FEDformer

FEDformer: Frequency Enhanced Decomposition transformer for long-term forecasting using Fourier transform and sparse attention in frequency domain.

The FEDformer model tackles the challenge of finding reliable dependencies on intricate temporal patterns of long-horizon forecasting.

The architecture has the following distinctive features:

• In-built progressive decomposition in trend and seasonal components based on a moving average filter.
• Frequency Enhanced Block and Frequency Enhanced Attention to perform attention in the sparse representation on basis such as Fourier transform.
• Classic encoder-decoder proposed by Vaswani et al. (2017) with a multi-head attention mechanism.

The FEDformer model utilizes a three-component approach to define its embedding:

• It employs encoded autoregressive features obtained from a convolution network.
• Absolute positional embeddings obtained from calendar features are utilized.

References

• Zhou, Tian, Ziqing Ma, Qingsong Wen, Xue Wang, Liang Sun, and Rong Jin.. “FEDformer: Frequency enhanced decomposed transformer for long-term series forecasting”

Figure 1. FEDformer Architecture.

1. FEDformer

FEDformer

FEDformer(
    h,
    input_size,
    stat_exog_list=None,
    hist_exog_list=None,
    futr_exog_list=None,
    decoder_input_size_multiplier=0.5,
    version="Fourier",
    modes=64,
    mode_select="random",
    hidden_size=128,
    dropout=0.05,
    n_head=8,
    conv_hidden_size=32,
    activation="gelu",
    encoder_layers=2,
    decoder_layers=1,
    MovingAvg_window=25,
    loss=MAE(),
    valid_loss=None,
    max_steps=5000,
    learning_rate=0.0001,
    num_lr_decays=-1,
    early_stop_patience_steps=-1,
    val_check_steps=100,
    batch_size=32,
    valid_batch_size=None,
    windows_batch_size=1024,
    inference_windows_batch_size=1024,
    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

FEDformer

The FEDformer model tackles the challenge of finding reliable dependencies on intricate temporal patterns of long-horizon forecasting.

The architecture has the following distinctive features:

• In-built progressive decomposition in trend and seasonal components based on a moving average filter.
• Frequency Enhanced Block and Frequency Enhanced Attention to perform attention in the sparse representation on basis such as Fourier transform.
• Classic encoder-decoder proposed by Vaswani et al. (2017) with a multi-head attention mechanism.

The FEDformer model utilizes a three-component approach to define its embedding:

• It employs encoded autoregressive features obtained from a convolution network.
• Absolute positional embeddings obtained from calendar features are utilized.

Parameters:

Name	Type	Description	Default
h	int	forecast horizon.	required
input_size	int	maximum sequence length for truncated train backpropagation.	required
stat_exog_list	List[str]	static exogenous columns.	None
hist_exog_list	List[str]	historic exogenous columns.	None
futr_exog_list	List[str]	future exogenous columns.	None
decoder_input_size_multiplier	float	multiplier for the input size of the decoder.	0.5
version	str	version of the model.	‘Fourier’
modes	int	number of modes for the Fourier block.	64
mode_select	str	method to select the modes for the Fourier block.	‘random’
hidden_size	int	units of embeddings and encoders.	128
dropout	float	dropout throughout Autoformer architecture.	0.05
n_head	int	controls number of multi-head’s attention.	8
conv_hidden_size	int	channels of the convolutional encoder.	32
activation	str	activation from [‘ReLU’, ‘Softplus’, ‘Tanh’, ‘SELU’, ‘LeakyReLU’, ‘PReLU’, ‘Sigmoid’, ‘GELU’].	‘gelu’
encoder_layers	int	number of layers for the TCN encoder.	2
decoder_layers	int	number of layers for the MLP decoder.	1
MovingAvg_window	int	window size for the moving average filter.	25
loss	PyTorch module	instantiated train loss class from losses collection.	MAE()
valid_loss	PyTorch module	instantiated validation loss class from losses collection.	None
max_steps	int	maximum number of training steps.	5000
learning_rate	float	Learning rate between (0, 1).	0.0001
num_lr_decays	int	Number of learning rate decays, evenly distributed across max_steps.	-1
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.	1024
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.

FEDformer.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

FEDformer.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 FEDformer
from neuralforecast.utils import AirPassengersPanel, augment_calendar_df

AirPassengersPanel, calendar_cols = augment_calendar_df(df=AirPassengersPanel, freq='M')

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

model = FEDformer(h=12,
                 input_size=24,
                 modes=64,
                 hidden_size=64,
                 conv_hidden_size=128,
                 n_head=8,
                 loss=MAE(),
                 futr_exog_list=calendar_cols,
                 scaler_type='robust',
                 learning_rate=1e-3,
                 max_steps=500,
                 batch_size=2,
                 windows_batch_size=32,
                 val_check_steps=50,
                 early_stop_patience_steps=2)

nf = NeuralForecast(
    models=[model],
    freq='ME',
)
nf.fit(df=Y_train_df, static_df=None, val_size=12)
forecasts = nf.predict(futr_df=Y_test_df)

Y_hat_df = forecasts.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])

if model.loss.is_distribution_output:
    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['FEDformer-median'], c='blue', label='median')
    plt.fill_between(x=plot_df['ds'][-12:], 
                    y1=plot_df['FEDformer-lo-90'][-12:].values, 
                    y2=plot_df['FEDformer-hi-90'][-12:].values,
                    alpha=0.4, label='level 90')
    plt.grid()
    plt.legend()
    plt.plot()
else:
    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['FEDformer'], c='blue', label='Forecast')
    plt.legend()
    plt.grid()

2. Auxiliary functions

AutoCorrelationLayer

AutoCorrelationLayer(
    correlation, hidden_size, n_head, d_keys=None, d_values=None
)

Bases: Module

Auto Correlation Layer

LayerNorm

LayerNorm(channels)

Bases: Module

Special designed layernorm for the seasonal part

Decoder

Decoder(layers, norm_layer=None, projection=None)

Bases: Module

FEDformer decoder

DecoderLayer

DecoderLayer(
    self_attention,
    cross_attention,
    hidden_size,
    c_out,
    conv_hidden_size=None,
    MovingAvg=25,
    dropout=0.1,
    activation="relu",
)

Bases: Module

FEDformer decoder layer with the progressive decomposition architecture

Encoder

Encoder(attn_layers, conv_layers=None, norm_layer=None)

Bases: Module

FEDformer encoder

EncoderLayer

EncoderLayer(
    attention,
    hidden_size,
    conv_hidden_size=None,
    MovingAvg=25,
    dropout=0.1,
    activation="relu",
)

Bases: Module

FEDformer encoder layer with the progressive decomposition architecture

FourierCrossAttention

FourierCrossAttention(
    in_channels,
    out_channels,
    seq_len_q,
    seq_len_kv,
    modes=64,
    mode_select_method="random",
    activation="tanh",
    policy=0,
)

Bases: Module

Fourier Cross Attention layer

FourierBlock

FourierBlock(
    in_channels, out_channels, seq_len, modes=0, mode_select_method="random"
)

Bases: Module

Fourier block

FourierBlock.compl_mul1d

compl_mul1d(input, weights)

FourierBlock.forward

forward(q, k, v, mask)

FourierBlock.index

index = get_frequency_modes(
    seq_len, modes=modes, mode_select_method=mode_select_method
)

FourierBlock.scale

scale = 1 / (in_channels * out_channels)

FourierBlock.weights1

weights1 = nn.Parameter(
    self.scale
    * torch.rand(
        8,
        in_channels // 8,
        out_channels // 8,
        len(self.index),
        dtype=(torch.cfloat),
    )
)

get_frequency_modes

get_frequency_modes(seq_len, modes=64, mode_select_method='random')