The DeepAR model produces probabilistic forecasts based on an autoregressive recurrent neural network optimized on panel data using cross-learning. DeepAR obtains its forecast distribution uses a Markov Chain Monte Carlo sampler with the following conditional probability: P(y[t+1:t+H]  y[:t],  x[:t+H](f),  x(s))\mathbb{P}(\mathbf{y}_{[t+1:t+H]}|\;\mathbf{y}_{[:t]},\; \mathbf{x}^{(f)}_{[:t+H]},\; \mathbf{x}^{(s)})

where x(s)\mathbf{x}^{(s)} are static exogenous inputs, x[:t+H](f)\mathbf{x}^{(f)}_{[:t+H]} are future exogenous available at the time of the prediction. The predictions are obtained by transforming the hidden states ht\mathbf{h}_{t} into predictive distribution parameters θt\theta_{t}, and then generating samples y^[t+1:t+H]\mathbf{\hat{y}}_{[t+1:t+H]} through Monte Carlo sampling trajectories.

ht=RNN([yt,xt+1(f),x(s)],ht1)θt=Linear(ht)y^t+1=sample(  P(yt+1    θt)  ) \begin{align} \mathbf{h}_{t} &= \textrm{RNN}([\mathbf{y}_{t},\mathbf{x}^{(f)}_{t+1},\mathbf{x}^{(s)}], \mathbf{h}_{t-1})\\ \mathbf{\theta}_{t}&=\textrm{Linear}(\mathbf{h}_{t}) \\ \hat{y}_{t+1}&=\textrm{sample}(\;\mathrm{P}(y_{t+1}\;|\;\mathbf{\theta}_{t})\;) \end{align}

References
- David Salinas, Valentin Flunkert, Jan Gasthaus, Tim Januschowski (2020). “DeepAR: Probabilistic forecasting with autoregressive recurrent networks”. International Journal of Forecasting.
- Alexander Alexandrov et. al (2020). “GluonTS: Probabilistic and Neural Time Series Modeling in Python”. Journal of Machine Learning Research.

Exogenous Variables, Losses, and Parameters Availability

Given the sampling procedure during inference, DeepAR only supports DistributionLoss as training loss.

Note that DeepAR generates a non-parametric forecast distribution using Monte Carlo. We use this sampling procedure also during validation to make it closer to the inference procedure. Therefore, only the MQLoss is available for validation.

Aditionally, Monte Carlo implies that historic exogenous variables are not available for the model.


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Decoder

 Decoder (in_features, out_features, hidden_size, hidden_layers)

Multi-Layer Perceptron Decoder

Parameters:
in_features: int, dimension of input.
out_features: int, dimension of output.
hidden_size: int, dimension of hidden layers.
num_layers: int, number of hidden layers.


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DeepAR

 DeepAR (h, input_size:int=-1, lstm_n_layers:int=2,
         lstm_hidden_size:int=128, lstm_dropout:float=0.1,
         decoder_hidden_layers:int=0, decoder_hidden_size:int=0,
         trajectory_samples:int=100, futr_exog_list=None,
         hist_exog_list=None, stat_exog_list=None,
         exclude_insample_y=False, loss=DistributionLoss(),
         valid_loss=MQLoss(), max_steps:int=1000,
         learning_rate:float=0.001, num_lr_decays:int=3,
         early_stop_patience_steps:int=-1, val_check_steps:int=100,
         batch_size:int=32, valid_batch_size:Optional[int]=None,
         windows_batch_size:int=1024, inference_windows_batch_size:int=-1,
         start_padding_enabled=False, step_size:int=1,
         scaler_type:str='identity', random_seed:int=1,
         num_workers_loader=0, drop_last_loader=False, optimizer=None,
         optimizer_kwargs=None, **trainer_kwargs)

DeepAR

Parameters:
h: int, Forecast horizon.
input_size: int, autorregresive inputs size, y=[1,2,3,4] input_size=2 -> y_[t-2:t]=[1,2].
lstm_n_layers: int=2, number of LSTM layers.
lstm_hidden_size: int=128, LSTM hidden size.
lstm_dropout: float=0.1, LSTM dropout.
decoder_hidden_layers: int=0, number of decoder MLP hidden layers. Default: 0 for linear layer.
decoder_hidden_size: int=0, decoder MLP hidden size. Default: 0 for linear layer.
trajectory_samples: int=100, number of Monte Carlo trajectories during inference.
stat_exog_list: str list, static exogenous columns.
hist_exog_list: str list, historic exogenous columns.
futr_exog_list: str list, future exogenous columns.
exclude_insample_y: bool=False, the model skips the autoregressive features y[t-input_size:t] if True.
loss: PyTorch module, instantiated train loss class from losses collection.
valid_loss: PyTorch module=loss, instantiated valid loss class from losses collection.
max_steps: int=1000, maximum number of training steps.
learning_rate: float=1e-3, Learning rate between (0, 1).
num_lr_decays: int=-1, Number of learning rate decays, evenly distributed across max_steps.
early_stop_patience_steps: int=-1, Number of validation iterations before early stopping.
val_check_steps: int=100, Number of training steps between every validation loss check.
batch_size: int=32, number of different series in each batch.
valid_batch_size: int=None, number of different series in each validation and test batch, if None uses batch_size.
windows_batch_size: int=1024, number of windows to sample in each training batch, default uses all.
inference_windows_batch_size: int=-1, number of windows to sample in each inference batch, -1 uses all.
start_padding_enabled: bool=False, if True, the model will pad the time series with zeros at the beginning, by input size.
step_size: int=1, step size between each window of temporal data.
scaler_type: str=‘identity’, type of scaler for temporal inputs normalization see temporal scalers.
random_seed: int, random_seed for pytorch initializer and numpy generators.
num_workers_loader: int=os.cpu_count(), workers to be used by TimeSeriesDataLoader.
drop_last_loader: bool=False, if True TimeSeriesDataLoader drops last non-full batch.
alias: str, optional, Custom name of the model.
optimizer: Subclass of ‘torch.optim.Optimizer’, optional, user specified optimizer instead of the default choice (Adam).
optimizer_kwargs: dict, optional, list of parameters used by the user specified optimizer.
**trainer_kwargs: int, keyword trainer arguments inherited from PyTorch Lighning’s trainer.

References
- David Salinas, Valentin Flunkert, Jan Gasthaus, Tim Januschowski (2020). “DeepAR: Probabilistic forecasting with autoregressive recurrent networks”. International Journal of Forecasting.
- Alexander Alexandrov et. al (2020). “GluonTS: Probabilistic and Neural Time Series Modeling in Python”. Journal of Machine Learning Research.


DeepAR.fit

 DeepAR.fit (dataset, val_size=0, test_size=0, random_seed=None,
             distributed_config=None)

Fit.

The fit method, optimizes the neural network’s weights using the initialization parameters (learning_rate, windows_batch_size, …) and the loss function as defined during the initialization. Within fit we use a PyTorch Lightning Trainer that inherits the initialization’s self.trainer_kwargs, to customize its inputs, see PL’s trainer arguments.

The method is designed to be compatible with SKLearn-like classes and in particular to be compatible with the StatsForecast library.

By default the model is not saving training checkpoints to protect disk memory, to get them change enable_checkpointing=True in __init__.

Parameters:
dataset: NeuralForecast’s TimeSeriesDataset, see documentation.
val_size: int, validation size for temporal cross-validation.
random_seed: int=None, random_seed for pytorch initializer and numpy generators, overwrites model.__init__’s.
test_size: int, test size for temporal cross-validation.


DeepAR.predict

 DeepAR.predict (dataset, test_size=None, step_size=1, random_seed=None,
                 **data_module_kwargs)

Predict.

Neural network prediction with PL’s Trainer execution of predict_step.

Parameters:
dataset: NeuralForecast’s TimeSeriesDataset, see documentation.
test_size: int=None, test size for temporal cross-validation.
step_size: int=1, Step size between each window.
random_seed: int=None, random_seed for pytorch initializer and numpy generators, overwrites model.__init__’s.
**data_module_kwargs: PL’s TimeSeriesDataModule args, see documentation.

Usage Example

from neuralforecast import NeuralForecast
from neuralforecast.losses.pytorch import MQLoss, DistributionLoss, GMM, PMM
from neuralforecast.tsdataset import TimeSeriesDataset
from neuralforecast.utils import AirPassengers, AirPassengersPanel, AirPassengersStatic
import pandas as pd
import pytorch_lightning as pl
import matplotlib.pyplot as plt

from neuralforecast import NeuralForecast
#from neuralforecast.models import DeepAR
from neuralforecast.losses.pytorch import DistributionLoss, HuberMQLoss
from neuralforecast.tsdataset import TimeSeriesDataset
from neuralforecast.utils import AirPassengers, AirPassengersPanel, AirPassengersStatic

#AirPassengersPanel['y'] = AirPassengersPanel['y'] + 10
Y_train_df = AirPassengersPanel[AirPassengersPanel.ds<AirPassengersPanel['ds'].values[-12]] # 132 train
Y_test_df = AirPassengersPanel[AirPassengersPanel.ds>=AirPassengersPanel['ds'].values[-12]].reset_index(drop=True) # 12 test

nf = NeuralForecast(
    models=[DeepAR(h=12,
                   input_size=48,
                   lstm_n_layers=3,
                   trajectory_samples=100,
                   loss=DistributionLoss(distribution='Normal', level=[80, 90], return_params=False),
                   learning_rate=0.005,
                   stat_exog_list=['airline1'],
                   futr_exog_list=['trend'],
                   max_steps=100,
                   val_check_steps=10,
                   early_stop_patience_steps=-1,
                   scaler_type='standard',
                   enable_progress_bar=True),
    ],
    freq='M'
)
nf.fit(df=Y_train_df, static_df=AirPassengersStatic, val_size=12)
Y_hat_df = nf.predict(futr_df=Y_test_df)

# Plot quantile predictions
Y_hat_df = Y_hat_df.reset_index(drop=False).drop(columns=['unique_id','ds'])
plot_df = pd.concat([Y_test_df, Y_hat_df], axis=1)
plot_df = pd.concat([Y_train_df, plot_df])

plot_df = plot_df[plot_df.unique_id=='Airline1'].drop('unique_id', axis=1)
plt.plot(plot_df['ds'], plot_df['y'], c='black', label='True')
#plt.plot(plot_df['ds'], plot_df['DeepAR'], c='purple', label='mean')
plt.plot(plot_df['ds'], plot_df['DeepAR-median'], c='blue', label='median')
plt.fill_between(x=plot_df['ds'][-12:], 
                 y1=plot_df['DeepAR-lo-90'][-12:].values, 
                 y2=plot_df['DeepAR-hi-90'][-12:].values,
                 alpha=0.4, label='level 90')
plt.legend()
plt.grid()
plt.plot()