We propose a methodology based on the standard ReLU Deep Neural Networks (DNN) to make predictions and quantify their uncertainty. Classically, people rely on linear, non-linear, or non-parametric kernel methods to fit and then predict the time series. As the universal approximation ability was revealed for DNN, its application has become more and more popular for prediction tasks in various scientific areas. However, the corresponding uncertainty quantification has not been studied thoroughly. Particularly, the uncertainty in prediction will consist of two parts: (1) the future variability; (2) the estimation variability within training data. To capture both variabilities, we build the so-called pertinent prediction interval (PPI) with the DNN model estimator. We first explore the consistency property of the DNN estimator with beta-mixing dependent data. Subsequently, we show that the implied forward bootstrap series is still beta-mixing and possesses the same stationary distribution as the original time series in probability, which is a key condition to enable the PPI. Lastly, the desired PPI is built after imposing minimal conditions on the limiting distribution of predictive roots. Simulations and real-data analysis are deployed to challenge our approach with standard non-parametric methods.
Figures & tables
Yt∗←f^n(Yt−1∗)+ϵt∗;fort=n+1,…,n+J.
Algorithm 1 Multi-step ahead point predictions and the corresponding QPI.
Ut∗←f^n(Ut−1∗)+ϵt∗;fort=2,…,m+n.
Algorithm 2 Multi-step ahead PPI based on forward bootstrap
Step
1
2
3
4
5
1
2
3
4
5
DNN
Kernel
Model-1; n = 50
CVR-QPI
0.927
0.898
0.901
0.884
0.889
0.886
0.906
0.923
0.902
0.906
LEN-QPI
4.105
4.352
4.353
4.355
4.354
3.533
4.385
4.552
4.607
4.635
CVR-PPI
0.941
0.902
0.904
0.893
0.897
0.915
0.913
0.921
0.910
0.913
LEN-PPI
4.276
4.376
4.377
4.376
4.381
4.068
4.552
4.669
4.717
4.739
Table 1: Simulation results with Model-1 data.
Step
1
2
3
4
5
1
2
3
4
5
DNN
Kernel
Model-2; n = 50
CVR-QPI
0.907
0.920
0.938
0.929
0.931
0.546
0.595
0.681
0.724
0.754
LEN-QPI
4.417
4.489
4.602
4.680
4.707
2.453
2.841
3.167
3.431
3.635
CVR-PPI
0.920
0.922
0.943
0.931
0.938
0.693
0.712
0.750
0.766
0.802
LEN-PPI
4.634
4.655
4.703
4.764
4.800
4.176
4.202
4.243
4.345
4.358
Table 2: Simulation results with Model-2 data.
Step
1
2
3
4
5
1
2
3
4
5
DNN
Kernel
Model-3; n = 50
CVR-QPI
0.897
0.912
0.918
0.894
0.933
0.021
0.018
0.022
0.035
0.034
LEN-QPI
4.343
4.407
4.441
4.483
4.534
0.078
0.104
0.116
0.146
0.169
CVR-PPI
0.918
0.920
0.934
0.899
0.936
0.150
0.143
0.157
0.183
0.163
LEN-PPI
4.600
4.601
4.589
4.603
4.643
1.160
1.211
1.208
1.257
1.330
Table 3: Simulation results with Model-3 data.
DNN
Kernel
w=50
d:
1
5
10
1
5
10
OCVR-QPI
0.904
0.839
0.765
0.901
0.667
0.140
OLEN-QPI
249.721
177.748
253.878
203.281
167.785
174.764
OCVR-PPI
0.911
0.862
0.803
0.917
0.748
0.182
OLEN-PPI
302.778
269.948
265.031
320.283
296.002
248.926
Table 4: The average coverage rate and interval length of different methods over each prediction horizon j and 7 selected time series, i.e., OCVR and OLEN defined in the main text.
DNN
Kernel
w=50
d:
1
5
10
1
5
10
StdCVR-QPI
0.014
0.021
0.069
0.013
0.075
0.133
StdLEN-QPI
309.195
245.363
444.964
285.897
275.712
385.182
StdCVR-PPI
0.014
0.019
0.059
0.017
0.061
0.159
StdLENPPI
402.349
406.473
436.904
448.045
467.784
521.170
Table 5: The sample standard deviation of the coverage rate and interval length from different methods over each prediction horizon j and 7 selected time series, i.e., the spread of the overall average coverage rate and the interval length.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 1: The illustration of a fully connected DNN with L=2 , H=4 and W=37 , and input dimension d=2 and output dimension 1.
Probabilistic time series forecasting estimates distributions of future values or trajectories to quantify uncertainty and support risk-aware decisions.
Deep neural networks achieve impressive results across diverse applications, yet their overconfidence on unseen inputs necessitates reliable epistemic uncertainty modeling. Existing methods for uncertainty modeling face a fundamental dilemma: Bayesian approaches provide principled estimates but remain computationally prohibitive, while efficient second-order predictors lack rigorous connections between their specific objectives and epistemic uncertainty quantification. To resolve this dilemma, we introduce Dirichlet-approximated possibilistic posterior predictions (DAPPr), a principled framework grounded in possibility theory. We define a possibilistic posterior over parameters, project it to the prediction space via supremum operators, and approximate the projected posterior using learnable Dirichlet possibility functions. This projection-and-approximation strategy yields a simple training objective with closed-form solutions. Despite its simplicity, extensive experiments across diverse benchmarks show that DAPPr achieves competitive or superior uncertainty quantification performance over state-of-the-art second-order predictors while maintaining both principled derivation and computational efficiency. Code is available at https://github.com/MaxwellYaoNi/DAPPr.
Yao Ni, Jeremie Houssineau, Yew-Soon Ong +1
Nanyang Technological University · A*STAR · The University of New South Wales +1
Uncertainty quantification (UQ) in deep learning regression is of wide interest, as it supports critical applications including sequential decision making and risk-sensitive tasks. In heteroskedastic regression, where the uncertainty of the target depends on the input, a common approach is to train a neural network that parameterizes the mean and the variance of the predictive distribution. Still, training deep heteroskedastic regression models poses practical challenges in the trade-off between uncertainty quantification and mean prediction, such as optimization difficulties, representation collapse, and variance overfitting. In this work we identify previously undiscussed fallacies and propose a simple and efficient procedure that addresses these challenges jointly by post-hoc fitting a variance model across the intermediate layers of a pretrained network on a hold-out dataset. We demonstrate that our method achieves on-par or state-of-the-art uncertainty quantification on several molecular graph datasets, without compromising mean prediction accuracy and remaining cheap to use at prediction time.
Mikkel Jordahn, Jonas Vestergaard Jensen, James Harrison +2
We investigate a forecasting framework based on a simple discrete-time dynamic model with coefficients varying in time. The parameters of the model are recovered within a deep learning framework, which makes it possible to retain a transparent parametric structure while simultaneously accounting for complex and nonstationary patterns in the observed phenomenon. Our analysis covers two specifications of the noise process. Besides the standard Gaussian setting, we also consider Laplace-distributed noise, which can offer a more adequate description in the presence of heavier tails and sharper local fluctuations. For both cases, we formulate the predictive scheme of the model and analyze the associated uncertainty quantification, including the construction of prediction intervals. The results illustrate that a relatively simple model, when combined with time-dependent parameter estimation, can serve as a mathematically tractable and practically flexible tool for forecasting complex dynamics under different noise assumptions. The general model is stated for TVAR(p), while the prediction-interval formulas and the numerical experiments are developed for the TVAR(1) case.
Agnieszka Kopeć, Paweł Przybyłowicz, Martyna Wiącek