Statistical forecasting with exogenous factors

A data-driven approach to sales forecasting, that uses exogenous temporal and weather encodings.

Table of contents

• General explanation
• Getting started
• Project structure
• Results
  • Weekly forecasting (long term)
    • Weekly average over all datasets
  • Daily forecasting (short term)
    • Daily average over all datasets
• Future Work

General explanation

This repository contains the code for a data-driven approach to sales forecasting, where a preliminary data analysis was done in order to statistically check for the validity of the hypothesis that the exogenous factors (time and weather) affect the total sold amount of products. Based on this analysis, different forecasting models were used, with a particular focus on the LSTM-RNN. Neural networks like the LSTM possess the ability to learn non-linear relationships in the data, which can make a difference when dealing with multivariate time series forecasting. You can see the results of this work here. In general, it seems to be the case then when using valid exogenous data, we can help a forecasting model understand not just the temporal relationship, but also understand the spatial plane of the phenomenon. In simpler words, adding valid exogenous data can help the forecasting model better understand what we are trying to predict. This approach was used for both long-term forecasting (weeks over seasons) and short-term forecasting (days over weeks), giving satisfactory results in both cases.

Getting started

You can find all the python packages that haven been used in this project inside the requirements.txt file. First you will need to install those packages or make sure your current environment has them in order to be able to run the code. A simple way to install all the packages is by running: pip -r requirements.txt

P.S: I recommend using Anacoda as an all-in-one scientific environment and just adding the missing packages.

Project structure

There are 2 main directories:

1. datasets: As the name suggests, in this directory you can find all the preprocessed data of the sales, along with the datasets which are augmented with the exogenous data. You can find the univariate datasets by looking at the daily/ or weekly/ folders, or the multivariate datasets by looking at the daily_aug/ or weekly_aug/ folders.
2. code: This is the main directory which contains the code for forecasting, models and data analysis. It is further divided into three subfolders:
   1. data_analysis: Inside you can find the analysis performed on the data of a single city (Milan). That analysis was then extended to all shops inside malls and streetshops. By checking both visually and statistically (hypothesis testing) it was derived that the weather and the time are usefulness and valid exogenous data.
   2. forecasting: In this folder you will find the two notebooks which contain the code for both daily and weekly forecasting.
   3. helpers: In this folder you will find the code for some auxiliary functions, as well as the code for the LSTM neural network.

The code itself is well documented and should hopefully be understandable.

Results

Weekly forecasting (long term)

Dataset	SARIMA RMSE	SARIMAX RMSE	MONO-LSTM RMSE	EXO1-LSTM RMSE	EXO2-LSTM RMSE	EXO3-LSTM RMSE	SARIMA SMAPE	SARIMAX SMAPE	MONO-LSTM SMAPE	EXO1-LSTM SMAPE	EXO2-LSTM SMAPE	EXO3-LSTM SMAPE
Milan	299.299334	320.983730	269.000266	259.300154	294.289433	276.278388	6.099036	8.041129	7.412993	7.515069	8.366195	8.250909
Rome	107.928035	103.974567	149.296986	95.339082	127.586807	117.149285	12.846811	12.232047	16.026479	10.029953	12.068225	12.251298
Turin	137.123418	142.099703	146.110333	137.757753	174.867309	163.300861	11.014910	12.382435	13.085100	12.106654	13.937898	13.659325

Weekly average over all datasets

SARIMA RMSE	SARIMAX RMSE	MONO-LSTM RMSE	EXO1-LSTM RMSE	EXO2-LSTM RMSE	EXO3-LSTM RMSE	SARIMA SMAPE	SARIMAX SMAPE	MONO-LSTM SMAPE	EXO1-LSTM SMAPE	EXO2-LSTM SMAPE	EXO3-LSTM SMAPE
181.450262	189.019333	188.135862	164.132330	198.914516	185.576178	9.986919	10.885204	12.174857	9.883892	11.457439	11.387177

Daily forecasting (short term)

Dataset	MONO-LSTM RMSE	EXO1-LSTM RMSE	EXO2-LSTM RMSE	EXO3-LSTM RMSE	MONO-LSTM SMAPE	EXO1-LSTM SMAPE	EXO2-LSTM SMAPE	EXO3-LSTM SMAPE
Milan	92.681814	97.419602	92.756764	87.843425	17.082440	19.892359	20.611555	16.801701
Rome	103.320297	101.16842	107.342307	99.044623	23.468583	21.877664	22.764811	22.043665
Turin	40.815775	34.641631	35.821523	36.963587	20.751887	18.803312	18.672907	17.939686

Daily average over all datasets

MONO-LSTM RMSE	EXO1-LSTM RMSE	EXO2-LSTM RMSE	EXO3-LSTM RMSE	MONO-LSTM SMAPE	EXO1-LSTM SMAPE	EXO2-LSTM SMAPE	EXO3-LSTM SMAPE
78.939295	77.743218	78.640198	74.617212	20.434304	20.191111	20.683091	18.928351

Future Work

1. Different exogenous data: Very interesting ones would be the use of visual features or textual features deriving from social media. This would be particularly useful for e-commerce.
2. Other model architectures: While this work focuses on "singular" approaches and mostly on the LSTM, it can be interesting to see how ensemble methods or different neural network architectures would perform.
3. Automatic feature extraction: I believe that another great thing to consider would be automating this process by having a way to extract meaningful features from the multivariate time series. This way the whole data analysis could be bypassed in a certain sense. Multi-modal forecasting is a big trend right now in forecasting research.