Conclusions and Future Trends” will conclude the chapter by discussing future trends in estimation. An extensive list of references is also provided.

Historical Overview on Estimation

A possibly incomplete, list of the major achievements on estimation theory and applications is reported in Table 1. The entries of the table, sorted in chronological order, provide for each contribution the name of the inventor (or inventors), the date, and a short description with main bibliographical references.

Estimation, Survey on, Table 1 Major developments on estimation
Full size table

Probably the first important application of estimation dates back to the beginning of the nineteenth century whenever least-squares estimation (LSE), invented by Gauss in 1795 (Gauss 1995; Legendre 1810), was successfully exploited in astronomy for predicting planet orbits (Gauss 1806). Least-squares estimation follows a deterministic approach by minimizing the sum of squares of residuals defined as differences between observed data and model-predicted estimates. A subsequently introduced statistical approach is maximum likelihood estimation (MLE), popularized by R. A. Fisher between 1912 and 1922 (Fisher 1912, 1922, 1925). MLE consists of finding the estimate of the unknown quantity of interest as the value that maximizes the so-called likelihood function, defined as the conditional probability density function of the observed data given the quantity to be estimated. In intuitive terms, MLE maximizes the agreement of the estimate with the observed data. Whenever the observation noise is assumed Gaussian (Kim and Shevlyakov 2008; Park et al. 2013), MLE coincides with LSE.

While estimation problems had been addressed for several centuries, it was not until the 1940s that a systematic theory of estimation started to be established, mainly relying on the foundations of the modern theory of probability (Kolmogorov 1933). Actually, the roots of probability theory can be traced back to the calculus of combinatorics (the Stomachion puzzle invented by Archimedes (Netz and Noel 2011)) in the third century B.C. and to the gambling theory (work of Cardano, Pascal, de Fermat, Huygens) in the sixteenth–seventeenth centuries.

Differently from the previous work devoted to the estimation of constant parameters, in the period 1940–1960 the attention was mainly shifted toward the estimation of signals. In particular, Wiener in 1940 (Wiener 1949) and Kolmogorov in 1941 (Kolmogorov 1941) formulated and solved the problem of linear minimum mean-square error (MMSE) estimation of continuous-time and, respectively, discrete-time stationary random signals. In the late 1940s and in the 1950s, Wiener-Kolmogorov’s theory was extended and generalized in many directions exploiting both time-domain and frequency-domain approaches. At the beginning of the 1960s Rudolf E. Kálmán made pioneering contributions to estimation by providing the mathematical foundations of the modern theory based on state-variable representations. In particular, Kálmán solved the linear MMSE filtering and prediction problems both in discrete-time (Kalman 1960b) and in continuous-time (Kalman and Bucy 1961); the resulting optimal estimator was named after him, Kalman filter (KF). As a further contribution, Kalman also singled out the key technical conditions, i.e., observability and controllability, for which the resulting optimal estimator turns out to be stable. Kalman’s work went well beyond earlier contributions of A. Kolmogorov, N. Wiener, and their followers (“frequency-domain” approach) by means of a general state-space approach. From the theoretical viewpoint, the KF is an optimal estimator, in a wide sense, of the state of a linear dynamical system from noisy measurements; specifically it is the optimal MMSE estimator in the Gaussian case (e.g., for normally distributed noises and initial state) and the best linear unbiased estimator irrespective of the noise and initial state distributions. From the practical viewpoint, the KF enjoys the desirable properties of being linear and acting recursively, step-by-step, on a noise-contaminated data stream. This allows for cheap real-time implementation on digital computers. Further, the universality of “state-variable representations” allows almost any estimation problem to be included in the KF framework. For these reasons, the KF is, and continues to be, an extremely effective and easy-to-implement tool for a great variety of practical tasks, e.g., to detect signals in noise or to estimate unmeasurable quantities from accessible observables. Due to the generality of the state estimation problem, which actually encompasses parameter and signal estimation as special cases, the literature on estimation since 1960 till today has been mostly concentrated on extensions and generalizations of Kalman’s work in several directions. Considerable efforts, motivated by the ubiquitous presence of nonlinearities in practical estimation problems, have been devoted to nonlinear and/or non-Gaussian filtering, starting from the seminal papers of Stratonovich (1959, 1960) and Kushner (1962, 1967) for continuous-time systems, Ho and Lee (1964) for discrete-time systems, and Jazwinski (1966) for continuous-time systems with discrete-time observations. In these papers, state estimation is cast in a probabilistic (Bayesian) framework as the problem of evolving in time the state conditional probability density given observations (Jazwinski 1970). Work on nonlinear filtering has produced over the years several nonlinear state estimation algorithms, e.g., the extended Kalman filter (EKF) (Schmidt 1966), the unscented Kalman filter (UKF) (Julier and Uhlmann 2004; Julier et al. 1995), the Gaussian-sum filter (Alspach and Sorenson 1972; Sorenson and Alspach 1970, 1971), the sequential Monte Carlo (also called particle) filter (SMCF) (Doucet et al. 2001; Gordon et al. 1993; Ristic et al. 2004), and the ensemble Kalman filter (EnKF) (Evensen 1994a,b, 2007) which have been, and are still now, successfully employed in various application domains. In particular, the SMCF and EnKF are stochastic simulation algorithms taking inspiration from the work in the 1940s on the Monte Carlo method (Metropolis and Ulam 1949) which has recently got renewed interest thanks to the tremendous advances in computing technology. A thorough review on nonlinear filtering can be found, e.g., in Daum (2005) and Crisan and Rozovskii (2011).

Other interesting areas of investigation have concerned smoothing (Bryson and Frazier 1963), robust filtering for systems subject to modeling uncertainties (Poor and Looze 1981), and state estimation for infinite-dimensional (i.e., distributed parameter and/or delay) systems (Balakrishnan and Lions 1967). Further, a lot of attention has been devoted to the implementation of the KF, specifically square-root filtering (Potter and Stern 1963) for improved numerical robustness and fast KF algorithms (Kailath 1973; Morf et al. 1974) for enhancing computational efficiency. Worth of mention is the work over the years on theoretical bounds on the estimation performance originated from the seminal papers of Rao (1945) and Cramér (1946) on the lower bound of the MSE for parameter estimation and subsequently extended in Tichavsky et al. (1998) to nonlinear filtering and in Hernandez et al. (2006) to more realistic estimation problems with possible missed and/or false measurements. An extensive review of this work on Bayesian bounds for estimation, nonlinear filtering, and tracking can be found in van Trees and Bell (2007). A brief review of the earlier (until 1974) state of art in estimation can be found in Lainiotis (1974).

Applications

Astronomy

The problem of making estimates and predictions on the basis of noisy observations originally attracted the attention many centuries ago in the field of astronomy. In particular, the first attempt to provide an optimal estimate, i.e., such that a certain measure of the estimation error be minimized, was due to Galileo Galilei that, in his Dialogue on the Two World Chief Systems (1632) (Galilei 1632), suggested, as a possible criterion for estimating the position of Tycho Brahe’s supernova, the estimate that required the “minimum amendments and smallest corrections” to the data. Later, C. F. Gauss mathematically specified this criterion by introducing in 1795 the least-squares method (Gauss 1806, 1995; Legendre 1810) which was successfully applied in 1801 to predict the location of the asteroid Ceres. This asteroid, originally discovered by the Italian astronomer Giuseppe Piazzi on January 1, 1801, and then lost in the glare of the sun, was in fact recovered 1 year later by the Hungarian astronomer F. X. von Zach exploiting the least-squares predictions of Ceres’ position provided by Gauss.

Statistics

Starting from the work of Fisher in the 1920s (Fisher 1912, 1922, 1925), maximum likelihood estimation has been extensively employed in statistics for estimating the parameters of statistical models (Bard 1974; Ghosh et al. 1997; Koch 1999; Lehmann and Casella 1998; Tsybakov 2009; Wertz 1978).

Telecommunications and Signal/Image Processing

Wiener-Kolmogorov’s theory on signal estimation, developed in the period 1940–1960 and originally conceived by Wiener during the Second World War for predicting aircraft trajectories in order to direct the antiaircraft fire, subsequently originated many applications in telecommunications and signal/image processing (Barkat 2005; Biemond et al. 1983; Elliott et al. 2008; Itakura 1971; Kay 1993; Kim and Woods 1998; Levy 2008; Najim 2008; Poor 1994; Tuncer and Friedlander 2009; Van Trees 1971; Wakita 1973; Woods and Radewan 1977). For instance, Wiener filters have been successfully applied to linear prediction, acoustic echo cancellation, signal restoration, and image/video de-noising. But it was the discovery of the Kalman filter in 1960 that revolutionized estimation by providing an effective and powerful tool for the solution of any, static or dynamic, stationary or adaptive, linear estimation problem. A recently conducted, and probably non-exhaustive, search has detected the presence of over 16,000 patents related to the “Kalman filter,” spreading over all areas of engineering and over a period of more than 50 years. What is astonishing is that even nowadays, more than 50 years after its discovery, one can see the continuous appearance of lots of new patents and scientific papers presenting novel applications and/or novel extensions in many directions (e.g., to nonlinear filtering) of the KF. Since 1992 the number of patents registered every year and related to the KF follows an exponential law.

Space Navigation and Aerospace Applications

The first important application of the Kalman filter was in the NASA (National Aeronautic and Space Administration) space program. As reported in a NASA technical report (McGee and Schmidt 1985), Kalman presented his new ideas while visiting Stanley F. Schmidt at the NASA Ames Research Center in 1960, and this meeting stimulated the use of the KF during the Apollo program (in particular, in the guidance system of Saturn V during Apollo 11 flight to the Moon), and, furthermore, in the NASA Space Shuttle and in Navy submarines and unmanned aerospace vehicles and weapons, such as cruise missiles. Further, to cope with the nonlinearity of the space navigation problem and the small word length of the onboard computer, the extended Kalman filter for nonlinear systems and square-root filter implementations for enhanced numerical robustness have been developed as part of the NASA’s Apollo program. The aerospace field was only the first of a long and continuously expanding list of application domains where the Kalman filter and its nonlinear generalizations have found widespread and beneficial use.

Control Systems and System Identification

The work on Kalman filtering (Kalman 1960b; Kalman and Bucy 1961) had also a significant impact on control system design and implementation. In Kalman (1960a) duality between estimation and control was pointed out, in that for a certain class of control and estimation problems one can solve the control (estimation) problem for a given dynamical system by resorting to a corresponding estimation (control) problem for a suitably defined dual system. In particular, the Kalman filter has been shown to be dual of the linear-quadratic (LQ) regulator, and the two dual techniques constitute the linear-quadratic-Gaussian (LQG) (Joseph and Tou 1961) regulator. The latter consists of an LQ regulator feeding back in a linear way the state estimate provided by a Kalman filter, which can be independently designed in view of the separation principle. The KF as well as LSE and MLE techniques are also widely used in system identification (Ljung 1999; Söderström and Stoica 1989) for both parameter estimation and output prediction purposes.

Tracking

One of the major application areas for estimation is tracking (Bar-Shalom and Fortmann 1988; Bar-Shalom et al. 2001, 2013; Blackman and Popoli 1999; Farina and Studer 1985, 1986), i.e., the task of following the motion of moving objects (e.g., aircrafts, ships, ground vehicles, persons, animals) given noisy measurements of kinematic variables from remote sensors (e.g., radar, sonar, video cameras, wireless sensors, etc.). The development of the Wiener filter in the 1940s was actually motivated by radar tracking of aircraft for automatic control of antiaircraft guns. Such filters began to be used in the 1950s whenever computers were integrated with radar systems, and then in the 1960s more advanced and better performing Kalman filters came into use. Still today it can be said that the Kalman filter and its nonlinear generalizations (e.g., EKF (Schmidt 1966), UKF (Julier and Uhlmann 2004), and particle filter (Gordon et al. 1993)) represent the workhorses of tracking and sensor fusion. Tracking, however, is usually much more complicated than a simple state estimation problem due to the presence of false measurements (clutter) and multiple objects in the surveillance region of interest, as well as for the uncertainty about the origin of measurements. This requires to use, besides filtering algorithms, smart techniques for object detection as well as for association between detected objects and measurements. The problem of joint target tracking and classification has also been formulated as a hybrid state estimation problem and addressed in a number of papers (see, e.g., Smeth and Ristic (2004) and the references therein).

Econometrics

State and parameter estimation have been widely used in econometrics (Aoki 1987) for analyzing and/or predicting financial time series (e.g., stock prices, interest rates, unemployment rates, volatility etc.).

Geophysics

Wiener and Kalman filtering techniques are employed in reflection seismology for estimating the unknown earth reflectivity function given noisy measurements of the seismic wavelet’s echoes recorded by a geophone. This estimation problem, known as seismic deconvolution (Mendel 1977, 1983, 1990), has been successfully exploited, e.g., for oil exploration.

Data Assimilation for Weather Forecasting and Oceanography

Another interesting application of estimation theory is data assimilation (Ghil and Malanotte-Rizzoli 1991) which consists of incorporating noisy observations into a computer simulation model of a real system. Data assimilation has widespread use especially in weather forecasting and oceanography. A large-scale state-space model is typically obtained from the physical system model, expressed in terms of partial differential equations (PDEs), by means of a suitable spatial discretization technique so that data assimilation is cast into a state estimation problem. To deal with the huge dimensionality of the resulting state vector, appropriate filtering techniques with reduced computational load have been suitably developed (Evensen 2007).

Global Navigation Satellite Systems

Global Navigation Satellite Systems (GNSSs), such as GPS put into service in 1993 by the US Department of Defense, provide nowadays a commercially diffused technology exploited by millions of users all over the world for navigation purposes, wherein the Kalman filter plays a key role (Bar-Shalom et al. 2001). In fact, the Kalman filter not only is employed in the core of the GNSS to estimate the trajectories of all the satellites, the drifts and rates of all system clocks, and hundreds of parameters related to atmospheric propagation delay, but also any GNSS receiver uses a nonlinear Kalman filter, e.g., EKF, in order to estimate its own position and velocity along with the bias and drift of its own clock with respect to the GNSS time.

Robotic Navigation (SLAM)

Recursive state estimation is commonly employed in mobile robotics (Thrun et al. 2006) in order to on-line estimate the robot pose, location and velocity, and, sometimes, also the location and features of the surrounding objects in the environment exploiting measurements provided by onboard sensors; the overall joint estimation problem is referred to as SLAM (simultaneous localization and mapping) (Dissanayake et al. 2001; Durrant-Whyte and Bailey 2006a,b; Mullane et al. 2011; Smith et al. 1986; Thrun et al. 2006).

Automotive Systems

Several automotive applications of the Kalman filter, or of its nonlinear variants, are reported in the literature for the estimation of various quantities of interest that cannot be directly measured, e.g., roll angle, sideslip angle, road-tire forces, heading direction, vehicle mass, state of charge of the battery (Barbarisi et al. 2006), etc. In general, one of the major applications of state estimation is the development of virtual sensors, i.e., estimation algorithms for physical variables of interest, that cannot be directly measured for technical and/or economic reasons (Stephant et al. 2004).

Miscellaneous Applications

Other areas where estimation has found numerous applications include electric power systems (Abur and Gómez Espósito 2004; Debs and Larson 1970; Miller and Lewis 1971; Monticelli 1999; Toyoda et al. 1970), nuclear reactors (Robinson 1963; Roman et al. 1971; Sage and Masters 1967; Venerus and Bullock 1970), biomedical engineering (Bekey 1973; Snyder 1970; Stark 1968), pattern recognition (Andrews 1972; Ho and Agrawala 1968; Lainiotis 1972), and many others.

Connection Between Information and Estimation Theories

In this section, the link between two fundamental quantities in information theory and estimation theory, i.e., the mutual information (MI) and respectively the minimum mean-square error (MMSE), is investigated. In particular, a strikingly simple but very general relationship can be established between the MI of the input and the output of an additive Gaussian channel and the MMSE in estimating the input given the output, regardless of the input distribution (Guo et al. 2005). Although this functional relation holds for general settings of the Gaussian channel (e.g., both discrete-time and continuous-time, possibly vector, channels), in order to avoid the heavy mathematical preliminaries needed to treat rigorously the general problem, two simple scalar cases, a static and a (continuous-time) dynamic one, will be discussed just to highlight the main concept.

Static Scalar Case

Consider two scalar real-valued random variables, x and y, related by

$$\displaystyle{ y = \sqrt{\sigma }x + v }$$
(1)

where v, the measurement noise, is a standard Gaussian random variable independent of x and σ can be regarded as the gain in the output signal-to-noise ratio (SNR) due to the channel. By considering the MI between x and y as a function of σ, i.e., \(I(\sigma ) = I\left (x,\sqrt{\sigma }x + v\right )\), it can be shown that the following relation holds (Guo et al. 2005):

$$\displaystyle{ \dfrac{d} {d\sigma }I(\sigma ) = \dfrac{1} {2}\ E\left [\left (x -\hat{ x}(\sigma )\right )^{2}\right ] }$$
(2)

where \(\hat{x}(\sigma ) = E\left [x\vert \sqrt{\sigma }x + v\right ]\) is the minimum mean-square error estimate of x given y. Figure 1 displays the behavior of both MI, in natural logarithmic units of information (nats), and MMSE versus SNR.

Estimation, Survey on, Fig. 1
figure 56figure 56

MI and MMSE versus SNR

Full size image

As mentioned in Guo et al. (2005), the above information-estimation relationship (2) has found a number of applications, e.g., in nonlinear filtering, in multiuser detection, in power allocation over parallel Gaussian channels, in the proof of Shannon’s entropy power inequality and its generalizations, as well as in the treatment of the capacity region of several multiuser channels.

Linear Dynamic Continuous-Time Case

While in the static case the MI is assumed to be a function of the SNR, in the dynamic case it is of great interest to investigate the relationship between the MI and the MMSE as a function of time.

Consider the following first-order (scalar) linear Gaussian continuous-time stochastic dynamical system:

$$\displaystyle{ \begin{array}{rcl} dx_{t}& =&ax_{t}dt + dw_{t} \\ dy_{t}& =&\sqrt{\sigma }x_{t}\,dt + dv_{t} \end{array} }$$
(3)

where a is a real-valued constant while w t and v t are independent standard Brownian motion processes that represent the process and, respectively, measurement noises. Defining by \(x_{0}^{t}\stackrel{\bigtriangleup }{=}\{x_{s},0 \leq s \leq t\}\) the collection of all states up to time t and analogously \(y_{0}^{t}\stackrel{\bigtriangleup }{=}\left \{y_{s},0 \leq s \leq t\right \}\) for the channel outputs (i.e., measurements) and considering the MI between x0t and y0t as a function of time t, i.e., \(I(t) = I\left (x_{0}^{t},y_{0}^{t}\right )\), it can be shown that (Duncan 1970; Mayer-Wolf and Zakai 1983)

$$\displaystyle{ \dfrac{d} {dt}I(t) = \dfrac{\sigma } {2}\ E\left [\left (x_{t} -\hat{ x}_{t}\right )^{2}\right ] }$$
(4)

where \(\hat{x}_{t} = E\left [x_{t}\vert y_{0}^{t}\right ]\) is the minimum mean-square error estimate of the state x t given all the channel outputs up to time t, i.e., y0t. Figure 2 depicts the time behavior of both MI and MMSE for several values of σ and a = 1.

Estimation, Survey on, Fig. 2
figure 57figure 57

MI and MMSE for different values of σ ( − 10 dB, 0 dB, +10 dB) and a = 1

Full size image

Conclusions and Future Trends

Despite the long history of estimation and the huge amount of work on several theoretical and practical aspects of estimation, there is still a lot of research investigation to be done in several directions. Among the many new future trends, networked estimation and quantum estimation (briefly overviewed in the subsequent parts of this section) certainly deserve special attention due to the growing interest on wireless sensor networks and, respectively, quantum computing.

Networked Information Fusion and Estimation

Information or data fusion is about combining, or fusing, information or data from multiple sources to provide knowledge that is not evident from a single source (Bar-Shalom et al. 2013; Farina and Studer 1986). In 1986, an effort to standardize the terminology related to data fusion began and the JDL (Joint Directors of Laboratories) data fusion working group was established. The result of that effort was the conception of a process model for data fusion and a data fusion lexicon (Blasch et al. 2012; Hall and Llinas 1997). Information and data fusion are mainly supported by sensor networks which present the following advantages over a single sensor:

  • Can be deployed over wide regions

  • Provide diverse characteristics/viewing angles of the observed phenomenon

  • Are more robust to failures

  • Gather more data that, once fused, provide a more complete picture of the observed phenomenon

  • Allow better geographical coverage, i.e., wider area and less terrain obstructions.

Sensor network architectures can be centralized, hierarchical (with or without feedback), and distributed (peer-to-peer). Today’s trend for many monitoring and decision-making tasks is to exploit large-scale networks of low-cost and low-energy consumption devices with sensing, communication, and processing capabilities. For scalability issues, such networks should operate in a fully distributed (peer-to-peer) fashion, i.e., with no centralized coordination, so as to achieve in each node a global estimation/decision objective through localized processing only.

The attainment of this goal actually requires several issues to be addressed like:

  • Spatial and temporal sensor alignment

  • Scalable fusion

  • Robustness with respect to data incest (or double counting), i.e., repeated use of the same information

  • Handling data latency (e.g., out-of-sequence measurements/estimates)

  • Communication bandwidth limitations

In particular, to counteract data incest the so-called covariance intersection (Julier and Uhlmann 1997) robust fusion approach has been proposed to guarantee, at the price of some conservatism, consistency of the fused estimate when combining estimates from different nodes with unknown correlations. For scalable fusion, a consensus approach (Olfati-Saber et al. 2007) can be undertaken. This allows to carry out a global (i.e., over the whole network) processing task by iterating local processing steps among neighboring nodes.

Several consensus algorithms have been proposed for distributed parameter (Calafiore and Abrate 2009) or state (Alriksson and Rantzer 2006; Kamgarpour and Tomlin 2007; Olfati-Saber 2007; Stankovic et al. 2009; Xiao et al. 2005) estimation. Recently, Battistelli and Chisci (2014) introduced a generalized consensus on probability densities which opens up the possibility to perform in a fully distributed and scalable way any Bayesian estimation task over a sensor network. As by-products, this approach allowed to derive consensus Kalman filters with guaranteed stability under minimal requirements of system observability and network connectivity (Battistelli et al. 2011, 2012; Battistelli and Chisci 2014), consensus nonlinear filters (Battistelli et al. 2012), and a consensus CPHD filter for distributed multitarget tracking (Battistelli et al. 2013). Despite these interesting preliminary results, networked estimation is still a very active research area with many open problems related to energy efficiency, estimation performance optimality, robustness with respect to delays and/or data losses, etc.

Quantum Estimation

Quantum estimation theory consists of a generalization of the classical estimation theory in terms of quantum mechanics. As a matter of fact, the statistical theory can be seen as a particular case of the more general quantum theory (Helstrom 1969, 1976). Quantum mechanics presents practical applications in several fields of technology (Personick 1971) such as, the use of quantum number generators in place of the classical random number generators. Moreover, manipulating the energy states of the cesium atoms, it is possible to suppress the quantum noise levels and consequently improve the accuracy of atomic clocks. Quantum mechanics can also be exploited to solve optimization problems, giving sometimes optimization algorithms that are faster than conventional ones. For instance, McGeoch and Wang (2013) provided an experimental study of algorithms based on quantum annealing. Interestingly, the results of McGeoch and Wang (2013) have shown that this approach allows to obtain better solutions with respect to those found with conventional software solvers. In quantum mechanics, also the Kalman filter has found its proper form, as the quantum Kalman filter. In Iida et al. (2010) the quantum Kalman filter is applied to an optical cavity composed of mirrors and crystals inside, which interacts with a probe laser. In particular, a form of a quantum stochastic differential equation can be written for such a system so as to design the algorithm that updates the estimates of the system variables on the basis of the measurement outcome of the system.

Cross-References