# Receding-Horizon Prediction of Vehicle Velocity Profile Using Deterministic and Stochastic Deep Neural Network Models

^{*}

## Abstract

**:**

^{2}score greater than 0.8 for the prediction horizon length of 10 s and remains to be solid (greater than 0.6) for the horizon lengths up to 25 s; (ii) the actual vehicle position and the velocity history are the most significant input features, where the optimal value of history interval length lies in the range from 30 to 50 s; (iii) the stochastic model have only slightly lower accuracy of predicting the velocity expectation along the receding horizon when compared to the deterministic model (the root mean square error is higher by 2.2%), and it outputs consistent standard deviation prediction.

## 1. Introduction

#### 1.1. Literature Review

^{2}score between the actual and predicted velocity profiles are commonly used as prediction accuracy indicators.

#### 1.2. Problem Statement

#### 1.3. Research Aim and Contributions

## 2. Description of the Recorded Dataset

#### 2.1. Recorded Driving Cycles

- Bus garage number;
- Timestamp;
- Bus geographical coordinates;
- Bus longitudinal velocity;
- Cumulative distance travelled (from odometer).

- (1)
- The driving cycle follows the reference route trajectory, i.e., no detours are acceptable;
- (2)
- The time difference between each pair of consecutive driving cycle samples is equal to the nominal sampling time, i.e., 1 s;
- (3)
- The initial and final velocity of the driving cycle is equal to zero;
- (4)
- The vehicle acceleration values are within the interval of [–3, 3] m/s
^{2}; - (5)
- The total number of numerically undetermined values (NaN) is zero;
- (6)
- The proportion of the dwell time samples to the total number of samples is less than 75%.

#### 2.2. Preparation of Training, Validation, and Test Datasets

## 3. Deterministic Vehicle Velocity Prediction Model

#### 3.1. Modelling of Deep Neural Network with Deterministic Output

#### 3.2. Analysis of Deterministic Model Prediction Accuracy

#### 3.2.1. Influence of Input Features

- (1)
- Current vehicle position ${d}_{k}$;
- (2)
- Current vehicle position ${d}_{k}$ and current vehicle velocity ${v}_{k}$;
- (3)
- Vehicle velocity history ${v}_{h,k}$;
- (4)
- Current vehicle position ${d}_{k}$ and vehicle velocity history ${v}_{h,k}$;
- (5)
- Current vehicle position ${d}_{k}$, vehicle velocity history ${v}_{h,k}$, and time of day ${t}_{k}$;
- (6)
- Current vehicle position ${d}_{k}$, vehicle velocity history ${v}_{h,k}$, time of day ${t}_{k}$, and day of week ${w}_{k}$.

_{n}, where $n$ denotes the ordinal number of the input combination as listed above. Note that the fixed prediction horizon length ${H}_{p}=10$ s (and the fixed velocity history interval length ${H}_{h}=20$ s, when applicable) are applied to each model to make the comparative analysis consistent.

^{2}/h

^{2}.

_{n}model, which are also given in Table 1 along the number of adjustable NN parameters. The vehicle velocity profile prediction accuracy saturates in the case of NN-DET-I

_{4}model ($RMS{E}_{v}=5.07$ km/h), which clearly indicates that both time of day ${t}_{k}$ and day of week ${w}_{k}$ have a negligible influence on the model prediction accuracy. The worst prediction accuracy is obtained for the NN-DET-I

_{1}model ($RMS{E}_{v}=11.84$ km/h), which relies on the current vehicle position ${d}_{k}$ as the only input feature. The reason for this is the insufficient conditionality of the model with respect to the input data, as previously confirmed when predicting individual vehicle velocity values in the case of NN-STC model (Figure 1, [22]). By adding the current velocity ${v}_{k}$ next to the vehicle position ${d}_{k}$ (NN-DET-I

_{2}model), a significant but still insufficient drop in $RMS{E}_{v}$ from 11.84 km/h to 7.28 km/h (≈40%) is achieved. Despite using the vehicle velocity history ${v}_{h,k}$ as only input, the NN-DET-I

_{3}model achieves the prediction accuracy close to the NN-DET-I

_{4}model ($RMS{E}_{v}=6.12\text{}$km/h vs. $5.07\text{}$km/h), which additionally uses the vehicle position input. It is important to note that none of the input feature combinations result in model overfitting, as evidenced by the absence of a gap between training and validation losses (Figure 5). Good model generalization properties are additionally confirmed through the closeness of the testing and validation losses (Table 1).

#### 3.2.2. Influence of Vehicle Velocity History Interval Length

_{h,T}, where $T$ denotes the history interval length in seconds. Again, the $RMS{E}_{v}$ loss values, defined by Equation (2) and calculated for the case of the validation dataset, are used to evaluate the prediction accuracy of the NN-DET-H

_{h,T}model. When training each of the NN-DET-H

_{h,T}models, the full set of candidate input features (${d}_{k}$, ${v}_{h,k}$, ${t}_{k}$, and ${w}_{k}$) is used along the fixed prediction horizon length ${H}_{p}=10$ s.

_{h,T}model, with the numerical values contained in Table 2. These results point out that the validation loss is locally saturated at ${H}_{h}=50$ s, where $RMS{E}_{v}$ equals 4.85 km/h, while in the case of longer velocity history (${H}_{h}=200$ s) $RMS{E}_{v}$ increases to 4.96 km/h. The reason for this is that the NN-DET-H

_{h,200}model is overly conditioned by the input data, which leads to model overfitting, i.e., poorer model generalization properties. This is reflected in the increasing gap between the training and validation loss plots in Figure 6, while the validation and testing loss values are close to each other (Table 2). The values of ${H}_{h}$ in the range of 30 to 50 s represent an optimal selection, as they provide minimal loss on validation (and test) dataset and reduce the complexity of NN-DET model.

#### 3.2.3. Influence of Vehicle Velocity Prediction Horizon Length

_{p,T}, where T denotes the receding horizon length in seconds. Each model is trained separately for the following input sets:

- Full set of candidate features (${d}_{k}$, ${t}_{k}$, ${w}_{k}$ i ${v}_{h,k}$);
- Vehicle velocity history only (${v}_{h,k}$),

_{p,T}model prediction accuracy is performed based on the following indicators calculated for the validation dataset:

- The loss value $RMS{E}_{v}$ defined by Equation (2);
- The ${R}_{j}^{2}$ score value, calculated for each $j$th discrete step of the prediction horizon length ${H}_{p}$:$${R}_{j}^{2}=1-\frac{{{\displaystyle \sum}}_{k=1}^{N}{\left({v}_{k,j}-{\widehat{v}}_{k,j}\right)}^{2}}{{{\displaystyle \sum}}_{k=1}^{N}{\left({v}_{k,j}-{\overline{v}}_{j}\right)}^{2}},\hspace{1em}\hspace{1em}j=1,\text{}2,\dots ,{H}_{p}.$$

_{a}and

_{b}correspond to the above-defined model input sets. These results indicate that the validation loss assumes almost linear upward trend until ${H}_{p}=20$ s for both input set cases, and then begin to saturate to the levels $RMS{E}_{v}=13.22$ km/h in the case of NN-DET-H

_{p,a}model (full set of inputs) and $RMS{E}_{v}=15.84$ km/h in the case of NN-DET-H

_{p,b}model (only the velocity history input), which are reached at ${H}_{p}=200$ s. The loss $RMS{E}_{v}$ is higher by 23% on average for NN-DET-H

_{p,b}model when compared to full model NN-DET-H

_{p,a}, reaffirming the importance of including the current city bus position as an additional input feature of the model.

_{p,a,200}model maintains a satisfactory prediction accuracy (${R}_{j}^{2}>0.25$, for all $j=1,2,\dots ,\text{}{H}_{p}$, ${H}_{p}=200$) which is not the case with the NN-DET-H

_{200,b}model, thus confirming once again the importance of including the current vehicle position ${d}_{k}$ in the input feature set. The prediction accuracy is very good (${R}_{j}^{2}\ge 0.8$) for the first 10 prediction steps ($j\le 10$) and remains to be solid (${R}_{j}^{2}>0.6$) up to first 25 prediction steps ($j\le 25$).

## 4. Stochastic Vehicle Velocity Prediction Model

#### 4.1. Modelling of Deep Neural Network with Stochastic Output

#### 4.2. Comparative Analysis of Prediction Accuracies of Stochastic and Deterministic Models

#### 4.2.1. Considered Metrics

- Loss function $RMS{E}_{v}$ defined by Equation (2);
- ${R}_{j}^{2}$ score defined by Equation (3), calculated for each $j$th discrete step along the prediction horizon ($j=1,\text{}2,\dots ,{H}_{p}=10$), as applied in Section 3;
- Mean value of ${\mathrm{RMSE}}_{j}$, also calculated for each $j$th discrete step along the prediction horizon:$$\overline{{\mathrm{RMSE}}_{j}}=\sqrt{\frac{1}{N}{\displaystyle \sum}_{k=1}^{N}{\left({\widehat{v}}_{k,j}-{v}_{k,j}\right)}^{2}};$$
- Mean prediction RMSE along the prediction horizon ${H}_{p}$ and for the given time step $k$ along the driving cycle:$${\mathrm{RMSE}}_{k}=\sqrt{\frac{1}{{H}_{p}}{\displaystyle \sum}_{j=1}^{{H}_{p}}{\left({\widehat{v}}_{k,j}-{v}_{k,j}\right)}^{2}};$$

#### 4.2.2. Analysis of Prediction Accuracy along the Prediction Horizon

#### 4.2.3. Analysis of Prediction Accuracy along the Route

## 5. Conclusions

- (a)
- Applying and examining the proposed vehicle prediction models within vehicle deterministic or stochastic model predictive control strategies (e.g., energy management strategy of a PHEV aimed at minimizing the vehicle fuel and electricity consumption for a wide range of driving cycles);
- (b)
- Considering other types of NNs, such as recurrent NNs, which can be more suitable for the task of dynamic system behaviour prediction and potentially bring further gains in model prediction accuracy;
- (c)
- Considering Markov chain-based stochastic velocity prediction method, which, in addition to the vehicle velocity (and acceleration), would also take information about the vehicle position when defining the Markov states;
- (d)
- Examining the proposed prediction models for other transport systems that are not characterized by fixed/repeating routes (unlike the city bus transport system considered herein), including different types of vehicles, as well;
- (e)
- Adding more relevant inputs to the model such as accelerator pedal opening and traffic related inputs (not available in the presented study) for potentially improved prediction performance;
- (f)
- Comparing the various developed prediction models to each other, as well as with respect to existing models.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ADAM | Adaptive Moment Estimation (optimization algorithm used for training of neural networks) |

CAN | Controller Area Network |

GPRS | General Packet Radio Service |

GPS | Global Positioning System |

MPC | Model Predictive Control |

MSE | Mean Squared Error |

NLL | Negative Log-Likelihood |

NN | Neural Network |

NN-DET | Deterministic Neural Network (model) |

NN-DSTC | Dynamic Stochastic Neural Network (model) |

NN-STC | Static Stochastic Neural Network (model) |

PHEV | Plug-in Hybrid Electric Vehicle |

RMSE | Root Mean Squared Error |

TPM | Transition Probability Matrix |

Table of Symbols | |

$\overline{MS{E}_{v}}$ | Mean squared error of predicted vs. recorded vehicle velocities (used as loss function for NN-DSTC model) |

$\overline{NLL}$ | Mean of negative log-likelihoods of predicted normal distributions of vehicle velocity ${\mathcal{N}}_{k,j}\left({\widehat{\mu}}_{k,j},{\widehat{\sigma}}_{k,j}\right)$ for each prediction step $j\in \left[1,\text{}{H}_{p}\right]$ (used as loss function for NN-DET model) |

j | Discrete step of prediction horizon, $j=1,2,\dots ,{H}_{p}$ |

k | kth discrete time instant (i.e., data sample) of given recorded driving cycle |

N | Total number of data samples used to train NN-DET and NN-DSTC models |

${H}_{h}$ | Vehicle velocity history interval length |

${H}_{p}$ | Prediction horizon length |

$\overline{RMS{E}_{k}}$ | Mean prediction RMSE calculated along the full prediction horizon ${H}_{p}$ for the kth time instant |

$\overline{RMS{E}_{j}}$ | Mean prediction RMSE calculated for jth discrete step of prediction horizon |

${R}_{j}^{2}$ | Coefficient of determination value calculated for jth discrete step of prediction horizon |

${d}_{k}$ | Vehicle position (i.e., distance travelled from reference route departure station) for kth time instant of given recorded driving cycle |

${t}_{k}$ | Time of day for kth time instant of given recorded driving cycle |

${v}_{k}$ | Vehicle velocity for kth time instant of given recorded driving cycle |

${w}_{k}$ | Day of week for kth time instant of given recorded driving cycle |

${v}_{h,k}$ | Vector of historical vehicle velocities of length ${H}_{h}$ |

${\sigma}_{k}$ | Vehicle velocity standard deviation calculated from test samples/population |

${\widehat{v}}_{k}$ | Vehicle velocity predicted by NN-DET model |

${\widehat{\mu}}_{k}$ | Vehicle velocity expectation predicted by NN-STC/NN-DSTC model |

${\widehat{\sigma}}_{k}$ | Vehicle velocity standard deviation predicted by NN-STC/NN-DSTC model |

$\mathcal{X}$ | Set of input (training) values |

$\mathcal{Y}$ | Set of output (target) values related to inputs $\mathcal{X}$ |

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**Figure 1.**Illustration of previously proposed static stochastic neural network (NN-STC) model [22] for predicting city bus velocity distribution along route (

**a**), including related testing results (

**b**).

**Figure 2.**Examples of ten randomly selected valid recorded driving cycles for the direction Babin kuk–Pile: velocity vs. time (

**a**) and velocity vs. travelled distance (

**b**) profiles.

**Figure 3.**Illustration of the process of generating input-output datasets $\mathcal{X}$ and $\mathcal{Y}$, necessary for training, validation, and testing of NN-based velocity prediction models.

**Figure 4.**Proposed architecture of deterministic deep feedforward neural network (NN-DET) for predicting vehicle velocity profile on receding time horizon ${\widehat{v}}_{k+j};j=1,2,\dots ,{H}_{p}$.

**Figure 5.**RMSE

_{v}loss values obtained for each NN-DET-I

_{n}model (defined by characteristic input feature sets) by using training and validation datasets.

**Figure 6.**RMSE

_{v}loss values obtained for different velocity history interval lengths H

_{h}of NN-DET model by using training and validation datasets.

**Figure 7.**RMSE

_{v}loss values obtained for different prediction horizon lengths H

_{p}of NN-DET-H

_{p,a}and NN-DET-H

_{p,b}models by using training and validation datasets.

**Figure 8.**${R}_{j}^{2}$ score values corresponding to jth discrete step of prediction horizon with length H

_{p}= 200 s and calculated for validation data set (note that for given sampling time of 1 s, the prediction in-terval of, e.g., j = 200 steps correspond to the length of 200 s).

**Figure 9.**Proposed architecture of dynamic stochastic deep feedforward neural network (NN-DSTC) for predicting parameters of normal vehicle velocity distribution ${\mathcal{N}}_{k,j}\left({\widehat{\mu}}_{k,j},{\widehat{\sigma}}_{k,j}\right);j=1,2,\dots ,{H}_{p}$ along receding time horizon of length ${H}_{p}$.

**Figure 11.**Predicted velocity ${\widehat{v}}_{k,j}$ and velocity expectation ${\widehat{\mu}}_{k,j}$ in relation to real velocity values ${v}_{k,j}$ for NN-DET model (

**a**) and NN-DSTC model (

**b**), respectively, where predictions for each jth time step along the prediction horizon H

_{p}are marked in different colours.

**Figure 12.**Comparative plots of prediction performance of NN-DET and NN-DSTC models, expressed in terms of ${R}_{j}^{2}$ (

**a**) and $\overline{{\mathrm{RMSE}}_{j}}$ (

**b**) metrics, determined for different prediction steps j along prediction horizon H

_{p}.

**Figure 13.**Comparative time response of vehicle velocity predictions provided by NN-DET and NN-DSTC models for one of driving cycles from test data set (

**a**) and characteristic response zoom-in detail (

**b**).

**Figure 14.**Boxplot of NN-DSTC model-predicted uncertainty parameter (i.e., standard deviation ${\widehat{\sigma}}_{j}$) of vehicle velocity prediction along steps $j=1,2,\dots ,{H}_{p}$ of prediction horizon.

**Figure 15.**Dependence of NN-DET model velocity prediction accuracy metrics ${R}_{j}^{2}$ (

**a**) and $\overline{RMS{E}_{j}}$ (

**b**) on mean of NN-DTSC model-predicted standard deviation ${\widehat{\sigma}}_{\mathrm{j}}$, where $\rho $ represents the corresponding Pearson correlation index value (100% means full correlation, and −100% denotes full anti-correlation).

**Figure 16.**Scattering of NN-DET and NN-DSTC models individual RMS velocity prediction errors along the selected bus route (

**a**), including the corresponding distributions given separately for NN-DET (

**b**) and NN-DSTC models (

**c**).

**Table 1.**Training, validation, and testing results for each NN-DET-I

_{n}models (defined by characteristic input feature sets), including the number of adjustable model parameters.

Model | $\mathbf{Loss}\text{}\mathbf{Value},\text{}\mathit{R}\mathit{M}\mathit{S}{\mathit{E}}_{\mathit{v}}\text{}[\mathbf{km}/\mathbf{h}]$ | Number of NN Parameters | ||
---|---|---|---|---|

Training Set | Validation Set | Test Set | ||

NN-DET-I_{1} | 11.84 | 11.84 | 11.83 | 44,074 |

NN-DET-I_{2} | 7.26 | 7.28 | 7.26 | 44,330 |

NN-DET-I_{3} | 6.01 | 6.12 | 6.11 | 48,938 |

NN-DET-I_{4} | 4.96 | 5.07 | 5.07 | 49,194 |

NN-DET-I_{5} | 4.92 | 5.03 | 5.05 | 49,450 |

NN-DET-I_{6} | 4.88 | 5.03 | 5.03 | 49,706 |

**Table 2.**Training, validation, and testing results of NN-DET models with different history interval length ${H}_{h}$ [s], including the number of adjustable model parameters.

Model | $\mathbf{Loss}\text{}\mathbf{Value},\text{}\mathit{R}\mathit{M}\mathit{S}{\mathit{E}}_{\mathit{v}}\text{}[\mathbf{km}/\mathbf{h}]$ | Number of NN Parameters | ||
---|---|---|---|---|

Training Set | Validation Set | Test Set | ||

NN-DET-H_{h,3} | 5.32 | 5.35 | 5.38 | 45,354 |

NN-DET-H_{h,7} | 5.13 | 5.21 | 5.23 | 46,378 |

NN-DET-H_{h,10} | 5.00 | 5.11 | 5.10 | 47,146 |

NN-DET-H_{h,15} | 4.93 | 5.06 | 5.05 | 48,426 |

NN-DET-H_{h,20} | 4.91 | 5.02 | 5.01 | 49,706 |

NN-DET-H_{h,30} | 4.76 | 4.91 | 4.92 | 52,266 |

NN-DET-H_{h,40} | 4.70 | 4.88 | 4.86 | 54,826 |

NN-DET-H_{h,50} | 4.61 | 4.85 | 4.84 | 57,386 |

NN-DET-H_{h,100} | 4.57 | 4.85 | 4.89 | 70,186 |

NN-DET-H_{h,200} | 4.53 | 4.96 | 4.93 | 95,786 |

**Table 3.**Training, validation, and testing results of NN-DET models with different prediction horizon length ${H}_{p}$ [s], including the number of adjustable model parameters.

Model | $\mathbf{Loss}\text{}\mathbf{Value},\text{}\mathit{R}\mathit{M}\mathit{S}{\mathit{E}}_{\mathit{v}}\text{}[\mathbf{km}/\mathbf{h}]$ | Number of NN Parameters | ||
---|---|---|---|---|

Training Set | Validation Set | Test Set | ||

NN-DET-H_{p,3} | 1.87 (2.05) * | 1.88 (2.05) | 1.90 (2.08) | 49,475 (48,707) |

NN-DET-H_{p,7} | 3.74 (4.45) | 3.83 (4.53) | 3.84 (4.52) | 49,607 (48,839) |

NN-DET-H_{p,10} | 4.84 (6.02) | 5.01 (6.13) | 5.00 (6.13) | 49,706 (48,938) |

NN-DET-H_{p,15} | 6.42 (8.14) | 6.56 (8.23) | 6.59 (8.26) | 49,871 (49,103) |

NN-DET-H_{p,20} | 7.42 (9.67) | 7.63 (9.82) | 7.58 (9.79) | 50,036 (49,268) |

NN-DET-H_{p,50} | 10.20 (13.65) | 10.34 (13.80) | 10.35 (13.76) | 51,026 (50,258) |

NN-DET-H_{p,100} | 12.04 (15.23) | 12.13 (15.33) | 12.15 (15.35) | 52,676 (51,908) |

NN-DET-H_{p,200} | 13.15 (15.77) | 13.20 (15.82) | 13.22 (15.84) | 55,976 (55,208) |

_{p})

_{b}model, while the ones given outside parentheses correspond to full-input-set (NN-DET-H

_{p})

_{a}models.

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**MDPI and ACS Style**

Topić, J.; Škugor, B.; Deur, J.
Receding-Horizon Prediction of Vehicle Velocity Profile Using Deterministic and Stochastic Deep Neural Network Models. *Sustainability* **2022**, *14*, 10674.
https://doi.org/10.3390/su141710674

**AMA Style**

Topić J, Škugor B, Deur J.
Receding-Horizon Prediction of Vehicle Velocity Profile Using Deterministic and Stochastic Deep Neural Network Models. *Sustainability*. 2022; 14(17):10674.
https://doi.org/10.3390/su141710674

**Chicago/Turabian Style**

Topić, Jakov, Branimir Škugor, and Joško Deur.
2022. "Receding-Horizon Prediction of Vehicle Velocity Profile Using Deterministic and Stochastic Deep Neural Network Models" *Sustainability* 14, no. 17: 10674.
https://doi.org/10.3390/su141710674