Forecasting With Exogenous Regressors

Consider a multivariate time series \(X^{(1)}, \ldots, X^{(t)}\), where each \(X^{(i)} \in \mathbb{R}^d\) is a d-dimensional vector. In multivariate forecasting, our goal is to predict the future values of the k’th univariate \(X_k^{(t+1)}, \ldots, X_k^{(t+h)}\).

Exogenous regressors \(Y^{(i)}\) are a set of additional variables whose values we know a priori. The task of forecasting with exogenous regressors is to predict our target univariate \(X_k^{(t+1)}, \ldots, X_k^{(t+h)}\), conditioned on - The past values of the time series \(X^{(1)}, \ldots, X^{(t)}\) - The past values of the exogenous regressors \(Y^{(1)}, \ldots, Y^{(t)}\) - The future values of the exogenous regressors \(Y^{(t+1)}, \ldots, Y^{(t+h)}\)

For example, one can consider the task of predicting the sales of a specific item at a store. Endogenous variables \(X^{(i)} \in \mathbb{R}^4\) may contain the number of units sold (the target univariate), the temperature outside, the consumer price index, and the current unemployemnt rate. Exogenous variables \(Y^{(i)} \in \mathbb{R}^6\) are variables that the store has control over or prior knowledge of. They may include whether a particular day is a holiday, and various information about the sort of markdowns the store is running.

To be more concrete, let’s show this with some real data.

# This is the same dataset used in the custom dataset tutorial
import os
from ts_datasets.forecast import CustomDataset
csv = os.path.join("..", "..", "data", "walmart", "walmart_mini.csv")
dataset = CustomDataset(rootdir=csv, index_cols=["Store", "Dept"], test_frac=0.10)
ts, md = dataset[0]
display(ts)

	Weekly_Sales	Temperature	Fuel_Price	MarkDown1	MarkDown2	MarkDown3	MarkDown4	MarkDown5	CPI	Unemployment	IsHoliday
Date
2010-02-05	24924.50	42.31	2.572	NaN	NaN	NaN	NaN	NaN	211.096358	8.106	False
2010-02-12	46039.49	38.51	2.548	NaN	NaN	NaN	NaN	NaN	211.242170	8.106	True
2010-02-19	41595.55	39.93	2.514	NaN	NaN	NaN	NaN	NaN	211.289143	8.106	False
2010-02-26	19403.54	46.63	2.561	NaN	NaN	NaN	NaN	NaN	211.319643	8.106	False
2010-03-05	21827.90	46.50	2.625	NaN	NaN	NaN	NaN	NaN	211.350143	8.106	False
...	...	...	...	...	...	...	...	...	...	...	...
2012-09-28	18947.81	76.08	3.666	3666.27	7.64	1.65	1417.96	4744.28	222.981658	6.908	False
2012-10-05	21904.47	68.55	3.617	8077.89	NaN	18.22	3617.43	3626.14	223.181477	6.573	False
2012-10-12	22764.01	62.99	3.601	2086.18	NaN	8.11	602.36	5926.45	223.381296	6.573	False
2012-10-19	24185.27	67.97	3.594	950.33	NaN	4.93	80.25	2312.85	223.425723	6.573	False
2012-10-26	27390.81	69.16	3.506	2585.85	31.75	6.00	1057.16	1305.01	223.444251	6.573	False

143 rows × 11 columns

from merlion.utils import TimeSeries

# Get the endogenous variables X and split them into train & test
endog = ts[["Weekly_Sales", "Temperature", "CPI", "Unemployment"]]
train = TimeSeries.from_pd(endog[md.trainval])
test = TimeSeries.from_pd(endog[~md.trainval])

# Get the exogenous variables Y
exog = TimeSeries.from_pd(ts[["IsHoliday", "MarkDown1", "MarkDown2", "MarkDown3", "MarkDown4", "MarkDown5"]])

Here, our task is to predict the weekly sales. We would like our model to also account for variables which may have an impact on consumer demand (i.e. temperature, consumer price index, and unemployment), as knowledge of these variables could improve the quality of our sales forecast. This would be a multivariate forecasting problem, covered here.

In principle, we could add markdowns and holidays to the multivariate model. However, as a retailer, we know a priori which days are holidays, and we ourselves control the markdowns. In many cases, we can get better forecasts by providing the future values of these variables in addition to the past values. Moreover, we may wish to model how changing the future markdowns would change the future sales. This is why we should model these variables as exogenous regressors instead.

All Merlion forecasters support an API which accepts exogenous regressors at both training and inference time, though only some models actually support the feature. Using the feature is as easy as specifying an optional argument exog_data to both train() and forecast(). We show how to use the feature for the popular Prophet model below, and demonstrate that adding exogenous regressors can improve the quality of the forecast.

from merlion.evaluate.forecast import ForecastMetric
from merlion.models.forecast.prophet import Prophet, ProphetConfig

# Train a model without exogenous data
model = Prophet(ProphetConfig(target_seq_index=0))
model.train(train)
pred, err = model.forecast(test.time_stamps)
smape = ForecastMetric.sMAPE.value(test, pred, target_seq_index=model.target_seq_index)
print(f"sMAPE (w/o exog) = {smape:.2f}")

# Train a model with exogenous data
exog_model = Prophet(ProphetConfig(target_seq_index=0))
exog_model.train(train, exog_data=exog)
exog_pred, exog_err = exog_model.forecast(test.time_stamps, exog_data=exog)
exog_smape = ForecastMetric.sMAPE.value(test, exog_pred, target_seq_index=exog_model.target_seq_index)
print(f"sMAPE (w/ exog)  = {exog_smape:.2f}")

12:31:47 - cmdstanpy - INFO - Chain [1] start processing
12:31:47 - cmdstanpy - INFO - Chain [1] done processing
12:31:48 - cmdstanpy - INFO - Chain [1] start processing
12:31:48 - cmdstanpy - INFO - Chain [1] done processing

sMAPE (w/o exog) = 8.21
sMAPE (w/ exog)  = 7.67

Before we wrap up this tutorial, we note that the exogenous variables contain a lot of missing data:

display(exog.to_pd())

	IsHoliday	MarkDown1	MarkDown2	MarkDown3	MarkDown4	MarkDown5
time
2010-02-05	0.0	NaN	NaN	NaN	NaN	NaN
2010-02-12	1.0	NaN	NaN	NaN	NaN	NaN
2010-02-19	0.0	NaN	NaN	NaN	NaN	NaN
2010-02-26	0.0	NaN	NaN	NaN	NaN	NaN
2010-03-05	0.0	NaN	NaN	NaN	NaN	NaN
...	...	...	...	...	...	...
2012-09-28	0.0	3666.27	7.64	1.65	1417.96	4744.28
2012-10-05	0.0	8077.89	NaN	18.22	3617.43	3626.14
2012-10-12	0.0	2086.18	NaN	8.11	602.36	5926.45
2012-10-19	0.0	950.33	NaN	4.93	80.25	2312.85
2012-10-26	0.0	2585.85	31.75	6.00	1057.16	1305.01

143 rows × 6 columns

Behind the scenes, Merlion models will apply an optional exog_transform to the exogenous variables, and they will then resample the exogenous variables to the same timestamps as the endogenous variables. This resampling is achieved using the exog_missing_value_policy and exog_aggregation_policy, which can be specified in the config of any model which accepts exogenous regressors. We can see the default values for each of these parameters by inspecting the config:

print(f"Default exog_transform:            {exog_model.config.exog_transform}")
print(f"Default exog_missing_value_policy: {exog_model.config.exog_missing_value_policy}")
print(f"Default exog_aggregation_policy:   {exog_model.config.exog_aggregation_policy}")

Default exog_transform:            Identity()
Default exog_missing_value_policy: MissingValuePolicy.ZFill
Default exog_aggregation_policy:   AggregationPolicy.Mean

So in this case, we first apply the identity transform to the exogenous data. Then, we impute missing values by filling them with zeros (ZFill), and we downsample the exogenous data by taking the Mean of any relevant windows.