Forecasting product Demand
library(xts)## Warning: package 'xts' was built under R version 4.0.3## Warning: package 'zoo' was built under R version 4.0.3library(forecast)## Warning: package 'forecast' was built under R version 4.0.3data <- read.csv("Bev.csv")
dates <- seq(as.Date("2014-01-19"), length = 176, by = "weeks")
bev_xts <- xts(data, order.by = dates)Manipulate XTS objects
# Create the individual region sales as their own objects
MET_hi <- bev_xts[,"MET.hi"]
MET_lo <- bev_xts[,"MET.lo"]
MET_sp <- bev_xts[,"MET.sp"]
# Sum the region sales together
MET_t <- MET_hi + MET_lo + MET_sp
# Plot the metropolitan region total sales
plot(MET_t)
## ARIMA models
# Split the data into training and validation
MET_t_train <- MET_t[index(MET_t) < "2017-01-01"]
MET_t_valid <- MET_t[index(MET_t) >= "2017-01-01"]
# Use auto.arima() function for metropolitan sales training data
MET_t_model <- auto.arima(MET_t_train)
MET_t_model## Series: MET_t_train
## ARIMA(1,0,0) with non-zero mean
##
## Coefficients:
## ar1 mean
## 0.6706 6252.8350
## s.e. 0.0594 181.7488
##
## sigma^2 estimated as 574014: log likelihood=-1238.86
## AIC=2483.72 AICc=2483.88 BIC=2492.83Forecasting and evaluating
- MAE Average measure of how far away is from the actual value
- MAPE: Average measure of how far away in absolute percentage terms is from the actual value
# Forecast the first 22 weeks of 2017
forecast_MET_t <- forecast(MET_t_model, h = 22)
# Plot this forecast #
plot(forecast_MET_t)
## Model performance
# Convert to numeric for ease
for_MET_t <- as.numeric(forecast_MET_t$mean)
v_MET_t <- as.numeric(MET_t_valid)
# Calculate the MAE
MAE <- mean(abs(for_MET_t - v_MET_t))
# Calculate the MAPE
MAPE <- 100*mean(abs((for_MET_t - v_MET_t)/v_MET_t))
# Print to see how good your forecast is!
print(MAE)## [1] 898.8403print(MAPE)## [1] 17.10332Visualizing the forecast
# Convert your forecast to an xts object
for_dates <- seq(as.Date("2017-01-01"), length = 22, by = "weeks")
for_MET_t_xts <- xts(forecast_MET_t$mean, order.by = for_dates)
# Plot the validation data set
plot(MET_t_valid, main = 'Forecast Comparison', ylim = c(4000, 8500))
# Overlay the forecast of 2017
lines(for_MET_t_xts, col = "blue")
Confidence intervals of the forecast
# Plot the validation data set
plot(MET_t_valid, main = 'Forecast Comparison', ylim = c(4000, 8500))
# Overlay the forecast of 2017
lines(for_MET_t_xts, col = "blue")
# Convert the limits to xts objects
lower <- xts(forecast_MET_t$lower[,2], order.by = for_dates)
upper <- xts(forecast_MET_t$upper[,2], order.by = for_dates)
# Adding confidence intervals of forecast to plot
lines(lower, col = "blue", lty = "dashed")
lines(upper, col = "blue", lty = "dashed")
Components of Demand
Price elasticity
- Is the economic measure of how much demand “reacts” to changes in price
- AS price changes, it is expected that demand changes as well, but how much?
\[ Price \space Elasticity = \frac{ \%Change \space in \space demand}{\% Change \space in \space price} \]
Elastic vs Inelastic
Elastic: Products have % changes in demand larger than the % change in price (Price elasticity > 1)
Inelastic: Products have % changes in demand smaller than the % change in price (Price elasticity < 1)
Unit elastic: (Price elasticity = 1)
We can estimate elasticity with linear regression:
bev_xts_train <- bev_xts[index(bev_xts) < "2017-01-01"]
MET_hi <- bev_xts[index(bev_xts) < "2017-01-01", "MET.hi"]
MET_hi_p <- bev_xts[index(bev_xts) < "2017-01-01", "MET.hi.p"]
# Save the prices of each product
l_MET_hi_p <- as.vector(log(bev_xts_train[,"MET.hi.p"]))
# Save as a data frame
MET_hi_train <- data.frame(as.vector(log(MET_hi)), l_MET_hi_p)
colnames(MET_hi_train) <- c("log_sales", "log_price")
# Calculate the regression
model_MET_hi <- lm(log_sales ~ log_price, data = MET_hi_train)
model_MET_hi##
## Call:
## lm(formula = log_sales ~ log_price, data = MET_hi_train)
##
## Coefficients:
## (Intercept) log_price
## 13.977 -1.517Seasonal /Holiday/ Promotional effects
- Seasonal effects: Winter coats, bathing suits, school supplies…
- Holiday effects: retail sales, holiday decorations, candy…
- Promotion effects: Digital marketing, shelf optimization
# Plot the product's sales
plot(MET_hi)
# Plot the product's price
plot(MET_hi_p)
# Create date indices for New Year's week
n.dates <- as.Date(c("2014-12-28", "2015-12-27", "2016-12-25"))
# Create xts objects for New Year's
newyear <- as.xts(rep(1, 3), order.by = n.dates)
# Create sequence of 154 weeks for merging
dates_train <- seq(as.Date("2014-01-19"), length = 154, by = "weeks")
# Merge training dates into New Year's object
newyear <- merge(newyear, dates_train, fill = 0)
# Create MET_hi_train_2 by adding newyear
MET_hi_train_2 <- data.frame(MET_hi_train, as.vector(newyear))
colnames(MET_hi_train_2)[3] <- "newyear"
# Build regressions for the product
model_MET_hi_full <- lm(log_sales ~ log_price + newyear, data = MET_hi_train_2)
model_MET_hi_full##
## Call:
## lm(formula = log_sales ~ log_price + newyear, data = MET_hi_train_2)
##
## Coefficients:
## (Intercept) log_price newyear
## 13.5083 -1.4036 0.3375Forecast with regression
bev_xts_valid <- bev_xts[index(bev_xts) >= '2017-01-01']
# Subset the validation prices #
l_MET_hi_p_valid <- as.vector(log(bev_xts_valid[,"MET.hi.p"]))
# Create a validation data frame #
MET_hi_valid <- data.frame(l_MET_hi_p_valid)
colnames(MET_hi_valid) <- "log_price"
# Predict the log of sales for your high end product
pred_MET_hi <- predict(model_MET_hi, newdata = MET_hi_valid)
# Convert predictions out of log scale
pred_MET_hi <- exp(pred_MET_hi)
# Convert to an xts object
dates_valid <- seq(as.Date("2017-01-01"), length = 22, by = "weeks")
pred_MET_hi_xts <- xts(pred_MET_hi, order.by = dates_valid)
# Plot the forecast
plot(pred_MET_hi_xts)
# Calculate and print the MAPE
MET_hi_v <- bev_xts_valid[,"MET.hi"]
MAPE <- 100*mean(abs((pred_MET_hi_xts - MET_hi_v)/MET_hi_v))
print(MAPE)## [1] 29.57455Blending regression model with times series
Examining residuals
REsiduals might be related across time. When this happens we can use times series models to predict them!
# Calculate the residuals from the model
MET_hi_full_res <- residuals(model_MET_hi_full)
# Convert the residuals to an xts object
MET_hi_full_res <- xts(MET_hi_full_res, order.by = dates_train)
hist(MET_hi_full_res)
# Plot the residuals over time
plot(MET_hi_full_res)
Forecasting residuals
# Build an ARIMA model on the residuals: MET_hi_arima
MET_hi_arima <- auto.arima(MET_hi_full_res)
# Look at a summary of the model
summary(MET_hi_arima)## Series: MET_hi_full_res
## ARIMA(3,0,1) with zero mean
##
## Coefficients:
## ar1 ar2 ar3 ma1
## 1.2150 -0.1758 -0.2945 -0.4675
## s.e. 0.1578 0.1885 0.0901 0.1574
##
## sigma^2 estimated as 0.03905: log likelihood=32.51
## AIC=-55.02 AICc=-54.61 BIC=-39.83
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.0004563225 0.1950202 0.1456822 -46.5269 224.0129 0.5251787
## ACF1
## Training set 0.001241359# Forecast 22 weeks with your model: for_MET_hi_arima
for_MET_hi_arima <- forecast(MET_hi_arima, h = 22)
# Print first 10 observations
head(for_MET_hi_arima, n = 10)## $method
## [1] "ARIMA(3,0,1) with zero mean"
##
## $model
## Series: MET_hi_full_res
## ARIMA(3,0,1) with zero mean
##
## Coefficients:
## ar1 ar2 ar3 ma1
## 1.2150 -0.1758 -0.2945 -0.4675
## s.e. 0.1578 0.1885 0.0901 0.1574
##
## sigma^2 estimated as 0.03905: log likelihood=32.51
## AIC=-55.02 AICc=-54.61 BIC=-39.83
##
## $level
## [1] 80 95
##
## $mean
## Time Series:
## Start = 1079
## End = 1226
## Frequency = 0.142857142857143
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## [21] 0.001270855 0.005465546
##
## $lower
## Time Series:
## Start = 1079
## End = 1226
## Frequency = 0.142857142857143
## 80% 95%
## 1079 -0.3127598 -0.4468164
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## 1212 -0.4307176 -0.6563862
## 1219 -0.4251130 -0.6508269
## 1226 -0.4211319 -0.6469590
##
## $upper
## Time Series:
## Start = 1079
## End = 1226
## Frequency = 0.142857142857143
## 80% 95%
## 1079 0.1937179 0.3277744
## 1086 0.1929708 0.3603392
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## 1184 0.4055358 0.6302188
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## 1205 0.4156214 0.6412900
## 1212 0.4218788 0.6475474
## 1219 0.4276547 0.6533686
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##
## $x
## Time Series:
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## Frequency = 0.142857142857143
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##
## $series
## [1] "MET_hi_full_res"
##
## $fitted
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## Frequency = 0.142857142857143
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##
## $residuals
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## Frequency = 0.142857142857143
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## [151] 0.1294880202 -0.0818058263 0.2667561544 -0.2230098056# Convert your forecasts into an xts object
dates_valid <- seq(as.Date("2017-01-01"), length = 22, by = "weeks")
for_MET_hi_arima <- xts(for_MET_hi_arima$mean, order.by = dates_valid)
# Plot the forecast
plot(for_MET_hi_arima)
Transfer functions and ensembling
Combining techniques:
- Transfer functions: everything gets built into one model
- Ensembling - “Average” multiple effects types of the model forecasts
- Combining two different techiniques into one mathematically:
\[ Log(Y_{t}) = \beta_{0} + \beta_{1} \space log(X_{1}) + \beta_{2} \space X_{2} + \cdots + \epsilon_{t} \]
\[ \epsilon_{t} = \alpha_{0} + \alpha_{1}\epsilon_{t-1} + \alpha_{2}\epsilon_{t-2} +\cdots + \epsilon \]
- Combining the forecasts into one mathematically
\[ log(Y_{t}) = log(\hat{Y_{t}}) + \hat{\epsilon_{t}} \]
\[ Y_{t} = \hat{Y_{t}} \space \times \space exp(\hat{\epsilon_{t}}) \]
Transfer function application
# Convert your residual forecast to the exponential version
for_MET_hi_arima <- exp(for_MET_hi_arima)
# Multiply your forecasts together!
for_MET_hi_final <- for_MET_hi_arima * pred_MET_hi_xts
# Plot the final forecast - don't touch the options!
plot(for_MET_hi_final, ylim = c(1000, 4300))
# Overlay the validation data set
lines(MET_hi_v, col = "blue")
Error metrics
# Calculate the MAE
MAE <- mean(abs(MET_hi_v - for_MET_hi_final))
print(MAE)## [1] 474.7013# Calculate the MAPE
MAPE <- 100*mean(abs((for_MET_hi_final - MET_hi_v)/MET_hi_v))
print(MAPE)## [1] 28.44671Ensemble application
# Build an ARIMA model using the auto.arima function
MET_hi_model_arima <- auto.arima(MET_hi)
# Forecast the ARIMA model you just built above
for_MET_hi <- forecast(MET_hi_model_arima, h = 22)
# Save the forecast as an xts object
dates_valid <- seq(as.Date("2017-01-01"), length = 22, by = "weeks")
for_MET_hi_xts <- xts(for_MET_hi$mean, order.by = dates_valid)
# Calculate the MAPE of the forecast
MAPE <- 100*mean(abs((for_MET_hi_xts - MET_hi_v)/MET_hi_v))
print(MAPE)## [1] 40.82418# Ensemble the two forecasts together
for_MET_hi_en <- 0.5*(pred_MET_hi_xts+for_MET_hi_xts)
# Calculate the MAE and MAPE
MAE <- mean(abs(for_MET_hi_en - MET_hi_v))
print(MAE)## [1] 572.8845MAPE <- 100*mean(abs((for_MET_hi_en - MET_hi_v)/MET_hi_v))
print(MAPE)## [1] 34.55672Hierarchical forecasting
- Can be used when different items that need to be forecasted can be arranged in a logical hierarchy
- Forecasts need to be reconciled up and down the hierarchy
mape <- function(yhat, y){
mean(abs((y - yhat)/y))*100}
MET_sp <- bev_xts[index(bev_xts) < '2017-01-01', "MET.sp"]
MET_sp_v <- bev_xts[index(bev_xts) >= '2017-01-01', "MET.sp"]
# Build a time series model
MET_sp_model_arima <- auto.arima(MET_sp)
# Forecast the time series model you just built for 22 periods
for_MET_sp <- forecast(MET_sp_model_arima, h = 22)
# Create an xts object on the forecast object you just created
for_MET_sp_xts <- xts(for_MET_sp$mean, order.by = dates_valid)
# Calculate the MAPE on your forecast with the validation data
MAPE <- mape(for_MET_sp_xts, MET_sp_v)
print(MAPE)## [1] 6.352973With regression…
# Build a regression model on the training data
model_MET_sp_full <- lm(log_sales ~ log_price + christmas + valentine + newyear + mother, data = MET_sp_train)
# Forecast the regression model using the predict function
pred_MET_sp <- predict(model_MET_sp_full, newdata = MET_sp_valid)
# Exponentiate your predictions and create an xts object
pred_MET_sp <- exp(pred_MET_sp)
pred_MET_sp_xts <- xts(pred_MET_sp, order.by = dates_valid)
# Calculate MAPE
MAPE <- mape(pred_MET_sp_xts, MET_sp_v)
print(MAPE)Ensemble forecasts
# Ensemble the two forecasts
for_MET_sp_en <- (for_MET_sp_xts + pred_MET_sp_xts)/2
# Calculate the MAPE
MAPE <- mape(for_MET_sp_en, MET_sp_v)
print(MAPE)