A Review of Urban Air Pollution Monitoring and Exposure Assessment Methods <s> Data Driven Spatial Prediction <s> The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries---stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). ::: Basis pursuit (BP) is a principle for decomposing a signal into an "optimal"' superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, abstract harmonic analysis, total variation denoising, and multiscale edge denoising. ::: BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear and quadratic programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver. <s> BIB001 </s> A Review of Urban Air Pollution Monitoring and Exposure Assessment Methods <s> Data Driven Spatial Prediction <s> Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces <s> BIB002 </s> A Review of Urban Air Pollution Monitoring and Exposure Assessment Methods <s> Data Driven Spatial Prediction <s> This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f. <s> BIB003 </s> A Review of Urban Air Pollution Monitoring and Exposure Assessment Methods <s> Data Driven Spatial Prediction <s> Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, sub-Nyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can complement conventional CS methods based on linear programming or greedy algorithms. We perform approximate Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast computation is obtained by reducing the size of the graphical model with sparse encoding matrices. To decode a length-N signal containing K large coefficients, our CS-BP decoding algorithm uses O(Klog(N)) measurements and O(Nlog^2(N)) computation. Finally, although we focus on a two-state mixture Gaussian model, CS-BP is easily adapted to other signal models. <s> BIB004 </s> A Review of Urban Air Pollution Monitoring and Exposure Assessment Methods <s> Data Driven Spatial Prediction <s> In this paper, we forecast the reading of an air quality monitoring station over the next 48 hours, using a data-driven method that considers current meteorological data, weather forecasts, and air quality data of the station and that of other stations within a few hundred kilometers. Our predictive model is comprised of four major components: 1) a linear regression-based temporal predictor to model the local factors of air quality, 2) a neural network-based spatial predictor to model global factors, 3) a dynamic aggregator combining the predictions of the spatial and temporal predictors according to meteorological data, and 4) an inflection predictor to capture sudden changes in air quality. We evaluate our model with data from 43 cities in China, surpassing the results of multiple baseline methods. We have deployed a system with the Chinese Ministry of Environmental Protection, providing 48-hour fine-grained air quality forecasts for four major Chinese cities every hour. The forecast function is also enabled on Microsoft Bing Map and MS cloud platform Azure. Our technology is general and can be applied globally for other cities. <s> BIB005 </s> A Review of Urban Air Pollution Monitoring and Exposure Assessment Methods <s> Data Driven Spatial Prediction <s> The Internet-of-Things (IoT) is the key enabling technology for transforming current urban environments into so-called Smart Cities. One of the goals behind making cities smarter is to provide a healthy environment that improves the citizens' quality of life and wellbeing. In this work, we introduce a novel data aggregation mechanism tailored to the application of large-scale air pollution monitoring with IoT devices. Our design exploits the intra- and inter-source correlations among air-pollution data using the framework of compressed sensing with side information. The proposed method delivers significant improvements in the data reconstruction quality with respect to the state of the art, even in the presence of noise when measuring and transmitting the data. <s> BIB006 </s> A Review of Urban Air Pollution Monitoring and Exposure Assessment Methods <s> Data Driven Spatial Prediction <s> Large-scale data collection by means of wireless sensor network and Internet-of-Things technology poses various challenges in view of the limitations in transmission, computation, and energy resources of the associated wireless devices. Compressive data gathering based on compressed sensing has been proven a well-suited solution to the problem. Existing designs exploit the spatiotemporal correlations among data collected by a specific sensing modality. However, many applications, such as environmental monitoring, involve collecting heterogeneous data that are intrinsically correlated. In this paper, we propose to leverage the correlation from multiple heterogeneous signals when recovering the data from compressive measurements. To this end, we propose a novel recovery algorithm—built upon belief-propagation principles—that leverages correlated information from multiple heterogeneous signals. To efficiently capture the statistical dependencies among diverse sensor data, the proposed algorithm uses the statistical model of copula functions. Experiments with heterogeneous air-pollution sensor measurements show that the proposed design provides significant performance improvements against the state-of-the-art compressive data gathering and recovery schemes that use classical compressed sensing, compressed sensing with side information, and distributed compressed sensing. <s> BIB007 </s> A Review of Urban Air Pollution Monitoring and Exposure Assessment Methods <s> Data Driven Spatial Prediction <s> We address the problem of compressed sensing (CS) with prior information: reconstruct a target CS signal with the aid of a similar signal that is known beforehand, our prior information . We integrate the additional knowledge of the similar signal into CS via $\ell _{1}$ - $\ell _{1}$ and $\ell _{1}$ - $\ell _{2}$ minimization. We then establish bounds on the number of measurements required by these problems to successfully reconstruct the original signal. Our bounds and geometrical interpretations reveal that if the prior information has good enough quality, $\ell _{1}$ - $\ell _{1}$ minimization improves the performance of CS dramatically. In contrast, $\ell _{1}$ - $\ell _{2}$ minimization has a performance very similar to classical CS, and brings no significant benefits. In addition, we use the insight provided by our bounds to design practical schemes to improve prior information. All our findings are illustrated with experimental results. <s> BIB008
Modern methods for air pollution data modeling and recovery try to capture dependencies among heterogeneous data such as different types of air pollutants and meteorological data. These statistical approaches are data driven and include models based on copula functions BIB007 and neural network models BIB005 . A first machine learning approach to model spatial correlations among different locations is presented in BIB005 and relies on a multi-view neural network architecture. The proposed model predicts the air quality of a location based on other stations' status consisting of air pollution measurements and meteorological data such as temperature and humidity. A critical step in their method is the partitioning of the spatial space into regions. The spatial neighbors of a station include not only nearby stations but also the stations located in adjacent cities. The spatial model is combined with a temporal model to provide a hybrid approach that aggregates spatial and temporal predictions. Another working direction in spatial prediction includes models that employ the Compressed Sensing (CS) paradigm BIB002 BIB003 . Let us denote with x ∈ R N a vector containing air pollution measurements at various geographical locations. Assume that only a subset of the entries of x is known. This partial observation mechanism is equivalent to a linear mapping of the form where Φ is a matrix defined as follows: let m be the number of known entries of the N-dimensional vector x. Let Ω be the m-length ordered set of indexes of the known entries of x. We construct a m × N matrix Φ such as the i-th row of Φ contains only one entry equal to 1 at the column position defined by the i-th element of Ω; all the rest of the entries are equal to zero. The subsampling process formulated in Label (2) is known as compressed sensing BIB002 BIB003 . Unique identification of a signal from a few measurements is feasible, if x has a sparse representation θ under a basis Ψ ∈ R N×N , that is, with θ 0 = s, where · 0 is the 0 quasi-norm counting the non-vanishing coefficients of the treated signal; s is the sparsity level of θ. Then, the unknown θ can be computed with well-known numerical algorithms BIB001 BIB004 . After recovering θ, the actual complete data x can be reconstructed from Equation (3). When, besides the incomplete measurements y, additional information related to x is available, then CS with side information BIB008 can lead to improved estimation accuracy. Recent methods, proposed in BIB007 BIB006 , exploit spatial correlations among multiple heterogeneous air pollution data sets to provide significant recovery performance improvements. In BIB006 , side information is employed in the reconstruction precess via an 1 -1 minimization algorithm. In BIB007 , the proposed model is based on a copula-based design to capture the correlation among the different types of data. Copula functions are used to model the marginal distributions and the dependence structure among data separately; to select the most appropriate marginal distribution, fitting tests are performed with training data. Then, a copula-based belief propagation method is developed to reconstruct unknown measurements. The main disadvantage of IDW and kriging interpolation approaches is that they do not take other environmental factors into account such as land use, terrain, traffic density, wind speed, etc. These techniques also suffer from the low data availability in the areas with low-density stations configuration, particularly when certain pollutants are known to vary significantly over small scales, e.g., NO 2 . The overview of spatial interpolation studies is given in Table 1 .