Fingerprint-based Location Tracking with Hodrick-Prescott Filtering Duc A Tran Ting Zhang Department of Computer Science University of Massachusetts, Boston, MA Email: {duc.tran, ting.zhang001}@umb.edu Abstract—Conventional location fingerprint techniques usually require a prebuilt training set of fingerprints sampled at known locations, so that locations of future fingerprints can be determined by comparing to this set For good accuracy, the training set should be large enough to appropriately cover the area However, it is not always feasible to obtain a quality training set in practice, and so recent studies have resorted to utilizing fingerprints that are available but without location information This paper investigates how these so-called unlabeled fingerprints can be useful for location tracking of a mobile device as it is moving Specifically, we propose a fingerprint-based tracking approach based on Hodrick-Prescott filtering and substantiate its potential via an evaluation study I I NTRODUCTION Location information is valuable to a myriad of applications of wireless networks In a surveillance sensor network, it is crucial to know the location of an incident caught by a sensor, such as fire in a building or oil spill in a coastal water The demand is also high for mobile apps providing navigation and other location-based services in hospitals, shopping malls, airport terminals, and campus buildings, to name a few GPS is the most effective way to get location information but does not work indoors Even for outdoor environments where this service is available, it is not energy-efficient to have to turn it on continuously all the time Consequently, numerous efforts have been made towards GPS-free localization solutions, most adopting the fingerprint approach This approach usually consists of two phases: training (offline) and positioning (online) In the offline phase, a number of sample locations are surveyed to build a map corresponding each location to a “fingerprint” which is a vector of measurements observed between the mobile device at this location and a set of “reference points” (RPs) For example, these RPs can be a set of Wi-Fi access points [1], FM broadcasting towers [2], or cellular towers [3], and measurements can be the received signal strength indices (RSSI) observed between them and the mobile device In the online phase, when we need to compute a location in real time, the current fingerprint of the device is compared against the fingerprint map to find the best location match Recently, fingerprint modalities other than radio have been suggested, such as sound [4] and geomagnetic field [5] By combining these different features where they apply, we can obtain a rich set of discriminative features for the fingerprint information 978-1-4799-3060-9/14/$31.00 c 2014 IEEE The localization accuracy of the fingerprint approach largely depends on the quality of the training data Due to the tedium and labor cost of the calibration task in the offline phase, it is not always feasible to obtain a quality fingerprint map In such a case, a viable approach [6]–[10] for better accuracy is to apply a semi-supervised learning method taking into account the availability of “unlabeled” fingerprints (whose location is unknown), not just the set of “labeled” fingerprints (whose location is known) Unlabeled fingerprints are abundant and can easily be obtained without manual location labeling This paper is focused on the problem of tracking a mobile device based on its sequentially obtained fingerprints We not require a prebuilt map of labeled fingerprints Instead, we only assume that once in a while a fingerprint is observed with a known location This assumption is necessary for otherwise it is impossible to make any inference about the device’s location It is noted that while there have been earlier research works on mobile location tracking, most of them make additional assumptions, such that those about special sensors built in the device (e.g., gyroscope, accelerometer, compass, camera) [11], [12], those about mobilityspecific constraints (e.g, speed, predefined map) [13] and those that are network-specific (e.g., vehicular or wireless sensor networks) [14], [15] In contrast, we are interested in fingerprint-based tracking and aim to devise a tracking framework with universal applicability in the sense that it can work orthogonally with any type of fingerprint space; i.e., applicable where fingerprint information can be of radio signals, acoustic, or geomagnetic, etc and can contain any other information that is location-discriminative Our intuition is that as the device is moving its fingerprints should satisfy two properties: spatial smoothness and temporal smoothness Fingerprints having similar values, regardless of their observation time, should correspond to nearby locations (spatial smoothness) and fingerprints observed in consecutive movements of the device should also correspond to nearby locations resulted from a constant speed (temporal smoothness) The spatial smoothness property has been exploited in earlier location fingerprint techniques [6]–[10] They commonly formulate the localization problem as a manifold regularization problem [16] which includes a regularizer term to maximize spatial smoothness In this paper, we want to investigate how useful the temporal smoothness property can be in order to improve localization accuracy Specifically, we formulate the fingerprint-based tracking problem as a regularization problem extended with a Hodrick-Prescott (HP) filter [17] term to regulate the temporal smoothness The location estimation algorithm, as a solution to this problem, is faster than the manifold regularization based algorithm, and as shown in our evaluation study, more accurate The remainder of the paper is structured as follows §II provides a brief survey of the related work §III presents the details of our proposed approach to the fingerprint-based tracking problem Evaluation results are discussed in §IV The paper is concluded in §V with pointers to our future work effectiveness of temporal smoothness in the fingerprint space for the location estimation, whereas the conventional manifold regularization approach focuses only on spatial smoothness HP is an effective tool for trend estimates in time series It has been used in the work of Rallapalli et al [28] for mobile tracking, which solves an optimization problem with constraints and assumptions about device-to-device distance In contrast, ours is the first effort to explore HP for fingerprintbased location tracking II R ELATED W ORK Suppose that we need to compute the instant location of a given fingerprint that is obtained in a stream manner, x1 , x2 , , xt , , where the time is discretized into time steps 1, 2, , t, Each fingerprint is a m-dimensional point, xt ∈ X ⊂ Rm , where m is the number of fingerprint features, e.g., RSSI from different Wi-Fi APs, readings from inertial measurement units (accelerometer, gyroscope, magnetometer), and any location-discriminative feature that is available with the device, etc Denote by yt ∈ Rd the location corresponding to xt , where d is the dimensionality of the location space For ease of presentation, let d = and so yt = yt is a real-valued number; we will discuss the case d > later We use the notation ht to represent whether a fingerprint xt is labeled with location (ht = 1) or unlabeled (ht = 0) The labeled fingerprints become available only once in a while, one at a time but totally unpredictable We want to find a real-time location estimator f : X → [0, 1] that, upon receipt of a new fingerprint xt at the current time t, needs to output its corresponding location f (xt ) We formulate this problem as a regularization problem which utilizes information about both labeled and unlabeled fingerprints that have been obtained by the current time In what follows, we present two approaches to this formulation The first approach is the conventional formulation based on manifold regularization The latter is the proposed formulation using Hodrick-Prescott filtering GPS-free localization in wireless networks has been a long-studied problem There exist many techniques to date, which differ in the type of network environment (e.g., sensor networks [18], wireless LANs [19], vehicular ad hoc networks [14]), the modality of information used to infer location (e.g., infrared [20], radio [1], sound [4], geo-magnetic [5], light [21]), or the type of algorithmic method (e.g., range-based [22], range-free [23]) Radar [1] is the world’s first Wi-Fi RSS-based indoor positioning system, which demonstrates the viability of using RSS information to locate a wireless device This system works using a radio map, a lookup table that maps building locations to the corresponding RSS fingerprints empirically observed at these locations The reference points are the WiFi access points within the user’s Wi-Fi range The radio map is searched to find the closest RSS readings and the centroid of the corresponding locations will be used as the estimate for the user’s location Radar represents the fingerprint approach where kNN is used to determine the location One can also employ other learning methods to relate a fingerprint to a location, such as probabilistically using Bayesian inference [24] or nonprobabilistically using an Artificial Neural Network (ANN) [25] or a Support Vector Machine (SVM) [26] When there are only a small number of sample fingerprints for training, we can utilize unlabeled fingerprints as a supplement to the original ones by solving a manifold regularization problem, a widely-used semi-supervised learning method of by Belkin et al in [16] in the area of Machine Learning, to propagate the labels for the unlabeled fingerprints based on their similarity with the labeled Pan et al [6], [7] apply manifold regularization with a Laplacian regularization term reflecting the intrinsic manifold structure of the fingerprints; here the manifold is a weighted graph of fingerprints in which the weight of an edge connecting two fingerprints represents the similarity between them Other regularization terms have also been investigated For example, e.g., Total Variation [27] considered in the recent work of Tran and Truong [10]; this work, however, suggests that manifold regularization with the Laplacian term offers better localization accuracy than TV Our research in this paper also applies a regularization framework for learning the location labels for the unlabeled fingerprints, but our regularization term is based on the Hodrick-Prescott filter [17] Our focus of attention is on the III F INGERPRINT- BASED L OCATION T RACKING A Manifold Regularization Ideally, the location estimator f if applied on a labeled fingerprint should result in an estimate that matches its given location Therefore, in search of f , a reasonable goal is to minimize the estimation error with respect to the labeled fingerprints This is quantified by minimizing f E(f ) = t t hi (f (xi ) − yi )2 (1) i=1 Another goal is to maximize the spatial smoothness in the fingerprint space As aforementioned, similar fingerprints, regardless of when they are observed, should correspond to nearby locations We quantify this by, first, organizing the fingerprints into an undirected weighted graph, where each vertex is a fingerprint and each edge has a weight |x −x |2 w(xi , xj ) = exp − i2σ2j (for some constant σ) reflecting the similarity between xi and xj , and, second, minimizing the Laplacian quadratic form of this graph: ⎧ ⎫ i t ⎨ ⎬ w(xi , xj )(f (xi ) − f (xj ))2 (2) S(f ) = ⎩ ⎭ f t i=1 j=1 To optimize these two goals, we combine them into a single objective function using Belkin et al.’s manifold regularization framework [16] Specifically, the location estimator f is the solution minimizing the following risk: {J(f ) = E(f ) + λS S(f )} , (3) f where coefficient λS > reflects the importance of spatial smoothness maximization The beauty of this approach is that we can easily derive a closed form for the location estimator f Denote the following vectors and matrices: f = [f (x1 ), f (x2 ), , f (xt )] , y = [y1 , y2 , , yt ] (yi is set to zero by default for unlabeled xi ), H = diag(h1 , h2 , , ht ), the identity matrix I = diag(1, 1, , 1), and the Laplacian matrix L of the t weighted fingerprint graph Then we can express the functionals in Eqs (1) and (2) in matrix form as follows, E(f ) = 1 t (f − y) H(f − y) and S(f ) = t2 f Lf Thus, the risk J(f ) in Eq (3) can be expressed in matrix form as J = = 1 (f − y) H(f − y) + λS f Lf t⎛ t ⎞ ⎟ 1⎜ ⎜f (H + λS L)f − 2y Hf + y Hy⎟ ⎠ t⎝ t Q To minimize J, set its derivative with respect to f to zero, ∂J = [(Q + Q )f − 2Hy] = (4) ∂f t Because of the symmetry of matrices H and L, we have Q + Q = 2Q and so Eq (4) leads to f= H+ λS L t of all these fingerprints are known, HP can be used to obtain a smoothed trajectory of locations, by solving the following problem: t f t (f (xi )−yi )2 +λT i=1 (f (xi )+f (xi−2 )−2f (xi−1 ))2 i=3 Here, the second term is added to penalize variations in the growth rate of the trend component The argument appearing in the second term, f (xi )+ f (xi−2 )− 2f (xi−1 ), is the second difference of the time series at time t; it is zero when and only when the points f (xi ), f (xi−1 ), and f (xi−2 ) are on a line The HP trend estimate, as the solution to this optimization problem, is, in matrix form, f = (I + λT DD )−1 y, where D is the second-order difference matrix ⎤ ⎡ 0 ⎢0 0 ⎥ ⎥ ⎢ ⎢ −2 ⎥ ⎥ ⎢ ⎢0 −2 ⎥ ⎥ ⎢ −2 ⎥ D=⎢ ⎥ ⎢0 ⎢ ⎥ ⎥ ⎢ ⎢ −2 0⎥ ⎥ ⎢ ⎣ −2 0⎦ −2 t×t Coefficient λT > controls the smoothness of the trend estimation If λT → 0, the trend converges to the original time series data On the other hand, if λT → ∞, the trend is the best straight line fit to the time series data HP assumes that every point in the time series is labeled, whereas we not know the location of every fingerprint in the fingerprint sequence To integrate HP, we revise the optimization problem to find the location estimator as follows: t hi (f (xi )−yi )2 +λT f i=1 t (f (xi )+f (xi−2 )−2f (xi−1 ))2 i=3 or −1 Hy {J(f ) = E(f ) + λT T (f )} , (5) fM R With this location estimator, the location estimate for fingerprint xt will be the corresponding element (last element) in vector f B Hodrick-Prescott Filtering The Hodrick-Prescott (HP) filter [17] is a mathematical tool used to obtain a smoothed-curve representation of a time series, one that is more sensitive to long-term than to shortterm fluctuations We propose to apply HP to our location tracking problem because the trajectory of a moving device should be smooth over time and should exhibit a trend; realworld mobility often exhibits moving at a constant velocity for a long period of time before changing speed [28] Treating the sequence of fingerprints as a time series, and if the locations f (6) where t T (f ) = (f (xi ) + f (xi−2 ) − 2f (xi−1 ))2 t i=3 (7) In matrix form, we have T (f ) = f DD f Thus, the risk J(f ) in Eq (6) is J = (f − y) H(f − y) + λT f DD f = f (H + λT DD )f − 2y Hf + y Hy ∂J = (H + λT DD ) f − 2Hy ⇒ ∂f Setting ∂J/∂f = 0, we have our location estimator as −1 f = (H + λT DD ) fHP Hy (8) (a) Floor plan (208 sample locations) (b) Trajectory (60 locations) (c) Trajectory (107 locations) (d) Trajectory (185 locations) Fig Floor plan (68m × 63m) with 208 sample locations, each represented by a black dot The hallways are marked with white color and stairs with green Black and blue areas are not penetrable Three trajectories of different patterns and path lengths are shown in red color We have so far assumed that the location is 1D For 2D or 3D localization, we simply apply the same algorithm separately for each coordinate The computation of fMR in Eq involves inverting matrices of t × t Its computational complexity, therefore, is O(t3 ) The HP-based approach is much faster though Note that ⎡ ⎤ −2 ⎢−2 −4 0⎥ ⎢ ⎥ ⎢ −4 −4 ⎥ ⎢ ⎥ ⎢0 ⎥ −4 −4 ⎢ ⎥ ⎢ −4 −4 0⎥ DD = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 −4 −4 ⎥ ⎢ ⎥ ⎣0 −4 −2⎦ −2 t×t is a pentadiagonal matrix and so is the matrix (H + λT DD ) The inverse of a pentadiagonal matrix can be computed in O(t) time; hence, O(t) time to compute fHP in Eq This is a clear advantage of the HP-based approach in comparison to the manifold regularization approach IV E VALUATION We discuss the evaluation results in this section We compare the HP filtering based approach to the manifold regularization approach, which for ease of presentation are referred to as HP and MR, respectively As HP is obviously better than MR in terms of computation time, we focus on their location error as metric for comparison This error (up to time t) is computed as the average “individual” location error for unlabeled fingerprints over the path traveled from the beginning (up to time t) The “individual” error corresponding to a fingerprint xt is the Euclidean distance between its location estimate at time t and its ground-truth location 60 trajectory 220 107 trajectory 250 HP MR 200 HP MR 180 200 Location Error Location Error 160 140 120 100 150 100 80 60 50 40 20 0.1 0.2 0.3 0.4 0.5 Label Rate 0.6 0.7 0.8 0.1 0.9 0.2 0.3 (a) Trajectory 0.5 Label Rate 0.6 0.7 0.9 0.9 HP MR 60 trajector 107 trajector 185 trajector 0.8 Location Error (Ratio) 200 Location Error 0.8 (b) Trajectory 185 trajectory 250 0.4 150 100 0.7 0.6 0.5 0.4 0.3 50 0.2 0.1 0.2 0.3 0.4 0.5 Label Rate 0.6 0.7 0.8 0.9 (c) Trajectory Fig 0.1 0.1 0.2 0.3 0.4 0.5 Label Rate 0.6 0.7 0.8 0.9 (d) HP’s error as a fraction of MR’s Location error of MR vs that of HP for the entire trajectory The unit for the y-axis is 10cm The evaluation was conducted with a dataset collected in an experiment on the floor of our Computer Science department (Figure 1(a)) This dataset consists of 208 WiFi RSSI fingerprints at 208 locations sampled throughout this floor, respectively There are in total 138 Wi-Fi access points and from those unreachable the corresponding RSSI is set to 100db At each sample location, the corresponding fingerprint is the average of the RSSIs observed at this location RSSI was measured by a person carrying an Android phone in no particular heading direction We consider three trajectories shown in Figure 1(b), Figure 1(c), and Figure 1(d), each being a path connecting sample locations For each trajectory, the labeling status of each point is determined based on a label rate pl ∈ {0.1, 0.3, 0.5, 0.7, 0.9} and, for each choice of pl , the results are averaged over five random runs We set γ = for the weight function For each case of study (sample trajectory, same label rate, same random run), we perform a cross-validation procedure to choose the best regularization coefficient λS for MR; i.e., that results in the best error Similarly we have a separate crossvalidation procedure to choose the coefficient λT for HP The range of possible coefficient values in the cross-validation is {10−7 , 10−6 , , 10−1 , 1, 10}, representing ten different scales The comparison is between HP using its best coefficient λT versus MR using its best coefficient λS Figure 2(a), Figure 2(b), and Figure 2(c) show the location errors of MR and HP for each of the three trajectories, respectively It is expected that the error should decrease as the label rate increases What is more noticeable, however, is the obvious superiority of HP’s accuracy compared to MR’s For example, when only 50% of the fingerprints are labeled, for all three trajectories, HP has an error of roughly 5m while MR’s error is more than 15m In most cases, compared to MR, HP consistently cuts the error down by a factor of 1.5 times or more This can be observed in Figure 2(d) showing the error of HP as a fraction of the error of MR Figure shows the evolution of location error of each approach over the time, here showing the result for the 185fingerprint trajectory for different cases of label rate pl It is observed again that throughout the travel path HP is obviously better than MR by a large margin, and this is regardless of whether the label rate is small (Figure 3(a)) or large (Figure 3(e)) Figure 3(f) plots the result averaging over all five cases of label rate Another observation favoring HP is that its error converges to a stable value quickly as more fingerprints are observed whereas MR’s error keeps increasing before showing any sign of convergence This observation is clearer for the case of large label rates (pl ≥ 0.3) than that for the case of small label rate (pl = 0.1) Figure draws the estimated trajectories resulted from pl = 0.1 250 pl = 0.3 200 HP MR HP MR 180 200 160 Location Error Location Error 140 150 100 120 100 80 60 50 40 20 0 20 40 60 80 100 Time 120 140 160 180 200 20 40 60 (a) 10% labeled 120 140 160 180 200 140 160 180 200 140 160 180 200 pl = 0.7 150 HP MR 160 100 Time (b) 30% labeled pl = 0.5 180 80 HP MR 120 Location Error Location Error 140 100 80 60 100 50 40 20 0 20 40 60 80 100 Time 120 140 160 180 200 20 40 60 (c) 50% labeled 100 Time 120 (d) 70% labeled pl = 0.9 120 80 185 trajectory 180 HP MR HP MR 160 100 Location Error Location Error 140 80 60 40 120 100 80 60 40 20 20 0 20 40 60 80 100 Time 120 140 160 180 200 (e) 90% labeled 0 20 40 60 80 100 Time 120 (f) Average over all cases of label rate Fig Evolution of location error over time for the 185-trajectory The unit for the y-axis is 10cm Not only HP is more accurate than MR throughout the trajectory, HP does not get worse as quickly as MR does Instead, HP converges to a stable error applying MR and HP to the 185-fingerprint sequence for three cases of label rates: 10% of the fingerprints are labeled (pl = 0.1), 50% labeled (pl = 0.5), and 90% labeled (pl = 0.9) Here, we show the location estimated for a fingerprint instantly at the time it is observed The first point is always put at the center because in the sequence generated it happens to be unlabeled and there is no labeled fingerprint available for learning As can be seen in this figure, in all cases of label rate, HP’s trajectory resembles the ground-truth trajectory more closely than MR does Even in the case only 50% of the fingerprints are labeled, HP results in a trajectory (Figure 4(d)) comparable to the trajectory produced by MR for the case 90% labeled (Figure 4(e)) It is noted that, after all the fingerprints in the sequence are observed, we can use the latest location estimator to obtain better estimates for all the unlabeled fingerprints in the past, including, for example, the first fingerprint These estimates are useful if there is a need for a posterior fix of the trajectory V C ONCLUSIONS We have shown convincingly that temporal smoothing is an important property we should take into account for fingerprint- 0 100 MR, pl = 0.1 200 300 27 30 29 28 26 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 100 74 64 300 75 95 185 400 500 600 700 100 0 200 49 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 164 166 170 168 169 109 181 179 177 178 180 173 174 175 184 183 176 300 115 129 400 500 500 182 165 172 600 171 163 167 600 700 100 63 700 (a) MR: 10% labeled 0 100 14 100 16 69 70 60 61 62 72 63 200 300 76 139 140 184 141 183 400 500 181 145 180 600 MR, pl = 0.5 200 300 400 (b) HP: 10% labeled 500 600 78 12 913 10 11 43 15 42 4437 28 18 34 30 41 48 17 221920 272625 49 33 29 45 36 38 53 24 58 31 32 55 39 21 71 747523 35 40 46 54 51 59 73 57 56 47 120 119 646768 808177 52 82 79 118117123122 125 144 66 143 129 130 146 50 142147 104 105 131 103 65 114 161 107 162 159 89102108 138 109 160 100101 99 88 95 96 87 98 137 90 152 179 112155 111 110158 8386 91 173 175 106 172 116 170 153 113 97 8485 92 115 164 185 93 169 94 78 136 135134 133 128 127 126 154 132 168 166 700 121 124 174 182 176 178 177148 149 150 151 171 157 156 167 165 163 700 100 13 12 200 11 200 300 400 500 600 800 10 700 100 MR, pl = 0.9 200 300 400 (d) HP: 50% labeled 500 43 42 37 44 27 28 45 14 30 29 36 38 41 31 32 39 35 46 40 15 60 57 5633 34 55 54 53 47 70 69 52 48 61 119 62 51 63 145 72 50 64 65 49 118 82 73 103 104 81 8588 89 99 100101 102130 87 90 117 109 132 80 8386 91 74 13698 112111 110 105 106 116 107 97 96113 79 84 92 115 75 93 76 77 114 108 94 95 78 150 135 138 137 134 131 129 128 127 126 180133 185139 140 184 141 142 183 143 182 157 144 156 166 181 167 155 146178147177148176149175 174151173152172153154 179 171 170 169168 13 100 300 HP, pl = 0.5 400 500 600 700 44 927 43 37 28 26 20 25 14 29 45 38 19 42 100 15 24 31 30 32 33 34 36 22 21 39 35 40 5341 46 16 23 55 54 47 60 69 59 58 57 56 71 70611768 52 48 18 49 62 67 200 120 72 63 66 50 64 65 119 51 121 122 123 73 105 118 88 89 100101102103104 90 117 300 74 112 8387 99111110 8691 98 106 116 107 97 113 108 75 848592 82 109 81 115 93 80 79 76 78 94 124 95 136 135134 133114 138137 132 131130 129 128 127 126 96 139 400 185 77 162 140 125 184 161 141 160 159 142 500 183 158 143 144 182 157166 165 163164 156 181 154 155 167 145 153 152 172 148 149 175 150 176 600 180 174 151 173 168 179 178 177 171 169 146 170 147 (c) MR: 50% labeled 62 200 HP, pl = 0.1 300 400 500 600 700 162 161 61 160 60 30 159 26 29 27 28 59 158 156 157 48 155 474658 31 45 44 43 38 42 154 37 32 39 40 41 33 36 34 57 153 35 151 150 149 152 56 148 7172 55 54 147 70 69 146 145 64 49 107 106 53 6568 144 105 50 6667 73 104 128 126 91 90 74 52 125 108 88 89 103 12751 124 87 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 123 92 143 122 102 8693 121 142 101 120 113 141 119 112 109 110 114 111 100 115 117118 116 75 99 140 137 9613497 98 139 138 95 136135 129 133132 130 131 94 185 76 181 77 180 179 78 178 85 177 184183 182 79 84 176 166 165 163 175 80 164 174 81 167 173 172 82 83 168 171 169 170 100 59 58 60 62 61 63 57 56 55 54 53 52 50 51 69 70 68 67 71 66 65 72 76 78 77 80 85 79 81 82 83 84 94 87 92 88 91 89 93 90 119 120 117 121 126 122 104 124 116 125 128 123 127 105 106 107 103 108 101 100 97114 118 99 98 96 102 144 145 146 143 142 147 151 154 155 141 156 153 152 157 148 158 150 159 149 161 160 162 140 112 110 111139 113 130 138 131 132 137 134 133 136 135 86 200 73 400 12 11 10 18 16 20 19 21 71 17 59 22 67 68 66 26 25 58 24 23 700 600 700 100 13 100 200 120 121 122 125 123 124 158 159160 161 165164 163 162 300 400 500 600 12 HP, pl = 0.9 200 11 10 300 26 25 24 23 58 400 500 43 44 37 42 27 28 30 29 45 36 38 41 31 32 39 35 40 46 57 5633 34 55 54 53 47 52 48 14 20 19 2118 15 69 60 17 59 22 71 70 61 16 62 68 51 67 72 63 50 64 65 6682 49 119 118 73 81 88 89 99 100101 102 103 104 87 90 117 80 8386 91 98 112111 110 74 116 107 106105 97 96113 79 84 92 115 75 93 76 77 114 108 130 94 95 78 136 132 135 109 138 137 134 133 131 129 128 127 126 85 185139 140 184 141 142 183 143 182 157 144 156 166 181145 167 155 146178147177148176149175 150 174151173152172153154 180 179 171 170 169168 600 700 120 121 122 125 123 124 158 159160 161 165164 163 162 700 (e) MR: 90% labeled (f) HP: 90% labeled Fig Drawing of the estimated 185-trajectory Red-colored points are location estimates for unlabeled fingerprints and blue-colored points are the ground-truth locations of the labeled fingerprints The numbers represent the ID of the fingerprints sorted in time of measurement based location tracking We have investigated the use of Hodrick-Prescott filtering as a way to integrate temporal smoothness in the location estimation Not only computationally faster, but this approach has consistently been shown in our evaluation study to be more accurate than the conventionally used manifold regularization approach which factors in only the spatial smoothness property For future work, we plan to investigate online/streaming algorithms that can locate each fingerprint in real time as it arrives without requiring to store the entire set of previously observed fingerprints We also want to evaluate with more comprehensive experiments over larger time and spatial scales ACKNOWLEDGMENTS This work was supported by the NSF award CNS-1116430 Any opinions, findings and conclusions or recommendations 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for localization in mobile networks,” in Proceedings of the sixteenth annual international conference on Mobile computing and networking, ser MobiCom ’10 New York, NY, USA: ACM, 2010, pp 161–172 [Online] Available: http://doi.acm.org/10.1145/1859995.1860015 ... have our location estimator as −1 f = (H + λT DD ) fHP Hy (8) (a) Floor plan (208 sample locations) (b) Trajectory (60 locations) (c) Trajectory (107 locations) (d) Trajectory (185 locations)... mobile tracking, which solves an optimization problem with constraints and assumptions about device-to-device distance In contrast, ours is the first effort to explore HP for fingerprintbased location. .. magnetometer), and any location- discriminative feature that is available with the device, etc Denote by yt ∈ Rd the location corresponding to xt , where d is the dimensionality of the location space