DeepAR - Nixtla

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:

\mathbb{P}(\mathbf{y}_{[t+1:t+H]}|\;\mathbf{y}_{[:t]},\; \mathbf{x}^{(f)}_{[:t+H]},\; \mathbf{x}^{(s)})

where

\mathbf{x}^{(s)}

are static exogenous inputs,

\mathbf{x}^{(f)}_{[:t+H]}

are future exogenous available at the time of the prediction. The predictions are obtained by transforming the hidden states

\mathbf{h}_{t}

into predictive distribution parameters

\theta_{t}

, and then generating samples

\mathbf{\hat{y}}_{[t+1:t+H]}

through Monte Carlo sampling trajectories.

\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

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.

Figure 1. DeepAR model, during training the optimization signal comes from likelihood of observations, during inference a recurrent multi-step strategy is used to generate predictive distributions.

1. DeepAR

`DeepAR`

DeepAR(
    h,
    input_size=-1,
    h_train=1,
    lstm_n_layers=2,
    lstm_hidden_size=128,
    lstm_dropout=0.1,
    decoder_hidden_layers=0,
    decoder_hidden_size=0,
    trajectory_samples=100,
    stat_exog_list=None,
    hist_exog_list=None,
    futr_exog_list=None,
    exclude_insample_y=False,
    loss=DistributionLoss(
        distribution="StudentT", level=[80, 90], return_params=False
    ),
    valid_loss=MAE(),
    max_steps=1000,
    learning_rate=0.001,
    num_lr_decays=3,
    early_stop_patience_steps=-1,
    val_check_steps=100,
    batch_size=32,
    valid_batch_size=None,
    windows_batch_size=1024,
    inference_windows_batch_size=-1,
    start_padding_enabled=False,
    training_data_availability_threshold=0.0,
    step_size=1,
    scaler_type="identity",
    random_seed=1,
    drop_last_loader=False,
    alias=None,
    optimizer=None,
    optimizer_kwargs=None,
    lr_scheduler=None,
    lr_scheduler_kwargs=None,
    dataloader_kwargs=None,
    **trainer_kwargs
)

Bases: BaseModel DeepAR Parameters:

Name	Type	Description	Default
`h`	`int`	Forecast horizon.	required
`input_size`	`int`	maximum sequence length for truncated train backpropagation. Default -1 uses 3 * horizon	`-1`
`h_train`	`int`	maximum sequence length for truncated train backpropagation. Default 1.	`1`
`lstm_n_layers`	`int`	number of LSTM layers.	`2`
`lstm_hidden_size`	`int`	LSTM hidden size.	`128`
`lstm_dropout`	`float`	LSTM dropout.	`0.1`
`decoder_hidden_layers`	`int`	number of decoder MLP hidden layers. Default: 0 for linear layer.	`0`
`decoder_hidden_size`	`int`	decoder MLP hidden size. Default: 0 for linear layer.	`0`
`trajectory_samples`	`int`	number of Monte Carlo trajectories during inference.	`100`
`stat_exog_list`	`str list`	static exogenous columns.	`None`
`hist_exog_list`	`str list`	historic exogenous columns.	`None`
`futr_exog_list`	`str list`	future exogenous columns.	`None`
`exclude_insample_y`	`bool`	the model skips the autoregressive features y[t-input_size:t] if True.	`False`
`loss`	`PyTorch module`	instantiated train loss class from losses collection.	`DistributionLoss(distribution=‘StudentT’, level=[80, 90], return_params=False)`
`valid_loss`	`PyTorch module`	instantiated valid loss class from losses collection.	`MAE()`
`max_steps`	`int`	maximum number of training steps.	`1000`
`learning_rate`	`float`	Learning rate between (0, 1).	`0.001`
`num_lr_decays`	`int`	Number of learning rate decays, evenly distributed across max_steps.	`3`
`early_stop_patience_steps`	`int`	Number of validation iterations before early stopping.	`-1`
`val_check_steps`	`int`	Number of training steps between every validation loss check.	`100`
`batch_size`	`int`	number of different series in each batch.	`32`
`valid_batch_size`	`int`	number of different series in each validation and test batch, if None uses batch_size.	`None`
`windows_batch_size`	`int`	number of windows to sample in each training batch, default uses all.	`1024`
`inference_windows_batch_size`	`int`	number of windows to sample in each inference batch, -1 uses all.	`-1`
`start_padding_enabled`	`bool`	if True, the model will pad the time series with zeros at the beginning, by input size.	`False`
`training_data_availability_threshold`	`Union[float, List[float]]`	minimum fraction of valid data points required for training windows. Single float applies to both insample and outsample; list of two floats specifies [insample_fraction, outsample_fraction]. Default 0.0 allows windows with only 1 valid data point (current behavior).	`0.0`
`step_size`	`int`	step size between each window of temporal data.	`1`
`scaler_type`	`str`	type of scaler for temporal inputs normalization see temporal scalers.	`‘identity’`
`random_seed`	`int`	random_seed for pytorch initializer and numpy generators.	`1`
`drop_last_loader`	`bool`	if True `TimeSeriesDataLoader` drops last non-full batch.	`False`
`alias`	`str`	optional, Custom name of the model.	`None`
`optimizer`	`Subclass of ‘torch.optim.Optimizer’`	optional, user specified optimizer instead of the default choice (Adam).	`None`
`optimizer_kwargs`	`dict`	optional, list of parameters used by the user specified `optimizer`.	`None`
`lr_scheduler`	`Subclass of ‘torch.optim.lr_scheduler.LRScheduler’`	optional, user specified lr_scheduler instead of the default choice (StepLR).	`None`
`lr_scheduler_kwargs`	`dict`	optional, list of parameters used by the user specified `lr_scheduler`.	`None`
`dataloader_kwargs`	`dict`	optional, list of parameters passed into the PyTorch Lightning dataloader by the `TimeSeriesDataLoader`.	`None`
`**trainer_kwargs`	`int`	keyword trainer arguments inherited from PyTorch Lighning’s trainer.

`DeepAR.fit`

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:

Name	Type	Description	Default
`dataset`	`TimeSeriesDataset`	NeuralForecast’s `TimeSeriesDataset`, see documentation.	required
`val_size`	`int`	Validation size for temporal cross-validation.	`0`
`random_seed`	`int`	Random seed for pytorch initializer and numpy generators, overwrites model.init’s.	`None`
`test_size`	`int`	Test size for temporal cross-validation.	`0`

Returns:

Type	Description
None

`DeepAR.predict`

predict(
    dataset,
    test_size=None,
    step_size=1,
    random_seed=None,
    quantiles=None,
    h=None,
    explainer_config=None,
    **data_module_kwargs
)

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

Name	Type	Description	Default
`dataset`	`TimeSeriesDataset`	NeuralForecast’s `TimeSeriesDataset`, see documentation.	required
`test_size`	`int`	Test size for temporal cross-validation.	`None`
`step_size`	`int`	Step size between each window.	`1`
`random_seed`	`int`	Random seed for pytorch initializer and numpy generators, overwrites model.init’s.	`None`
`quantiles`	`list`	Target quantiles to predict.	`None`
`h`	`int`	Prediction horizon, if None, uses the model’s fitted horizon. Defaults to None.	`None`
`explainer_config`	`dict`	configuration for explanations.	`None`
`**data_module_kwargs`	`dict`	PL’s TimeSeriesDataModule args, see documentation.

Returns:

Type	Description
None

Usage Example

import pandas as pd
import matplotlib.pyplot as plt

from neuralforecast import NeuralForecast
from neuralforecast.models import DeepAR
from neuralforecast.losses.pytorch import DistributionLoss, MQLoss
from neuralforecast.utils import AirPassengersPanel, AirPassengersStatic
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=24,
                   lstm_n_layers=1,
                   trajectory_samples=100,
                   loss=DistributionLoss(distribution='StudentT', level=[80, 90], return_params=True),
                   valid_loss=MQLoss(level=[80, 90]),
                   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='ME'
)
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-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()

2. Auxiliary functions

`Decoder`

Decoder(in_features, out_features, hidden_size, hidden_layers)

Bases: Module Multi-Layer Perceptron Decoder Parameters:

Name	Type	Description	Default
`in_features`	`int`	dimension of input.	required
`out_features`	`int`	dimension of output.	required
`hidden_size`	`int`	dimension of hidden layers.	required
`hidden_layers`	`int`	number of hidden layers.	required

​1. DeepAR

​DeepAR

​DeepAR.fit

​DeepAR.predict

​Usage Example

​2. Auxiliary functions

​Decoder