# Online Prediction by ExpeRt Aggregation

### Description

The package `opera`

performs, for regression-oriented time-series,
predictions by combining a finite set of forecasts provided by the user.
More formally, it considers a sequence of observations `Y`

(such as
electricity consumption, or any bounded time series) to be predicted step
by step. At each time instance `t`

, a finite set of experts
(basicly some based forecasters) provide predictions `x`

of the next
observation in `y`

. This package proposes several adaptive and robust
methods to combine the expert forecasts based on their past performance.

### Author(s)

Pierre Gaillard <pierre@gaillard.me>

### References

Prediction, Learning, and Games. N. Cesa-Bianchi and G. Lugosi.

Forecasting the electricity consumption by aggregating specialized experts;
a review of sequential aggregation of specialized experts, with an
application to Slovakian an French contry-wide one-day-ahead (half-)hourly
predictions, Machine Learning, in press, 2012. Marie Devaine, Pierre
Gaillard, Yannig Goude, and Gilles Stoltz

Contributions to online robust aggregation: work on the approximation error and on
probabilistic forecasting. Pierre Gaillard. PhD Thesis, University Paris-Sud, 2015.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | ```
#'
library('opera') # load the package
set.seed(1)
# Example: find the best one week ahead forecasting strategy (weekly data)
# packages
library(mgcv)
# import data
data(electric_load)
idx_data_test <- 620:nrow(electric_load)
data_train <- electric_load[-idx_data_test, ]
data_test <- electric_load[idx_data_test, ]
# Build the expert forecasts
# ##########################
# 1) A generalized additive model
gam.fit <- gam(Load ~ s(IPI) + s(Temp) + s(Time, k=3) +
s(Load1) + as.factor(NumWeek), data = data_train)
gam.forecast <- predict(gam.fit, newdata = data_test)
# 2) An online autoregressive model on the residuals of a medium term model
# Medium term model to remove trend and seasonality (using generalized additive model)
detrend.fit <- gam(Load ~ s(Time,k=3) + s(NumWeek) + s(Temp) + s(IPI), data = data_train)
electric_load$Trend <- c(predict(detrend.fit), predict(detrend.fit,newdata = data_test))
electric_load$Load.detrend <- electric_load$Load - electric_load$Trend
# Residual analysis
ar.forecast <- numeric(length(idx_data_test))
for (i in seq(idx_data_test)) {
ar.fit <- ar(electric_load$Load.detrend[1:(idx_data_test[i] - 1)])
ar.forecast[i] <- as.numeric(predict(ar.fit)$pred) + electric_load$Trend[idx_data_test[i]]
}
# Aggregation of experts
###########################
X <- cbind(gam.forecast, ar.forecast)
colnames(X) <- c('gam', 'ar')
Y <- data_test$Load
matplot(cbind(Y, X), type = 'l', col = 1:6, ylab = 'Weekly load', xlab = 'Week')
# How good are the expert? Look at the oracles
oracle.convex <- oracle(Y = Y, experts = X, loss.type = 'square', model = 'convex')
plot(oracle.convex)
oracle.convex
# Is a single expert the best over time ? Are there breaks ?
oracle.shift <- oracle(Y = Y, experts = X, loss.type = 'percentage', model = 'shifting')
plot(oracle.shift)
oracle.shift
# Online aggregation of the experts with BOA
#############################################
# Initialize the aggregation rule
m0.BOA <- mixture(model = 'BOA', loss.type = 'square')
# Perform online prediction using BOA There are 3 equivalent possibilities 1)
# start with an empty model and update the model sequentially
m1.BOA <- m0.BOA
for (i in 1:length(Y)) {
m1.BOA <- predict(m1.BOA, newexperts = X[i, ], newY = Y[i])
}
# 2) perform online prediction directly from the empty model
m2.BOA <- predict(m0.BOA, newexpert = X, newY = Y, online = TRUE)
# 3) perform the online aggregation directly
m3.BOA <- mixture(Y = Y, experts = X, model = 'BOA', loss.type = 'square')
# These predictions are equivalent:
identical(m1.BOA, m2.BOA) # TRUE
identical(m1.BOA, m3.BOA) # TRUE
# Display the results
summary(m3.BOA)
plot(m1.BOA)
``` |

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