Modern traffic management should benefit from the diverse sensors, smart phones, and social networks data that offer the potential of enhanced services. In disaster scenarios, it is no longer guaranteed that a central server and reliable communication is always available. This motivates a distributed computing setting with restricted communication. Also in distributed High Performance Computing communication costs have to be reduced to the minimum and costly broadcast to all compute nodes hould be avoided. We want to learn local models with high communication efficiency. They still require the exchange of label information in a setting of supervised learning. The transmission of all labels among the nodes can be as costly as communicating all observations. Sophisticated methods are required to trade-off prediction performance against communication costs. We hereby present an in-network algorithm based on local models that only sends label counts to neighboring nodes. method is a novel approach that transfers no data about individual observations, but just agProceedings of the 2 nd International Workshop on Mining Urban
gregated label information. We outline its MPI implementation. And evaluate our approach on real world data in a traffic monitoring scenario. Tests reveal that in comparison to sending all labels, the algorithm is scalable. authors. Copying permitted for private and academic purposes.
Traffic flow prediction is an important task for traffic managers. It allows performance assessment of major traffic infrastructure, like roads and junctions. Individual mobility benefits from predictions, as they provide necessary data for proactive, smart decisions on individual travel plans, e.g. predictive situation-aware trip planning by avoidance of likely traffic hazards (Niu et al., 2015; Liebig et al., 2014). Traffic flow models are based on sensor observations of current traffic gained by a mesh of (mostly static) presence sensors. While existing learning methods centralize and process measurements on a dedicated traffic management server, they have some major drawbacks, particularly in cases of disaster: The need for a reliable communication infrastructure reduces sustainability in case of natural hazards.
First, the server-side collection causes high communication costs, decreasing the system’s ability to process all sensor data, in time. Second, the area of traffic prediction systems is limited by the political area of homogeneous regulations for sending the data through the network. Third, increasing the network’s density bares the risk of re-identification of individual persons and tracking them throughout the network. Existing systems are therefore limited by communication bandwidths, processing capabilities and political regulations.
We tackle these limitations by a distributed spatio-temporal in-network learning algorithm, where sensors compute local models and efficiently communicate label counts with their topological neighbors.
Our approach sends space-time aggregated values that, by design, provide kanonymity. Hence, our method is privacy preserving and can be applied for large-scale traffic management scenarios. Our particular focus is on the prediction of future traffic flow at junctions throughout the region of interest (e.g. a city, a state or even areas at European scale). Possible applications comprise, for instance, • distributed car-to-car scenarios where cars or trucks communicate at junctions the number of observed vehicles at the road to estimate traffic flow and alter their individual transportation plans based on predicted traffic conditions, or, • large scale traffic flow prediction that processes massive local observations on a high performance computer.
Scalable in-network algorithms belong to the field of distributed data mining. Existing work mostly focuses on horizontally partitioned data. There, full observations, i.e. all features and labels, are stored on different nodes in a network. However, network states representing the current traffic flow are vertically partitioned. Here, only partial information about observations is stored on different nodes. Learning and prediction therefore either require the transmission of observations or labels to other nodes. Previous work (Das et al., 2011; Lee et al., 2012; Stolpe et al., 2013) has focused on sending less information about observations to a central coordinator. Here, we deal with reducing the amount of labels sent to neighboring peer nodes. Communication-efficient algorithms for vertical distributed learning are not just relevant for traffic flow prediction, but for applications as diverse as intrusion detection, monitoring production processes or smart grid management. The main contributions of our work are the following: 1. We introduce a privacy-preserving approach for the distributed learning of spatio-temporal prediction models which transfers only aggregated label information, but no data about individual observations. 2. A connection is drawn between the task of learning from label proportions and reducing communication costs in distributed environments, and it is evaluated on real-world data. 3. We introduce a fast search strategy for the LLP algorithm (Stolpe & Morik, 2011) and demonstrate its prediction performance in the context of traffic flow prediction.
The next section reviews related work. Section 3 details our problem setting and introduces a novel approach for the innetwork training of local models. Section 4 discusses learning from aggregated label information, discusses its implementation in using message passing interface (MPI), analyses its communication cost and aspects of privacy. Evaluations of our approach can be found in Sect. 5. We finish with conclusions and outlook on future work.
Many distributed data mining algorithm learn from horizontally partitioned data, whereas our data is vertically partitioned.
like (Yunhong et al., 2009) are not scalable, since they
send quadratic kernel matrices to a central server.
Distributed optimizati
Also in traffic flow prediction, most literature describes
processes on central servers. There are two major ways
to model traffic: using a simulation (Raney & Nagel, 2006)
or applying an imputation model, trained on previous
sensor measurements. Models are required for the estimation
of traffic flow at locations not being observed at all. Such
imputation is not the focus of our study, but the
prediction of traffic flow at sensor locations. We point the
interested reader to methods of simulation (e.g. cellular
automaton (Raney & Nagel, 2006)) and model-based imputation
(e.g. (Liebig et al., 2012)). Most learning-based traffic flow
prediction methods analyse time series, where a popular
model is based on auto-regressive integrated moving
average (ARIMA) (Ahmed et al., 1979). Recently, an
application of a Gaussian Markov Model was proposed in
(Schnitzler et al., 2014), and more advanced graphical models,
namely Spatio-Temporal-Random-Fields (STRFs), were
applied to traffic modeling in (Piatkowski et al., 2013).
Distributed approaches comprise an approach that applies
kNN and Gaussian Process Regressi
The task of learning from aggregated label information was
first introduced in (Ku¨ck & de Freitas, 2005).
Theoretical bounds have only recently been proven in (Yu et al.,
2014). (Musicant et al., 2007) propose variants of
existing algorithms. The SVM optimization problem has been
adapted to the setting (Ru¨ping, 2010; Yu et al., 2013).
Mean Map (Quadrianto et al., 2009) estimates the mean
operator solving a system of linear equati
Given are m distributed sensor nodes P1, . . . , Pm. Each sensor node Pi delivers an infinite series of real-valued measurements . . . , vt(−i)1, vt , vt+1, . . . for different time (i) (i) points . . . , t − 1, t, t + 1, . . ..
Time spans between two measurements are equidistant, given a constant sample rate. Let t denote the current time of measurement, while t − a and t+a are time points a steps in the past and future. Each sensor node also has a spatial location.
Many traffic flow management tasks require the prediction of traffic flow categories, that are achieved by a discretization of raw values into distinct intervals (e.g. risk level assignment or decision for emergency traffic signals).The task, given the current time point t, is therefore to predict a label y from a set Y = {Y1, . . . , Yl} of distinct categories at some arbitrary node Pi at future time point t + r, based on the current and previous (raw) sensor readings at all or a subset of nodes P1, . . . , Pm.
We assume that for learning, measurements and labels are somehow recorded (see below) over a fixed-length time period.
models, each node Pi thus provides a sequence Vi = hv1(i), . . . , vn(i)i of measurements, vj Li = hy1(i), . . . , yn(i)i of labels yj (i) ∈ Y .
(i) ∈ R, and a sequence DISTRIBUTED LEARNING OF LOCAL MODELS Instead of centralizing all data, we propose that each Pi records and stores its own measurements and labels. For predicting future traffic flow categories at node Pi, we restrict learning to Pi itself and c topological neighboring nodes around Pi. For instance, to learn and predict the future type of traffic flow at some street junction, considered are only measurements and labels recorded at the junction itself and at c junctions closest to it.
Before training, each Pi preprocesses measurements Vi as follows.
A window of size p is slided over the series Vi with step size 1, storing all thereby created windows x(i) = {vt(−i)p+1, . . . , vt(i)}, t = p, . . . , n as rows in a t dataset Di. Let N (i) = {n(1i), . . . , n(ci)} be the set of indices for the c neighboring nodes around Pi. Based on the datasets Di, Dn(i) , . . . , Dn(ci) and labels Li, we want 1 to learn a local function (model) f (i) that, given windows n(i) xt(i), xt 1 , . . . , xtn(ci) of sensor readings from node Pi and its neighbors, predicts the label yt(+i)r at node Pi with horizon r correctly.
n(i) Interpreting windows xt(i), xt 1 , . . . , xtn(ci) single observation x that should be classified, the data is vertically partitioned, since each neighboring node of Pi only stores partial information about x, i.e. a subset of feaas features of a tures.
An obvious choice for the training of f (i) at Pi is to ask for the recorded measurements at each neighboring node, concatenate their columns at Pi and join the labels stored at Pi to the new dataset. The approach is more scalable than centralizing all data, since the number c of neighbors is fixed, avoiding the bottleneck problem of limited bandwidth. However, each node still needs to transmit all measurements to each of its neighbors, consuming at least as much energy per node as sending all data to a single server. Therefore, we propose to send only label information from node Pi to its neighbors and to train models f0 and fn(i()i) , . . . , fn(i(c)i) at its neighbors. As model f (i) at node 1 Pi, we propose a majority vote over predictions from itself and its neighboring nodes. All models are local, since they (i) at node Pi only consider measurements and labels of a fixed number of close topological neighboring nodes around Pi. Moreover, the approach works fully in-network without a central coordinator, since each node only communicates with its neighboring peer nodes. As learners at each node, one may consider supervised learners, like kNN, Decision Trees or SVMs. Considering the limited computational resources of sensor nodes, however, our evaluation in Sect. 5 is solely based on kNN.
Since the number of bits to encode all labels is often less than an encoding of all measurements, communication is saved by sending labels from Pi instead of measurements to Pi. However, supervised learning still requires individual labels for all observations. The question is if communication can be reduced even further, by sending fewer labels or aggregated label information to each neighboring node. Semi-supervised (Chapelle et al., 2006) and active learning (Balcan et al., 2010) show that training on fewer labels may achieve a similar performance as training on all labels. However, such methods do not preserve the privacy of the data, since they need individual labels of observations (see Sect. 4). Instead, we propose to send only aggregated label information, i.e. label counts, to neighboring nodes for learning.
Before sending label information to each of its neighboring nodes, Pi divides its time-related sequence Li of labels into consecutive batches C1(i), . . . , C(i) of a fixed size b h (see Fig. ??). It respects the prediction horizon r, such that each C(i) consists of labels from time point t + (j − 1)b + r j to t + jb + r and align correctly with time points of observations (i.e. windows of measurements) at other nodes. Let n be from here on the size of datasets Di, i.e. the number of windows stored. Then, h is ⌈n/bs⌉. For each batch j, labels y ∈ Y are aggregated by counting how often they occur, and stored in a h × l matrix of label counts Q(i) = (qj(id)), where qj(id) = |{y ∈ Cj |y = Yd}| . n(i)
n(i) Let Pn(ei) be a neighboring node receiving label counts from Pi. Pn(ei) transforms Q(i) into a label proportion matrix Π(i) = (πj(id)) = qj(id)/b, i.e. the counts of labels are divided by batch size b. Since every node knows b and r, Pn(ei) can partition its own windows x1 e , . . . , xnn(ei) of measurements n(i) into batches B1 e , . . . , Bhe . Since the sender respects r, the time spans used for aggregating the labels align correctly with the windows of measurements stored at Pn(ei) . The learning task at node Pn(ei) now consists of learning a model fn(i(e)i) , only based on its batches of (unlabeled) measurements and the label information from node Pi, stored in the label proportion matrix Π(i), such that the expected prediction error over individual observations is minimized.
Several methods have been developed to solve the task (see Sect. 2). Considering the limited computational resources of sensor nodes, the LLP algorithm (Stolpe & Morik, 2011) looked most promising for our evaluation in Sect. 5, since LLP has a linear running time and its centroid model a small memory footprint.
Moreover, it can handle multiclass classification problems as they arise in traffic monitoring. However, we found that it still needs to be improved for scalability issues and performance. The next section describes LLP shortly, while Sect. 4 introduces a new local search method.
LLP learns from label proportions by first clustering all observations and then assigning labels to each cluster. The task of cluster analysis consists of partitioning a set of observations into a set C of k disjunct groups (clusters) C1, . . . , Ck, such that the similarity of observations in each cluster is minimized. LLP relies on the idea that observations having the same class also share similar features, i.e. that clusters somehow correspond to classes. LLP allows for several clusters per class and assumes that the majority of elements of a cluster belongs to the same class. Once given a clustering the only remaining problem is to assign correct labels to each cluster. defined as μ(x) = argminCk∈C ||x − ck||2 .
More formally, let μ : X → C be a mapping that assigns an arbitrary observation x ∈ X to a cluster C ∈ C. For centroids c1, . . . , ck found with k-Means, μ(x) would be n(i) Further, let ℓ : C → Y be a mapping which assigns a label λ ∈ Y to each cluster C ∈ C. For ease of notation, let f denote model fn(i(e)i) to be learned at node Pn(ei) , Bi denote the batch B e and Π denote matrix Π(i). f is the composition i of mappings ℓ and μ, i.e. f = ℓ ◦ μ.
With prediction model f , entries γjd of a model-based proportion matrix Γf = (γjd) can be calculated as 1 |Bj | x∈Bj γjd =
X I(f (x), Yd), I = 1 : f (x) = Yd 0 : f (x) 6= Yd .
The LLP algorithm now minimizes the mean squared error MSE(Π, Γf ) =
X(πjd − γjd)2 , 1 h
X hl j=1 d=1 l (1) (2) between the given label proportion matrix Π and the modelbased proportion matrix Γf by trying different label mappings ℓ.
A LOCAL SEARCH STRATEGY WITH MULTISTARTS The LLP algorithm as introduced in (Stolpe & Morik, 2011) can work with different cluster algorithms and labeling strategies. LLP with an exhaustive labeling strategy, called LLPexh in the following, tries all possible labelings of the clusters.
evaluations done in Sect. 5. The greedy strategy proposed in (Stolpe & Morik, 2011) didn’t achieve sufficient accuracies for traffic prediction. Hence, a better search strategy is demanded.
We propose a local search that is started multiple times with different random combinations of labels. LLP with this search strategy will be called LLPlsm in the following. The local search greedily improves on the current labeling of clusters by trying all possible labels at each component of a labeling vector λ. Fitness measures how well the modelbased label proportion matrix Γf , as calculated from the current labeling, matches the given label proportions. If the fitness improves, the search starts from the first component of the labeling vector λ, again. Otherwise, it resets the label at the current position kpos to the label of the best (local) solution found so far. Returned is the best labeling found over all starts of the different greedy searches. In each iteration, the greedy search runs until no further improvement is possible. Moreover, at each step of the algorithm, the fitness either improves or is staying the same (which is a stopping criterion). Therefore, each search finds a local minimum. Since the number of searches is finite, the returned labeling vector is also locally minimal. In comparison to LLPexh, it cannot be guaranteed that a globally optimal solution is found. However, with regard to the prediction results presented in Sect. 5, we found that a local search performed sufficient enough, despite a much lower running time.
LLP as introduced in (Stolpe & Morik, 2011) combines the MSE with two other error measures. However, we found that the use of these additional measures decreases the accuracy in the traffic monitoring scenario. Hence, all experiments in Sect. 5 are based on the MSE, only. Similarly, we abstain from the evolutionary feature weighting presented in (Stolpe & Morik, 2011), since it would heavily increase the algorithm’s running time.
We explicitly focus on the implementation with the Message Passing Interface (MPI) (Message Passing Forum, 1994) as message passing in the top super computers in the top500 list1 base on the MPI standard. MPI implements the single program multiple data paradigm (Darema, 2001) whereas every node of a distributed system executes the same program but uses different data. To coordinate this architecture a MPI program consists of 1 master node and multiple slave nodes that perform computations, the results are collected at the master. The art of MPI programming is to divide the problem into multiple tasks that are transferred to the slaves and processed thereby. Usually, the number of tasks exceeds the number of slave nodes and the distribution of the tasks has to be organized by the master. The work in (Nupairoj & Ni, 1994) analyses the performance of such MPI systems and reveals that communication is major bottleneck in MPI programs. A succeeding publication (Piernas et al., 1997) provides empirical estimates for computation of communication costs depending on message lengths. Based on this publications two major conclusions ban be made: (1) The shorter the messages the lower the communication cost, and(2) broadcast messages 1http://www.top500.org/lists/2014/11/, last accessed May, 1st should be avoided. Our algorithm respects both findings and therefore seems suitable for an MPI implementation. For ease of development we decided for the Cran-R package Rmpi (Yu, 2002) which provides basic MPI functionalities in Cran-R, thus matrix operators can be applied to the data. Our program comprises the following generic steps: 1. Load Rmpi, and spawn slaves 2. Definition of the functions for the master 3. Definition of necessary functions for the LLP algorithm at the slave
5. Send required data and functions to the slaves
7. Communicate with the slaves to perform computation
For the learning from label proportions, our implementation presumes a shared network file system and initially processes the data at the master such that the sliding windows of the measurements are stored as Robjects on the file system. Every task gets its pointer to the corresponding slice of data and the label proportions of neighbouring nodes. The LLP algorithm is executed in every task and the trained models (cluster centers and their labels) are again stored physically for later re-use. This also allows the deployment of the parallel learned models in embedded devices or the future application of the label proportion models in high performance computation settings. Next subsection analyses the communication cost of LLP in comparison to kNN algorithm.
Each node Pi transmits a matrix Q to each of its neighboring nodes, consisting of counts for each label Yd ∈ Y and batch. Such counts may be assumed to be integers. The maximum value of each integer is b, which means we need to reserve at most ⌈log2 b⌉ bits for each label. The number of batches, given n observations, is ⌈n/b⌉. The total number of bits zAGG for encoding matrix Q is therefore b zAGG = l n m ⌈log2 b⌉|Y | .
In comparison, the number of bits zALL required to encode all labels of n observations, for |Y | different labels, is at most (3) (4) zALL = n⌈log2 |Y |⌉ . The total costs are then either zAGG or zALL, multiplied by the number of nodes m. Here we assume that label information is broadcast to each neighboring node, which is not unrealistic for sensors in topologically close regions. All payloads reported in Sect. 5 base on this assumption.
The vulnerable data are the original sensor readings. These traffic flow measurements bare the risk of re-identification of individual vehicles. For example in a dense sensor network with sparse observations of vehicles, their occurrence may be tracked throughout the network. As mobility often is a regular behaviour and contains patterns this risk is even higher. In this section we show that our LLPlsm-based algorithm transforms the data such that re-identification risk is at most 1/s.
In our distributed setting, adversaries of a particular sensor node are malicious sensors that could use received measurements of neighboring sensors for deduction of individual mobility traces. The following attack model is possible: The adversary analyses differences among neighboring sensor readings and deduces individual movement. If the difference among two neighboring sensor readings is zero and both traffic flow counts are w, it is (depending on network topology) likely that w vehicles moved between the two sensors. In case of three neighboring sensors Pa, Pb, Pc their measurements va, vb, vc can be combined as follows: If va − vb = w = vc it may be deduced that on the way from Pa to Pb w vehicles turned to Pc, in case va − vb = −w = −vc w vehicles originated from the location Pa.
With our new LLPlsm-based approach we process discretized traffic flow values and just communicate counts of these value ranges. We denote the minimal (nonzero) interval width by s. Thus, measurements may not be distinguished up to a granularity of s vehicles and w is bounded by s, w
≥ s. In turn, the risk of re-identification with the hereby described attack model is at most 1/s. Our approach therefore provides s-anonymity by design. The aggregation of label information reduces the remaining risk for disclosure of neighboring labels at a malicious sensor node.
The solely transmission of label counts prevents
doubtless reconstructi
We perform tests of the method on data of the city of Dublin. The Sydney Coordinated Adaptive Traffic System (SCATS) provides information on vehicular traffic at over 750 fixed sensor locations as spatio-temporal time series (McCann, 2014). The data we use2 is a snapshot from 2Data is publicly available at http://dublinked.ie . 01/01/2013 till 14/05/2013, consisting of tuples (t, u, w), where u is the location of the observation and consists of an index for the junction, the arm and the lane number at which the sensor is located at. The metric w contains the aggregated vehicle count at sensor location since last measurement. The time stamp t denotes the recording time. 95 Accuracy of kNN vs. LLP-lsm with dif erent aggregations Payload of kNN vs. LLP with dif erent aggregations 90 65 kNN LLP-25 LLP-50 LLP-75 LLP-100 0 kNN LLP-25 LLP-50 LLP-75 LLP-100 model model 15, 50 different random starting points, 500 iterations at maximum) and label the clusters with LLPlsm (with 150 starts of the local greedy search) at each node for different batch sizes b = 25, 50, 75 and 100. The accuracy of each method is assessed by a 10-fold cross validation, i.e. all models are trained and evaluated for different hold-out sets 10 times. In total 296 × 7 × 10 = 20, 720 models for kNN need to be evaluated and 296 × 7 × 10 × 4 = 82, 880 models trained and evaluated for LLPlsm. The evaluation has been done offline in parallel on different machines (about 36 CPU cores).
Figure 1 shows the trade-off between accuracy and payload sent for kNN and LLPlsm trained on differently sized batches of aggregated labels. Besides the average accuracy over all 10-fold cross-validations at each node, the bars in Fig. 1 (left) also depict the standard deviation of accuracy over all nodes.
In general, LLPlsm performs slightly worse than kNN. Nevertheless, there are still many junctions for which the traffic flow is predicted quite well with LLPlsm. Some locations have bad performance with both methods, a comparison to the map reveals that these are locations of parking areas e.g. inner-city parking houses and recreational areas where many vehicles stay for a long period of time.
The task of scalable traffic flow prediction involves a tradeoff between the accuracy of models and the amount of communication between networked nodes. Especially in high performance computation and embedded devices commu
nication is costly.
In this paper we presented a novel approach for local models that trades-off communication costs to prediction accuracy which is suitable for in-network deployment and cluster computations.
Future work will focus on examining more sophisticated aggregation strategies for labels. We will study how to include dynamic distributed traffic flow prediction in stateof-the-art (multi-modal) route planning methods proposed by (Bast et al., 2014).
This research has received funding from the European Union’s Seventh Framework Programme under grant agreement number FP7-318225, INSIGHT. Additionally, this work has been supported by Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center SFB 876, project B3. We thank Jan Czogalla for data preprocessing. mental. Analysis of Freeway Traffic Time Series Data
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