An incremental approach to forecasting and classification of taxi demand based on evolving fuzzy systems

Abstract

Private transport has become a viable and increasingly popular alternative to urban transportation. However, with this growth, an old and recurring problem becomes more latent: the relationship between passenger demands and taxi supply. This problem suggests the creation and use of techniques which make it possible to reduce the gap between the demand for taxi passengers and the effective contingent of vehicles needed to meet this demand. This work introduces a new approach to forecasting and classifying taxi passengers’ demands. The proposed approach uses historical data from taxi rides and meteorological data. The Kruskal-Wallis method identifies the most relevant variables, and an evolving fuzzy system performs demand forecasting/classification. Five evolving systems are evaluated with our approach: Autonomous Learning Multi-Model (ALMMo), evolving Multivariable Gaussian Fuzzy System (eMG), evolving Fuzzy with Multivariable Gaussian Participatory Learning and Recursive Maximum Correntropy (eFCE), evolving Fuzzy with Multivariable Gaussian Participatory Learning and Multi-Innovations Recursive Weighted Least Squares (eFMI), and evolving Neo-Fuzzy Neuron (eNFN). In addition, computational experiments using real-world data were conducted to evaluate and compare the performance of the proposed approach. The results revealed that it obtained performance superior or comparable to state-of-the-art ones. Therefore, the experimental results suggest that the proposed approach is promising as an alternative for forecasting and classifying taxi passenger demand.

Keywords

Taxi demand forecasting classification evolving systems fuzzy systems

1 Introduction

Taxi rides are an attractive alternative to traditional public transport, as they offer efficient and convenient services to passengers, allowing for a personalized and easily accessible experience [1]. However, a pertinent and constant problem for this transport system has not yet been resolved: the supply and demand of passengers. Passengers may be left without service due to a high demand in a specific region, and, on the other hand, there are places where the offer of vehicles is large, but few customers will request rides [2]. By knowing in advance what the demand of a particular region is, taxi companies can intelligently organize their fleet, increasing or decreasing the offer according to the need. As a result, the waiting time for passengers and the idleness of taxis is reduced, and consequently, taxi drivers can serve more customers and optimize their profits [3]. In addition, vehicle depreciation and time spent in transit are also reduced, leading to reduced fuel and maintenance costs [4]. Research shows that if the supply of taxis were able to meet the desired demand, the inefficiency of the private transport market would fall by more than 60% [5].

Various machine-learning techniques have been used to address the issue of passenger supply and demand. Autoregressive Integrated Moving Average (ARIMA) [6, 7], Convolutional Neural Networks (CNN) [8, 9], Long Short-Term Memory (LSTM) [10, 11], and Support Vector Machine (SVM) [1, 12] are examples of some of them. However, these techniques, for the most part, present certain limitations inherent to their nature, namely:

needing periodic revisions of the models, training them, and adapting them again to the reality of the data [13];

not dealing with situations in which there are changes in environmental conditions, typical of online configurations with non-stationary data [14];

rarely being able to handle complex systems where there are multiple modes of operation [15].

These limitations generate methodological gaps that make it challenging to predict taxi demand accurately. This is mainly because the demand is related simultaneously to several exogenous factors with high volatility. Temperature, time of day, important events in the region, and traffic are some examples among them [16].

Furthermore, given its nature, extracting knowledge from datasets with these characteristics is a complex and unintuitive task, favoring uncertainty in interpreting such information [17]. Given this context, it is desirable that the techniques used to forecast taxi demand have continuous and incremental learning, adapting to external factors related to traffic. Additionally, these systems must, preferably, supply the restrictions inherent to the aforementioned traditional techniques.

Evolving Fuzzy Systems (EFS) have been proposed for the incremental processing of large data streams, in which samples are presented only once to the model and then discarded [15]. These systems have as their main characteristic the development of their structure (in this case, neurons, clusters, data clouds, and/or fuzzy rules) and parameters update online, eventually in real-time, as new information is received from a continuous data stream [18, 19]. In addition, the characteristic of the EFS in dealing with the uncertainty of information is highlighted, and these are inherent to the problem in question. EFS have been successfully used in several applications, such as forecasting in the financial market [18 , 21], handling missing data [22, 23], urban mobility and traffic management [24, 25], forecasting and monitoring weather data [26, 27], pattern recognition [19, 28], and diagnostics in health care [29], to name a few.

In this context, this work introduces a new approach to forecasting and classifying taxi passenger demand. The proposed approach to forecast taxi passenger demand can be summarized in the following steps:

The first step is the dataset build. The dataset contains taxi rides information and meteorological data of a city or region. After obtaining the data, the area is split into microregions, named zones. Then, for each zone, the rides are aggregated to characterize the taxi demand.

The second step is feature construction. The input variables set are built in using historical values of taxi demand and meteorological data. A set of 26 input variables are obtained, of which 21 are extracted from taxi rides and 5 from meteorological data.

The third step is the feature selection. The selection is performed by Kruskal-Wallis statistical method.

Finally, the fourth step is demand forecasting. The forecasting is carried out by an evolving fuzzy system that performs the forecast one step ahead.

Taxi demand classification is performed by transforming demand forecasting into a classification problem. In other words, the classification is carried out using the outputs of the demand forecast. So, the steps to create the dataset, feature construction, and feature selection are the same used to forecast demand, so they will not be described again. The below steps brief the proposed approach to classifying taxi demand.

The first step is the classes definition. Based on the dataset created in the first steps, a division of rides is made into 4 classes (Very Low, Low, Medium, and High).

After the classes definition, the classification is performed by converting the results obtained in the forecasting step into classes.

The third and last step is the class’s visualization into a heatmap, updated at every new prediction.

In summary, the main contributions of this work are:

The description of a new approach to forecasting taxi passenger demand using evolving fuzzy systems, historical ride data, and weather information.

The proposition of a new approach to the classification of taxi demand. The proposed approach uses the demand predicted by the evolving model to perform the classification.

The proposed approach was developed for online configurations with volatile environmental conditions and non-stationary data. In these environments, data arrives as a stream, has unlimited size, and is subject to variations and changes in the concept. These are typical characteristics of the problem at hand.

The representation of taxi demand in heatmaps. This visualization is more intuitive, allowing a quick visualization of the demand in different city areas and facilitating decision-making.

Subsequently, the rest of the article is organized as follows. Section 2 provides a brief literature review concerning works related to demand forecasting. Section 3 presents the concepts of evolving fuzzy systems and describes the evolving models used in this work. In Section 4, the proposed approach is introduced and detailed. The computational experiments and their respective results are presented in Section 5s. Finally, Section 6 illustrates the final considerations and proposals for future studies.

2 Literature review

Taxi passenger demand forecasting has been the subject of several types of research, encouraging the modeling and application of various techniques to solve this problem. Initial studies used consolidated statistical methods for time series and historical data from the rides for demand forecasting. For example, in Moreira-Matias et al. [6] a hybrid statistical model is created to forecast passenger demand for the next 30 minutes at 63 stands 1 in Porto, Portugal. The model proposed by Moreira-Matias et al. [6] is incremented in [7] to simulate real-time demand forecasts. Zander’s research [30] forecasts taxi demand in Stockholm, Sweden, using Artificial Neural Networks (ANN). In Qu et al. [31], historical data and Least Squares Support Vector Machine (LS-SVM) are used to demand forecasts in the next 10 minutes in China.

Some studies suggest including exogenous variables, such as time of day, weather conditions, regional events, and day of the week, to improve the models’ performance [10, 32]. Ke et al. [10] performed the taxi demand forecast in Hangzhou, China, using a model based on deep learning named Fusional Convolutional Long Short-Term Memory (FCL-Net). The FCL-Net uses historical data and meteorological information as input. Vanichrujee et al. [32] proposed an approach that combines the results of three models, Gated Recurrent Unit (GRU), eXtreme Gradient Boosting (XGBoost), and Long Short-Term Memory (LSTM), to perform demand forecasting in 7 regions of Bangkok, Thailand. The hybrid model uses taxi rides and weather conditions (rainy, sunny, etc.) as input. In work conducted by Tong et al. [33], a linear regression model named Linear Unit Original Taxi Demand (LinOUTD) is proposed. LinOUTD was designed to process variables from different sources, such as points of interest, discounts applied to rides, and weather conditions, for example. The experiments were conducted in Beijing and Hangzhou, China, performing demand forecasts with 1-hour intervals.

Yu et al. [34] developed a modified density-based spatial clustering algorithm with noise (DBSCAN) model, using points of interest and historical demand to perform demand forecasts at 1-hour intervals in Beijing (China). Rodrigues et al. [35] introduced a framework based on Convolutional Neural Networks (CNN) and Long Short-Term Memory (LSTM). The framework uses taxi rides and event information to predict demand at two major entertainment centers in New York City. In studies by Liu et al. [36], a model was developed using Random Forest and Ridge Regression techniques to predict passenger demand at points of interest in Xian, China, with a 1-hour interval. Finally, Kong et al. [37] developed a framework named TBI2Flow, which aggregates information from passenger applications, weather, local traffic, and vehicle location to perform the forecast in Shangai at 1-hour intervals.

Other approaches present in the literature perform knowledge extraction through maps (spatial) combined with historical rides data (temporal) to create hybrid models, known as Spatio-Temporal models. Zhao et al. [38] proposed the Unified Spatial-Temporal Network (USTN), which uses maps processed in convolutional layers, along with historical data from Uber 2 and information from New York yellow taxis, to predict demand with 1-hour intervals. Liu et al. [39] created a model named the Context-Aware Attention-Based Convolutional Recurrent Neural Network (CACRNN), combining Points Of Interest (POI) with meteorological data to predict demands in the next 15 minutes in Chengdu and New York. In Luo et al. [40], a feature selector is used based on a statistical method known as Augmented Dickey-Fuller (ADF). After selection, the Multi-Task Deep Learning (MTDL) model forecasts demand in the next 10 minutes in New York. A model called Multi-Level Recurrent Neural Networks (MLRNN), based on Long Short-Term Memory (LSTM), is the subject of the work by Zhang et al. [41]. The model receives meteorological information and historical data to predict demands in the next 30 minutes in Manhattan, New York.

In the studies by Zhu et al. [42], a framework is proposed combining Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM) to capture the temporal dependencies of rides. At the same time, convolutional layers are used to interpret event data and convert it into information to predict demand in New York, in two centers where significant events occur in the city (Barclays Center e Terminal 5). Lin et al. [43] use weather data, moving averages, and historical rides to feed a model which uses three deep learning techniques: Convolutional Neural Networks (CNN), Long Short-Term Memory (LSTM), and Gated Recurrent Unit (GRU). This study predicts demands in points of interest (tourist attraction sites, hospitals, etc.) in Kaohsiung City, Taiwan.

Complementary Table 1 summarizes all the works described in this section: techniques and models used, time intervals, input variables, and cities where the forecasts occur.

Table 1
Overview of the state-of-the-art

Year Author Models/Techniques Intervals Types of data City

2012 [6] ARIMA and PR 30 min. Historical data Porto (Portugal)

2013 [7] ARIMA and PR 30 min. Historical data Porto (Portugal)

2017 [10] FCL-net 1 hour Historical and weather data Hangzhou (China)

2017 [30] ANN 1 hour Historical data Stockholm (Sweden)

2017 [33] LinOUTD 1 hour Historical, event and weather data, among others Beijing, Hangzhou (China)

2018 [32] GRU, XGBoost, LSTM 1 hour Historical and weather data Bangkok (Thailand)

2019 [31] LS-SVM 10 min. Historical data (China)

2019 [34] Modified DBSCAN 1 hour Historical data and points of interest Beijing (China)

2019 [35] LSTM 1 hour Historical and event data New York (USA)

2020 [36] RF and RR 1 hour Historical data Xian (China)

2020 [37] TBI2Flow 1 hour Historical and weather data Shanghai (China)

2020 [38] USTN 1 hour Historical data and map images New York (USA)

2020 [39] CACRNN 15 min. Historical and event data Chengdu (China), New York (USA)

2021 [40] MTDL 10 min. Historical data New York (USA)

2021 [41] MLRNN 30 min. Historical and weather data New York (USA)

2021 [42] MLP and LSTM 30 min. Historical and event data New York (USA)

2021 [43] CNN and LSTM 30 min. Historical and weather data Kaohsiung (Taiwan)

Year	Author	Models/Techniques	Intervals	Types of data	City
2012	[6]	ARIMA and PR	30 min.	Historical data	Porto (Portugal)
2013	[7]	ARIMA and PR	30 min.	Historical data	Porto (Portugal)
2017	[10]	FCL-net	1 hour	Historical and weather data	Hangzhou (China)
2017	[30]	ANN	1 hour	Historical data	Stockholm (Sweden)
2017	[33]	LinOUTD	1 hour	Historical, event and weather data, among others	Beijing, Hangzhou (China)
2018	[32]	GRU, XGBoost, LSTM	1 hour	Historical and weather data	Bangkok (Thailand)
2019	[31]	LS-SVM	10 min.	Historical data	(China)
2019	[34]	Modified DBSCAN	1 hour	Historical data and points of interest	Beijing (China)
2019	[35]	LSTM	1 hour	Historical and event data	New York (USA)
2020	[36]	RF and RR	1 hour	Historical data	Xian (China)
2020	[37]	TBI2Flow	1 hour	Historical and weather data	Shanghai (China)
2020	[38]	USTN	1 hour	Historical data and map images	New York (USA)
2020	[39]	CACRNN	15 min.	Historical and event data	Chengdu (China), New York (USA)
2021	[40]	MTDL	10 min.	Historical data	New York (USA)
2021	[41]	MLRNN	30 min.	Historical and weather data	New York (USA)
2021	[42]	MLP and LSTM	30 min.	Historical and event data	New York (USA)
2021	[43]	CNN and LSTM	30 min.	Historical and weather data	Kaohsiung (Taiwan)

3 Evolving fuzzy systems

Evolving fuzzy systems were conceived to fill a methodological gap in the context of adaptive modeling, considering the online processing of a data stream [44]. Generally, this kind of data has the following characteristics:

the samples are presented only once and are discarded [45];

the data stream has unlimited size, and its processing is online [46];

systems which process this type of data must deal with environmental changes, such as system drifts and shifts [15];

the processing of a large number of samples [21].

Taxi demand can be understood as a data stream because ride information is constantly generated (Pick-Up and Drop-Off) and is subjected to environmental changes (transit, wheater, etc.). Therefore, the evolving fuzzy systems are suitable for taxi demand forecasting and classification.

After presenting a brief introduction to evolving fuzzy systems and their promising applications in taxi demand data, the subsequent sections regard a technical description of the five models used in this work: ALMMo (Autonomous Learning Multi-Model System), eMG (evolving Multivariable Gaussian Fuzzy System), eFCE (evolving Fuzzy with Multivariable Gaussian Participatory Learning and Recursive Maximum Correntropy), eFMI (evolving Fuzzy with Multivariable Gaussian Participatory Learning and Multi-Innovations Recursive Weighted Least Squares), and eNFN (evolving Neo-Fuzzy Neuron).

3.1 ALMMO - autonomous learning multi-model system

ALMMo, proposed in [18], is an evolving multi-model fuzzy system with fuzzy rules of the AnYa type [47]. The antecedent of the AnYa rules is based on the concept of data clouds. The principle of data clouds is similar to that of clustering algorithms. In other words, each cloud represents a set of samples with similar characteristics. Each cloud is shaped by its constituent data, and focal points represent its centroids. ALMMo rules can be described by:

$IF (j^{*} = \underset{i = 1, 2, \dots, G}{argmin} (∥ x_{t} - c_{t}^{i} ∣ ∣)) THEN (Ξ_{j *} \leftarrow x_{t}),$

in which x_t = [x₁, x₂, …, x_M] ^T is a data sample in Euclidean space defined as R ^M, argmin the method which searches for the nearest data cloud, Ξ_i a i-th data cloud and $c_{t}^{i}$ their corresponding focal point. The number of data clouds is represented by G in the observed universe R ^M, and also represents the number of rules.

ALMMo structure evolves based on the data clouds, creating or deleting them as new samples are added to the model. With the arrival of new samples, the scalar means of products ${\bar{X}}_{t}$ and the global mean μ_t at time t are calculated recursively by:

$\begin{matrix} {\bar{X}}_{t} = \frac{t - 1}{t} {\bar{X}}_{t - 1} + \frac{1}{t} {∥ x_{t} ∥}^{2}, \end{matrix}$ (1)

and

$\begin{matrix} μ_{t} = \frac{t - 1}{t} μ_{t - 1} + \frac{1}{t} x_{t} . \end{matrix}$ (2)

Then the unimodal density $\bar{D}$ is calculated between the current sample x_t and all identified focal points of the i-th class by:

$\begin{matrix} {\bar{D}}_{t} (x_{t}) = \frac{1}{1 + \frac{∣ x_{t} - μ_{t} ∥^{2}}{{\bar{X}}_{t} - {∥ μ_{t} ∥}^{2}}} . \end{matrix}$ (3)

At each x_t sample, the ALMMo structure is evaluated, and you can create a new rule, merge two rules, delete a rule or update the parameters. Thus, with each sample received, it is verified through Condition 1 whether a new rule should be created.

$\begin{matrix} Condition 1 : IF ({\bar{D}}_{t + 1} (x_{t + 1}) \\ > max_{i = 1, 2, \dots, G_{t}} ({\bar{D}}_{t + 1} (c_{t}^{i}))) \\ OR ({\bar{D}}_{t + 1} (x_{t + 1}) \\ < min_{i = 1, 2, \dots, G_{t}} ({\bar{D}}_{t + 1} (c_{t}^{i}))) . \end{matrix}$

If Condition 1 is not accepted, the sample is assigned to the closest data cloud with updated parameters. Otherwise, if Condition 1 is satisfied, a new rule is created with the focal point at x_t, and Condition 2 is tested. This condition checks if the created cloud overlaps with any existing clouds.

$Condition 2 : : IF :: ({\bar{D}}_{t + 1, i} \geq (\frac{1}{1 + n^{2}})) .$

If Condition 2 is satisfied, the overlapped cloud will be replaced by the newly created one, and the new cloud will inherit the parameters of the consequent overlapping cloud. Otherwise, if Condition 2 is not satisfied, the consequent parameters of the new cloud are initialized.

Afterward, the quality of the rule base is evaluated by a utility measure which aims to exclude unrepresentative rules. The utility measure is calculated based on the cumulative sum of the rule’s contribution to the output calculation from its creation to the current sample. In other words, it is the measure of the importance of a respective fuzzy rule concerning the other rules (i = 1, 2, . . . , G_t+1). Its calculation is obtained by:

$\begin{matrix} η_{t + 1}^{i} = \frac{1}{t + 1 - {activated}^{i}} \sum_{l = {activated}^{i}}^{t + 1} δ_{l}^{i}, \end{matrix}$ (4)

where activatedⁱ represents the time that the rule/data cloud was activated and $δ_{l}^{i}$ is the activation level of the i-th rule/data cloud at the instant l. The $δ_{l}^{i}$ activation level is calculated for each rule (i = 1, 2, . . . , C_t), in:

$\begin{matrix} δ_{l}^{i} = \frac{{\bar{D}}_{l}^{i} (x_{l})}{\sum_{j = 1}^{G^{l}}} . \end{matrix}$ (5)

Let η0 be a constant which defines a tolerance value. Rules with low representation will be excluded as per Condition 3. If Condition 3 is met, the j-th rule/data cloud will be deleted, together with their consequent parameters.

$\begin{matrix} Condition 3 : & IF (η_{t}^{j} + 1 < η 0) . \end{matrix}$

In addition to rule exclusion, ALMMo implements a method for excluding inter-correlated entries to improve processing time, reduce memory usage, and improve the algorithm’s overall performance. The proposed method is based on the normalized cumulative sum of the input parameter values.

3.2 eMG - evolving multivariable gaussian fuzzy system

eMG [48] is an evolving fuzzy model that uses first-order Takagi-Sugeno rules and multivariable Gaussian membership functions. The rule base is built under an evolving clustering algorithm with participatory learning. The main characteristic of participatory learning is to use what has already been learned to evaluate the impact of including a new sample in the model. In short, the relevance of a new sample is evaluated concerning what has already been consolidated as knowledge [49].

The eMG structure evolves by updating, creating, or merging clusters/rules, and the output is obtained by the weighted average of the contributions of each rule. The rules are represented by their respective clusters, and each new sample presented to eMG updates its structure by a compatibility measure computed for each cluster. The compatibility measure $p_{t}^{i} \in [0, 1]$ is calculated through the distance between the current input x_t and the centers of the existing clusters $c_{t}^{i}$ , that is, $p_{t}^{i}$ is obtained by:

$p_{t}^{i} = exp [- \frac{1}{2} M (x_{t}, c_{t}^{i})],$ (6)

where the distance M is calculated by:

$M (x_{t}, c_{t}^{i}) = (x_{t} - c_{t}^{i}) {(\sum_{i}^{t})}^{- 1} {(x_{t} - c_{t}^{i})}^{T},$ (7)

in which x_t represents the current sample and $c_{t}^{i}$ the center of the i-th cluster. A threshold is defined for the compatibility measure T_p, which is obtained by:

$T_{p} = exp [- \frac{1}{2} χ_{m, α}^{2}],$ (8)

where $χ_{m, a}^{2}$ is a Chi-Square distribution with m degrees of freedom and α the one-way confidence interval. After calculating the measure of compatibility between the new sample and all the clusters, the cluster’s index with the highest degree of compatibility is found i^*.

An alert mechanism is used to identify when the structure of the current clusters does not adequately represent the current knowledge, and this needs revision [48]. For each new sample x_t, the alert index $a_{t}^{i} \in [0, 1]$ is calculated for all clusters and estimated through the cumulative probability of V_t, that is:

$a_{t}^{i} = \bar{p} (V_{t} < \bar{z}),$ (9)

where $\bar{p} (V_{t} = \bar{z})$ is a binomial distribution. A threshold of the alert index T_a is also defined, which is used in creating clusters. The alert index T_a is calculated by:

$T_{a} = 1 - \frac{α}{ω},$ (10)

α being a parameter that defines the level of significance and ω the observation window for calculating the alert index.

A new cluster is created when the compatibility measure $p_{t}^{i}$ is less than the compatibility threshold T_p for all clusters and the alert index for the cluster with the highest degree of compatibility $a_{t}^{i^{*}}$ is greater than its corresponding threshold T_a. In other words, a new cluster will be created if

$p_{t}^{i} < T_{p}, \forall i = 1, ..., c_{i}^{t} and a_{t}^{i^{*}} > T_{a} for i^{*} = \max_{i} (p_{t}^{i}) .$

Otherwise, if

$p_{t}^{i} > T_{p}, \forall i = 1, ..., c_{t}^{i} and a_{t}^{i^{*}} < T_{a} for i^{*} = \max_{i} (p_{t}^{i}),$

then the sample is assigned to the most compatible cluster, which has its center updated using:

$c_{t + 1}^{i^{*}} = c_{t}^{i^{*}} + λ (p_{t}^{i^{*}})^{1 - a_{t}^{i^{*}}} (x_{t} - c_{t}^{i^{*}}),$ (11)

where λ ∈ [0, 1] is the learning rate. The eMG consequent parameters are updated by a weighted least squares recursive algorithm.

3.3 eFCE - evolving fuzzy with multivariable gaussian participatory learning and recursive maximum correntropy

eFCE [50] builds its structure based on a recursive clustering algorithm with participatory learning and multivariable Gaussian membership functions. The eFCE structure evolves by including, merging, or excluding clusters and rules. As in eMG, clusters are created using a compatibility measure and an alert index. However, in eFCE, the compatibility measure is computed by Euclidean and Mahalanobis distances. Using two distances avoids the singularity problem when calculating the inverse of the dispersion matrix of clusters with a small number of samples. The distance measure is defined as follows:

If $n_{t}^{i}$ < N_max, the Euclidean distance is used, that is,

$D = (x_{t} - μ_{t}^{i})^{T} I_{mxm} (x_{t} - μ_{t}^{i}) .$ (12)

If $n_{t}^{t}$ > N_max, the Mahalanobis distance is employed, i.e,

$D = (x_{t} - μ_{t}^{i})^{T} (\sum)^{- 1} (x_{t} - μ_{t}^{i}),$ (13)

in which $n_{t}^{i}$ is the number of samples from the i-th cluster, N_max is the limit for creating microclusters or clusters, I_mxm is an identity matrix m x m, ∑ is a scatter matrix (Mahalanobis distance) or an identity matrix (Euclidean distance).

The exclusion of clusters and rules is based on the concept of age and population. The cluster’s age is used to define the inactivity time of the cluster, and the population represents the number of samples attributed to a cluster. Thus, we find $\bar{i}$ , the index of the least active cluster for each sample. The cluster indexed by $\bar{i}$ is excluded if:

${age}_{t}^{\bar{i}} > ω and \frac{n_{t}^{\bar{i}}}{t} < 0.01,$

in which t is the number of samples.

The procedure for the consequent parameters update of the eFCE is based on the Recursive Maximum Correntropy. Correntropy is a generalized similarity measure between two random variables that minimize the output error by maximizing the probability density of the source error.

3.4 eFMI - evolving fuzzy with multivariable gaussian participatory learning and multi-innovations recursive weighted least squares

The evolving Fuzzy with Multivariable Gaussian Participatory Learning and Multi-Innovations Recursive Weighted Least Squares (eFMI) [51], as the eMG and the eFCE, is an evolving system that uses a clustering algorithm based on participatory learning and multivariable Gaussian membership functions. The clustering algorithm uses the compatibility measure computed recursively to add a new cluster. In addition, the age and population concepts are used to exclude inactive clusters and rules.

The eFMI uses a merge procedure based on the remarkably overlapping of a cluster pair. In the method, if two clusters i^* and i have the norm of the difference in distance between their centers, μ_{i
^*} and μ_i, less than or equal to a predefined threshold, then they are merged. More formally,

$| | μ_{t}^{i^{*}} - μ_{t}^{i} | | \leq ρ (i = 1 . . . c_{t} and i \neq i^{*}),$ (14)

in which ρ is the threshold for merging clusters. The center of the new cluster is defined by the weighted average as follows:

$μ_{t}^{i^{*} \cup i} = μ_{t}^{i^{*}} - \frac{n_{t}^{i}}{n_{t}^{i^{*}} + n_{t}^{i}} (μ_{t}^{i^{*}} - μ_{t}^{i}),$ (15)

in which n is the number of samples of the clusters. The new cluster depends on the number of samples of the merged clusters. The dispersion matrix of the resulting cluster is updated to the average value of the dispersion matrices of the merged clusters by:

$\sum_{t}^{i^{*} \cup i} = \frac{\sum_{t}^{i^{*}} + \sum_{t}^{i}}{2} .$ (16)

The eFMI consequent parameters are updated by the MI-WRLS (Multi-Innovation Weighted Recursive Least Squares). The MI-WRLS introduces the concept of innovation length that guides the amount of innovation vector estimates and represents the errors passed used to update the consequent parameters.

3.5 eNFN - evolving Neo-Fuzzy neuron

eNFN [44] has a structure composed of n zero-order Takagi-Sugeno models, one for each input variable. The domain of the input variables is partitioned by triangular and complementary membership functions, and its structure evolves based on the modeling error computed recursively. The consequent parameters are updated by an algorithm based on gradient descent with an optimal learning rate.

The eNFN starts its learning with two membership functions for each input variable. New functions can be added based on the relationship between the global average error of the model and the average local error of the most active membership function. The mean value ${\hat{μ}}_{t}$ and the global error variance ${\hat{σ}}_{t}^{2}$ are recursively calculated for input x_t at time t by:

${\hat{μ}}_{t} = {\hat{μ}}_{t - 1} - β ({\hat{μ}}_{t - 1} - e_{t}),$ (17)

and

${\hat{σ}}_{t}^{2} = (1 - β) ({\hat{o}}_{t - 1}^{2} + β {({\hat{μ}}_{t} - e_{t})}^{2}),$ (18)

where $e_{t} = y_{t} - \hat{y_{t}}$ is the error in t and β is the learning rate. The average value of the local error $μ_{b_{ti}^{*}}$ that corresponds to the most active membership function $b_{i}^{*}$ is calculated recursively for the input x_ti by:

$μ_{b_{ti}^{*}} = μ_{b_{ti - 1}^{*}} - β (μ_{b_{ti - 1}^{*}} - e_{t}) .$ (19)

A limiter τ is used to allow control over the number of rules, avoiding complex models and overfitting. In eNFN, this limiter is compared to the smallest distance (dist) allowed between the modal value of the function to be created and adjacent functions. A new membership function will be created and inserted into the model if

$μ_{b_{ti}^{*}} > {\hat{μ}}_{t} + {\hat{σ}}_{t}^{2} e dist > τ .$

On the other hand, the exclusion of membership functions follows the concept of age. A function will be excluded from the model if it remains inactive for a long time [14]. The age of a particular membership function is calculated by:

${age}_{j} = t - {activated}_{j},$ (20)

where t is the current step and activated_j represents the step where the activation of the j-th membership function took place. Consider $b_{i}^{-}$ an index of the least active membership function at time t. It will be excluded if

$b_{i}^{-} > ω,$

where ω is a parameter that indicates the time limit for deleting a function.

3.6 Comparison of the evolving algorithms

In this Section, we show in Table 2 a short comparison between the evolving algorithms used in this work. The table presents the membership functions, fuzzy rule type, the methods to add, merge, and delete rules and/or clusters, and the approach to updating the consequent parameters. In Table 2, WRLS is Weighted Recursive Least Squares, FWRLS is Fuzzy Weighted Recursive Least Squares, MI-WRLS is Multi-Innovation Weighted Recursive Least Squares, RMC is Recursive Maximum Correntropy, and GD is Gradient Descent with Optimal Learning Rate.

Table 2
Comparison between the evolving models

Models Membership Functions Fuzzy Rules Add Rules and/or Clusters Merge Rules and/or Clusters Delete Rules and/or Clusters Update Consequent

ALMMo [18] Data clouds AnYa Density measure - Utility measure FWRLS

eMG [48] Gaussian First-Order TS Compatibility measure Degree of compatibility - WRLS

eFMI [51] Gaussian First-Order TS Compatibility measure Degree of compatibility Age and Population MI-WRLS

eFCE [50] Gaussian First-Order TS Compatibility measure Degree of compatibility Age and Population RMC

eNFN [44] Triangular Zero-Order TS Modeling error - Age GD

Models	Membership Functions	Fuzzy Rules	Add Rules and/or Clusters	Merge Rules and/or Clusters	Delete Rules and/or Clusters	Update Consequent
ALMMo [18]	Data clouds	AnYa	Density measure	-	Utility measure	FWRLS
eMG [48]	Gaussian	First-Order TS	Compatibility measure	Degree of compatibility	-	WRLS
eFMI [51]	Gaussian	First-Order TS	Compatibility measure	Degree of compatibility	Age and Population	MI-WRLS
eFCE [50]	Gaussian	First-Order TS	Compatibility measure	Degree of compatibility	Age and Population	RMC
eNFN [44]	Triangular	Zero-Order TS	Modeling error	-	Age	GD

4 Proposed approach

This section presents the proposed approach to taxi demand forecasting and classification. Section 4.1 details the approach for demand forecasting, while Section 4.2 describes the methodology for demand classification.

4.1 Demand forecasting

The proposed approach for taxi demand forecasting is illustrated in Figure 1. As can be seen, the approach consists of 4 steps. The first step is extracting the rides database, detailed in Section 4.1.1. Then, Section 4.1.2 describes the input variables and the construction of the dataset. Section 4.1.3 details the third step, which is the feature selection. Finally, the fourth and final step is the demand forecasting presented in Section 1.1.4

Fig. 1

Steps of the proposed approach to demand forecasting.

4.1.1 Dataset definition

The rides database contains information on taxi rides completed in a given city or region, provided by companies which operate in the field. Generally, this information is extracted from the monitoring systems installed in the cars (GPS devices and other mobile applications) and presented in the following format: Pick-Up DateTime, Drop-Off DateTime, Pick-Up Location, and Drop-Off Location. Additionally, these datasets can provide complementary information such as the route of the rides, the amount paid, payment type (credit card, cash, etc.), number of passengers, and type of vehicle.

After obtaining the rides dataset, the extraction of information used in the following steps is performed. Thus, only the following information is selected: Pick-Up DateTime, Drop-Off DateTime, Pick-Up Location, and Drop-Off Location.

The next step divides the region of interest into microregions, called zones. This division aims to frame the study region in pre-defined limits and cluster the rides’ locations. Several studies make this division by delimiting the maximum latitudes and longitudes of a region R and dividing the region into K similar polygons, with rectangles or hexagons being the commonly used shapes [8 , 53]. A set of latitudes and longitudes defines each vertex of the polygon. Therefore, a zone z can be described as R_z, where 1 ≤ z ≤ K. There are other ways of delimiting zones, such as the selection of tourist attractions (stadiums, museums, etc.) [32, 35], the spatial division by neighborhoods, or irregular shapes pre-defined, for example, by the local government [9]. In this work, to perform the zones division, the QGIS 3 tool was used. However, it is possible to find several other software to assist in the delimitation and division of zones, such as ArcGIS 4 and the library GeoPandas 5

Once the region was divided into zones, the locations (longitude and latitude of Pick-Up and Drop-Off) were converted to their respective zone number. A pair of coordinates will belong to a zone z, if its latitude and longitude are within the limits of the coordinates of all vertices of the polygon referring to that zone. It is noteworthy that the conversion of latitudes and longitudes into zones can also be performed with the support of the previously mentioned software, which can create polygons.

Considering a scenario in which the demand forecast for Pick-Up and Drop-Off is performed, a Pick-Up and a Drop-Off dataset are created for each of the k zones. For example, the Pick-Up dataset from the z zone is composed of the Pick-Up DateTime of all rides which occurred within the boundaries of that zone. After defining each ride’s zone and constructing the dataset for each taxi zone, the next step is the aggregation of rides to characterize taxi demands. First, the continuous-time (days and hours) is initially partitioned into identical sequential periods, defined as time intervals. Then, all rides completed in a given time interval are grouped to characterize the taxi demand for that interval. In other words, the taxi demand D in a zone z is obtained by the number of rides in that zone in a given time interval. Thus, at the end of this step, there is a dataset containing the time intervals and their respective taxi demands for each zone.

4.1.2 Feature construction

A time series is defined by a set of observations generated sequentially over time [54]. Therefore, given its characteristics, taxi demand forecasting can be defined as a time series, and this can be described as a nonlinear dynamic problem and represented as in [51, 55] by:

${\hat{y}}_{t} = f (y_{t - 1}, . . ., y_{t - n_{y}}, {\bar{a}}_{1}, . . ., {\bar{a}}_{n a}, {\bar{h}}_{t}, . . ., {\bar{h}}_{t - n_{h}}),$ (21)

in which the forecast at step t is performed using n_y lags to model the series, t being the current step, y the lagged values of the series, $\bar{a}$ the auxiliary variables, $\bar{h}$ the exogenous variables, n_a the number of auxiliary variables, n_h the number of exogenous variables and f a nonlinear function suitable for modeling the series.

In this study, the historical data used to model the series are characterized by:

Historical Demand of Intervals ( hdi ): taxi demand of the last 8-time intervals 6 . The historical demand of the z zone intervals is represented by:

${hdi}_{z, t} = {hdi}_{z, t - 1}, {hdi}_{z, t - 2}, . . ., {hdi}_{z, t - 8} .$ (22)

Daily Historical Demand ( dhd ): demand from 8 previous days 7 . The historical demand for days in the z zone can be described as:

${dhd}_{z, tr} = {dhd}_{z, tr - 1}, {dhd}_{z, tr - 2}, . . ., {dhd}_{z, tr - 8},$ (23)

for the same relative time interval, in which tr represents the previous day’s relative time interval of the demand to be predicted.

Average of the Historical Demand of the Intervals ( ahdi ): average of the last 8 lags of historical demand (hdi). The ahdi_z is obtained for the z zone by:

${ahdi}_{z, t} = 1 / 8 \sum_{i = 1}^{8} {hdi}_{z, t - i} .$ (24)

Average of the Daily Historical Demand 8 ( adhd ): average of the last 8 lags of previous days historical demand (dhd_z). Its calculation is represented as follows:

${adhd}_{z, tr} = 1 / 8 \sum_{i = 1}^{8} {dhd}_{z, tr - i} .$ (25)

Thus, 18 variables extracted from historical values of taxi demand (series lags) are generated. In addition, using information extracted from taxi rides, the following 3 variables are obtained:

Day of Week ( dw ): Sunday (1), Monday (2),..., Saturday (7).

Weekend ( wk ): weekday (0) ou weekend (1).

Time of the day ( td ): divided into three periods: rest time (0), normal time (1) and peak time (2). Rest time is from 20:00 to 06:00. Peak hours include the following periods: [06:00 - 09:00, 12:00 - 15:00, 17:00 - 20:00]. The remaining times of the day are determined as normal times 9 .

The exogenous variables are obtained from meteorological information, as suggested in [10, 33]. The meteorological data were extracted from the platform Wheater Underground 10 . On this platform, meteorological data are available every hour, being used in the dataset in the interval before the demand is predicted. The following list details the 5 variables used in this work:

Temperature (te): measurement in Fahrenheit.

Relative Humidity (rh): measurement in percentage.

Precipitation ( pr ): measurement in millimeters.

Wind Speed ( ws ): measurement in miles per hour (mph).

Time Condition ( tc ): textual description that was represented by indexes from 1 (Fog) to 50 (Windy).

Therefore, there are 26 variables, 21 of which are extracted from information contained in the rides dataset and 5 exogenous variables obtained from meteorological information. Thus, at the end of this step, there is a dataset containing the indexes of the intervals, the 26 input variables, and their respective taxi demands for each zone.

4.1.3 Feature selection

Once the creation of the datasets with the input variables and their respective desired outputs (taxi demands), a feature selection step is performed. This step identifies and selects the most relevant input variables for demand forecasting. In this work, the selection is performed using the Kruskal-Wallis [57] statistical method, which orders the variables according to their degree of relevance.

The selection of variables is necessary since a high number of dimensions presented to the models causes two main problems: (i) longer execution time of the models and; (ii) curse of dimensionality, a term used to explain the problem caused by the exponential increase in the volume of data associated with the inclusion of extra dimensions in the euclidean space [58]. In practical terms, once the number of variables exceeds a certain threshold, the models will likely lose performance, consequently affecting their results.

Once sorted, the most relevant N variables which will compose the new datasets are selected. Unfortunately, the Kruskal-Wallis method does not explicitly indicate which N is optimal for the best performance. Therefore, this value must be found empirically or by some other method. At the end of the feature selection step, for each zone, there is a dataset with the indexes of the intervals, the most relevant N variables, and their respective taxi demands.

4.1.4 Demand forecasting

In the proposed approach, demand forecasting is performed by an evolving fuzzy system which performs the forecast one step ahead, that is, for the next time interval. Then, the forecast is obtained individually for each Pick-Up and/or Drop-Off zone using the set of variables selected in the previous phase as input.

Therefore, if the most relevant variables for a given dataset are hdi_t-1, hdi_t-2, dhd_tr-1, dhd_tr-5, wk and td, the model for this dataset will be defined by: $\begin{matrix} {\hat{y}}_{t} = & f ({hdi}_{t - 1}, {hdi}_{t - 2}, {dhd}_{tr - 1}, {dhd}_{tr - 5}, wk, td) . \end{matrix}$ (26)

4.2 Demand classification and heatmaps

Figure 2 summarizes the demand classification approach, illustrating its main steps. The steps of creating the dataset and demand forecasting are, mutatis mutandis, as described in Section 4.1 and therefore will not be described again. Section 4.2.1 describes the definition of classes. In this work, taxi demand classification is performed by transforming the demand forecasting task into a classification problem, as suggested in [59]. Thus, the classification is performed using the outputs of the forecast step. The demand classification step is detailed in Section 4.2.2. Finally, Section 4.2.3 illustrates the process of heatmaps construction.

Fig. 2

Steps for classification and heatmaps generation.

4.2.1 Classes definition

Initially, the number of classes is defined. We chose to work with 4 classes of demand, namely: (i) Very Low; (ii) Low; (iii) Medium; (iv) High. Next, it is necessary to specify the domain of each class, i.e., the range of values which determine its lower and upper limits. It is desirable that these are defined, for each class, keeping the balance between the number of rides between classes. Thus, the problems caused by unbalanced classes are avoided [60]. Finally, class boundaries are identified based on the demands of all zones. In this work, the boundaries of all classes are always multiple of 5. The identification of limits can be performed with a histogram composed of the demands of all zones of a dataset of the rides.

4.2.2 Demand classification

As described in the previous section, classification is performed by transforming a forecasting task into a classification task. Thus, the outputs obtained in the forecasting step (see Section 4.1.4) are used in the demand classification. The values predicted by the evolving model are interpreted and converted into one of the classes. The conversion is defined as follows:

$Class = {\begin{matrix} Very Low, If 0 \leq {\hat{y}}_{t} \leq {UB}_{Very Low}, \\ Low, If {LB}_{Low} \leq {\hat{y}}_{t} \leq {UP}_{Low}, \\ Medium If {LB}_{Medium} \leq {\hat{y}}_{t} \leq {UP}_{Medium}, \\ High If {LB}_{High} \leq {\hat{y}}_{t} \leq {UP}_{High}, \end{matrix}$ (27)

where LB and UP are the lower and upper bounds of the class, respectively.

4.2.3 Heatmaps

Subsequently, the demands converted into classes will compose a heatmap. Heatmaps are graphical representations of taxi demands in a city or region. Through heatmaps, it is possible to visualize and interpret the demand for taxis simultaneously in all areas of a city or region in a simple, intuitive, and easy way.

Each zone is represented by colors on the map, with the highest demands identified by warmer colors and those with lower demands by cooler colors. In this work, the heatmap is generated considering the 4 classes defined in Section 4.2.1.

5 Experiments and results

This section presents the computational experiments to evaluate and compare the performance of the proposed approach in Pick-Up and Drop-Off demand forecasting and classification tasks. Section 5.1 illustrates demand forecasting and classification for Chengdu (China). The forecasting and demand classification experiments for New York City (United States of America) are shown in Section 5.2. The datasets used in experiments are available at the URL: https://shorturl.at/nPX89. The experiments were conducted using the following methodology:

Time Intervals: 15 and 30 minutes time intervals were defined, as in [6 –9]. Thus, for each z zone, 4 datasets are created: (i) Pick-Up with 15 minutes; (ii) Drop-Off with 15 minutes; (iii) Pick-Up with 30 minutes and; (iv) Drop-Off with 30 minutes.

Feature Selection: for the experiments, datasets containing the 5, 10, and 20 most relevant variables indicated by the Kruskal-Wallis method were used, that is, N = {5, 10, 20}. In addition, the dataset with the 26 variables was also used. For objectivity purposes, in the results described in tables 6, 4, 6, 7, 8, 9, 11, and 12, only the best result obtained by each model is presented, with the respective number of variables.

Evolving Fuzzy Models: the algorithms ALMMo [18], eFCE [50], eFMI [51], eMG [48], and eNFN [44] were used to fulfill the experiments. The models are deterministic, i.e., their results do not change with each run. Therefore, each algorithm was run only once. The samples’ processing is performed online, in which the models evolve their structure and parameters for all samples. It is noteworthy that no procedure was performed to fine-tune the model’s hyperparameters. It was decided to use the same parameters in all experiments to maintain the generalist approach. The hyperparameter values for each evolving model followed the metric limits defined by their respective authors, being listed as follows:

ALMMo: forgettingfactor = 0.1; densitythreshold = 0.8; ω = 10.

eFCE: α = 0.01; λ = 0.05; w = 40, Σ = 10^-1, Nmax = 3, ρ = 0.12.

eFMI: α = 0.01; λ = 0.05; w = 40, Σ = 10^-1, Nmax = 200, ρ = 0.1, v_r = 5.

eMG: α = 0.01; λ = 0.05; w = 40, Σ = 10^-1.

eNFN: η = 10; β = 0.01; ω = 100.

Forecasting Experiments: in these experiments, the performance of the proposed approach was evaluated and compared with alternative approaches by RMSE (Root Mean Square Error), NDEI (Non-Dimensional Error Index), and R² (R Squared), which can be obtained by:

$RMSE = \sqrt{\frac{1}{S} \sum_{t = 1}^{S} | {\hat{y}}_{t} - y_{t} |^{2}},$ (28)

$NDEI = \frac{RMSE}{σ ({\hat{y}}_{t})},$ (29)

and

$R^{2} = 1 - \frac{\sum_{t = 1}^{S} | {\hat{y}}_{t} - y_{t} |^{2}}{\sum_{t = 1}^{S} | y_{t} - \bar{y} |^{2}},$ (30)

in which ${\hat{y}}_{t}$ is the predicted demand, y_t is the desired demand, $\bar{y}$ is the mean of the desired demand, S is the number of samples used to evaluate the performance, and σ is the Standard Deviation of the system output. The results are assessed by the mean errors of all zones.

Classification Experiments: the first step for classification experiments is the definition of classes. As discussed in Section 4.2.1, four classes were defined (Very Low, Low, Medium, and High). The Accuracy and F1-Score are used to evaluate the performance in classification. The Accuracy and F1-Score are given by:

$Accuracy = \frac{cc}{S} * 100,$ (31)

$F 1 - Score = 2 * \frac{Precision * Recall}{Precision - Recall} * 100,$ (32)

where cc is the number of samples correctly classified, and S is the total number of samples. The Precision is defined by (33) and the Recall by (34):

$Precision = \frac{TP}{TP + FP},$ (33)

$Recall = \frac{TP}{TP + FN},$ (34)

in which TP is True Positive, FP is False Positive, and FN is False Negative. In addition to Accuracy and F1-Score, the performance of the best classifier is represented by a Confusion Matrix and a Heatmap.

5.1 Demand forecasting and classification in chengdu (China)

This section describes the computational experiments performed on the dataset regarding rides completed in Chengdu, China. The database is provided by Didi Chuxing 11 . The available files contain rides in the range from 11/01/2016 to 11/30/2016.

The study site is located in a specific region of Chengdu, with approximately 65 km². The region’s boundaries range from 104.043 to 104.129 degrees East in longitude and 30.653 to 30.726 North in latitude. The area in question was partitioned into 25 rectangular zones with approximately 1.6 km on each side, as illustrated in Figure 3. The dataset was divided into 80% (11/01/2016 to 11/23/2016) for feature selection by Kruskal-Wallis and the definition of classes of values. The remaining 20% (11/24/2016 to 11/30/2016) are reserved for performance evaluation and comparison. Data sets were normalized between [-1 and 1]. To compare results, the settings mentioned above were the same used by Zhang et al. [8].

Fig. 3

Division of 25 zones in Chengdu.

In his work, Zhang et al. [8] evaluated demand forecasting performance by comparing 11 models with offline learning: SVM, XGBoost, STL-LSTM, STL-ConvLSTM, STL-TCNN, MTL-LSTM, MTL-ConvLSTM, FLC-Net, FTCNN-Net, MTL-TCNN e MTL-TCNN (ST-DTW). The best results were obtained by MTL-TCNN (ST-DTW), which will be used as a benchmark in this work.

5.1.1 Demand forecasting in chengdu

Table 3 illustrates the performance of the models in forecasting Pick-Up and Drop-Off with 15-minute intervals. The table presents the RMSE, NDEI, and R² of the best results obtained by each models. It suggests that the best performance was achieved by ALMMo (with 5 input variables), followed by Zhang et al. [8], eFCE (with 5 input variables), eMG (with 5 variables input), eNFN (with 10 input variables), and eFMI (with 5 input variables). Figure 4 presents the forecast of ALMMo (with 5 input variables) at 15-minute intervals in zone 15, in which we see its fast convergence.

Table 3
Performance on demand forecast at 15-minute intervals for Chengdu

Model Pick-Up Drop-Off

RMSE NDEI R² RMSE NDEI R²

ALMMo {N=5} 8.720 0.332 0.881 8.669 0.339 0.873

Zhang et al. [8] 12 8.935 —— —— —— —— ——

eFCE - {N=5} 9.190 0.357 0.859 9.591 0.372 0.848

eMG - {N=5} 9.673 0.370 0.850 9.948 0.402 0.815

eNFN - {N=10} 10.210 0.391 0.834 10.001 0.397 0.824

eFMI - {N=5} 10.618 0.398 0.827 10.855 0.401 0.828

Model	Pick-Up	Drop-Off
ALMMo {N=5}	8.720	0.332	0.881	8.669	0.339	0.873
Zhang et al. [8] 12	8.935	——	——	——	——	——
eFCE - {N=5}	9.190	0.357	0.859	9.591	0.372	0.848
eMG - {N=5}	9.673	0.370	0.850	9.948	0.402	0.815
eNFN - {N=10}	10.210	0.391	0.834	10.001	0.397	0.824
eFMI - {N=5}	10.618	0.398	0.827	10.855	0.401	0.828

Fig. 4

ALMMo forecasts (with 5 input variables) at 15-minute intervals, for zone 15, from 11/24/2016 to 11/30/2016.

Table 4 presents the best results of each model in predicting Pick-Up and Drop-Off for the 30-minute intervals. In this experiment, the best performance was achieved by ALMMo, followed by eFMI, both with 5 input variables. The eMG (with 5 input variables) obtained the third best performance for Pick-Up and the eFCE (with 5 input variables) for Drop-Off. The model with the worse performance was eNFN (with 10 input variables). Figure 5 illustrates the forecast of Pick-Up and Drop-Off of ALMMo (with 5 input variables) with 30-minute intervals for zone 15. As in Figure 4, in Figure 5, we see the fast convergence of the ALMMo.

Table 4

Performance on demand forecast at 30-minute intervals for Chengdu.

Model 13	Pick-Up			Drop-Off
	RMSE	NDEI	R²	RMSE	NDEI	R²
ALMMo {N=5}	15.172	0.288	0.911	15.640	0.293	0.907
eFMI {N=5}	15.911	0.303	0.899	16.247	0.295	0.907
eMG {N=5}	17.339	0.339	0.872	17.508	0.338	0.870
eFCE {N=5}	17.366	0.341	0.869	17.397	0.336	0.874
eNFN {N=10}	17.490	0.333	0.881	17.514	0.341	0.872

Fig. 5

ALMMo forecasts (with 5 input variables) at 30-minute intervals, for zone 15, from 11/24/2016 to 11/30/2016.

5.1.2 Demand classification in chengdu

Table 5 shows, for Pick-Up and Drop-Off, the range of values of each class for the 15 and 30-minute intervals, in which D represents the demand. In addition to the range of values, the table highlights (in parentheses) the number of samples in each class.

The performance of the models in classifying demand by value range with 15-minute intervals for Chengdu was illustrated, in Table 6, by Accuracy and F1-Score as well as its respective Standard Deviation. The best performance of Pick-Up and Drop-Off was achieved by ALMMo (with 5 input variables), followed by eFCE (with 5 input variables). The eMG achieved the third best performance for Pick-Up and the eFMI for Drop-Off, both with 5 input variables. The eNFN (with 10 input variables) obtained the worse performance.

Table 5
Definition of demands in value ranges for Chengdu

Type Int. Ranges and Quantity

Very Low Low Medium High

Pick-Up 15 D ≤ 10 10 < D ≤ 35 35 < D ≤ 70 D > 70

(8607) (8452) (10283) (8658)

Pick-Up 30 D ≤ 20 20 < D ≤ 70 70 < D ≤ 140 D > 140

(4196) (4164) (5320) (4320)

Drop-Off 15 D ≤ 10 10 < D ≤ 35 35 < D ≤ 70 D > 70

(8788) (9432) (9542) (8238)

Drop-Off 30 D ≤ 20 20 < D ≤ 70 70 < D ≤ 140 D > 140

(4275) (4713) (4869) (4143)

Type	Int.	Ranges and Quantity
Pick-Up	15	D ≤ 10	10 < D ≤ 35	35 < D ≤ 70	D > 70
		(8607)	(8452)	(10283)	(8658)
Pick-Up	30	D ≤ 20	20 < D ≤ 70	70 < D ≤ 140	D > 140
		(4196)	(4164)	(5320)	(4320)
Drop-Off	15	D ≤ 10	10 < D ≤ 35	35 < D ≤ 70	D > 70
		(8788)	(9432)	(9542)	(8238)
Drop-Off	30	D ≤ 20	20 < D ≤ 70	70 < D ≤ 140	D > 140
		(4275)	(4713)	(4869)	(4143)

Table 6

Performance on demand classification at 15-minute interval for Chengdu

Model	Pick-Up				Drop-Off
	Acc.	Std.	F1 Sc.	Std.	Acc.	Std.	F1 Sc.	Std.
ALMMo {N=5}	83.845	4.77	84.326	4.65	82.738	5.60	83.016	5.38
eFCE {N=5}	83.820	4.57	84.431	4.40	82.601	6.28	82.808	6.15
eMG {N=5}	83.065	5.55	83.595	5.39	81.435	7.12	81.648	6.98
eFMI {N=5}	82.613	4.96	82.754	4.92	82.172	6.06	82.331	5.87
eNFN {N=10}	81.571	5.20	81.701	5.12	80.268	6.79	80.304	6.65

Figure 6 shows the confusion matrices for Pick-Up and Drop-Off at 15-minute intervals. The ALLMo (with 5 input variables) achieved the best performance in the Very Low and High classes, with 89% and 91%, respectively. In the intermediate classes, the assertiveness rate was equal to or above 74%, with the highest hit rate obtained in the Medium class for Pick-Ups, with 82%. In addition, note that incorrect classifications, when they occur, are only in adjacent classes.

Fig. 6

Confusion Matrices at 15-minute intervals for ALMMo.

Figure 7 illustrates the heatmaps for the 25 zones of Chengdu. Figure 7 (A) shows the map generated based on the ALMMo classification for Pick-Up, and Figure 7 (B) shows the desired map. On the other hand, Figure 7 (C) depicts the map for Drop-Off obtained based on the ALMMo results, and Figure 7 (D) the desired one. As can be seen, there are few visual differences between the predicted and desired maps, thus showing the good performance of the evolving model in question. The black highlights illustrate the zones where the predicted class does not match the desired class. These highlights exemplify that, even though the model did not perform the classification correctly, the difference in this error is only one adjacent class.

Fig. 7

Heatmaps in the 25 zones in Chengdu - day 11/30/2016 in the range from 07:00 to 07:15: (A) map obtained by ALMMo {N = 5} for Pick-Up; (B) desired map for Pick-Up; (C) map obtained by ALMMo {N = 5} for Drop-Off and; (D) desired map for Drop-Off.

Table 7 shows the Accuracy and F1-Score, as well as its Standard Deviation for Chengdu, in intervals of 30 minutes. The best performance for Pick-Up and Drop-Off was obtained by ALMMo (with 5 input variables), followed by eFMI (with 5 input variables). Subsequently, eMG (with 5 input variables) achieved the best performance for Pick-Up and eFCE (with 5 input variables) by Drop-Off.

Table 7

Performance on demand classification at 30-minute interval for Chengdu

Model	Pick-Up				Drop-Off
	Acc.	Std.	F1 Sc.	Std.	Acc.	Std.	F1 Sc.	Std.
ALMMo {N = 5}	86.952	4.11	87.227	3.98	85.286	5.67	85.396	5.51
eFMI {N = 5}	86.773	4.47	86.865	4.28	85.205	5.86	85.312	5.60
eMG {N = 5}	86.274	4.53	86.447	4.38	84.964	6.11	84.985	5.86
eFCE {N = 5}	86.190	4.51	86.294	4.46	85.107	6.22	85.149	6.01
eNFN {N = 10}	84.869	4.80	84.819	4.67	83.143	6.06	83.144	5.87

Figure 15 illustrates the confusion matrices for Pick-Up and Drop-Off for 30-minute intervals. Again, the best hits obtained by ALMMo (with 5 input variables) were in the Very Low and High classes, with a minimum percentage of 88% and a maximum of 94%. The hit rate is equal to or above 78% in the intermediate classes, with better accuracy achieved in the Medium class, with 86%. Again, as with 15-minute matrices, the errors are present only in adjacent classes.

Fig. 8

Confusion Matrices at 30-minute intervals for ALMMo.

The heatmaps generated for the 25 zones of Chengdu, considering the interval from 07:00 to 07:30 a.m. on 11/30/2016, are presented in Figure 9. The composite map using the ALMMo results for Pick-Up is shown in Figure 9 (A) and the desired one in Figure 9 (B). On the other hand, the map for Drop-Off generated based on the ALMMo forecasting is illustrated in Figure 9 (C), and the desired map for Drop-Off is in Figure 9 (D). As in the maps in Figure 9, it can be seen that the predicted and desired maps are similar. Once more, the black highlights show examples of zones where there is a divergence between the predicted and the desired classification. The difference in ranks is also just one adjacent class.

Fig. 9

Heatmaps in the 25 zones in Chengdu - day 11/30/2016 in the range from 07:00 to 07:30 a.m.: (A) map obtained by ALMMo {N = 5} for Pick-Up; (B) desired map for Pick-Up; (C) map obtained by ALMMo {N = 5} for Drop-Off and; (D) desired map for Drop-Off.

5.2 Demand forecasting and classification in new york (united states of america)

This section presents the computational experiments on the rides dataset from New York City in the United States of America. The database comes from the NYC Taxi and Limousine Commission 14 and has daily records of the trips of three types of taxis: yellow, green, and FHV (For-Hire Vehicle). In this dataset, the city of New York is divided into 263 zones (microregions) already pre-defined by the city government. The experiments were carried out in a subset of 63 zones of Manhattan, New York [8], as illustrated in Figure 10. The experiments were carried out from 01/07/2017 to 30/06/2020. The first 20% (07/01/2017 to 12/31/2017) were reserved for feature selection by Kruskal-Wallis and the classification of the values range. The samples’ remaining 80% (01/01/2018 to 06/30/2020) were used to evaluate the models. This period also includes the COVID-19 pandemic 15 , which has negatively affected demand for private rides services. Datasets were normalized by the interval [1].

Fig. 10

Division of 63 zones in New York.

5.2.1 Demand forecasting in new york

Table 8 shows the best performance of the models in New York for the 15-minute interval in Pick-Up and Drop-Off. The best result was presented by ALMMo (with 20 input variables), followed by eMG (with 10 input variables), eFCE (with 10 input variables), eNFN (with 5 input variables), and eFMI (with 26 input variables). Figure 11 illustrates ALMMo’s Pick-Up and Drop-Off forecasts (with 20 input variables), from 12/10/2019 to 06/30/2020 at Yorkville West (zone 263), with 15-minute intervals. The graph shows a drastic decrease in demands at instant 9000. This decrease is due to the decree establishing the lockdown due to the COVID-19 pandemic. It is relevant to highlight that the model was able to adapt quickly to changes in data dynamics.

Table 8
Performance on demand forecast at 15-minute intervals for New York

Model Pick-Up Drop-Off

RMSE NDEI R² RMSE NDEI R²

ALMMo {N=20} 7.705 0.391 0.812 7.145 0.378 0.829

eMG {N=10} 8.041 0.408 0.798 7.518 0.397 0.813

eFCE {N=10} 8.111 0.410 0.796 7.524 0.396 0.814

eNFN {N=5} 8.897 0.478 0.701 8.334 0.462 0.729

eFMI {N=26} 9.651 0.499 0.680 8.959 0.475 0.735

Model	Pick-Up	Drop-Off
ALMMo {N=20}	7.705	0.391	0.812	7.145	0.378	0.829
eMG {N=10}	8.041	0.408	0.798	7.518	0.397	0.813
eFCE {N=10}	8.111	0.410	0.796	7.524	0.396	0.814
eNFN {N=5}	8.897	0.478	0.701	8.334	0.462	0.729
eFMI {N=26}	9.651	0.499	0.680	8.959	0.475	0.735

Fig. 11

ALMMo forecasts (with 20 input variables) at 15-minute intervals for Yorkville West (zone 263) from 12/10/2019 to 06/30/2020.

Table 9 illustrates the best results of the models for New York for the 30-minute intervals in Pick-Up and Drop-Off. The ALMMo (with 20 input variables) was the model which presented the best results. The eMG (with 10 input variables), eFCE (with 10 input variables), eFMI (with 26 input variables), and eNFN (with 5 input variables) describe the best results in the sequence. Figure 12 depicts the ALMMo Pick-Up and Drop-Off forecasts (with 20 input variables) from 12/10/2019 to 06/30/2020 at Yorkville West (zone 263), with 30-minute intervals. As seen in Figure 11, there is a drastic change in the trend pattern of demands. Again, the model maintained the forecast adequate to the new dynamics of the data, indicating good adaptability and accuracy in both time intervals.

Table 9

Performance on demand forecast at 30-minute intervals for New York

Model	Pick-Up			Drop-Off
	RMSE	NDEI	R²	RMSE	NDEI	R²
ALMMo {N = 20}	14.087	0.354	0.845	12.709	0.336	0.863
eMG {N = 10}	14.536	0.364	0.837	13.177	0.346	0.856
eFCE {N = 10}	14.611	0.366	0.835	13.199	0.347	0.854
eFMI {N = 26}	16.069	0.409	0.795	14.293	0.384	0.821
eNFN {N = 5}	16.105	0.434	0.749	14.415	0.407	0.782

Fig. 12

ALMMo forecasts (with 20 input variables) with 30-minute intervals for Yorkville West (zone 263) from 12/10/2019 to 06/30/2020.

5.2.2 Demand classification in new york

Table 10 illustrates the ranges and the respective values contained in each of the 4 classes for the New York dataset, where the values in parentheses represent the amount in each class, and D, the demand.

Table 10
Definition of demands in value ranges for New York

Type Int. Ranges and Quantity

Very Low Low Medium High

Pick-Up 15 D ≤ 10 10 < D ≤ 40 40 < D ≤ 80 D > 80

(315197) (309900) (224854) (214497)

Pick-Up 30 D ≤ 20 20 < D ≤ 60 60 < D ≤ 120 D > 120

(154801) (114231) (108218) (154974)

Drop-Off 15 D ≤ 10 10 < D ≤ 40 40 < D ≤ 80 D > 80

(301136) (341414) (231584) (190314)

Drop-Off 30 D ≤ 20 20 < D ≤ 60 60 < D ≤ 120 D > 120

(146694) (129418) (111408) (144704)

Type	Int.	Ranges and Quantity
Pick-Up	15	D ≤ 10	10 < D ≤ 40	40 < D ≤ 80	D > 80
		(315197)	(309900)	(224854)	(214497)
Pick-Up	30	D ≤ 20	20 < D ≤ 60	60 < D ≤ 120	D > 120
		(154801)	(114231)	(108218)	(154974)
Drop-Off	15	D ≤ 10	10 < D ≤ 40	40 < D ≤ 80	D > 80
		(301136)	(341414)	(231584)	(190314)
Drop-Off	30	D ≤ 20	20 < D ≤ 60	60 < D ≤ 120	D > 120
		(146694)	(129418)	(111408)	(144704)

The classification performance for Pick-Up and Drop-Off obtained by each model for the 15-minute interval in New York are described in Table 11 by the Accuracy, F1-Score, and its respective Standart Deviation. The best performance was obtained by ALMMo (with 20 variables), followed by eMG (with 20 input variables), eFCE (with 20 input variables), eNFN (with 5 input variables), and eFMI (with 26 input variables).

Table 11

Performance on demand classification at 15-minute interval for New York

Model	Pick-Up				Drop-Off
	Acc.	Std.	F1 Sc.	Std.	Acc.	Std.	F1 Sc.	Std.
ALMMo{N=20}	87.154	5.31	87.263	5.35	86.524	4.69	86.594	4.74
eMG {N=20}	86.537	5.50	86.647	5.51	85.899	4.92	85.967	4.94
eFCE {N=20}	86.509	5.52	86.618	5.53	85.782	4.91	85.822	4.95
eNFN {N=5}	85.171	5.87	85.090	5.82	84.430	5.23	84.351	5.23
eFMI {N=26}	83.761	6.79	83.951	6.41	82.317	5.98	82.147	6.08

Figure13 illustrates the confusion matrices for Pick-Up and Drop-Off at 15-minute intervals. The ALLMo (with 20 input variables) performed better in the Very Low class with a hit rate of 95%, after in the High class, with 88%. In the Low and Medium classes, the ALLMo had hit rates equal to or above 78%, demonstrating its good assertiveness. It is noteworthy that the incorrect classifications have a low percentage in the matrix, and the errors are found only in adjacent classes.

Fig. 13

Confusion Matrices at 15-minute intervals for ALMMo.

Figure 14 shows the heatmaps for the 63 zones of New York on 01/01/2020 for the range from 6:00 to 6:15 p.m.. Figure 14 (A) illustrates the map obtained based on the ALMMo results for Pick-Up, and Figure 14 (B) the desired map for Pick-Up. On the other hand, Figure 14 (C) depicts the map for Drop-Off built based on the results of ALMMo, and Figure 14 (D) the desired one for Drop-Off. The black highlights show some areas where there were differences in classification. As can be seen, the few errors in the classification remain in adjacent classes.

Fig. 14

Heatmaps for 63 zones in New York - 01/01/2020 in the range of 18:00 to 18:15: (A) map obtained by ALMMo {N = 20} for Pick-Up; (B) desired map for Pick-Up; (C) map obtained by ALMMo {N = 20} for Drop-Off and; (D) desired map for Drop-Off.

The results of the 30-minute interval in New York are shown in Table 12. ALMMo (with 20 input variables) was the model with the best performance, followed by eFCE (with 10 input variables), eMG (with 10 input variables), eNFN (with 5 input variables), and eFMI (with 26 input variables).

Table 12

Performance on demand classification at 30-minute interval for New York

Model	Pick-Up				Drop-Off
	Acc.	Std.	F1 Sc.	Std.	Acc.	Std.	F1 Sc.	Std.
ALMMo {N=20}	88.060	5.12	88.094	5.15	87.449	4.46	87.284	4.48
eFCE {N=10}	87.809	5.11	87.863	5.12	87.090	4.49	87.124	4.48
eMG {N=10}	87.806	5.11	87.859	5.12	87.065	4.51	87.098	4.50
eNFN {N=5}	86.252	5.72	86.191	5.69	85.520	5.16	85.436	5.15
eFMI {N=26}	85.509	6.23	85.803	6.26	85.088	5.25	85.247	5.09

Figure 15 illustrates the confusion matrices for Pick-Up and Drop-Off at 30-minute intervals. The greatest hits of the ALMMo (with 20 input variables) occurred in the Very Low and High classes, with 96% (Pick-Up) and 95% (Drop-Up) in the Very Low class, and 91% (Pick-Up and Drop-Off) in the High class. In the intermediate classes, the ALLMo obtained assertiveness equal to or above 78%. Here we can see the same pattern as in the other experiments, i.e., the low rate of errors and these errors happening only in the adjacent classes.

Fig. 15

Confusion Matrices at 30-minute intervals for ALMMo.

The heatmaps generated for the 63 zones of New York, covering the range from 6:00 to 6:30 p.m. on 01/01/2020, are shown in Figure16. The map constructed using ALMMO results for Pick-Up is shown in Figure 16 (A) and the desired one in Figure 16 (B). On the other hand, the map for Drop-Off generated based on the ALMMo forecasts is illustrated in Figure 16 (C), and the desired map for Drop-Off is in Figure16 (D). As with previous heatmaps, the black markings exemplify zones with a divergence between classifications. Again, as in the other maps, the classification error is between adjacent classes.

Fig. 16

Heatmaps for 63 zones in New York - 01/01/2020 in the range of 18:00 to 18:30: (A) map obtained by ALMMo {N = 20} for Pick-Up; (B) desired map for Pick-Up; (C) map obtained by ALMMo {N = 20} for Drop-Off and; (D) desired map for Drop-Off.

6 Conclusions

This work presented a new approach to taxi demand forecasting and classification. The proposed approach considers historical data from taxi rides and meteorological information and uses the Kruskal-Wallis statistical method to identify the most relevant variables. Then, the forecast is performed by evolving fuzzy systems, and the classification is carried out using the forecast outputs.

Computational experiments were carried out to evaluate the performance of the proposed approach considering ALMMo, eMG, eFCE, eFMI, and eNFN, to predict and classify demand for Pick-Up and Drop-Off with 15 and 30-minute intervals in Chengdu and New York. The computational results of the evolving algorithms were compared with each other and state-of-the-art. The algorithms of the proposed approach proved to be competitive with state-of-the-art, achieving superior or comparable performance. The comparison was performed only between the evolving models in the classification experiments. Among the evolving models, ALMMo presented the best results.

The computational results suggested that the proposed approach is promising for the forecasting and classification of passenger demand since: (i) it presents online data processing, different from most works with the same theme in the literature; (ii) it obtained better or similar results compared to state-of-the-art approaches; (iii) it showed consistency in forecasts over extended periods. In addition, the proposal uses heatmaps as a more intuitive way of visualizing the data, thus facilitating decision-making more assertively.

The limitations of the research include: the lack of similar works on passenger demand involving evolving systems for purposes of comparison with the proposed approach; the lack of real-time experiments to assess the evolving characteristics of the systems, and; the inaccessibility of ride databases with more recent data, due to the privacy of information by taxi companies.

As a perspective for future work, it is suggested to apply new techniques of region division to the proposed approach. Another possible future work would incorporate methodologies that analyze and decode images to capture spatial information from the regions studied. Finally, evaluating the proposed approach’s robustness and scalability is important.

Footnotes

Acknowledgment

The authors acknowledge CAPES, Brazilian Ministry of Education, code 001. In addition, the authors would like to thank Didi Chuxing for the dataset. Data source: DiDi Chuxing GAIA Open Dataset Initiative.

Places reserved to taxi requests.

Private rides company ().

The time window of the historical data has been set to 8 past intervals as in [9 , ].

The historical number chosen was 8, as it also includes all the days of the previous week, allowing the models to learn the specific trends of each day.

Most works identify the moving average of previous days as the historical average. Therefore, this work will follow the same nomenclature.

The time division was based on results obtained from empirical observations of local traffic in the regions, provided by municipal governments.

Available at:

In Zhang et al. [] only theRMSEis presented and there were no experiments that performed the demand forecast for Drop-Off.

In [] there were no experiments that performed the demand forecast with a 30-minute interval for Pick-Up.

Available at:

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