Reconciling Predictions in the Regression Setting: An Application to Bus Travel Time Prediction (original) (raw)

Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives

International Journal of Forecasting

Forecast reconciliation is a post-forecasting process aimed to improve the quality of the base forecasts for a system of hierarchical/grouped time series (Hyndman et al., 2011). Contemporaneous (cross-sectional) and temporal hierarchies have been considered in the literature, but-except for Kourentzes and Athanasopoulos (2019)-generally these two features have not fully considered together. Adopting a notation able to simultaneously deal with both forecast reconciliation dimensions, the paper shows two new results: (i) an iterative cross-temporal forecast reconciliation procedure which extends, and overcomes some weaknesses of, the two-step procedure by Kourentzes and Athanasopoulos (2019), and (ii) the closed-form expression of the optimal (in least squares sense) point forecasts which fulfill both contemporaneous and temporal constraints. The feasibility of the proposed procedures, along with first evaluations of their performance as compared to the most performing 'single dimension' (either cross-sectional or temporal) forecast reconciliation procedures, is studied through a forecasting experiment on the 95 quarterly time series of the Australian GDP from Income and Expenditure sides considered by Athanasopoulos et al. (2019).

Hierarchical Forecasting and Reconciliation in The Context of Temporal Hierarchy

IRJET, 2022

The purpose of this study is to find a suitable forecast aggregation strategy for forecasting temporally aggregated hierarchical data series when the base level data exhibits a seasonal pattern. The study employs 10-year monthly data of foreign tourists visited in Kerala. Forecasting is essential for the four levels of hierarchy; the monthly, quarterly, half yearly and annual foreign tourist visit data. The forecasting strategies deliberated in the project are; bottomup approach, top-down approach, and the optimal combination approach with Ordinary least square (OLS) for reconciliation. The performance of different strategies is compared using the Mean Absolute Percentage Error (MAPE). The exponential smoothing techniques; single exponential smoothing, double exponential smoothing and triple exponential smoothing are used for forecasting individual series. The study concludes that the suitable forecast aggregation strategy for forecasting temporally aggregated hierarchical data series when the base level data exhibits a seasonal pattern is bottom-up approach. Bottom-up approach outperform all top-down approaches and optimal combination approaches on average and across all levels.

Bus travel time prediction: a log-normal auto-regressive (AR) modelling approach

Transportmetrica A: Transport Science, 2020

Providing real-time arrival time information of the transit buses has become inevitable in urban areas to improve the efficiency of the public transportation system. However, accurate prediction of arrival time of buses is still a challenging problem in dynamically varying traffic conditions especially under heterogeneous traffic condition without lane discipline. One broad approach researchers have adopted over the years is to divide the entire bus route into sections and model the correlations of section travel times either spatially or temporally. The proposed study adopts this approach of working with section travel times and developed two predictive modelling methodologies namely (a) classical time-series approach employing a seasonal AR model with possible integrating non-stationary effects and (b) linear non-stationary AR approach, a novel technique to exploit the notion of partial correlation for learning from data to exploit the temporal correlations in the bus travel time data. Many of the reported studies did not explore the distribution of travel time data and incorporated their effects into the modelling process while implementing time series approach. The present study conducted a detailed analysis of the marginal distributions of the data from Indian conditions (that we use for testing in this paper). This revealed a predominantly log-normal behaviour which was incorporated into the above proposed predictive models. Towards a complete solution, the study also proposes a multi-section ahead travel time prediction algorithm based on the above proposed classes of temporal models learnt at each section to facilitate real time implementation. Finally, the predicted travel time values were corroborated with the actual travel time values. From the results, it was found that the proposed method was able to perform better than historical average, exponential smoothing, ARIMA, and ANN methods and the methods that considered either temporal or spatial variations alone.

Clustering and Forecasting Urban Bus Passenger Demand with a Combination of Time Series Models

Mathematics

The present paper focuses on the analysis of large data sets from public transport networks, more specifically, on how to predict urban bus passenger demand. A series of steps are proposed to ease the understanding of passenger demand. First, given the large number of stops in the bus network, these are divided into clusters and then different models are fitted for a representative of each of the clusters. The aim is to compare and combine the predictions associated with traditional methods, such as exponential smoothing or ARIMA, with machine learning methods, such as support vector machines or artificial neural networks. Moreover, support vector machine predictions are improved by incorporating explanatory variables with temporal structure and moving averages. Finally, through cointegration techniques, the results obtained for the representative of each group are extrapolated to the rest of the series within the same cluster. A case study in the city of Salamanca (Spain) is presen...

Probabilistic forecast reconciliation: Properties, evaluation and score optimisation

European Journal of Operational Research

We develop a framework for prediction of multivariate data that follow some known linear constraints, such as the example where some variables are aggregates of others. This is particularly common when forecasting time series (predicting the future), but also arises in other types of prediction. For point prediction, an increasingly popular technique is reconciliation, whereby predictions are made for all series (so-called 'base' predictions) and subsequently adjusted to ensure coherence with the constraints. This paper extends reconciliation from the setting of point prediction to probabilistic prediction. A novel definition of reconciliation is developed and used to construct densities and draw samples from a reconciled probabilistic prediction. In the elliptical case, it is proven that the true predictive distribution can be recovered from reconciliation even when the location and scale matrix of the base prediction are chosen arbitrarily. To find reconciliation weights, an objective function based on scoring rules is optimised. The energy and variogram scores are considered since the log score is improper in the context of comparing unreconciled to reconciled predictions, a result also proved in this paper. To account for the stochastic nature of the energy and variogram scores, optimisation is achieved using stochastic gradient descent. This method is shown to improve base predictions in simulation studies and in an empirical application, particularly when the base prediction models are severely misspecified. When misspecification is not too severe, extending popular reconciliation methods for point prediction can result in a similar performance to score optimisation via stochastic gradient descent. The methods described here are implemented in the ProbReco package for R.

Effects of Data Characteristics on Bus Travel Time Prediction: A Systematic Study

Sustainability

The prediction of bus travel time with accuracy is a significant step toward improving the quality of public transportation. Drawing meaningful inferences from the data and using these to aid in prediction tasks is always an area of interest. Earlier studies predicted bus travel times by identifying significant regressors, which were identified based on chronological factors. However, travel time patterns may vary depending on time and location. A related question is whether the prediction accuracy can be improved with the choice of input variables. The present study analyzes this question systematically by presenting the input data in different ways to the prediction algorithm. The prediction accuracy increased when the dataset was grouped, and separate models were trained on them, the highest accurate case being the one where the data-derived clusters were considered. This demonstrates that understanding patterns and groups within the dataset helps in improving prediction accuracy.

Comparing state-of-the-art regression methods for long term travel time prediction

Intelligent Data Analysis, 2012

Long-term travel time prediction (TTP) can be an important planning tool for both freight transport and public transport companies. In both cases it is expected that the use of long-term TTP can improve the quality of the planned services by reducing the error between the actual and the planned travel times. However, for reasons that we try to stretch out along this paper, long-term TTP is almost not mentioned in the scientific literature. In this paper we discuss the relevance of this study and compare three non-parametric state-of-the-art regression methods: Projection Pursuit Regression (PPR), Support Vector Machine (SVM) and Random Forests (RF). For each one of these methods we study the best combination of input parameters. We also study the impact of different methods for the pre-processing tasks (feature selection, example selection and domain values definition) in the accuracy of those algorithms. We use bus travel time's data from a bus dispatch system. From an off-the-shelf point-of-view, our experiments show that RF is the most promising approach from the three we have tested. However, it is possible to obtain more accurate results using PPR but with extra pre-processing work, namely on example selection and domain values definition.

Forecast combination based forecast reconciliation: insights and extensions

2021

In a recent paper, while elucidating the links between forecast combination and cross-sectional forecast reconciliation, Hollyman et al. (2021) have proposed a forecast combination-based approach to the reconciliation of a simple hierarchy. A new Level Conditional Coherent (LCC) point forecast reconciliation procedure was developed, and it was shown that the simple average of a set of LCC, and bottom-up reconciled forecasts (called Combined Conditional Coherent, CCC) results in good performance as compared to those obtained through the state-of-the-art cross-sectional reconciliation procedures. In this paper, we build upon and extend this proposal along some new directions. (1) We shed light on the nature and the mathematical derivation of the LCC reconciliation formula, showing that it is the result of an exogenously linearly constrained minimization of a quadratic loss function in the differences between the target and the base forecasts with a diagonal associated matrix. (2) Endo...

Bus Travel Time Predictions Using Additive Models

2014 Ieee International Conference on Data Mining, 2014

Many factors can affect the predictability of public bus services such as traffic, weather and local events. Other aspects, such as day of week or hour of day, may influence bus travel times as well, either directly or in conjunction with other variables. However, the exact nature of such relationships between travel times and predictor variables is, in most situations, not known. In this paper we develop a framework that allows for flexible modeling of bus travel times through the use of Additive Models. In particular, we model travel times as a sum of linear as well as nonlinear terms that are modeled as smooth functions of predictor variables. The proposed class of models provides a principled statistical framework that is highly flexible in terms of model building. The experimental results demonstrate uniformly superior performance of our best model as compared to previous prediction methods when applied to a very large GPS data set obtained from buses operating in the city of Rio de Janeiro.