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lmForc
In lmForc: Linear Model Forecasting
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
library(lmForc)
{width=100%}
Sections
[Forecast Class]
[is_forc]
[oos_realized_forc]
[oos_lag_forc]
[oos_vintage_forc]
[conditional_forc]
[historical_average_forc]
[random_walk_forc]
[autoreg_forc]
[performance_weighted_forc]
[states_weighted_forc]
lmForc Motivation
Linear forecasting models have two main features: simplicity and
interpretability. These features allow the performance of linear forecasting
models to be tested in multiple ways, both in-sample and out-out-sample. The
lmForc package brings these performance tests to the R language with a
philosophy that compliments the features of linear models: One, the syntax for
creating linear models is simple and the syntax for testing them should be
equally simple. Two, performance tests should be realistic and when possible
replicate what it would have been like to forecast in real-time.
Forecast Class
At the heart of the lmForc package is the Forecast class. Base R does not
provide a good format for working with forecasts, so lmForc addresses this by
introducing a new class for storing forecasts that is simple and rigorous. The
Forecast class is paramount to the lmForc philosophy of simple syntax and
realistic tests. Forecast is an S4 class that contains equal length vectors
with the following data:
origin \ \ \ The time when the forecast was made.future \ \ \ The time that is being forecasted.forecast The forecast itself.realized \ If available, the realized value at the time being forecasted.
The Forecast class also includes an additional length-one slot h_ahead for
representing how many periods ahead are being forecasted. This slot is optional,
but becomes useful for documentation and performing out-of-sample forecast
tests.
We demonstrate the Forecast class by constructing a simple Forecast object.
This forecast contains four observations at the quarterly frequency.
my_forecast <- Forecast(
origin = as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31")),
future = as.Date(c("2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31")),
forecast = c(4.21, 4.27, 5.32, 5.11),
realized = c(4.40, 4.45, 4.87, 4.77),
h_ahead = 4L
)
Note that we have chosen Date objects at the quarterly frequency for our
origin and future slots. This forecast is for four quarters ahead, so we
fill in the h_ahead slot with the integer four.
The origin, future, and h_ahead slots can be filled with any values. This
is where the general nature of the Forecast class shines. In the origin and
future slots we could use dates at a daily or yearly frequency, the POSIXct
class to include minutes and seconds, or integers to represent discrete periods.
We can also store different types of forecasts. In the example above, we have
forecasts made at different origin times for a constant four quarter ahead
forecast horizon. We could also store a forecast made at a single origin time
for a horizon of future times by setting all of the origin values to one
time and using future to represent the horizon of times that we are
forecasting. In this case the h_ahead slot becomes irrelevant and it is left
as NULL. The flexibility of these slots allows us to represent any type of
numeric forecast.
The forecast and realized slots take numeric vectors. In the forecast slot
we see the forecast that was made at each origin time and in the realized
slot we see the true value that was realized at each future time. The realized
values may not exist yet, so this slot may be partially populated or not populated
at all. If the realized slot is set to NULL then it will be
populated with a vector of NA values.
The Forecast class strikes a balance between simplicity and rigor. It is
simple enough to store any numeric forecast, but it is rigorous enough to create
a useful data structure. For example, we can quickly calculate a number of
forecast accuracy metrics for the Forecast object we created above using only
one argument.
mse(my_forecast)
rmse(my_forecast)
mae(my_forecast)
mape(my_forecast)
R2(my_forecast)
Because the forecast and realized slots must be numeric vectors, and all
slots must be of the same length, we can calculate forecast accuracy metrics
without having to worry about input validation or coercing multiple vectors to
the correct format. The forecast accuracy metrics available in the lmForc package
are calculated as follows where:
$$
n = \text{forecast vector length}\
\quad Y_i = \text{realized values}\
\hat{Y_i} = \text{forecast values}
$$
MSE is calculated as:
$$
MSE = \frac{1}{n} \sum_{i=1}^{n}{(Y_i - \hat{Y_i})^2}
$$
RMSE is calculated as:
$$ RMSE = \sqrt{MSE} $$
MAE is calculated as:
$$ MAE = \frac{1}{n} \sum_{i=1}^{n}{|\hat{Y_i} - Y_i|} $$
MAPE is calculated as:
$$ MAPE = \frac{1}{n} \sum_{i=1}^{n}\frac{|Y_i - \hat{Y_i}|}{{Y_i}} $$
R2 is calculated as:
$$ R^2 = cor(\hat{Y_i}, \ Y_i)^2 $$
Note that these equations require two inputs, but because both inputs are
already stored in the Forecast object we only need to pass one argument to the
mse() and rmse() functions. Calculating forecast accuracy is a simple use
case of the Forecast class. More complex use cases exist in the lmForc
package, where many of the functions require inputs to be of the Forecast
class. When weighting multiple forecasts or testing a linear model that is
conditional on another forecast, the consistent structure of the class results
in simple functions that execute correctly, no matter the type of forecast
passed to the function. Furthermore, all lmForc functions return objects of
the Forecast class which creates synergy between functions. One
can take two linear models, test their performance out-of-sample with the
oos_realized_forc() function which returns Forecast objects, and then pass
these two objects to the performance_weighted_forc() function to find the weighted
out-of-sample performance of both models.
One fear may be that the novel Forecast class will not play well with
functions and packages that already exist in the R language. The lmForc
package provides methods for accessing all of the vectors stored in a Forecast
object as well as the forc2df() function which returns one or multiple
Forecast objects as a data frame. With these methods, one can easily pass the
data in a Forecast object to other functions.
forc2df(my_forecast)
origin(my_forecast)
future(my_forecast)
forc(my_forecast)
realized(my_forecast)
Example Dataset
Examples throughout the rest of the vignette will use a stylized dataset with a
date column of ten quarterly dates, a dependent variable y, and two
independent variables x1 and x2. Equations are also written in terms of the
variables y, x1, and x2.
date <- as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31",
"2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31",
"2012-03-31", "2012-06-30"))
y <- c(1.09, 1.71, 1.09, 2.46, 1.78, 1.35, 2.89, 2.11, 2.97, 0.99)
x1 <- c(4.22, 3.86, 4.27, 5.60, 5.11, 4.31, 4.92, 5.80, 6.30, 4.17)
x2 <- c(10.03, 10.49, 10.85, 10.47, 9.09, 10.91, 8.68, 9.91, 7.87, 6.63)
data <- data.frame(date, y, x1, x2)
head(data)
is_forc
The is_forc() function produces an in-sample forecast based on a linear model.
The function takes a linear model call and an optional vector of time data
associated with the linear model as inputs. The linear model is estimated once over the
entire sample period and the coefficients are multiplied by the realized values
in each period of the sample. This is functionally identical to the predict()
function in the stats package, but it returns a Forecast object instead of a
numeric vector.
For all observations i in the sample, coefficients are estimated as:
$$
Y_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i + \epsilon_i
\qquad \text{for all } i
$$
And forecasts are estimated as:
$$
forecast_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i \qquad
$$
is_forc(
lm_call = lm(y ~ x1 + x2, data),
time_vec = data$date
)
Note that because we are creating an in-sample forecast, h_ahead is set to 0
and the origin time equals the future time. This test evaluates how well a
linear forecast model fits the historical data.
oos_realized_forc
The oos_realized_forc() function produces an h period ahead out-of-sample
forecast that is conditioned on realized values. The function takes a linear
model call, an integer number of periods ahead to forecast, a period to end the
initial coefficient estimation and begin forecasting, an optional vector of
time data associated with the linear model, and an optional integer number of
past periods to estimate the linear model over. The linear model is originally
estimated with data up to estimation_end minus the number of periods specified
in the estimation_window argument. For instance, if the linear model is being
estimated on quarterly data and the estimation_window is set to 20L, coefficients
will be estimated using five years of data up to estimation_end. If
estimation_window is set to NULL then the linear model is estimated with all
available data up to estimation_end. Coefficients are multiplied by realized
values of the covariates h_ahead periods ahead to create an h_ahead period
ahead forecast. This process is iteratively repeated for each period after
estimation_end with coefficients updating in each period as more information
would have become available to the forecaster. In each period, coefficients are
updated based on all available information if estimation_window is set to
NULL, or a rolling window of past periods if estimation_window is set to an
integer value.
In the sample i, for each period p greater than or equal to
estimation_end, coefficients are updated as:
$$
Y_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i + \epsilon_i
\qquad \text{for all} \quad p-w \leq i \leq \text{p}
$$
Where w is the estimation_window. h_ahead h forecasts are estimated as:
$$
forecast_{p+h} = \beta_0 + \beta_1 X1_{p+h} + \beta_2 X2_{p+h} \qquad
$$
oos_realized_forc(
lm_call = lm(y ~ x1 + x2, data),
h_ahead = 2L,
estimation_end = as.Date("2011-03-31"),
time_vec = data$date,
estimation_window = NULL,
return_betas = FALSE
)
Note that the oos_realized_forc function returns an out-of-sample forecast
that conditions on realized values that would not have been available to the
forecaster at the forecast origin. This test evaluates the performance of a
linear forecast model had it been conditioned on perfect information.
oos_lag_forc
The oos_lag_forc() function produces an h period ahead out-of-sample
forecast conditioned on present period values. The function takes a linear model
call, an integer number of periods ahead to forecast, a period to end the
initial coefficient estimation and begin forecasting, an optional vector of
time data associated with the linear model, and an optional integer number of
past periods to estimate the linear model over. The linear model data is lagged by
h_ahead periods and the linear model is re-estimated with data up to
estimation_end minus the number of periods specified in the estimation_window
argument to create a lagged linear model. If estimation_window is left NULL
then the linear model is estimated with all available lagged data up to
estimation_end. Coefficients are multiplied by present period realized values
of the covariates to create a forecast for h_ahead periods ahead. This process
is iteratively repeated for each period after estimation_end with coefficients
updating in each period as more information would have become available to the
forecaster. In each period, coefficients are updated based on all available
information if estimation_window is set to NULL, or a rolling window of past
periods if estimation_window is set to an integer value.
In the sample i, for each period p greater than or equal to
estimation_end, coefficients are updated as:
$$
Y_i = \beta_0 + \beta_1 X1_{i-h} + \beta_2 X2_{i-h} + \epsilon_i
\qquad \text{for all} \quad p-w \leq i \leq \text{p}
$$
Where w is the estimation_window and h is h_ahead. h_ahead forecasts are estimated as:
$$
forecast_{p+h} = \beta_0 + \beta_1 X1_{p} + \beta_2 X2_{p} \qquad
$$
oos_lag_forc(
lm_call = lm(y ~ x1 + x2, data),
h_ahead = 2L,
estimation_end = as.Date("2011-03-31"),
time_vec = data$date,
estimation_window = NULL,
return_betas = FALSE
)
This test evaluates the performance of a lagged linear model had it been
conditioned on present values that would have been available to the
forecaster at the forecast origin. This is in contrast to conditioning on
realized values or a forecast of the covariates.
oos_vintage_forc
The oos_vintage_forc() function produces an out-of-sample forecast conditioned
on h period ahead forecasts of the linear model covariates. The function takes
a linear model call, a vector of time data associated with the linear model,
a vintage forecast for each covariate in the linear model, and an optional
integer number of past periods to estimate the linear model over. For each period
in the vintage forecasts, coefficients are updated based on information that would
have been available to the forecaster at the forecast origin. Coefficients are
estimated over information from the last estimation_window number of periods.
If estimation_window is left NULL then coefficients are estimated over all
of the information that would have been available to the forecaster. Coefficients
are then multiplied by vintage forecast values to produce a replication of real
time forecasts.
In the sample i, for each period p in the vintage forecasts VF1 and VF2,
coefficients are updated as:
$$
Y_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i + \epsilon_i
\qquad \text{for all} \quad p-w \leq i \leq \text{p}
$$
And h_ahead h forecasts are estimated as:
$$
forecast_{p+h} = \beta_0 + \beta_1 VF1_p + \beta_2 VF2_p \qquad
$$
We introduce stylized vintage forecasts of X1 and X2 to demonstrate the
oos_vintage_forc() function. Using four quarter ahead forecasts of the
covariates X1 and X2, we create an out-of-sample forecast based on the
coefficients and covariate forecasts that the forecaster would have used in
each period.
x1_forecast_vintage <- Forecast(
origin = as.Date(c("2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30")),
future = as.Date(c("2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30")),
forecast = c(6.30, 4.17, 5.30, 4.84),
realized = c(4.92, 5.80, 6.30, 4.17),
h_ahead = 4L
)
x2_forecast_vintage <- Forecast(
origin = as.Date(c("2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30")),
future = as.Date(c("2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30")),
forecast = c(7.32, 6.88, 6.82, 6.95),
realized = c(8.68, 9.91, 7.87, 6.63),
h_ahead = 4L
)
oos_vintage_forc(
lm_call = lm(y ~ x1 + x2, data),
time_vec = data$date,
x1_forecast_vintage, x2_forecast_vintage,
estimation_window = NULL,
return_betas = FALSE
)
This test replicates the forecasts that a linear model conditional on forecasts
of covariates would have produced in real-time. Here we see the strength of the
Forecast class. Because the vintage forecasts of X1 and X2 share the same data
structure, we can calculate a forecast that is conditional on these objects
without fearing inconsistency across inputs.
conditional_forc
The conditional_forc() function produces a forecast conditioned on forecasts
of the linear model covariates. The function takes a linear model call, a vector
of time data associated with the linear model, and a forecast for each covariate
in the linear model. The linear model is estimated once over the entire sample
period and the coefficients are multiplied by the forecasts of each covariate.
For all observations i in the sample, coefficients are estimated as:
$$
Y_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i + \epsilon_i
\qquad \text{for all } i
$$
And for all periods p in the covariate forecasts F1 and F2, forecasts are estimated as:
$$
forecast_{p+h} = \beta_0 + \beta_1 F1_p + \beta_2 F2_p \qquad
$$
The difference between conditional_forc() and oos_vintage_forc() is that in
the conditional_forc() function coefficients are only estimated once over all
observations. Coefficients do not update based on what information would have been
available to the forecaster at any given point in time. We introduce stylized
forecasts of X1 and X2 to demonstrate the conditional_forc() function. Because
in this example we are making a conditional forecast for the future instead of testing past forecasts, we can condition on a horizon of forecasts. This is in contrast
to the oos_vintage_forc() example where we test the performance of four quarter
ahead vintage forecasts.
x1_forecast <- Forecast(
origin = as.Date(c("2012-06-30", "2012-06-30", "2012-06-30", "2012-06-30")),
future = as.Date(c("2012-09-30", "2012-12-31", "2013-03-31", "2013-06-30")),
forecast = c(4.14, 4.04, 4.97, 5.12),
realized = NULL,
h_ahead = NULL
)
x2_forecast <- Forecast(
origin = as.Date(c("2012-06-30", "2012-06-30", "2012-06-30", "2012-06-30")),
future = as.Date(c("2012-09-30", "2012-12-31", "2013-03-31", "2013-06-30")),
forecast = c(6.01, 6.05, 6.55, 7.45),
realized = NULL,
h_ahead = NULL
)
conditional_forc(
lm_call = lm(y ~ x1 + x2, data),
time_vec = data$date,
x1_forecast, x2_forecast
)
This function is used to create a forecast for the present period or replicate
a forecast made at a specific period in the past. Note that because we are
forecasting into the future, realized is NULL. Also, because we are
forecasting a horizon of dates, h_ahead is NULL.
historical_average_forc
The historical_average_forc() function produces an h period ahead forecast based
on the historical average of the series that is being forecasted. The function takes
an average function, a vector of realized values, an integer number of periods ahead
to forecast, a period to end the initial average estimation and begin forecasting,
an optional vector of time data associated with the realized values, and an optional
integer number of past periods to estimate the average over. The historical average
is originally calculated with realized values up to estimation_end minus the number
of periods specified in estimation_window. If estimation_window is left NULL
then the historical average is calculated with all available realized values up to
estimation_end. In each period the historical average is set as the h_ahead period
ahead forecast. This process is iteratively repeated for each period after estimation_end
with the historical average updating in each period as more information would have become
available to the forecaster.
If avg_function is set to mean, in the sample i, for each period p greater than or
equal to estimation_end, h_ahead h forecasts are calculated as:
$$
forecast_{p+h} = \frac{1}{p-w} \sum_{i=p-w}^{p}{Y_i} \qquad
$$
Where Y is the series being forecasted and w is the estimation_window.
historical_average_forc(
avg_function = "mean",
realized_vec = data$y,
h_ahead = 2L,
estimation_end = as.Date("2011-03-31"),
time_vec = data$date,
estimation_window = 4L
)
historical_average_forc() returns a historical average forecast where the h_ahead
period ahead forecast is simply the historical average or rolling window average of the
series being forecasted. This replicates the historical average forecast that would
have been produced in real-time and can serve as a benchmark for other forecasting models.
random_walk_forc
The random_walk_forc() function produces an h period ahead forecast based
on the last realized value in the series that is being forecasted. The function
takes a vector of realized values, an integer number of periods ahead to forecast,
and an optional vector of time data associated with the realized values. In each
period, the current period value of the realized_vec series is set as the
h_ahead period ahead forecast.
In the sample i, for each period p, h_ahead h forecasts are calculated as:
$$
forecast_{p+h} = Y_p
$$
Where Y is the series being forecasted.
random_walk_forc(
realized_vec = data$y,
h_ahead = 6L,
time_vec = data$date
)
random_walk_forc() returns a random walk forecast where the h_ahead
period ahead forecast is simply the present value of the series being forecasted.
This replicates the random walk forecast that a forecaster would have produced
in real-time and can serve as a benchmark for other forecasting models.
autoreg_forc
The autoreg_forc() function produces an h period ahead forecast based
on an autoregressive (AR) model. The function takes a vector of realized values,
an integer number of periods ahead to forecast, an integer number of lags to include
in the autoregressive model, a period to end the initial model estimation and begin
forecasting, an optional vector of time data associated with the realized values,
and an optional integer number of past periods to estimate the autoregressive model
over. An AR(ar_lags) autoregressive model is originally estimated with realized values up to
estimation_end minus the number of periods specified in estimation_window. If
estimation_window is left NULL then the autoregressive model is estimated with
all realized values up to estimation_end. The AR(ar_lags) model is estimated by
regressing the vector of realized values on vectors of the same realized values that
have been lagged by one to ar_lags steps. The AR coefficients of this model are
multiplied by lagged values and the present period realized value to create a one
period ahead forecast. If h_ahead is greater than one, the one period ahead forecasting
process is iteratively repeated so that the two period ahead forecast conditions on
the one period ahead forecasted value. This process of rolling one period ahead forecasts
forward continues until an h_ahead forecast is obtained. The h_ahead forecasting process
is repeated for each period after estimation_end with AR model coefficients updating as
more information would have become available to the forecaster. In each period, coefficients
are updated based on all available realized values if estimation_window is set to NULL,
or a rolling window of past periods if estimation_window is set to an integer value.
In the sample i with ar_lags set to two and h_ahead set to two. For each period p
greater than or equal to estimation_end, coefficients are calculated as:
$$
Y_i = \beta_0 + \beta_1 Y_{i-1} + \beta_2 Y_{i-2}
\qquad \text{for all} \quad p-w \leq i \leq \text{p}
$$
Where Y is the series being forecasted and w is the estimation_window.
h_ahead two step ahead forecasts are estimated as:
$$
Y_{p+1} = \beta_0 + \beta_1 Y_p + \beta_2 Y_{p-1} \
forecast_{p+h} = Y_{p+2} = \beta_0 + \beta_1 Y_{p+1} + \beta_2 Y_{p}
$$
autoreg_forc(
realized_vec = data$y,
h_ahead = 2L,
ar_lags = 2L,
estimation_end = as.Date("2011-06-30"),
time_vec = data$date,
estimation_window = NULL,
return_betas = FALSE
)
autoreg_forc() returns an autoressive forecast based on information that would
have been available at the forecast origin. This function replicates the forecasts
than an AR model would have produced in real-time and can serve as a
benchmark for other forecasting models.
performance_weighted_forc
The performance_weighted_forc() function produces a weighted average of multiple
forecasts based on the recent performance of each forecast. The function takes
two or more forecasts of the Forecast class, an evaluation window, and an
error function. For each forecast period, the error function is used to calculate
forecast accuracy over the past eval_window number of periods. The forecast
accuracy of each forecast is then used to weight forecasts based on a weighting function.
In each period, weights are calculated and used to create a weighted average forecast.
We use a stylized example in which we create a weighted forecast of two forecasts:
Y1 and Y2.
For all periods p in the k number of forecasts Y, weights W are calculated over the eval_window e as:
$$
W_k = \frac{1/MSE(Y_{ki})}{1/\sum_{k=1}^{k}MSE(Y_{ki})} \qquad \text{where}
\quad i = p-e \leq i \leq p
$$
Forecasts are estimated as:
$$
forecast_p = Y1_pW_1 + Y2_pW_2
$$
y1_forecast <- Forecast(
origin = as.Date(c("2009-03-31", "2009-06-30", "2009-09-30", "2009-12-31",
"2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31",
"2011-03-31", "2011-06-30")),
future = as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31",
"2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31",
"2012-03-31", "2012-06-30")),
forecast = c(1.33, 1.36, 1.38, 1.68, 1.60, 1.55, 1.32, 1.22, 1.08, 0.88),
realized = c(1.09, 1.71, 1.09, 2.46, 1.78, 1.35, 2.89, 2.11, 2.97, 0.99),
h_ahead = 4L
)
y2_forecast <- Forecast(
origin = as.Date(c("2009-03-31", "2009-06-30", "2009-09-30", "2009-12-31",
"2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31",
"2011-03-31", "2011-06-30")),
future = as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31",
"2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31",
"2012-03-31", "2012-06-30")),
forecast = c(0.70, 0.88, 1.03, 1.05, 1.01, 0.82, 0.95, 1.09, 1.07, 1.06),
realized = c(1.09, 1.71, 1.09, 2.46, 1.78, 1.35, 2.89, 2.11, 2.97, 0.99),
h_ahead = 4L
)
performance_weighted_forc(
y1_forecast, y2_forecast,
eval_window = 2L,
errors = "mse",
return_weights = FALSE
)
The performance_weighted_forc() function returns a weighted forecast of the Y1 and Y2
forecasts based on performance in recent periods. The weights used in each
period can be returned to the Global Environment by setting return_weights to
TRUE. Note that although we were only weighting performance over the past two
periods, we have five NA forecasts. This reflects the lmForc philosophy of
replicating what it would be like to forecast in real-time. If a forecaster was
making a forecast at 2010-06-30, they would only have access to realized values
up to 2010-06-30, in this case the first two rows. This is why a weighted forecast
with an eval_window of two can only be computed once the forecast origin becomes
2010-06-30 and the forecaster has access to two realized values. This function
can be used to compute a weighted forecast for the present period or to test how
a weighted forecast would have performed historically. The weighted forecasts are
based on information that would have been available to the forecaster at the
forecast origin.
states_weighted_forc
The states_weighted_forc() function produces a weighted average of multiple
forecasts based on how each forecast performed during the past state of the world that is
most similar to the current state of the world. The function takes two or more
forecasts, a data frame, matrix, or array of matching variables, an optional
vector of time data associated with the matching variables, a matching window
size, a matching function, and an error function. The first step of the weighted
forecast process is to match the current state of the world to a similar past
state of the world. For each forecast period, the matching_vars are standardized
and the past matching_window periods of the matching variables are considered
as the current state of the world. This current state of the world is compared to
all past matching_window size periods of the matching variables. The current
state is matched to the past state that minimizes the user selected matching function.
For example, if matching is set to euclidean then the matched past state is
the past state which has the minimum euclidean distance to the current state of
the world. The objective is to select a past period that is similar to the current
state of the world as given by the matching variables. Once a past state has been
matched, the accuracy of each forecast is calculated over the periods of the past
state according to the user selected error function. Forecast weights are then computed
based on forecast accuracy during the past state. The objective is to give more weight to
the forecasts that perform better in conditions that reflect the current state
of the world. The forecast weights are then used to create a weighted forecast for
the current period.
We use a stylized example with one matching variable x as well as two forecasts
Y1 and Y2.
The matching variable x is first standardized using the function:
$$
x_i = \frac{x_i - mean(x)}{sd(x)}
$$
For all periods p, the current state of the world c and past states of the
world p are calculated as:
$$
c_i = x_i \qquad \text{where} \quad p-w \leq i \leq p \
\qquad \qquad \qquad \qquad p_{id} = x_i \qquad \text{where} \quad d-w \leq i \leq d \quad \text{for all} \quad d \lt p-w
$$
Where w is the matching_window size and d are all periods that occur before
the beginning of the current state.
All possible past states are passed through the matching function and the matched
past state is selected as the past state that minimizes the matching function.
If matching is set to euclidean, the matched past state p is the past
state that satisfies the following:
$$
p = \min \sqrt{\sum_{i=1}^{n}(c_i - p_{id})^2} \qquad \text{for all past states } d
$$
Forecast accuracy and forecast weights are computed over observations from the
matched past state p. If errors is set to mse then the forecast weights
W for each forecast k are calculated as.
$$
W_k = \frac{1/MSE(Y_{kp})}{1/\sum_{k=1}^{k}MSE(Y_{kp})}
$$
The current period forecast is then calculated as:
$$
forecast_p = Y1_pW_1 + Y2_pW_2
$$
date <- as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31",
"2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31",
"2012-03-31", "2012-06-30"))
future <- as.Date(c("2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31",
"2012-03-31", "2012-06-30", "2012-09-30", "2012-12-31",
"2013-03-31", "2013-06-30"))
y <- c(1.09, 1.71, 1.09, 2.46, 1.78, 1.35, 2.89, 2.11, 2.97, 0.99)
x1 <- c(4.22, 3.86, 4.27, 5.60, 5.11, 4.31, 4.92, 5.80, 6.30, 4.17)
x2 <- c(10.03, 10.49, 10.85, 10.47, 9.09, 10.91, 8.68, 9.91, 7.87, 6.63)
data <- data.frame(date, y, x1, x2)
matching_vars <- data[, c("x1", "x2")]
y1_forecast <- Forecast(
origin = date,
future = future,
forecast = c(1.33, 1.36, 1.38, 1.68, 1.60, 1.55, 1.32, 1.22, 1.08, 0.88),
realized = c(1.78, 1.35, 2.89, 2.11, 2.97, 0.99, 1.31, 1.41, 1.02, 1.05),
h_ahead = 4L
)
y2_forecast <- Forecast(
origin = date,
future = future,
forecast = c(0.70, 0.88, 1.03, 1.05, 1.01, 0.82, 0.95, 1.09, 1.07, 1.06),
realized = c(1.78, 1.35, 2.89, 2.11, 2.97, 0.99, 1.31, 1.41, 1.02, 1.05),
h_ahead = 4L
)
states_weighted_forc(
y1_forecast, y2_forecast,
matching_vars = matching_vars,
time_vec = data$date,
matching_window = 2L,
matching = "euclidean",
errors = "mse",
return_weights = FALSE
)
The states_weighted_forc() function returns a weighted forecast of the Y1 and
Y2 forecasts based on how these forecasts performed in past states of the world
that resemble the current state of the world. The weights used in each period
and a list of the matched states can be returned to the Global Environment by
setting return_weights to TRUE. This function can be used to compute a
states weighted forecast for the present period or to test how a states weighted
forecast would have performed historically. The states weighted forecasts are
based on information that would have been available to the forecaster
at the forecast origin.
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
library(lmForc)
{width=100%}
Sections
[Forecast Class]
[is_forc]
[oos_realized_forc]
[oos_lag_forc]
[oos_vintage_forc]
[conditional_forc]
[historical_average_forc]
[random_walk_forc]
[autoreg_forc]
[performance_weighted_forc]
[states_weighted_forc]
lmForc Motivation
Linear forecasting models have two main features: simplicity and interpretability. These features allow the performance of linear forecasting models to be tested in multiple ways, both in-sample and out-out-sample. The lmForc package brings these performance tests to the R language with a philosophy that compliments the features of linear models: One, the syntax for creating linear models is simple and the syntax for testing them should be equally simple. Two, performance tests should be realistic and when possible replicate what it would have been like to forecast in real-time.
Forecast Class
At the heart of the lmForc package is the Forecast class. Base R does not
provide a good format for working with forecasts, so lmForc addresses this by
introducing a new class for storing forecasts that is simple and rigorous. The
Forecast class is paramount to the lmForc philosophy of simple syntax and
realistic tests. Forecast is an S4 class that contains equal length vectors
with the following data:
origin\ \ \ The time when the forecast was made.future\ \ \ The time that is being forecasted.forecastThe forecast itself.realized\ If available, the realized value at the time being forecasted.
The Forecast class also includes an additional length-one slot h_ahead for
representing how many periods ahead are being forecasted. This slot is optional,
but becomes useful for documentation and performing out-of-sample forecast
tests.
We demonstrate the Forecast class by constructing a simple Forecast object.
This forecast contains four observations at the quarterly frequency.
my_forecast <- Forecast( origin = as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31")), future = as.Date(c("2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31")), forecast = c(4.21, 4.27, 5.32, 5.11), realized = c(4.40, 4.45, 4.87, 4.77), h_ahead = 4L )
Note that we have chosen Date objects at the quarterly frequency for our
origin and future slots. This forecast is for four quarters ahead, so we
fill in the h_ahead slot with the integer four.
The origin, future, and h_ahead slots can be filled with any values. This
is where the general nature of the Forecast class shines. In the origin and
future slots we could use dates at a daily or yearly frequency, the POSIXct
class to include minutes and seconds, or integers to represent discrete periods.
We can also store different types of forecasts. In the example above, we have
forecasts made at different origin times for a constant four quarter ahead
forecast horizon. We could also store a forecast made at a single origin time
for a horizon of future times by setting all of the origin values to one
time and using future to represent the horizon of times that we are
forecasting. In this case the h_ahead slot becomes irrelevant and it is left
as NULL. The flexibility of these slots allows us to represent any type of
numeric forecast.
The forecast and realized slots take numeric vectors. In the forecast slot
we see the forecast that was made at each origin time and in the realized
slot we see the true value that was realized at each future time. The realized
values may not exist yet, so this slot may be partially populated or not populated
at all. If the realized slot is set to NULL then it will be
populated with a vector of NA values.
The Forecast class strikes a balance between simplicity and rigor. It is
simple enough to store any numeric forecast, but it is rigorous enough to create
a useful data structure. For example, we can quickly calculate a number of
forecast accuracy metrics for the Forecast object we created above using only
one argument.
mse(my_forecast) rmse(my_forecast) mae(my_forecast) mape(my_forecast) R2(my_forecast)
Because the forecast and realized slots must be numeric vectors, and all
slots must be of the same length, we can calculate forecast accuracy metrics
without having to worry about input validation or coercing multiple vectors to
the correct format. The forecast accuracy metrics available in the lmForc package
are calculated as follows where:
$$
n = \text{forecast vector length}\
\quad Y_i = \text{realized values}\
\hat{Y_i} = \text{forecast values}
$$
MSE is calculated as: $$ MSE = \frac{1}{n} \sum_{i=1}^{n}{(Y_i - \hat{Y_i})^2} $$
RMSE is calculated as: $$ RMSE = \sqrt{MSE} $$ MAE is calculated as: $$ MAE = \frac{1}{n} \sum_{i=1}^{n}{|\hat{Y_i} - Y_i|} $$ MAPE is calculated as: $$ MAPE = \frac{1}{n} \sum_{i=1}^{n}\frac{|Y_i - \hat{Y_i}|}{{Y_i}} $$ R2 is calculated as: $$ R^2 = cor(\hat{Y_i}, \ Y_i)^2 $$
Note that these equations require two inputs, but because both inputs are
already stored in the Forecast object we only need to pass one argument to the
mse() and rmse() functions. Calculating forecast accuracy is a simple use
case of the Forecast class. More complex use cases exist in the lmForc
package, where many of the functions require inputs to be of the Forecast
class. When weighting multiple forecasts or testing a linear model that is
conditional on another forecast, the consistent structure of the class results
in simple functions that execute correctly, no matter the type of forecast
passed to the function. Furthermore, all lmForc functions return objects of
the Forecast class which creates synergy between functions. One
can take two linear models, test their performance out-of-sample with the
oos_realized_forc() function which returns Forecast objects, and then pass
these two objects to the performance_weighted_forc() function to find the weighted
out-of-sample performance of both models.
One fear may be that the novel Forecast class will not play well with
functions and packages that already exist in the R language. The lmForc
package provides methods for accessing all of the vectors stored in a Forecast
object as well as the forc2df() function which returns one or multiple
Forecast objects as a data frame. With these methods, one can easily pass the
data in a Forecast object to other functions.
forc2df(my_forecast) origin(my_forecast) future(my_forecast) forc(my_forecast) realized(my_forecast)
Example Dataset
Examples throughout the rest of the vignette will use a stylized dataset with a
date column of ten quarterly dates, a dependent variable y, and two
independent variables x1 and x2. Equations are also written in terms of the
variables y, x1, and x2.
date <- as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30")) y <- c(1.09, 1.71, 1.09, 2.46, 1.78, 1.35, 2.89, 2.11, 2.97, 0.99) x1 <- c(4.22, 3.86, 4.27, 5.60, 5.11, 4.31, 4.92, 5.80, 6.30, 4.17) x2 <- c(10.03, 10.49, 10.85, 10.47, 9.09, 10.91, 8.68, 9.91, 7.87, 6.63) data <- data.frame(date, y, x1, x2) head(data)
is_forc
The is_forc() function produces an in-sample forecast based on a linear model.
The function takes a linear model call and an optional vector of time data
associated with the linear model as inputs. The linear model is estimated once over the
entire sample period and the coefficients are multiplied by the realized values
in each period of the sample. This is functionally identical to the predict()
function in the stats package, but it returns a Forecast object instead of a
numeric vector.
For all observations i in the sample, coefficients are estimated as:
$$ Y_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i + \epsilon_i \qquad \text{for all } i $$ And forecasts are estimated as:
$$ forecast_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i \qquad $$
is_forc( lm_call = lm(y ~ x1 + x2, data), time_vec = data$date )
Note that because we are creating an in-sample forecast, h_ahead is set to 0
and the origin time equals the future time. This test evaluates how well a
linear forecast model fits the historical data.
oos_realized_forc
The oos_realized_forc() function produces an h period ahead out-of-sample
forecast that is conditioned on realized values. The function takes a linear
model call, an integer number of periods ahead to forecast, a period to end the
initial coefficient estimation and begin forecasting, an optional vector of
time data associated with the linear model, and an optional integer number of
past periods to estimate the linear model over. The linear model is originally
estimated with data up to estimation_end minus the number of periods specified
in the estimation_window argument. For instance, if the linear model is being
estimated on quarterly data and the estimation_window is set to 20L, coefficients
will be estimated using five years of data up to estimation_end. If
estimation_window is set to NULL then the linear model is estimated with all
available data up to estimation_end. Coefficients are multiplied by realized
values of the covariates h_ahead periods ahead to create an h_ahead period
ahead forecast. This process is iteratively repeated for each period after
estimation_end with coefficients updating in each period as more information
would have become available to the forecaster. In each period, coefficients are
updated based on all available information if estimation_window is set to
NULL, or a rolling window of past periods if estimation_window is set to an
integer value.
In the sample i, for each period p greater than or equal to
estimation_end, coefficients are updated as:
$$
Y_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i + \epsilon_i
\qquad \text{for all} \quad p-w \leq i \leq \text{p}
$$
Where w is the estimation_window. h_ahead h forecasts are estimated as:
$$ forecast_{p+h} = \beta_0 + \beta_1 X1_{p+h} + \beta_2 X2_{p+h} \qquad $$
oos_realized_forc( lm_call = lm(y ~ x1 + x2, data), h_ahead = 2L, estimation_end = as.Date("2011-03-31"), time_vec = data$date, estimation_window = NULL, return_betas = FALSE )
Note that the oos_realized_forc function returns an out-of-sample forecast
that conditions on realized values that would not have been available to the
forecaster at the forecast origin. This test evaluates the performance of a
linear forecast model had it been conditioned on perfect information.
oos_lag_forc
The oos_lag_forc() function produces an h period ahead out-of-sample
forecast conditioned on present period values. The function takes a linear model
call, an integer number of periods ahead to forecast, a period to end the
initial coefficient estimation and begin forecasting, an optional vector of
time data associated with the linear model, and an optional integer number of
past periods to estimate the linear model over. The linear model data is lagged by
h_ahead periods and the linear model is re-estimated with data up to
estimation_end minus the number of periods specified in the estimation_window
argument to create a lagged linear model. If estimation_window is left NULL
then the linear model is estimated with all available lagged data up to
estimation_end. Coefficients are multiplied by present period realized values
of the covariates to create a forecast for h_ahead periods ahead. This process
is iteratively repeated for each period after estimation_end with coefficients
updating in each period as more information would have become available to the
forecaster. In each period, coefficients are updated based on all available
information if estimation_window is set to NULL, or a rolling window of past
periods if estimation_window is set to an integer value.
In the sample i, for each period p greater than or equal to
estimation_end, coefficients are updated as:
$$
Y_i = \beta_0 + \beta_1 X1_{i-h} + \beta_2 X2_{i-h} + \epsilon_i
\qquad \text{for all} \quad p-w \leq i \leq \text{p}
$$
Where w is the estimation_window and h is h_ahead. h_ahead forecasts are estimated as:
$$ forecast_{p+h} = \beta_0 + \beta_1 X1_{p} + \beta_2 X2_{p} \qquad $$
oos_lag_forc( lm_call = lm(y ~ x1 + x2, data), h_ahead = 2L, estimation_end = as.Date("2011-03-31"), time_vec = data$date, estimation_window = NULL, return_betas = FALSE )
This test evaluates the performance of a lagged linear model had it been conditioned on present values that would have been available to the forecaster at the forecast origin. This is in contrast to conditioning on realized values or a forecast of the covariates.
oos_vintage_forc
The oos_vintage_forc() function produces an out-of-sample forecast conditioned
on h period ahead forecasts of the linear model covariates. The function takes
a linear model call, a vector of time data associated with the linear model,
a vintage forecast for each covariate in the linear model, and an optional
integer number of past periods to estimate the linear model over. For each period
in the vintage forecasts, coefficients are updated based on information that would
have been available to the forecaster at the forecast origin. Coefficients are
estimated over information from the last estimation_window number of periods.
If estimation_window is left NULL then coefficients are estimated over all
of the information that would have been available to the forecaster. Coefficients
are then multiplied by vintage forecast values to produce a replication of real
time forecasts.
In the sample i, for each period p in the vintage forecasts VF1 and VF2, coefficients are updated as:
$$ Y_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i + \epsilon_i \qquad \text{for all} \quad p-w \leq i \leq \text{p} $$
And h_ahead h forecasts are estimated as:
$$ forecast_{p+h} = \beta_0 + \beta_1 VF1_p + \beta_2 VF2_p \qquad $$
We introduce stylized vintage forecasts of X1 and X2 to demonstrate the
oos_vintage_forc() function. Using four quarter ahead forecasts of the
covariates X1 and X2, we create an out-of-sample forecast based on the
coefficients and covariate forecasts that the forecaster would have used in
each period.
x1_forecast_vintage <- Forecast( origin = as.Date(c("2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30")), future = as.Date(c("2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30")), forecast = c(6.30, 4.17, 5.30, 4.84), realized = c(4.92, 5.80, 6.30, 4.17), h_ahead = 4L ) x2_forecast_vintage <- Forecast( origin = as.Date(c("2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30")), future = as.Date(c("2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30")), forecast = c(7.32, 6.88, 6.82, 6.95), realized = c(8.68, 9.91, 7.87, 6.63), h_ahead = 4L ) oos_vintage_forc( lm_call = lm(y ~ x1 + x2, data), time_vec = data$date, x1_forecast_vintage, x2_forecast_vintage, estimation_window = NULL, return_betas = FALSE )
This test replicates the forecasts that a linear model conditional on forecasts
of covariates would have produced in real-time. Here we see the strength of the
Forecast class. Because the vintage forecasts of X1 and X2 share the same data
structure, we can calculate a forecast that is conditional on these objects
without fearing inconsistency across inputs.
conditional_forc
The conditional_forc() function produces a forecast conditioned on forecasts
of the linear model covariates. The function takes a linear model call, a vector
of time data associated with the linear model, and a forecast for each covariate
in the linear model. The linear model is estimated once over the entire sample
period and the coefficients are multiplied by the forecasts of each covariate.
For all observations i in the sample, coefficients are estimated as:
$$ Y_i = \beta_0 + \beta_1 X1_i + \beta_2 X2_i + \epsilon_i \qquad \text{for all } i $$
And for all periods p in the covariate forecasts F1 and F2, forecasts are estimated as:
$$ forecast_{p+h} = \beta_0 + \beta_1 F1_p + \beta_2 F2_p \qquad $$
The difference between conditional_forc() and oos_vintage_forc() is that in
the conditional_forc() function coefficients are only estimated once over all
observations. Coefficients do not update based on what information would have been
available to the forecaster at any given point in time. We introduce stylized
forecasts of X1 and X2 to demonstrate the conditional_forc() function. Because
in this example we are making a conditional forecast for the future instead of testing past forecasts, we can condition on a horizon of forecasts. This is in contrast
to the oos_vintage_forc() example where we test the performance of four quarter
ahead vintage forecasts.
x1_forecast <- Forecast( origin = as.Date(c("2012-06-30", "2012-06-30", "2012-06-30", "2012-06-30")), future = as.Date(c("2012-09-30", "2012-12-31", "2013-03-31", "2013-06-30")), forecast = c(4.14, 4.04, 4.97, 5.12), realized = NULL, h_ahead = NULL ) x2_forecast <- Forecast( origin = as.Date(c("2012-06-30", "2012-06-30", "2012-06-30", "2012-06-30")), future = as.Date(c("2012-09-30", "2012-12-31", "2013-03-31", "2013-06-30")), forecast = c(6.01, 6.05, 6.55, 7.45), realized = NULL, h_ahead = NULL ) conditional_forc( lm_call = lm(y ~ x1 + x2, data), time_vec = data$date, x1_forecast, x2_forecast )
This function is used to create a forecast for the present period or replicate
a forecast made at a specific period in the past. Note that because we are
forecasting into the future, realized is NULL. Also, because we are
forecasting a horizon of dates, h_ahead is NULL.
historical_average_forc
The historical_average_forc() function produces an h period ahead forecast based
on the historical average of the series that is being forecasted. The function takes
an average function, a vector of realized values, an integer number of periods ahead
to forecast, a period to end the initial average estimation and begin forecasting,
an optional vector of time data associated with the realized values, and an optional
integer number of past periods to estimate the average over. The historical average
is originally calculated with realized values up to estimation_end minus the number
of periods specified in estimation_window. If estimation_window is left NULL
then the historical average is calculated with all available realized values up to
estimation_end. In each period the historical average is set as the h_ahead period
ahead forecast. This process is iteratively repeated for each period after estimation_end
with the historical average updating in each period as more information would have become
available to the forecaster.
If avg_function is set to mean, in the sample i, for each period p greater than or
equal to estimation_end, h_ahead h forecasts are calculated as:
$$ forecast_{p+h} = \frac{1}{p-w} \sum_{i=p-w}^{p}{Y_i} \qquad $$
Where Y is the series being forecasted and w is the estimation_window.
historical_average_forc( avg_function = "mean", realized_vec = data$y, h_ahead = 2L, estimation_end = as.Date("2011-03-31"), time_vec = data$date, estimation_window = 4L )
historical_average_forc() returns a historical average forecast where the h_ahead
period ahead forecast is simply the historical average or rolling window average of the
series being forecasted. This replicates the historical average forecast that would
have been produced in real-time and can serve as a benchmark for other forecasting models.
random_walk_forc
The random_walk_forc() function produces an h period ahead forecast based
on the last realized value in the series that is being forecasted. The function
takes a vector of realized values, an integer number of periods ahead to forecast,
and an optional vector of time data associated with the realized values. In each
period, the current period value of the realized_vec series is set as the
h_ahead period ahead forecast.
In the sample i, for each period p, h_ahead h forecasts are calculated as:
$$ forecast_{p+h} = Y_p $$
Where Y is the series being forecasted.
random_walk_forc( realized_vec = data$y, h_ahead = 6L, time_vec = data$date )
random_walk_forc() returns a random walk forecast where the h_ahead
period ahead forecast is simply the present value of the series being forecasted.
This replicates the random walk forecast that a forecaster would have produced
in real-time and can serve as a benchmark for other forecasting models.
autoreg_forc
The autoreg_forc() function produces an h period ahead forecast based
on an autoregressive (AR) model. The function takes a vector of realized values,
an integer number of periods ahead to forecast, an integer number of lags to include
in the autoregressive model, a period to end the initial model estimation and begin
forecasting, an optional vector of time data associated with the realized values,
and an optional integer number of past periods to estimate the autoregressive model
over. An AR(ar_lags) autoregressive model is originally estimated with realized values up to
estimation_end minus the number of periods specified in estimation_window. If
estimation_window is left NULL then the autoregressive model is estimated with
all realized values up to estimation_end. The AR(ar_lags) model is estimated by
regressing the vector of realized values on vectors of the same realized values that
have been lagged by one to ar_lags steps. The AR coefficients of this model are
multiplied by lagged values and the present period realized value to create a one
period ahead forecast. If h_ahead is greater than one, the one period ahead forecasting
process is iteratively repeated so that the two period ahead forecast conditions on
the one period ahead forecasted value. This process of rolling one period ahead forecasts
forward continues until an h_ahead forecast is obtained. The h_ahead forecasting process
is repeated for each period after estimation_end with AR model coefficients updating as
more information would have become available to the forecaster. In each period, coefficients
are updated based on all available realized values if estimation_window is set to NULL,
or a rolling window of past periods if estimation_window is set to an integer value.
In the sample i with ar_lags set to two and h_ahead set to two. For each period p
greater than or equal to estimation_end, coefficients are calculated as:
$$
Y_i = \beta_0 + \beta_1 Y_{i-1} + \beta_2 Y_{i-2}
\qquad \text{for all} \quad p-w \leq i \leq \text{p}
$$
Where Y is the series being forecasted and w is the estimation_window.
h_ahead two step ahead forecasts are estimated as:
$$ Y_{p+1} = \beta_0 + \beta_1 Y_p + \beta_2 Y_{p-1} \ forecast_{p+h} = Y_{p+2} = \beta_0 + \beta_1 Y_{p+1} + \beta_2 Y_{p} $$
autoreg_forc( realized_vec = data$y, h_ahead = 2L, ar_lags = 2L, estimation_end = as.Date("2011-06-30"), time_vec = data$date, estimation_window = NULL, return_betas = FALSE )
autoreg_forc() returns an autoressive forecast based on information that would
have been available at the forecast origin. This function replicates the forecasts
than an AR model would have produced in real-time and can serve as a
benchmark for other forecasting models.
performance_weighted_forc
The performance_weighted_forc() function produces a weighted average of multiple
forecasts based on the recent performance of each forecast. The function takes
two or more forecasts of the Forecast class, an evaluation window, and an
error function. For each forecast period, the error function is used to calculate
forecast accuracy over the past eval_window number of periods. The forecast
accuracy of each forecast is then used to weight forecasts based on a weighting function.
In each period, weights are calculated and used to create a weighted average forecast.
We use a stylized example in which we create a weighted forecast of two forecasts: Y1 and Y2.
For all periods p in the k number of forecasts Y, weights W are calculated over the eval_window e as:
$$ W_k = \frac{1/MSE(Y_{ki})}{1/\sum_{k=1}^{k}MSE(Y_{ki})} \qquad \text{where} \quad i = p-e \leq i \leq p $$
Forecasts are estimated as:
$$ forecast_p = Y1_pW_1 + Y2_pW_2 $$
y1_forecast <- Forecast( origin = as.Date(c("2009-03-31", "2009-06-30", "2009-09-30", "2009-12-31", "2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30")), future = as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30")), forecast = c(1.33, 1.36, 1.38, 1.68, 1.60, 1.55, 1.32, 1.22, 1.08, 0.88), realized = c(1.09, 1.71, 1.09, 2.46, 1.78, 1.35, 2.89, 2.11, 2.97, 0.99), h_ahead = 4L ) y2_forecast <- Forecast( origin = as.Date(c("2009-03-31", "2009-06-30", "2009-09-30", "2009-12-31", "2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30")), future = as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30")), forecast = c(0.70, 0.88, 1.03, 1.05, 1.01, 0.82, 0.95, 1.09, 1.07, 1.06), realized = c(1.09, 1.71, 1.09, 2.46, 1.78, 1.35, 2.89, 2.11, 2.97, 0.99), h_ahead = 4L ) performance_weighted_forc( y1_forecast, y2_forecast, eval_window = 2L, errors = "mse", return_weights = FALSE )
The performance_weighted_forc() function returns a weighted forecast of the Y1 and Y2
forecasts based on performance in recent periods. The weights used in each
period can be returned to the Global Environment by setting return_weights to
TRUE. Note that although we were only weighting performance over the past two
periods, we have five NA forecasts. This reflects the lmForc philosophy of
replicating what it would be like to forecast in real-time. If a forecaster was
making a forecast at 2010-06-30, they would only have access to realized values
up to 2010-06-30, in this case the first two rows. This is why a weighted forecast
with an eval_window of two can only be computed once the forecast origin becomes
2010-06-30 and the forecaster has access to two realized values. This function
can be used to compute a weighted forecast for the present period or to test how
a weighted forecast would have performed historically. The weighted forecasts are
based on information that would have been available to the forecaster at the
forecast origin.
states_weighted_forc
The states_weighted_forc() function produces a weighted average of multiple
forecasts based on how each forecast performed during the past state of the world that is
most similar to the current state of the world. The function takes two or more
forecasts, a data frame, matrix, or array of matching variables, an optional
vector of time data associated with the matching variables, a matching window
size, a matching function, and an error function. The first step of the weighted
forecast process is to match the current state of the world to a similar past
state of the world. For each forecast period, the matching_vars are standardized
and the past matching_window periods of the matching variables are considered
as the current state of the world. This current state of the world is compared to
all past matching_window size periods of the matching variables. The current
state is matched to the past state that minimizes the user selected matching function.
For example, if matching is set to euclidean then the matched past state is
the past state which has the minimum euclidean distance to the current state of
the world. The objective is to select a past period that is similar to the current
state of the world as given by the matching variables. Once a past state has been
matched, the accuracy of each forecast is calculated over the periods of the past
state according to the user selected error function. Forecast weights are then computed
based on forecast accuracy during the past state. The objective is to give more weight to
the forecasts that perform better in conditions that reflect the current state
of the world. The forecast weights are then used to create a weighted forecast for
the current period.
We use a stylized example with one matching variable x as well as two forecasts Y1 and Y2.
The matching variable x is first standardized using the function:
$$ x_i = \frac{x_i - mean(x)}{sd(x)} $$
For all periods p, the current state of the world c and past states of the world p are calculated as:
$$ c_i = x_i \qquad \text{where} \quad p-w \leq i \leq p \ \qquad \qquad \qquad \qquad p_{id} = x_i \qquad \text{where} \quad d-w \leq i \leq d \quad \text{for all} \quad d \lt p-w $$
Where w is the matching_window size and d are all periods that occur before
the beginning of the current state.
All possible past states are passed through the matching function and the matched
past state is selected as the past state that minimizes the matching function.
If matching is set to euclidean, the matched past state p is the past
state that satisfies the following:
$$ p = \min \sqrt{\sum_{i=1}^{n}(c_i - p_{id})^2} \qquad \text{for all past states } d $$
Forecast accuracy and forecast weights are computed over observations from the
matched past state p. If errors is set to mse then the forecast weights
W for each forecast k are calculated as.
$$ W_k = \frac{1/MSE(Y_{kp})}{1/\sum_{k=1}^{k}MSE(Y_{kp})} $$
The current period forecast is then calculated as:
$$ forecast_p = Y1_pW_1 + Y2_pW_2 $$
date <- as.Date(c("2010-03-31", "2010-06-30", "2010-09-30", "2010-12-31", "2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30")) future <- as.Date(c("2011-03-31", "2011-06-30", "2011-09-30", "2011-12-31", "2012-03-31", "2012-06-30", "2012-09-30", "2012-12-31", "2013-03-31", "2013-06-30")) y <- c(1.09, 1.71, 1.09, 2.46, 1.78, 1.35, 2.89, 2.11, 2.97, 0.99) x1 <- c(4.22, 3.86, 4.27, 5.60, 5.11, 4.31, 4.92, 5.80, 6.30, 4.17) x2 <- c(10.03, 10.49, 10.85, 10.47, 9.09, 10.91, 8.68, 9.91, 7.87, 6.63) data <- data.frame(date, y, x1, x2) matching_vars <- data[, c("x1", "x2")] y1_forecast <- Forecast( origin = date, future = future, forecast = c(1.33, 1.36, 1.38, 1.68, 1.60, 1.55, 1.32, 1.22, 1.08, 0.88), realized = c(1.78, 1.35, 2.89, 2.11, 2.97, 0.99, 1.31, 1.41, 1.02, 1.05), h_ahead = 4L ) y2_forecast <- Forecast( origin = date, future = future, forecast = c(0.70, 0.88, 1.03, 1.05, 1.01, 0.82, 0.95, 1.09, 1.07, 1.06), realized = c(1.78, 1.35, 2.89, 2.11, 2.97, 0.99, 1.31, 1.41, 1.02, 1.05), h_ahead = 4L ) states_weighted_forc( y1_forecast, y2_forecast, matching_vars = matching_vars, time_vec = data$date, matching_window = 2L, matching = "euclidean", errors = "mse", return_weights = FALSE )
The states_weighted_forc() function returns a weighted forecast of the Y1 and
Y2 forecasts based on how these forecasts performed in past states of the world
that resemble the current state of the world. The weights used in each period
and a list of the matched states can be returned to the Global Environment by
setting return_weights to TRUE. This function can be used to compute a
states weighted forecast for the present period or to test how a states weighted
forecast would have performed historically. The states weighted forecasts are
based on information that would have been available to the forecaster
at the forecast origin.
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