Higher-Order Convolutional Neural Networks for Essential Climate Variables Forecasting
Abstract
:1. Introduction
- The theoretical extension of lower-order Convolutional Neural Networks (CNNs) to their N-D analogs by providing a rigorous mathematical foundation for the forward and backward passes;
- The demonstration of issues and workarounds for incorporating this extension into a working DL framework;
- The design and development of a novel 4D-ETCN to tackle the ECVs’ forecasting problem and its evaluation on properly designed datasets for the cause;
- The introduction of two novel datasets encoding satellite-derived geophysical parameters, specifically soil temperature at different levels, obtained on monthly periodicity over 40 years.
2. Materials and Methods
2.1. Proposed Method: Higher-Order Convolutional Neural Networks
2.1.1. Preliminary Definitions and Tensor Algebra
2.1.2. Forward Propagation in ND CNNs
- 1.
- Perform Tucker decomposition on tensor , and obtain its core tensor and its factor matrices :
- 2.
- Perform Tucker decomposition on tensor , and obtain its core tensor and its factor matrices :
- 3.
- Derive the core tensor of the N-D Tucker convolution, , by combining the core tensors and via their Kronocker product:
- 4.
- Derive the factor matrices of the N-D Tucker convolution, , by combining the factor matrices and via their partial (mode 1) convolution:
- 5.
- Derive the output tensor of the N-D Tucker convolution, , by combining the core tensor and the factor matrices via their Tucker composition:
2.1.3. Backpropagation in ND CNNs
- Since is the output of the previous layer, becomes the loss gradient for the previous layer.
- is used to update the filters of the convolutional layer via the gradient step:
- Loss gradient from the next layer:
- Gradient with respect to the layer’s input:
- Gradient with respect to the layer’s weights:
- Gradient with respect to the layer’s bias:
2.1.4. Efficient Implementation: Stacking (N-1)D Convolutions
2.2. Proposed Simultaneous Spatiotemporal ECVs’ Forecasting Architecture
2.2.1. Embedded Temporal Convolutional Network—ETCN
2.2.2. 4D-ETCN for ECVs’ Forecasting Modeling
2.2.3. 4D-ETCN Modules for ECVs’ Feature Learning
2.2.4. 4D-ETCN Architectures for ECVs’ Forecasting
3. Results
3.1. Dataset Description
3.2. Experimental Setup
3.3. Evaluation Metrics for Essential Climate Variables Forecasting
3.4. 4D-ETCN Architecture Ablation Study
3.5. Convergence Analysis
3.6. Evaluation against State-of-the-Art Methods
4. Discussion
4.1. Impact of Training Set Size on Forecasting Accuracy
4.2. Effect of Temporal Information on Forecasting Accuracy
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
ECV | Essential Climate Variable |
GCOS | Global Climate Observing System |
ML | Machine Learning |
DL | Deep Learning |
RS | Remote Sensing |
ETCN | Embedded Temporal Convolutional Network |
CNN | Convolutional Neural Network |
RNN | Recurrent Neural Network |
LSTM | Long Short-Term Memory |
TCN | Temporal Convolutional Network |
GRU | Gated Recurrent Unit |
CDS | Climate Data Store |
ROI | Region Of Interest |
MSE | Mean Square Error |
MAE | Mean Absolute Error |
RMSE | Root Mean Square Error |
ubRMSE | Unbiased Root Mean Square Error |
PSNR | Peak Signal-to-Noise Ratio |
SSIM | Structural Similarity Index Measure |
PLCC | Pearson Linear Correlation Coefficient |
Appendix A. Forward Propagation in ND CNNs
Appendix A.1. Tucker Convolution Model Sanity Checks
- (i)
- Direct: The convolution is determined directly from sums, which are the definition of convolution.
- (ii)
- Fourier: The (Fast) Fourier Transform (FFT) is used to perform the convolution.
- (iii)
- Tucker: The convolution is computed via Tucker decomposition, as explained in the respective section of the paper.
- Perform N-D convolution at each simulation (i.e., by avoiding a selection of size equal to one across a specific mode, which indicates a trailing singleton dimension).
- Perform the maximum compression possible via and (i.e., since in the Tucker decomposition format the sizes of and have to be at most equal to these of and ).
- (i)
- The bottom and top of each box are the 25th and 75th percentiles of the sample, respectively. The distance between the bottom and top of each box is the interquartile range, which in the 1D and 2D cases is wider for the Fourier–Tucker models’ comparison. On the contrary, for the 3D and 4D cases it is clearly narrower.
- (ii)
- The red line in the middle of each box is the sample median. Even in the cases where the median is not centered in the box (i.e., the plot shows sample skewness), all median NMSE values are of order , thus indicating the equivalence of computations.
- (iii)
- The whiskers are lines extending above and below each box. Whiskers go from the end of the interquartile range to the furthest observation within the whisker length (the adjacent value). We observe that the whiskers for the Fourier–Tucker models’ comparison are shorter in nearly every case, thus indicating tighter error distributions.
- (iv)
- Observations beyond the whisker length are marked as outliers. By default, an outlier is a value that is more than 1.5 times the interquartile range away from the bottom or top of the box. Once more, we highlight the fact that the Fourier–Tucker models’ comparison is the one with the least number of outliers, which are more to be found in the 4D convolution case.
- (v)
- In all cases, the relative approximation error is of order , which indicates the validity of the convolution generalization via tensor decompositions.
- (i)
- The interquartile range of the Tucker models is wider in the 1D and 2D cases and narrower for the 3D and 4D cases.
- (ii)
- The median value of the Tucker model is slightly higher than these of the direct and Fourier ones in the 1D and 2D cases, whereas it is nearly the same or even lower in the 3D and 4D cases.
- (iii)
- The whiskers for the Tucker model are shorter in the 1D and 2D cases and wider in the 3D and 4D cases.
- (iv)
- The Tucker model seems quite robust to outliers in the 4D convolution case, which indicates faster computations in cases where the size across a specific mode of the convolution tensor is high.
- (v)
- In all cases, the computational time required by each convolution model is low, except the 4D case, where the proposed convolution generalization via tensor decompositions scales better than its competitors.
Padding | Order | Direct | Fourier | Tucker |
---|---|---|---|---|
Full | 1 | |||
2 | ||||
3 | ||||
4 | ||||
Same | 1 | |||
2 | ||||
3 | ||||
4 | ||||
Valid | 1 | |||
2 | ||||
3 | ||||
4 |
Appendix A.2. Stacked Convolution Model Sanity Checks
- (i)
- The interquartile range is clearly narrower for the Fourier–Stacked models’ comparison in every dimensionality and padding case.
- (ii)
- All median NMSE values are of order , thus indicating the equivalence of computations.
- (iii)
- The whiskers for the Fourier–Stacked models’ comparison are generally shorter, thus indicating tighter error distributions.
- (iv)
- In nearly every case there are no outlier values, except the 4D one where once more the Fourier–Stacked models’ comparison is the one with the least number of outliers.
- (v)
- In all cases, the relative approximation error is of order , which indicates the validity of the proposed convolution generalization via stacking lower-dimensional convolutions.
- (i)
- The interquartile range of the Stacked model is wider in the 2D and 3D cases and nearly equally narrow to its competitors for the 4D case.
- (ii)
- The median value of the Stacked model is slightly higher than these of the direct and Fourier ones in the 2D and 3D cases, whereas it is nearly the same in the 4D case.
- (iii)
- The whiskers for the Stacked model are greater in every convolution cases except the “valid” ones, where the Fourier ones are more significant.
- (iv)
- The Stacked model seems as robust to outliers as the direct one, with both being slightly inferior to the Fourier convolution model.
- (v)
- In all cases, the computational time required by each convolution model is low, thus indicating the speed and efficiency of the performed computations.
Stacked Convolution Algorithm
Algorithm A1 Stacked Convolution Algorithm |
|
Appendix B. Backpropagation in ND CNNs
Appendix C. 4D-ETCN Architectures Parameter Tuning
Filter Size—Encoder | # Filters—Encoder | MSE—Crete | MSE—Italy |
---|---|---|---|
(2, 2, 2, 2) | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 | |||
(3, 3, 3, 3) | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128-128 | |||
128–256–256 | |||
256–512–512 | |||
(4, 4, 4, 4) | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 |
Filter Size—TCN | # Filters—Encoder | MSE—Crete | MSE—Italy |
---|---|---|---|
2 | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 | |||
3 | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 | |||
4 | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128-128 | |||
128-256–256 | |||
256–512–512 |
Filter Size—Decoder | # Filters—Encoder | MSE—Crete | MSE—Italy |
---|---|---|---|
(2, 2, 2, 2) | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 | |||
(3, 3, 3, 3) | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 | |||
(4, 4, 4, 4) | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 |
Dropout Rate | # Filters—Encoder | MSE—Crete | MSE—Italy |
---|---|---|---|
0.3 | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 | |||
0.4 | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 | |||
0.5 | 4–8–8 | ||
8–16–16 | |||
16–32–32 | |||
32–64–64 | |||
64–128–128 | |||
128–256–256 | |||
256–512–512 |
Filter Size—Encoder | Filter Size—ETCN | Filter Size—Decoder | Dropout Rate | # Filters—Encoder | MSE—Crete |
---|---|---|---|---|---|
(4, 4, 4, 4) | 3 | (4, 4, 4, 4) | 0.3 | 4–8–8 | |
8–16–16 | |||||
16–32–32 | |||||
32–64–64 | |||||
64–128–128 | |||||
128–256–256 | |||||
256–512–512 |
Filter Size—Encoder | Filter Size—ETCN | Filter Size—Decoder | Dropout Rate | # Filters—Encoder | MSE—Italy |
---|---|---|---|---|---|
(3, 3, 3, 3) | 4 | (3, 3, 3, 3) | 0.4 | 4–8–8 | |
8–16–16 | |||||
16–32–32 | |||||
32–64–64 | |||||
64–128–128 | |||||
128–256–256 | |||||
256–512–512 |
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Dataset | # Trainable Parameters | Builtin | Tucker |
---|---|---|---|
MNIST-2D | 40 | ||
MNIST-3D | 112 |
Total Samples | Training Set | Validation Set | Test Set |
---|---|---|---|
473 | 342 | 114 | 17 |
Dataset | Model | MSE | MAE | SSIM | PSNR | PLCC | ubRMSE | MAE (K) |
---|---|---|---|---|---|---|---|---|
Crete | Conv-LSTM [9] | |||||||
ETCN [9] | ||||||||
4D-ETCN-Proposed | ||||||||
Italy | Conv-LSTM [9] | |||||||
ETCN [9] | ||||||||
4D-ETCN-Proposed |
Dataset | Model | # Parameters | Time Training (Minutes) | Time Testing/Inference (Seconds) |
---|---|---|---|---|
Crete | Conv-LSTM [9] | 459,012 | ||
ETCN [9] | ||||
4D-ETCN-Proposed | 34,599,367 | 1 | ||
Italy | Conv-LSTM [9] | 459,012 | ||
ETCN [9] | ||||
4D-ETCN-Proposed | 47,104,583 | 1 |
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Giannopoulos, M.; Tsagkatakis, G.; Tsakalides, P. Higher-Order Convolutional Neural Networks for Essential Climate Variables Forecasting. Remote Sens. 2024, 16, 2020. https://doi.org/10.3390/rs16112020
Giannopoulos M, Tsagkatakis G, Tsakalides P. Higher-Order Convolutional Neural Networks for Essential Climate Variables Forecasting. Remote Sensing. 2024; 16(11):2020. https://doi.org/10.3390/rs16112020
Chicago/Turabian StyleGiannopoulos, Michalis, Grigorios Tsagkatakis, and Panagiotis Tsakalides. 2024. "Higher-Order Convolutional Neural Networks for Essential Climate Variables Forecasting" Remote Sensing 16, no. 11: 2020. https://doi.org/10.3390/rs16112020