20260317 meeting

Finalizing work v2.0

Published 2026-03-15 Updated 2026-03-17 3073 words 15 minutes views

The purpose of this experiment was originally to reproduce the results of Weather2K and NASA GES DISC MERRA-2 under the same data range as the previous experiment, but without performing time reshaping. However, when running the experiment on the National Center for High-performance Computing (NCHC), we found that the model exceeded the vRAM limitation of a single GPU (the model was too large). Therefore, in this experiment, all experiments were adjusted by reducing the number of Residual layers and then re-running them in order to produce usable experimental and control groups.

In addition, this experiment refactored all previously used experimental code (the SSSDS4 + AFRK implementation was not modified). At the same time, the experimental range of both datasets was unified to a six-month period. The time-series prediction horizon was set to 10% of the sequence, approximately 18 days, and the unobserved locations accounted for 20% of all locations.

While investigating why the original SSSD code could not run directly on multiple GPUs, we found that several steps explicitly specify the computation device as CUDA, such as

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.device("cuda")

This approach restricts the SSSD code to NVIDIA GPUs only, preventing execution on CPU, MPS, TPU, or other devices. Moreover, hard-coding .device("cuda") forces the program to run only on the first GPU ("cuda:0"), which prevents data communication across multiple GPUs. Since rewriting the entire implementation would be extremely time-consuming, we currently do not plan to refactor it. Instead, as mentioned earlier, we reduce certain model layers in order to run the experiments.

Experiment Overview

The following experiments were conducted. Detailed settings can be found in the next section, Experiment Settings.

Control Variable	Experiment 1	Experiment 2	Experiment 3	Experiment 4
Iterations	4,000	4,000	4,000	4,000
Training Strategy	$SSSD^{S4 + AFRK}$	$SSSD^{S4}$	$SSSD^{S4 + AFRK}$	$SSSD^{S4}$
Time Reshaping	false	false	true	true
Merged Time Steps $p$	—	—	8 / 24	8 / 24
Input Channels	4 / 1	4 / 1	32 / 24	32 / 24
S4 Max Seq. Length	1,176 / 4,608	1,176 / 4,608	147 / 192	147 / 192
Missing $k$	88 / 144	88 / 144	11 / 6	11 / 6

Experiment naming based on the above table is as follows:

Experiment 1
- Weather2k-1var-S4+AFRK
- MERRA2-1var-S4+AFRK
Experiment 2
- Weather2k-1var-S4
- MERRA2-1var-S4
Experiment 3
- Weather2k-S4+AFRK
- MERRA2-S4+AFRK
Experiment 4
- Weather2k-S4
- MERRA2-S4

Experiment Settings

The configuration used in this experiment is shown below. Only the input/output channels, time series length (S4 max sequence length), missing values (missing $k$), whether AFRK is enabled (enable spatial training), and directory paths vary depending on the specific experiment.

The following example shows the configuration for Weather2k-S4+AFRK.

model.yaml

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wavenet:
  # WaveNet model parameters
  input_channels:  32  # Number of input channels
  output_channels: 32  # Number of output channels
  residual_layers: 20  # Number of residual layers
  residual_channels: 20  # Number of channels in residual blocks
  skip_channels: 20  # Number of channels in skip connections

  # Diffusion step embedding dimensions
  diffusion_step_embed_dim_input:  64  # Input dimension
  diffusion_step_embed_dim_hidden: 128  # Middle dimension
  diffusion_step_embed_dim_output: 128  # Output dimension

  # Structured State Spaces sequence model (S4) configurations
  s4_max_sequence_length: 166  # Maximum sequence length
  s4_state_dim: 128  # State dimension
  s4_dropout: 0.2  # Dropout rate
  s4_bidirectional: true  # Whether to use bidirectional layers
  s4_use_layer_norm: true  # Whether to use layer normalization

diffusion:
  # Diffusion model parameters
  T: 100  # Number of diffusion steps
  beta_0: 0.0001  # Initial beta value
  beta_T: 0.01  # Final beta value

AFRK:
# AutoFRK model parameters, for more details, please refer to https://pypi.org/project/autoFRK/
  method: "fast"  # autoFRK method to use (e.g., "fast")
  tps_method: "rectangular"  # autoFRK's TPS method to use (e.g., "rectangular")

training.yaml

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# Training configuration
batch_size: 100  # Batch size
output_directory: "/home/u6025091/SSSD_CP/results/Weather2k-S4+AFRK"  # Output directory for checkpoints and logs
ckpt_iter: "max"  # Checkpoint mode (max or min)
iters_per_ckpt: 500  # Checkpoint frequency (number of epochs)
iters_per_logging: 200  # Log frequency (number of iterations)
n_iters: 4000  # Maximum number of iterations
learning_rate: 0.0005  # Learning rate

# Additional training settings
only_generate_missing: true  # Generate missing values only
use_model: 2  # Model to use for training
masking: "forecast"  # Masking strategy for missing values
missing_k: 18  # Number of missing values

# Data paths
data:
  train_path: "/home/u6025091/SSSD_CP/datasets/Weather2k/data_train_known_real.npy"  # Path to training data

# autoFRK config
enable_spatial_training: true  # Enable spatial training step
location_path: "/home/u6025091/SSSD_CP/datasets/Weather2k/stations_known_locations.npy"  # Path to known locations

inference.yaml

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# Inference configuration
batch_size: 100  # Batch size for inference
output_directory: "/home/u6025091/SSSD_CP/results/Weather2k-S4+AFRK/inference"  # Output directory for inference results
ckpt_path: "/home/u6025091/SSSD_CP/results/Weather2k-S4+AFRK"  # Path to checkpoint for inference
trials: 1 # Replications

# Additional training settings
only_generate_missing: true  # Generate missing values only
use_model: 2  # Model to use for training
masking: "forecast"  # Masking strategy for missing values  # inference mode need to fix in the code, or using other masking strategy will be failed
missing_k: 18  # Number of missing values

# Data paths
data:
  test_path: "/home/u6025091/SSSD_CP/datasets/Weather2k/data_train_known_real.npy"  # Path to test data

# autoFRK config
enable_spatial_inference: true  # Enable spatial prediction step
enable_spatial_normalization: true  # Enable spatial normalization before and after autoFRK
known_location_path: "/home/u6025091/SSSD_CP/datasets/Weather2k/stations_known_locations.npy"  # Path to known locations
unknown_location_path: "/home/u6025091/SSSD_CP/datasets/Weather2k/stations_unknown_locations.npy"  # Path to unknown locations

The selected experimental locations are shown below.

https://raw.githubusercontent.com/Josh-test-lab/website-assets-repository/refs/heads/main/posts/1150317%20meeting/Weather2k.png — Weather2k Location Distribution.

https://raw.githubusercontent.com/Josh-test-lab/website-assets-repository/refs/heads/main/posts/1150317%20meeting/MERRA2.png — MERRA-2 Location Distribution.

The results of each experiment are presented in the following sections.

Experiment 1

Weather2k-1var-S4+AFRK

Training time: 35h 25m 14.82s (Note: This experiment was not trained on a dedicated machine. It shared the same machine with Weather2k-1var-S4, Weather2k-S4, Weather2k-S4+AFRK during training.)
Number of MRTS bases used by AFRK during the inference stage: 382

Metric	ALL Locs & All Time	Known Locs & All Time	Unknown Locs & All Time	ALL Locs & Future	Known Locs & Future	Unknown Locs & Future	ALL Locs & Past	Known Locs & Past	Unknown Locs & Past
MSPE	1.494639e+01	1.160173e+01	2.828929e+01	7.756372e+01	7.868970e+01	7.307184e+01	8.156561e+00	4.327126e+00	2.343335e+01
RMSPE	3.866056e+00	3.406131e+00	5.318767e+00	8.807027e+00	8.870722e+00	8.548207e+00	2.855969e+00	2.080175e+00	4.840800e+00
MSPE%	1.380822e+10	3.299012e+09	5.573267e+10	2.246783e+10	2.608424e+10	8.040864e+09	1.286923e+10	8.283242e+08	6.090407e+10
RMSPE%	1.175084e+05	5.743703e+04	2.360777e+05	1.498927e+05	1.615062e+05	8.967086e+04	1.134426e+05	2.878062e+04	2.467875e+05
MAPE	2.332151e+00	2.120980e+00	3.174576e+00	5.441194e+00	5.450313e+00	5.404813e+00	1.995026e+00	1.759968e+00	2.932743e+00
MAPE%	7.822800e+08	5.627048e+08	1.658233e+09	5.047446e+08	5.480241e+08	3.320892e+08	8.123742e+08	5.642966e+08	1.802031e+09

MERRA2-1var-S4+AFRK

Training time: 20h 24m 12.59s
Number of MRTS bases used by AFRK during the inference stage: 301

Metric	ALL Locs & All Time	Known Locs & All Time	Unknown Locs & All Time	ALL Locs & Future	Known Locs & Future	Unknown Locs & Future	ALL Locs & Past	Known Locs & Past	Unknown Locs & Past
MSPE	160.204191	160.390768	159.457883	1628.647842	1631.785612	1616.096765	0.975361	0.841929	1.509089
RMSPE	12.657179	12.664548	12.627663	40.356509	40.395366	40.200706	0.987604	0.917567	1.228450
MSPE%	0.611571	0.612622	0.607367	6.222452	6.237316	6.162993	0.003162	0.002715	0.004950
RMSPE%	0.782030	0.782702	0.779338	2.494484	2.497462	2.482538	0.056233	0.052108	0.070357
MAPE	3.875349	3.801459	4.170910	34.341577	34.369929	34.228172	0.571782	0.486805	0.911689
MAPE%	0.014226	0.013981	0.015206	0.128176	0.128351	0.127477	0.001869	0.001579	0.003032

Experiment 2

Weather2k-1var-S4

Training time: 10h 3m 41.84s (Note: This experiment was not trained on a dedicated machine. It shared the same machine with Weather2k-1var-S4+AFRK during training.)
Number of MRTS bases used by AFRK during the inference stage: 382

Metric	ALL Locs & All Time	Known Locs & All Time	Unknown Locs & All Time	ALL Locs & Future	Known Locs & Future	Unknown Locs & Future	ALL Locs & Past	Known Locs & Past	Unknown Locs & Past
MSPE	1.160546e+01	8.335466e+00	2.465048e+01	8.308004e+01	8.508390e+01	7.508604e+01	3.855207e+00	1.334625e-02	1.918156e+01
RMSPE	3.406679e+00	2.887121e+00	4.964925e+00	9.114825e+00	9.224093e+00	8.665220e+00	1.963468e+00	1.155260e-01	4.379676e+00
MSPE%	1.214263e+10	2.615445e+09	5.014950e+10	2.295537e+10	2.672051e+10	7.935082e+09	1.097017e+10	1.642926e+06	5.472697e+10
RMSPE%	1.101936e+05	5.114142e+04	2.239408e+05	1.515103e+05	1.634641e+05	8.907908e+04	1.047386e+05	1.281767e+03	2.339380e+05
MAPE	1.034689e+00	6.119608e-01	2.721080e+00	5.516763e+00	5.541266e+00	5.419014e+00	5.486807e-01	7.745780e-02	2.428533e+00
MAPE%	3.535854e+08	7.255939e+07	1.474684e+09	4.961357e+08	5.373860e+08	3.315754e+08	3.381282e+08	2.215650e+07	1.598635e+09

MERRA2-1var-S4

Training time: 16h 48m 57.28s
Number of MRTS bases used by AFRK during the inference stage: 301

Metric	ALL Locs & All Time	Known Locs & All Time	Unknown Locs & All Time	ALL Locs & Future	Known Locs & Future	Unknown Locs & Future	ALL Locs & Past	Known Locs & Past	Unknown Locs & Past
MSPE	153.309263	153.324779	153.247198	1564.356985	1566.560730	1555.542007	0.304088	0.082327	1.191135
RMSPE	12.381812	12.382438	12.379305	39.551953	39.579802	39.440360	0.551442	0.286926	1.091391
MSPE%	0.587130	0.587618	0.585178	5.992495	6.004252	5.945470	0.001006	0.000272	0.003941
RMSPE%	0.766244	0.766563	0.764969	2.447957	2.450357	2.438333	0.031721	0.016505	0.062781
MAPE	3.651232	3.550918	4.052489	33.980246	33.997542	33.911060	0.362544	0.249477	0.814813
MAPE%	0.013523	0.013196	0.014834	0.127042	0.127182	0.126480	0.001214	0.000836	0.002728

Experiment 3

Weather2k-S4+AFRK

Training time: (Note: This experiment was not trained on a dedicated machine. It shared the same machine with Weather2k-1var-S4+AFRK during training.)
Number of MRTS bases used by AFRK during the inference stage:

MERRA2-S4+AFRK

Training time:
Number of MRTS bases used by AFRK during the inference stage:

Experiment 4

Weather2k-S4

Training time: 4h 31m 17.40s (Note: This experiment was not trained on a dedicated machine. It shared the same machine with Weather2k-1var-S4+AFRK during training.)
Number of MRTS bases used by AFRK during the inference stage: 178

Metric	ALL Locs & All Time	Known Locs & All Time	Unknown Locs & All Time	ALL Locs & Future	Known Locs & Future	Unknown Locs & Future	ALL Locs & Past	Known Locs & Past	Unknown Locs & Past
MSPE	2.729977e+01	2.688188e+01	2.896683e+01	2.397620e+02	2.747339e+02	1.002486e+02	4.261693e+00	6.368540e-03	2.123748e+01
RMSPE	5.224918e+00	5.184774e+00	5.382084e+00	1.548425e+01	1.657510e+01	1.001242e+01	2.064387e+00	7.980313e-02	4.608414e+00
MSPE%	1.143239e+10	2.267235e+09	4.799498e+10	2.030899e+10	2.313886e+10	9.019811e+09	1.046987e+10	4.046914e+06	5.222121e+10
RMSPE%	1.069224e+05	4.761549e+04	2.190776e+05	1.425096e+05	1.521146e+05	9.497268e+04	1.023224e+05	2.011694e+03	2.285196e+05
MAPE	1.437579e+00	1.047657e+00	2.993092e+00	9.437545e+00	1.014651e+01	6.609263e+00	5.701122e-01	6.103477e-02	2.600977e+00
MAPE%	3.696454e+08	7.675813e+07	1.538062e+09	4.665294e+08	4.950431e+08	3.527794e+08	3.591399e+08	3.140192e+07	1.666586e+09

MERRA2-S4

Training time: 2h 14m 26.41s
Number of MRTS bases used by AFRK during the inference stage: 111

Metric	ALL Locs & All Time	Known Locs & All Time	Unknown Locs & All Time	ALL Locs & Future	Known Locs & Future	Unknown Locs & Future	ALL Locs & Past	Known Locs & Past	Unknown Locs & Past
MSPE	183.060926	194.459146	137.468045	1864.618866	1987.368928	1373.618619	0.723318	0.047242	3.427622
RMSPE	13.530001	13.944861	11.724677	43.181233	44.579916	37.062361	0.850481	0.217352	1.851384
MSPE%	0.680973	0.722399	0.515269	6.939079	7.383059	5.163158	0.002383	0.000159	0.011281
RMSPE%	0.825211	0.849941	0.717822	2.634213	2.717179	2.272258	0.048819	0.012604	0.106213
MAPE	3.821658	3.698350	4.314889	35.220395	36.220564	31.219723	0.416976	0.171845	1.397498
MAPE%	0.013882	0.013486	0.015466	0.129019	0.132507	0.115069	0.001397	0.000580	0.004666

Bonus

All of the aforementioned experiments used a six-month time span as the time-series length. However, reducing the number of residual layers in the model appears to have weakened its ability to capture spatiotemporal dependencies, with particularly poor performance observed in the MERRA-2 experiments.

Therefore, in this round of experiments, two additional experiments were conducted. Both use the MERRA-2 dataset as the test dataset (as it is relatively large and can easily reach the vRAM limit within the same time range; during training, vRAM usage exceeded 30 GB in all experiments below). These experiments aim to evaluate the performance of the following parameter settings under reduced input batch_size:

Experiment 5 - MERRA2-1var-S4+AFRK-low-batch

The training set for this experiment is the same as that in MERRA2-1var-S4+AFRK. However, several parameters are reduced, while the model depth is increased to be closer to the configuration used in the previous meeting. Due to vRAM constraints, the settings cannot be made exactly identical. The detailed parameters are as follows:

model.yaml

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wavenet:
  # WaveNet model parameters
  input_channels:  1  # Number of input channels
  output_channels: 1  # Number of output channels
  residual_layers: 20  # Number of residual layers
  residual_channels: 36  # Number of channels in residual blocks
  skip_channels: 36  # Number of channels in skip connections

  # Diffusion step embedding dimensions
  diffusion_step_embed_dim_input:  128  # Input dimension
  diffusion_step_embed_dim_hidden: 256  # Middle dimension
  diffusion_step_embed_dim_output: 256  # Output dimension

  # Structured State Spaces sequence model (S4) configurations
  s4_max_sequence_length: 3984  # Maximum sequence length
  s4_state_dim: 128  # State dimension
  s4_dropout: 0.2  # Dropout rate
  s4_bidirectional: true  # Whether to use bidirectional layers
  s4_use_layer_norm: true  # Whether to use layer normalization

diffusion:
  # Diffusion model parameters
  T: 100  # Number of diffusion steps
  beta_0: 0.0001  # Initial beta value
  beta_T: 0.01  # Final beta value

AFRK:
# AutoFRK model parameters, for more details, please refer to https://pypi.org/project/autoFRK/
  method: "fast"  # autoFRK method to use (e.g., "fast")
  tps_method: "rectangular"  # autoFRK's TPS method to use (e.g., "rectangular")

training.yaml

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# Training configuration
batch_size: 60  # Batch size
output_directory: "/home/u6025091/SSSD_CP/results/MERRA2-1var-S4+AFRK-lowbatch"  # Output directory for checkpoints and logs
ckpt_iter: "max"  # Checkpoint mode (max or min)
iters_per_ckpt: 500  # Checkpoint frequency (number of epochs)
iters_per_logging: 200  # Log frequency (number of iterations)
n_iters: 4000  # Maximum number of iterations
learning_rate: 0.0005  # Learning rate

# Additional training settings
only_generate_missing: true  # Generate missing values only
use_model: 2  # Model to use for training
masking: "forecast"  # Masking strategy for missing values
missing_k: 432  # Number of missing values

# Data paths
data:
  train_path: "/home/u6025091/SSSD_CP/datasets/MERRA2-1var/data_train_known_real.npy"  # Path to training data

# autoFRK config
enable_spatial_training: true  # Enable spatial training step
location_path: "/home/u6025091/SSSD_CP/datasets/MERRA2-1var/stations_known_locations.npy"  # Path to known locations

Training time: 35h 11m 12.33s
Number of MRTS bases used by AFRK during the inference stage: 301

Metric	ALL Locs & All Time	Known Locs & All Time	Unknown Locs & All Time	ALL Locs & Future	Known Locs & Future	Unknown Locs & Future	ALL Locs & Past	Known Locs & Past	Unknown Locs & Past
MSPE	164.784290	164.888406	164.367824	1668.971215	1671.274799	1659.756880	1.679684	1.545304	2.217204
RMSPE	12.836833	12.840888	12.820602	40.853044	40.881228	40.740114	1.296026	1.243102	1.489028
MSPE%	0.632449	0.633265	0.629184	6.413790	6.426202	6.364142	0.005557	0.005116	0.007321
RMSPE%	0.795267	0.795780	0.793211	2.532546	2.534995	2.522725	0.074542	0.071523	0.085561
MAPE	4.193186	4.136441	4.420163	34.907394	34.918514	34.862917	0.862729	0.798626	1.119142
MAPE%	0.015369	0.015185	0.016103	0.130691	0.130813	0.130204	0.002864	0.002647	0.003731

Experiment 6 - MERRA2-1var-S4+AFRK-low-day

In this experiment, several parameters are similarly reduced, while the model depth is increased to more closely match the configuration used in the previous meeting. Additionally, the time-series length of the training set is shortened to three months. The detailed parameter settings are as follows:

model.yaml

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wavenet:
  # WaveNet model parameters
  input_channels:  1  # Number of input channels
  output_channels: 1  # Number of output channels
  residual_layers: 28  # Number of residual layers
  residual_channels: 40  # Number of channels in residual blocks
  skip_channels: 40  # Number of channels in skip connections

  # Diffusion step embedding dimensions
  diffusion_step_embed_dim_input:  128  # Input dimension
  diffusion_step_embed_dim_hidden: 256  # Middle dimension
  diffusion_step_embed_dim_output: 256  # Output dimension

  # Structured State Spaces sequence model (S4) configurations
  s4_max_sequence_length: 1992  # Maximum sequence length
  s4_state_dim: 128  # State dimension
  s4_dropout: 0.2  # Dropout rate
  s4_bidirectional: true  # Whether to use bidirectional layers
  s4_use_layer_norm: true  # Whether to use layer normalization

diffusion:
  # Diffusion model parameters
  T: 100  # Number of diffusion steps
  beta_0: 0.0001  # Initial beta value
  beta_T: 0.01  # Final beta value

AFRK:
# AutoFRK model parameters, for more details, please refer to https://pypi.org/project/autoFRK/
  method: "fast"  # autoFRK method to use (e.g., "fast")
  tps_method: "rectangular"  # autoFRK's TPS method to use (e.g., "rectangular")

training.yaml

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# Training configuration
batch_size: 80  # Batch size
output_directory: "/home/u6025091/SSSD_CP/results/MERRA2-1var-S4+AFRK-final-low-days"  # Output directory for checkpoints and logs
ckpt_iter: "max"  # Checkpoint mode (max or min)
iters_per_ckpt: 500  # Checkpoint frequency (number of epochs)
iters_per_logging: 200  # Log frequency (number of iterations)
n_iters: 4000  # Maximum number of iterations
learning_rate: 0.0005  # Learning rate

# Additional training settings
only_generate_missing: true  # Generate missing values only
use_model: 2  # Model to use for training
masking: "forecast"  # Masking strategy for missing values
missing_k: 216  # Number of missing values

# Data paths
data:
  train_path: "/home/u6025091/SSSD_CP/datasets/MERRA2-1var-final-low-days/data_train_known_real.npy"  # Path to training data

# autoFRK config
enable_spatial_training: true  # Enable spatial training step
location_path: "/home/u6025091/SSSD_CP/datasets/MERRA2-1var-final-low-days/stations_known_locations.npy"  # Path to known locations

Training time: 24h 25m 30.20s
Number of MRTS bases used by AFRK during the inference stage: 301

Metric	ALL Locs & All Time	Known Locs & All Time	Unknown Locs & All Time	ALL Locs & Future	Known Locs & Future	Unknown Locs & Future	ALL Locs & Past	Known Locs & Past	Unknown Locs & Past
MSPE	1.346637	1.077605	2.422764	10.404323	10.459679	10.182895	0.364479	0.060272	1.581304
RMSPE	1.160447	1.038078	1.556523	3.225573	3.234143	3.191065	0.603721	0.245504	1.257499
MSPE%	0.004831	0.003865	0.008696	0.037348	0.037534	0.036602	0.001305	0.000214	0.005670
RMSPE%	0.069507	0.062171	0.093250	0.193256	0.193738	0.191316	0.036130	0.014640	0.075297
MAPE	0.517593	0.406171	0.963280	2.524476	2.528199	2.509585	0.299979	0.176072	0.795608
MAPE%	0.001852	0.001454	0.003446	0.009074	0.009087	0.009024	0.001069	0.000626	0.002841

Conclusion

At present, the MERRA2-1var-S4+AFRK-low-day experiment using the parameter settings from Experiment 6 achieves the best performance across all evaluation metrics. In particular, for future time-step predictions, metrics such as MSPE, RMSPE, and MAPE are significantly reduced, while the performance for past time-step predictions is also satisfactory. Therefore, future experiments should be based on this parameter configuration, with additional runs conducted to further verify its stability and reliability.

The current issue is whether it is necessary to simultaneously reduce the time-series length of the Weather2k dataset. Further guidance from the advisor would be greatly appreciated.

Some experimental results are currently missing as the experiments are still in progress. We kindly ask for your understanding.

https://raw.githubusercontent.com/Josh-test-lab/website-assets-repository/refs/heads/main/posts/1150317%20meeting/not%20yet%20finish.png — Experiments in progress.

Experimental Snapshots

A total of four machines from the National Center for High-performance Computing (NCHC) were used in this experiment. Below are snapshots from three of these machines.

https://raw.githubusercontent.com/Josh-test-lab/website-assets-repository/refs/heads/main/posts/1150317%20meeting/Experimental%20snapshot.png — Experimental snapshots.

Draft Manuscript

References

Zhu X, Xiong Y, Wu M, et al. Weather2K: A Multivariate Spatio-Temporal Benchmark Dataset for Meteorological Forecasting Based on Real-Time Observation Data from Ground Weather Stations[C]//International Conference on Artificial Intelligence and Statistics. PMLR, 2023: 2704-2722.
Juan Lopez Alcaraz, Nils Strodthoff (2022). Diffusion-based time series imputation and forecasting with structured state space models. Transactions on Machine Learning Research. Retrieved from https://openreview.net/forum?id=hHiIbk7ApW
SSSD (2022). GitHub. Retrieved from https://github.com/AI4HealthUOL/SSSD