Sales Prediction with Time Series Modeling Gautam Shine, Sanjib Basak Nonlinearity induced by hidden layer = + =1 Parameters and learned from data Autoregression can be included through lagged inputs Optimization is non-convex, averaging needed ARIMA β’ Order (5,2,0) chosen by minimizing Akaike information, which is a regularized maximum likelihood estimate β’ Fourier terms used to introduce multiple-seasonality β’ Predictions are too smooth to capture sales spikes Neural Net β’ 10 autoregression lags and 14 hidden layers, chosen by minimizing generalization error β’ Averaged prediction of 100 nets initialized with random seeds β’ Qualitatively captures sales spikes with lower MSE than ARIMA, but low daily accuracy ARIMA + Regression β’ Regression with indicator vectors for special days like Black Friday and Christmas greatly improves ARIMA β’ Captures sales spikes while retaining daily accuracy β’ Lower MSE than the neural net Autoregression (AR) = + + =1 β β Moving Average (MA) = + + =1 β β Autoregressive Integrated Moving Average (ARIMA) = + + =1 β β + =1 β β Conventional Time Series Models Feed-Forward Neural Networks Sales forecasting is critical for retailers β’ optimal stocking of products β’ website stability under peak traffic β’ planning, customer support, and marketing Anomalies like Black Friday are especially difficult to capture with models Example data from online sales of tech products: Forecast Results Predicting Sales Hidden Node Count and Autoregression Order β’ Erratic behavior can be observed when the number of hidden nodes is low β’ Autoregression order made little difference beyond 5 or so β’ Both chosen to minimize test MSE, but nearby values could also have been reasonable Learning Curve β’ Curve is non-monotonic since data is not i.i.d. and set size affects prediction interval β’ Neural nets require more data to achieve the same forecast accuracy, but can exceed ARIMA with large data sets Model Fitting
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Sales Prediction with Time Series ModelingGautam Shine, Sanjib Basak
Nonlinearity induced by hidden layer
π¦π = ππ +
π=1
π
π€ππ π₯π
Parameters ππ and π€ππ learned from data
Autoregression can be included through lagged inputs
Optimization is non-convex, averaging needed
ARIMA
β’ Order (5,2,0) chosen by minimizing Akaike information, which is a regularized maximum likelihood estimate
β’ Fourier terms used to introduce multiple-seasonality
β’ Predictions are too smooth to capture sales spikes
Neural Net
β’ 10 autoregression lags and 14 hidden layers, chosen by minimizing generalization error
β’ Averaged prediction of 100 nets initialized with random seeds
β’ Qualitatively captures sales spikes with lower MSE than ARIMA, but low daily accuracy
ARIMA + Regression
β’ Regression with indicator vectors for special days like Black Friday and Christmas greatly improves ARIMA
β’ Captures sales spikes while retaining daily accuracy
β’ Lower MSE than the neural net
Autoregression (AR)
ππ‘ = π + ππ‘ +
π=1
π
ππ‘βπ ππ‘βπ
Moving Average (MA)
ππ‘ = π + ππ‘ +
π=1
π
ππ‘βπ ππ‘βπ
Autoregressive Integrated Moving Average (ARIMA)
ππ‘ = π + ππ‘ +
π=1
π
ππ‘βπ ππ‘βπ +
π=1
π
ππ‘βπ ππ‘βπ
Conventional Time Series Models
Feed-Forward Neural Networks
Sales forecasting is critical for retailers
β’ optimal stocking of products
β’ website stability under peak traffic
β’ planning, customer support, and marketing
Anomalies like Black Friday are especially difficult to capture with models
Example data from online sales of tech products:
Forecast ResultsPredicting Sales
Hidden Node Count and Autoregression Order
β’ Erratic behavior can be observed when the number of hidden nodes is low
β’ Autoregression order made little difference beyond 5 or so
β’ Both chosen to minimize test MSE, but nearby values could also have been reasonable
Learning Curve
β’ Curve is non-monotonic since data is not i.i.d. and set size affects prediction interval
β’ Neural nets require more data to achieve the same forecast accuracy, but can exceed ARIMA with large data sets
Model Fitting
Related Documents
