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Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

Various statistical model specifications for describing spatiotemporal processes have been proposed over the years, including the space–time autoregressive integrated moving average (STARIMA) and its various extensions. These model specifications assume that the correlation in data can be adequately described by parameters that are globally fixed spatially and/or temporally. They are inadequate for cases in which the correlations among data are dynamic and heterogeneous, such as network data. The aim of this article is to describe autocorrelation in network data with a dynamic spatial weight matrix and a localized STARIMA model that captures the autocorrelation locally (heterogeneity) and dynamically (nonstationarity). The specification is tested with traffic data collected for central London. The result shows that the performance of estimation and prediction is improved compared with standard STARIMA models that are widely used for space–time modeling.

En los últimos años, se han propuesto diversas especificaciones de modelado estadístico para describir procesos espacio-temporales. Esto incluye el modelo espacio-temporal autorregresivo integrado de media móvil (STARIMA) y sus varios derivados. Estas especificaciones de modelo asumen que la correlación de los datos puede ser adecuadamente descrita por parámetros que se fijan a nivel global en el espacio y/o tiempo. Dichos parámetros son inadecuados para los casos en los que las correlaciones entre los datos son dinámicas y heterogéneas, como en el contexto de los datos de la red. El objetivo de este artículo es describir la autocorrelación en los datos de red con una matriz de ponderación espacial dinámica y un modelo STARIMA localizado (LSTARIMA) que captura la autocorrelación local (heterogeneidad) de forma dinámica (no estacionariedad). La especificación del modelo es evaluada con datos de tráfico recolectados en el centro de Londres. Los resultados demuestran que los rendimientos de estimación y predicción mejoran con el método propuesto en comparación con los modelos STARIMA estándar que son ampliamente utilizados para el modelado de espacio-temporal.

通过设定多种统计模型来描述地理时空过程已提出多年,包括时空自回归移动平均(STARIMA)及其变形。此类模型通过假设数据相关性可由在时间域或者空间域上全局不变的参数加以充分描述。因此,上述模型不适用于具有动态或异质相关性的数据,如网络数据。本文试图采用一个动态空间权重矩阵与局部时空自回归移动平均(LSTARIMA)模型来描述数据的自相关程度,以此捕捉局域自相关(异质性)和动态自相关(非平稳性)。以伦敦市中心的交通数据作为模型的实证案例的测试结果显示,相对于广泛应用于时空过程分析的标准STARIMA模型,本文的模型在参数估计和预测性能上均有提升。


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

Modeling of spatiotemporal processes requires the use of specialized models that account for its special properties, the most widely studied of which is spatiotemporal autocorrelation. Autocorrelation is the tendency for near observations to be more similar than distant observations in space and/or time (see Cliff and Ord 1969; Box, Jenkins, and Reinsel 1970). Its presence violates the assumption of stationary, that is, that the data have constant mean and variance. Autocorrelation is inherent in spatiotemporal data and renders traditional statistical techniques such as ordinary least squares inefficient. However, to date, little research has been carried out about how autocorrelation varies in a spatiotemporal context and its implications for space–time modeling (Cheng, Haworth, and Wang 2012). This is reflected in the range of models that currently are applied to spatiotemporal data, with structures that are globally fixed both spatially and temporally (Pfeifer and Deutsch 1980; Giacinto 2006; Cheng, Wang, and Li 2011; Cressie and Wikle 2011).

In autoregressive integrated moving average (ARIMA) models (Tiao and Box 1981) and space–time ARIMA (STARIMA) models (Pfeifer and Deutsch 1980, 1981), the spatiotemporal series must be transformed to stationarity, usually by differencing. It then can be modeled using a combination of autoregressive (AR) and/or moving average (MA) terms. STARIMA is a generalization of a family of space–time models, with space–time AR MA (STARMA; Hooper and Hewings 1981), space–time MA, and space–time AR (STAR) models (Griffith and Heuvelink 2009) being special cases. The latter, which is more widely applied than STARIMA, does not include a MA term, and correlation in space and time is captured only by the AR parameters that are fixed globally both spatially and temporally. Elhorst (2001) provides a general framework for identifying AR-distributed lag models in space and time.

Spatial panel data models (Elhorst 2003; Elhorst, Piras, and Arbia 2010) can be specified to account for autocorrelation in one of two ways: either with a spatial AR process in the error term—a spatial error model (equivalent to a spatial MA)—or with a spatial AR-dependent variable—a spatial lag model. Parameter estimates for panel data models can be heterogeneous in space to account for local variations in autocorrelation properties. More recently, models have been put forward for continuous space–time panels (Oud et al. 2012). In space–time geostatistical models (Gething et al. 2008; Heuvelink and Griffith 2010), the spatiotemporal process is decomposed into a deterministic trend and a space–time stochastic residual, and a space–time covariance function is fitted to the residuals. Then, kriging is used for forecasting (Heuvelink and Griffith 2010). Characterization of the covariance function is usually simplified by assuming spatiotemporal stationarity, which may be restrictive. Furthermore, geostatistical models are less accurate when extrapolating than when interpolating. Griffith (2010) provides a good overview of the progress to date in statistical space–time modeling.

The assumption of a stationary spatiotemporal process has been shown to be unrealistic in some circumstances. For example, traffic theories say that the current conditions on a section of a road can be influenced by the previous conditions of adjacent road sections in either upstream or downstream1 directions depending on the degree of congestion (see, e.g., Lighthill and Whitham 1955; Richards 1956). In congested conditions, the influence comes mainly from downstream, whereas in free-flowing conditions, the influence comes from upstream. On a road network comprising hundreds or thousands of links, such spatiotemporal autocorrelation, structure is dynamic (in time) and heterogeneous (in space; Cheng, Haworth, and Wang 2012). For example, Cheng, Haworth, and Wang (2012) reveal that the spatiotemporal autocorrelation between unit journey times recorded on a 22-link subset of London's road network is spatially heterogeneous and temporally dynamic. Spatial heterogeneity results from variation in the level of correlation between individual road links across the study area. Temporal dynamics result from changes in the strength of correlation between locations over time, with stronger correlation apparent in the morning peak period than the interpeak and evening peak. Furthermore, the size of the spatial neighborhood also changes with time, becoming smaller in congested conditions and larger in free-flowing conditions. Capturing the most relevant information at any point in time is a problem of determining the instantaneous forecastability of neighborhood data using a dynamic spatial weight and a dynamic spatial neighborhood.

Progress has been made in introducing spatial heterogeneity and/or temporal dynamics to space–time models. Giacinto (2006) propose a generalized STARMA model that generalizes the STARMA model to include spatially varying AR and MA coefficients. The generalized STARIMA model of Min, Hu, and Zhang (2010) also allows the AR and MA parameters to vary by location and outperforms a standard STARIMA model in terms of forecasting accuracy. Although these methods allow for spatially dynamic parameter estimates, their spatial structure is fixed to an extent, as the size of the spatial neighborhood considered is the same for each location. They are also fixed temporally. Min et al. (2009) present a dynamic form of the STARIMA that accounts for temporal dynamics. They replace the traditional distance-weighted spatial weight matrix with a temporally dynamic matrix that reflects the current traffic turn ratios observed at each road intersection. The weight matrix can be updated in real time based on current conditions, but the method is limited to intersection-based flow data and is fixed spatially.

A simpler and more generalizable approach that accounts for both spatial heterogeneity and temporal nonstationarity draws from a set of models, each of which pertains to a certain traffic state. Min and Wynter (2011) devise a multivariate STAR model in which the weight matrix is selected from a set of templates reflecting typical traffic states. Average speeds are used to calculate a dynamic spatial neighborhood based on the number of links that can deliver traffic to the current location within the forecast horizon. This approach shows impressive forecasting performance for multiple steps ahead, and the authors claim it is scalable to large networks. Ding et al. (2011) propose a similar approach in which the weight matrix varies based on the current level of service. The approaches of Min et al. and Ding et al. go some way in accounting for the spatial and temporal variability in conditions on a road network. However, calculation of the templates they use is based on historical traffic conditions. Therefore, a natural tendency exists for them to perform better when conditions are close to average conditions. How well they perform when the conditions differ from the typical conditions captured in the templates is unclear.

The aim of this article was to describe the modeling of dynamic (transient) and heterogeneous autocorrelation in network data with improved traditional models that constitute a generic dynamic model capable of capturing the autocorrelation locally (spatial heterogeneity) and dynamically (temporal nonstationarity), providing an improvement over the traditional space–time series models. We incorporate the concepts of dynamic spatial weights and dynamic spatial neighborhood using a dynamic spatial weight matrix to model the spatial heterogeneity and temporal nonstationarity in network data.

Dynamic spatial weight matrices

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

This section describes the construction of a dynamic spatial weight matrix for road network data. The weight matrix has an adjacency and weight structure that is dynamic in time and space, which is updated based on the current traffic condition.

Spatial weight matrix for networks

A spatial weight matrix W encapsulates what we know or hypothesize about the structure and behavior of some phenomenon over space. When a temporal dimension is included, W must also encapsulate our knowledge or hypotheses about spatial structure and behavior over time. W comprises two components: a spatial adjacency structure and a spatial weighting structure.

Spatial adjacency

Drawing from graph theory, an arrangement of spatial units may be viewed as a graph G = (N, E) with a set N of n nodes and a set E of e edges joining pairs of nodes. The incidence structure of this graph is defined by the presence or absence of an edge (i, j) linking nodes i and j, and can be represented by an N × N binary 0–1 adjacency matrix in which nonzero elements signify edges (Peeters and Thomas 2009). Two nodes directly linked by an edge are adjacent and termed first-order spatial neighbors. The adjacency matrix containing all first-order relations between spatial units is termed its first-order adjacency matrix. Second-order spatial neighbors of a node are the first-order neighbors of its first-order neighbors (excluding itself) and so on. By following the paths between nodes in the graph, adjacency matrices W1, W2, … Wk of orders up to k can be defined. In networks, an alternative definition of adjacency is often used in which i and j are edges where a variable is observed, and the nodes represent connections between them. Black (1992) proposes this formulation, which has been applied to transport networks (Black and Thomas 1998) and migration flows (Chun 2008), among other phenomena, and is the definition we adopt in this study. Because they are often used to represent flows, networks can include another dimension, namely, direction. Transport networks fall somewhere between directed and undirected networks because vehicles can move only in one direction on an expressway while exerting influence in two directions. Therefore, the influence of upstream and downstream may be different, necessitating a different formulation for the spatial weight of each.

Spatial weights

Spatial weights Wij is the element of an adjacency matrix W that describes the perceived influence on spatial unit i of its neighbor j. A weight can be chosen in various ways, the simplest of which is the binary scheme previously outlined. Application-specific schemes include the length of shared border or distance between centroids in areal data (Cliff and Ord 1969), the length of road links in network data (Kamarianakis and Prastacos 2005), and the resources of actors in social networks (Leenders 2002). Row normalizing a spatial weight matrix (making all rows sum to one) is common practice. However, in some studies, column normalization has been used, allowing the matrix to represent influence exerted by i rather than accepted influence from j (Leenders 2002).

The choice of weighting scheme is nontrivial and can be very important because different weight matrices often lead to different inferences being drawn and can introduce bias into an analysis. The effect of this bias has been explored in the spatial literature (see Stetzer 1982; Florax and Rey 1995; Griffith 1996; Griffith and Lagona 1998) and the network literature (Páez, Scott, and Volz 2008). In spatiotemporal data, assuming that the relative contributions of the spatial neighbors of a unit remain the same across all times may not be reasonable. For example, weather conditions are spatiotemporally correlated, but the direction of dependence relates to wind direction, which constantly is changing. On road networks, the direction of dependence relates to traffic conditions (Chandra and Al-Deek 2009; Cheng, Haworth, and Wang 2012).

In the next section, we introduce the dynamic spatial weight matrix, which accounts for spatiotemporal nonstationarity by extending existing spatial weight matrix structures in the context of dynamic network processes.

A dynamic spatial weight matrix

A dynamic spatial weight matrix is a spatial weighting scheme that has the flexibility to account for autocorrelation structures that are nonstationary in time and/or heterogeneous in space. This is achieved by incorporating two key concepts: a dynamic spatial neighborhood, which means the size of a spatial neighborhood can change at each instance of time, and a dynamic spatial weight, which means that the influence of each neighbor can change at each time step. These two concepts can be incorporated into existing space–time model structures by modifying the spatial weight matrix W at link i with a time-varying matrix, W(h,t), where t is a time index, and h = 1, 2, … , m is the spatial order at time t. In this specification, a spatial neighborhood is updated by changing the value of h; m is the maximum size of a spatial neighborhood, and a spatial weight is updated by changing its i, j element in W(h,t), where i, j = 1, 2, … , N are the spatial units. Updating these values is application specific and depends on a researcher's view of the nature of the autocorrelation between locations. To illustrate this point, we use the example of road networks, which are networks of flows where spatiotemporal correlation depends on the traffic state.

A dynamic spatial neighborhood of road networks

Assuming a set of link (road section) based traffic data are available, sampled at a regularly spaced time interval Δt (e.g., five minutes). The effective spatial neighborhood m(t) of a link i can be defined as the neighborhood of links that can deliver traffic to link i within Δt, based on the information available at time t – 1. Links that fall outside the range of Δt cannot have an influence and hence are excluded. This definition ensures a parsimonious specification of a spatial neighborhood. Such a neighborhood is larger in free-flowing conditions and smaller in congested conditions.

A dynamic spatial weight of road networks

Here, we assume that prediction is based on historic travel time, and the dynamic spatial weight between a pair of links is calculated as a function of their relative traffic speeds. If we see a drop in speed on one link, we expect this decrease to translate into a drop in speed on an adjacent link, and vice versa. For all link pairs (i, j) that are of spatial lag h, the corresponding inline image is defined as

  • display math(1)
  • display math(2)

where vi(t) and vj(t) are the respective average traffic speeds on links i and j at time t. The entry wij takes the value of zero if the spatial lag between i and j is not h.

Here, the neighborhood matrix tends to equilibrate the speed differentials over space. Given a link pair (i, j), suppose that traffic on link i proceeds at a lower speed than on an upstream link j (i.e., vi(t) < vj(t)), which gives wij > 0. The contribution of traffic from link j increases the travel time on link i due to the arrival of higher speed upstream flow. In contrast, suppose that traffic on link i proceeds at a lower speed than on a downstream link j (i.e., vi(t) < vj(t)), which gives wi,j < 0. The contribution of traffic from link j decreases the travel time on link i due to its higher speed downstream flow.

A localized space–time model: LSTARIMA

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

In this section, we define a new space–time model, the localized STARIMA (LSTARIMA) model. Like the traditional STARIMA model, LSTARIMA makes use of a spatial weight matrix W to model the influence of the spatiotemporal neighborhood. However, it relaxes the globally fixed parametric structure of STARIMA by allowing the AR and MA parameters to vary by location, which allows it to account for spatial heterogeneity. Furthermore, it accounts for temporal nonstationarity by allowing the size of the spatial neighborhood to vary with time. In this sense, it is a locally dynamic space–time model.

Model specification

Let z(t) be an N-dimensional column vector containing the observations zi(t) on each link i, where i = 1, 2, … , N, during each time interval t, where t = 1, 2, … , T. The conventional STARIMA model can be defined as

  • display math(3)

The first term in equation (3) is the AR component, whereas the second term is the MA. The parameters p and q are the AR and MA orders, respectively. The term ε(·) is an N-dimensional column vector of residuals on each link, and h is the spatial order, which represents the order of spatial separation between two locations. The parameters mk and nl are the spatial orders associated with the kth and lth temporally lagged terms in the AR and MA components, respectively. They specify the size of the spatial neighborhood that could influence the link of interest i within temporal lags k and l. The notation ϕkh and θlh are the AR and MA parameters, respectively, to be calibrated for the entire network. The matrix W(h) is an N × N spatial weight matrix for spatial lag h, containing the set of weights wij specifying the assumed relationship between i and j (see Kamarianakis and Prastacos 2005; Getis 2009). The number of parameters to be calibrated in equation (3) is p × mk + q × nl.

We extend the standard STARIMA model to account for spatial heterogeneity and temporal nonstationarity using the following formulation, which we call LSTARIMA:

  • display math(4)

where W(h,t–k,i) and W(h,t–l,i) are the elements of dynamic spatial weight matrix W(h,t) pertaining to link i at temporal lags k and l. LSTARIMA has a separate set of AR and MA parameters for each link i, which are stored in N × N diagonal matrices Φkh and Θlh such that

  • display math(5)

where [ϕi,kh] and [θi,lh] are the parameters for each link i (i = 1, 2, … , N). The number of parameters that needs to be calibrated in equation (4) is (p × mk(t−k,i) + q × nl(t−l,i), although the whole model needs to calibrate N links of the entire network.

The STARIMA and ARIMA models can be viewed as special cases of the LSTARIMA model. For example, if p1 = p2 = pN and q1 = q2 = qN (i.e., p and q are spatially fixed), mk(t − 1, i) = mk(t − 2, i) = … = mk(t − k, i) and nl(t − 1, i) = nl(t − 2, i) = … = nl(t − l, i) (i.e., the spatial influence of adjacent links does not change over time), and [ϕ1,kh] = [ϕ2,kh] = [ϕN,kh] and [θ1,lh] = [θ2,lh] = … = [θN,lh] (i.e., all parameters are the same for all of the links), then LSTARIMA (equation (4)) becomes a STARIMA model (equation (3)). This is unlikely for road network data, but for other data sets such as annual temperature, spatial autocorrelations may not change rapidly over time. Moreover, if mk(t − k, i) = 0 and nl(t − l, i) = 0 (i.e., the adjacent links produce no spatial influence), then the LSTARIMA becomes an ARIMA. This reduction might happen when traffic is flowing freely (or possibly is highly congested), resulting in the speeds on all the links being more or less the same.

In the next sections, we use LSTARIMA (pi, qi), ARIMA (pi, di, qi), and STARIMA (p, d, q) specifications to represent the models because differencing (which is described by parameter d) is not needed for the LSTARIMA model.

Model calibration

Equation (4) is a time series model that considers the spatial influence of adjacent links in networks. Thus, its parameters can be estimated by means of standard time series calibration algorithms. The procedure of parameter estimation in equation (4) can be regarded as the minimization of the following sum of squared errors function:

  • display math(6)

where T is the number of observations in time, zi(t) is the observation vector at time t and link i, εi(t) is the random error vector at time t, and inline image.

Equation (6) presents a nonlinear least squares minimization problem because εi(t) is required for the calibration of the MA parameter θlh, but is unknown a priori. Thus, εi(t) must be estimated first in order to determine ϕkh and θlh. Furnishing appropriate starting values is important to ensure convergence of the optimization procedure. Although trial and error can be used to implement the parameter optimization process, it cannot guarantee convergence. Moreover, an exhaustive search is needed, which is time consuming.

Hannan and Rissanen (1982) demonstrated that the Hannan–Rissanen (HR) algorithm is an effective approach for parameter optimization of time series models. It makes use of the residuals of a high-order AR model to feed the εi(t) as initialized values. Then, parameters ϕkh and θlh are calibrated using the linear least squares method. The methodology has been proven valid in their work and has been broadly accepted in practice. Here, we use the same procedure for LSTARIMA model calibration by considering εi(t) to be an independent random variable after the spatiotemporal autocorrelation has been fully modeled by the dynamic weight matrix. This result is verified by testing for spatial, temporal, and spatiotemporal autocorrelation in the predictive residuals in the subsequent case study.

Although Box and Jenkins's algorithm (Box, Jenkins, and Reinsel 1970, pp. 498–505) was proposed earlier than the HR algorithm, it does not provide the details of how to furnish appropriate initial values (epsilon) to ensure convergence of the optimization procedure of model calibration. Given the implementation method and that mathematical proof has been provided (Hannan and Rissanen 1982), HR algorithm is commonly used in practice and is chosen here to calibrate our model.

A case study

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

The LSTARIMA model is designed specifically to deal with highly heterogeneous and nonstationary spatiotemporal (network) processes, an example of which is road traffic. On traffic networks, the spatiotemporal relationship between observations recorded at detector locations is dependent on the traffic state, which is constantly changing. In this section, we use an empirical example to demonstrate the model-building procedure for a LSTARIMA model in the context of travel time prediction on London's road network.

The study area and road traffic network data

The case study area is identical to that used in Cheng, Haworth, and Wang (2012). It consists of 22 road links located in central London, United Kingdom, and is shown in Fig. 1a. The topological representation of the test network can be seen in Fig. 1b. The average length of road links in the test network is 1.4 km, with minimum and maximum lengths of 0.473 and 3.85 km, respectively.

figure

Figure 1. Selected road networks in central London: (a) spatial location of selected links in central London and (b) network diagram of links; arrows represent traffic flow direction, and numbers are link IDs (Cheng, Haworth, and Wang 2012).

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The data are travel times collected using automatic number plate recognition technology by Transport for London (TfL) as part of its London Congestion Analysis Project. The raw observations are five-minute aggregated travel times (in seconds). The data set spans 166 successive days (Monday to Sunday), from May 24, 2010 to November 5, 2010, which were selected after discussions with TfL. To reduce noise in the data, only the period between 6:00 am and 9:00 pm is used because capture rates often are low overnight.

The traffic pattern differs between weekends and working days, and within the working week, it differs from Monday to Friday. Wednesday is considered to be a neutral day during the week. Strictly speaking, traffic is different everyday (even for a single link), even between Saturday and Sunday. A recent investigation into the autocorrelation structure of traffic data in a study by Cheng, Haworth, and Wang (2012) demonstrate that the autocorrelation of the traffic road network is nonstationary in both time and space, revealing that the data violate the assumption of spatial homogeneity and temporal stationarity of the STARIMA model. Many ways exist to deal with these kinds of differences in a traffic study, for example, modeling each day of the week separately or grouping them into working days and weekends. Differencing either daily and/or weekly normally is required to transform the time series into stationary data so that it can be accommodated by an ARIMA or STARIMA model. The proposed LSTARIMA model tackles this challenge by using a dynamic weight matrix, which requires no such transformation.

To train and predict the LSTARIMA model, the data set was separated into two subsets: the training set (May 24 to October 24, 2010; 154 days) for calibrating the model parameters and the testing set (October 25 to November 5, 2010; 12 days) for evaluating the prediction performance of the model. The raw travel time data have been converted to unit travel times (seconds/kilometer) to allow comparability between travel times on links of differing lengths.

Construction of a dynamic spatial weight matrix

As discussed, the impact of the spatial neighborhood can be modeled using a dynamic spatial weight matrix. The following three steps achieve this purpose: (1) build a spatial adjacency matrix; (2) determine the dynamic spatial order; and (3) calculate the dynamic spatial weights.

Step 1: build a spatial adjacency matrix

The first step is to build a spatial adjacency matrix based on the topological structure of the network, which appears in Fig. 1. Spatial adjacency matrices of spatial order up to three were constructed using the method described in this article. Fig. 2 shows the sketch map of W(1), where 1 indicates that two links are spatially adjacent, and “-” (zeros have been replaced with “-” for clarity of presentation) shows that they are not. This matrix is used in both the STARIMA and LSTARIMA model specifications. Several nodes in the adjacency matrix have entries of zero (in the respective column/row). This is the border effect. The second order of adjacency of these nodes is much more connected. When this adjacency definition is applied to a large network, such boundary effects are not substantial.

figure

Figure 2. Sketch map of spatial adjacency matrix W(1).

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Step 2: determine the dynamic spatial order

The second step is to determine the dynamic spatial order for each link in the network using the method described in this article. Fig. 3a and b shows sketch maps of the dynamic spatial order of L463 between 6:00 am and 9:00 pm on May 24, 2010. Fig. 3a reveals that, in most cases, L463 can deliver its traffic only to its first-order spatial neighbor within the forecasting horizon and rarely reaches its second-order spatial neighbor. In contrast, Fig. 3b portrays the situation upstream, revealing that, in most cases, L2301 (a second-order spatial neighbor of L463) can deliver its traffic to L463. Investigation of the dynamic spatial order for all 22 links reveals considerable variation across links, with a maximum spatial order of three.

figure

Figure 3. Sketch map of dynamic spatial adjacency of L463 at (a) downstream and (b) upstream.

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Step 3: calculate the dynamic spatial weights

The third step is to calculate dynamic spatial weights by means of equation (1), which is the difference of the average speeds of two adjacent links divided by the speed of the target link (i.e., L463). Here, the average speed of each link is the reciprocal of unit travel time. Table 1 shows a snapshot of the dynamic spatial weight matrix of L463 for 7:00–9:00 am on May 24, 2010. The weights, which are updated every five min, represent the strength of spatial influence of adjacent links on the target link. In this case, L463 involves three upstream links (L2301, L2007, and L1616) and two downstream links (L1593 and L2324). Table 1 shows that the weights vary across time, reflecting the dynamics of the spatial influence of the road traffic network.

Table 1. Snapshot of the Dynamic Spatial Weight of L463
Spatial order hUpstream linksDownstream links
FirstSecondFirstSecond
Temporal order kL2007 (len = 0.6 km)L2301 (len = 3.7 km)L1616 (len = 0.5 km)L1593 (len = 1.2 km)L2324 (len = 0.6 km)
7:0008.710.21−1.820
7:0508.80−0.25−3.130
7:1002.31−0.070.000
7:15010.290.87−1.420
7:2006.391.56−0.290
7:2503.702.000.340
7:3002.920.210.400
7:3503.571.180.060
7:4003.000.83−0.700
7:4501.870.25−0.230
7:500.3700.24−0.450
7:55−0.1400.320.210
8:000.0600.03−0.040
8:050.001.530.980.070
8:100.1200.91−0.180
8:150.001.960.540.00−0.08
8:200.2500.320.040
8:250.3600.510.200
8:300.2500.43−0.360
8:350.2000.49−0.060
8:400.2400.65−0.050
8:450.3800.26−0.760
8:5002.320.62−0.570
8:5503.521.27−1.690
9:0003.811.21−0.970

Model training

The LSTARIMA model is trained using the HR calibration algorithm (Hannan and Rissanen 1982). One model must be trained for each AR order p and MA order q. This training can be time consuming, so the search range must be constrained to realistic values. In this case, p and q are varied from one to five because we expect the correlation to decline after five time lags (30 min at 5-min aggregation intervals) based on spatiotemporal autocorrelation analysis. Then, the best-performing model is chosen from these possibilities.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

This section summarizes the experimental results. First, we introduce three models that we use to benchmark the model. Then, we examine the results in terms of predictive accuracy, model structure, and residual autocorrelation analysis.

Benchmark models

To assess the performance of the LSTARIMA model, we compare its accuracy with three other models: a naïve model, the ARIMA model, and the STARIMA model.

The naïve model, also called a random walk model, has the following form (Thomakos and Guerard 2004):

  • display math(7)

where z(t – 1) is a historical observation at time t – 1, inline image is the predicted value at time t, and εt is a zero-mean residual. The forecast of the naïve model is the previously observed data point in the series (nothing changes). This model is the simplest forecasting model in that the minimum requirement for any more complicated model is to outperform it.

The ARIMA and STARIMA models are two popular models that can be used for benchmarking. Both have proved reliable in the traffic-forecasting context (Thomakos and Guerard 2004; Kamarianakis and Prastacos 2005; Min and Wynter 2011). For consistency, the HR algorithm that is used to train the LSTARIMA model is used to train the ARIMA and STARIMA models. Again, AR and MA orders of one to five are tested. For the STARIMA model, spatial orders of one to three are fixed globally and tested. After an extensive search, the preferred model is found to be a STARIMA (4, 0, 3) at spatial order two. Because the LSTARIMA model extends the principles of the ARIMA and STARIMA models, it should outperform both models in the majority of cases in order to be preferable.

Prediction accuracy

To assess the prediction accuracy of each of the models, the root mean squared error (RMSE) index is used. Fig. 4a shows a bar graph of RMSE of the best-performing models for each link at five-minute intervals. The more sophisticated models perform only slightly better than the naïve model at this aggregate level. The traffic in the test data set generally does not change much during five-minute time increments, making the naïve model a strong predictor.

figure

Figure 4. Predictive accuracy (RMSE) of LSTARIMA vis-à-vis the benchmark models at (a) 5-, (b) 15-, and (c) 30-min intervals.

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However, we do observe that the LSTARIMA model has the lowest average RMSE of the models for this time interval. The second best-performing model is the ARIMA model, followed by the naïve model. The most surprising result is that the STARIMA model performs worse than the naïve model. The STARIMA model uses global parameter coefficients and a fixed spatial adjacency structure to explain spatiotemporal autocorrelation and thus cannot capture the spatiotemporal nonstationarity and heterogeneity of travel time on the road traffic network, even after differencing. Fig. 4b and c show the bar graphs of RMSE at 15- and 30-min intervals. The results are similar to those for the five-minute interval, with the LSTARIMA being the best-performing model. The naïve model, however, shows a sharp decrease in performance at these levels of aggregation.

To assess the differences in predictive performance of the various models, a pairwise F-test was carried out with their residuals. An F-test (Snedecor and Cochran 1989) is a test of statistical significance of the ratio of two sample variances and is used here to test if the residual variance of one model is significantly less than that of another (one-tailed test). Because the residuals of the fitted models have zero mean, their variances can be used as a measure of model performance, with smaller values being preferred. The ratio of the variances of the residuals of a pair of models is used to calculate the F-value using the following expression:

  • display math(8)

where inline image and inline image are the error variances. The quotient of equation (8) would have the F-distribution if the error terms were independent in time and space; failing this, Pitman's t-test (Pitman 1939) can be used, although in the present case, the results from the more straightforward but approximate F-test are so strong as to be conclusive. In this case, we test the null hypotheisis that “model 1” (i.e., LSTARIMA) is no better than “model 2” (i.e., STARIMA). This is a one-tailed (left-side) F-test because the alternative hypothesis is that model 1 performs better as indicated by a smaller error variance with an F-value less than 1. The significance level is set to 0.05. If the probability value of the F-test is less than 0.05, there is sufficient evidence to reject the null hypothesis that model 1 performs no better than model 2.

The F-test of residuals between LSTARIMA and STARIMA shows that LSTARIMA performs better than STARIMA on all 22 links at 5-, 15-, and 30-min intervals (with P-values less than 0.05). The same tests are applied to other group comparisons (i.e., LSTARIMA and the naïve, and LSTARIMA and ARIMA). The F-test results for LSTARIMA and the naïve model also indicate that LSTARIMA performs significantly better. Although the comparison between LSTARIMA and ARIMA does not indicate a significant difference in predictive accuracy, LSTARIMA is superior to ARIMA in terms of RMSE link by link, achieving better accuracy on 22, 18, and 12 links at the 5-, 15-, and 30-min intervals, respectively.

Model structure

For the ARIMA and LSTARIMA models, one model must be built for each link, whereas only one STARIMA model is required for the entire network. Both ARIMA and STARIMA involve preprocessing as part of model fitting to remove trend and cyclical patterns. The strategy adopted here involves testing combinations of differencing and logarithmic transformations. Examples of the preferred models include ARIMA (3, 0, 1) with a logarithm transformation for L463, and ARIMA (2, 0, 1) and ARIMA (4, 0, 2) without a transformation for links L1593 and L2324, respectively. Table 2 summarizes the AR and MA orders for each of the best models across the test network.

Table 2. AR and MA Orders (p and q) of the LSTARIMA and ARIMA Models for All 22 Road Links
LSTARIMA
ParsL1025L2301L2007L1616L524L463L1593L2324L2085L432L1592
p12321422332
q13441212441
ParsL425L2140L1384L2079L1419L474L1447L1623L2052L448L2055
p32324111111
q23411122341
ARIMA
ParsL1025L2301L2007L1616L524L463L1593L2324L2085L432L1592
p42413324332
d00200000001
q13241112421
ParsL425L2140L1384L2079L1419L474L1447L1623L2052L448L2055
p44333244424
d21010001000
q13212211111

Table 2 reveals that different p and q values exist for LSTARIMA and ARIMA models on the same links. For example, at link L1025, ARIMA (4, 0, 1) achieves the best accuracy, whereas the preferred LSTARIMA model is LSTARIMA (1, 0, 1). This discrepancy may be caused by the spatial contribution of adjacent links in the LSTARIMA specification. Statistical results for the 22 road links show that, generally, a simpler LSTARIMA model with smaller p and q values can perform better than a complicated ARIMA model with larger p and q values. The best STARIMA model is STARIMA (4, 0, 3), which also has larger p and q values than most LSTARIMA models.

Residual autocorrelation analysis

Pure predictive accuracy is not the only measure of a good spatiotemporal model. The model also must be able to capture the dynamics of the underlying spatiotemporal process and to account for spatiotemporal autocorrelation and nonstationarity in data. One way of evaluating this is to test the predictive residuals of a model for autocorrelation. If a model truly captures the underlying process, then its residuals should be independent and identically distributed random variables. To assess this, we examine the temporal, spatial, and spatiotemporal autocorrelation in the residuals of the LSTARIMA, STARIMA, and ARIMA models. Strictly speaking, we do not think autocorrelation functions (ACFs) and Moran's I are good indicators for space–time data because, unlike the space–time ACF (ST-ACF), they cannot capture the interaction between the spatial and temporal dimensions. However, they are presented here for reference because most temporal and/or spatial models use these indicators.

Fig. 5a–c shows snapshots of the residual profile (three days from October 25 to 27, 2010) on L1025 for each of the models at a five-minute interval. Visual inspection suggests a cyclical pattern in the residuals of the ARIMA (Fig. 5b) and STARIMA (Fig. 5c) models that is stronger than one in the LSTARIMA residuals. The largest residual of the LSTARIMA model (around 150 s) is smaller than the other two (around 200 s). AR conditional heteroskedastic processes exist in the data (stochastic volatility in the variances), which are very common in series of high frequency, such as the traffic data here.

figure

Figure 5. Snapshot of profile of residuals for three models (three days from October 25 to 27, 2010) on L1025. (a) LSTARIMA, (b) ARIMA, and (c) STARIMA.

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Temporal autocorrelation

We investigate the statistical significance of this pattern using the temporal ACF. Fig. 6a–c shows the ACF plots of the entire residual series of each model.

figure

Figure 6. ACF analysis of residual series of the three models on L1025. (a) LSTARIMA, (b) ARIMA, and (c) STARIMA.

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The most striking observation is that the ACF plot of the STARIMA model (Fig. 6c) shows significant autocorrelation up to temporal lag 3 (two negative correlations, one positive correlation). Similar results are observed when examining the residuals of the STARIMA model on other links, indicating that the STARIMA specification cannot fully capture the temporal autocorrelation in the network. This problem is caused by the global parametric structure of the STARIMA model and has been avoided in the LSTARIMA (and ARIMA) model because it relaxes the assumption of fixed temporal dependence for all locations, allowing the AR parameters p and MA parameters q to change across space (links). The ACF plots for the LSTARIMA and ARIMA models appear broadly similar, although the ARIMA model has more significant spikes, indicating that the LSTARIMA model performs slightly better in dealing with temporal autocorrelation. Although the coefficients keep on changing from positive to negative, or vice versa, there is no regular wave-shaped pattern. This outcome indicates that the ARIMA model and the LSTARIMA model, in particular, can account for most of the temporal autocorrelation in the data. However, all three models exhibit significant residual autocorrelation at a lag of one day (180), indicating that the daily cyclical pattern is not fully accounted for. Although this feature does not affect the performance of the models, it will be addressed in further research.

Spatial autocorrelation

Spatial autocorrelation is tested for by calculating the value of local Moran's I (abbreviated to LISA [local indicator of spatial autocorrelation], after Anselin 1995) for each hour between 6:00 am and 9:00 pm on October 25, 2010. Fig. 7a–c shows level plots of the LISA of all 22 links at spatial order one. In general, Fig. 7a–c reveals that positive and negative autocorrelation is staggered over space and time, and the minority of LISA values is statistically significant, indicating that the residuals of the three models seem to approximate a random distribution in space. However, the number of statistically significant cells increases from Fig. 7a–c. Specifically, there are 35 statistically significant cell values in Fig. 7c, indicating that the spatial autocorrelation at those times and positions (link) cannot be fully captured by the STARIMA specification. The LSTARIMA model has the fewest cells that are statistically significant—only 10 compared with 13 for ARIMA—indicating that the LSTARIMA model is the most effective model for capturing spatial autocorrelation.

figure

Figure 7. Map of LISA of all 22 links at spatial order one on October 25, 2010 for the three models, where the links label the X-axis and time labels the Y-axis (6:00 am–9:00 pm). (a) LSTARIMA, (b) ARIMA, and (c) STARIMA. Gray scale is the strength of LISA; cells with black borders indicate statistically significant LISA values (P-value < 0.05).

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Spatiotemporal autocorrelation

Finally, we evaluate the residual spatiotemporal autocorrelation for each of the models. The indicator we use is the spatiotemporal ACF (STACF; Martin and Oeppen 1975). Fig. 8a–c shows the STACF plots for the three models at spatial order one and the five-minute interval. Strong, significant spatiotemporal autocorrelation is observed at temporal lags 1–3 in the residuals of the STARIMA model. The STACF plots of the LSTARIMA and ARIMA models are comparatively closer to a random distribution over space and time. However, the ARIMA model displays moderately higher spatiotemporal autocorrelation at temporal lags 1–10.

figure

Figure 8. STACF plots for three models at first-order spatial weight matrix. (a) LSTARIMA, (b) ARIMA, and (c) STARIMA.

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Summary of results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

In summary, the F-test results confirm that the LSTARIMA model achieves the best overall performance across all time frames, followed closely by the ARIMA model. The STARIMA model does not perform as well and only performs better than the naïve model when the temporal interval of prediction increased to 30 min. The LSTARIMA appears to be a better model than the STARIMA because of greater parameter flexibility (dynamic spatial neighborhood and dynamic spatial weight). Although the F-test indicates that the LSTARIMA and ARIMA models are equivalent, higher predictive accuracy for individual links is obtained with the LSTARIMA model. The principle of ARIMA seems simpler than LSTARIMA, but the structure of the model is more complicated than that for the LSTARIMA model because larger p and q values are obtained for the estimated ARIMA models.

The residuals from the three models are not fully independent (in time and in space). This implies that preprocessing for the ARIMA and STARIMA models may be inappropriate. This possibility supports our previous supposition in Cheng, Haworth, and Wang (2012) that the extraction of a globally stationary spatiotemporal process through differencing may not be realistic. Although some preprocessing for the LSTARIMA model may also be needed, its predictive residuals display almost negligible temporal, spatial, and spatiotemporal autocorrelation, suggesting that preprocessing seems unnecessary. Therefore, we maintain that preprocessing is not needed for the LSTARIMA model, especially for practical purposes because finding an adequate transformation given the results for the ARIMA and STARIMA models is very difficult. No data preprocessing is a great advantage of the LSTARIMA specification, as is its simpler structure (low p and q values) compared with the ARIMA and STARIMA specifications.

Discussion and conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

Existing space–time models such as STARIMA are designed to account for the effects of autocorrelation in spatiotemporal data. Their global parametric structure does not equip them to deal with spatial heterogeneity and temporal nonstationarity, even though these are characteristic of many spatiotemporal data sets, with networks of traffic flows being just one example. As we demonstrate in this article, extracting a stationary spatiotemporal process from highly nonstationary data can be unrealistic, limiting the efficacy of existing global models. This article addresses the shortcomings of such models by introducing a new concept: the dynamic spatial weight matrix. This matrix comprises the following three components that enable local dynamics to be incorporated into established model structures: an adjacency structure, a dynamic spatial neighborhood, and a dynamic spatial weight.

We demonstrate application of the dynamic spatial weight matrix by extending the STARIMA model specification to create a new space–time model—the LSTARIMA model—which is aimed at improving the ability of the STARIMA model to cope with the dynamics and heterogeneity of spatiotemporal data on networks. This new model specification relaxes the assumption of fixed temporal dependence for all locations, allowing the AR parameters p and MA parameters q to change across space. By introducing space-varying Φkh and coefficients for each location, spatial heterogeneity is substantially accounted for in the specification. This model framework is generic, and we demonstrate that the ARIMA and STARIMA models are special cases of it.

We examine the efficacy of the LSTARIMA model with the case of road traffic networks in London. Unlike the traditional (ST)ARIMA models, our experiment demonstrates that the LSTARIMA model works well without the need for data preprocessing (e.g., a logarithmic transformation and differencing), used widely with (ST)ARIMA models, suggesting that the LSTARIMA model is capable of dealing with traffic time series better than the (ST)ARIMA models. With smaller p and q values, the LSTARIMA model has also a simpler structure than the (ST)ARIMA model. This results from the innovative dynamic spatial weight matrix used in the LSTARIMA specification, which allows it to capture unstable traffic states in a space–time series.

Although we examine only the case of road traffic networks in this study, one of the key strengths of our methodology is its modularity. Each of the three components of the dynamic spatial weight matrix (i.e., an adjacency structure, a dynamic spatial neighborhood, and a dynamic spatial weight) can be modified according to the application of interest. For example, within the transport setting, this model could be used to predict traffic flows and speeds by replacing the speed-based dynamic spatial weight with a flow-based weight. This substitution could be linked to, for example, kinematic wave theory (Lighthill and Whitham 1955). Modification of the spatial adjacency structure and the dynamic spatial neighborhood may not even be needed in this case. Furthermore, the method could be applied to flows on other networks, such as the Internet, where the heterogeneity may result from global and local shifts in usage. This application would require domain-specific knowledge to be incorporated into each of the three elements of the dynamic spatial weight matrix but would not change the overall model structure. Moreover, the method is not limited to networks, and a range of other spatiotemporal processes exist that may benefit from its application, such as environmental monitoring and house price forecasting. In the latter case, the update of dynamic spatial weights may be driven by knowledge of exogenous factors in the wider (local) economy. As an aside, the dynamic spatial weight matrix framework is not dependent on the LSTARIMA estimator algorithm used here.

The algorithm can be seen as a fourth module, and there is scope to replace the linear model used here with nonlinear models from the field of machine learning, such as kernel-based approaches (Wang, Cheng, and Haworth 2012). Combining this with geographically weighted regression (GWR) could lead to a localized dynamic GWR, that is, a spatiotemporal GWR. This will be the direction for our future research.

In conclusion, the results presented here demonstrate that the combination of a dynamic spatial weight matrix and the LSTARIMA model specification is a viable approach for space–time modeling of highly dynamic, heterogeneous network processes. Further studies will be carried out on a city-wide scale to demonstrate the effectiveness of the approach in a real-time, applied setting. Furthermore, the general framework presented here lends itself to applications in a wide range of network and spatial processes. All that is required is to define an application-specific adjacency structure, a dynamic spatial weight matrix, and a dynamic spatial neighborhood that can replace the traffic-specific formulations given here.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References

We thank Transport for London for providing the journey time data. The research for this article was carried out under the STANDARD project, sponsored by the U.K. Engineering and Physical Sciences Research Council (EP/G023212/1). Comments from the editor and the anonymous reviewers are greatly appreciated and improved the readability of the article. The authors bear sole responsibility for any mistakes that may appear.

Note
  1. 1

    In this context, downstream refers to the direction in which traffic flows, and upstream the opposite direction.

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  2. Abstract
  3. Introduction
  4. Dynamic spatial weight matrices
  5. A localized space–time model: LSTARIMA
  6. A case study
  7. Results
  8. Summary of results
  9. Discussion and conclusions
  10. Acknowledgements
  11. References
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