1 Introduction
Probabilities are notoriously difficult to communicate effectively to lay audiences (Spiegelhalter et al. 2011). Fan charts provide one such method to illustrate either forecasts or past results that are based on probabilistic distributions. Using shading fan charts focus the attention of the reader on the whole distribution away from a single central estimate. Visualising the distribution can aid in communicating the degree of underlying uncertainty in probabilistic forecasts to non-specialists, that might not have been apparent in basic plots and summary statistics.
Fan charts were first introduced by the Bank of England for their inflation forecasts in February 1996 (Britton et al. 1998). Since their initial development fan charts have become a standard method to display uncertainty of future economic indicators by many central banks (Julio 2007). Their use has also spread to other fields such as climate science (McShane and Wyner 2011) and demography (Gerland et al. 2014).
Fan charts can be created using various software. Within R, the
vars package (Pfaff 2008)
has a fanchart function for forecasts of confidence regions. It is
based solely on “varpred” class objects, i.e., on the predictions of
Vector Autoregressive models fitted using other functions within the
vars package. Similarly, the
forecast package
(Automatic time series forecasting: The forecast package for R 2008) produces fan charts for forecasts based on time series
models from the “forecast” class. (Julio 2009) provides VBA code in
order to plot fan charts for quarterly GDP data in Excel. Alternatively
one could use point and click methods in Excel to build customised fan
charts based on stacked area charts.1 (Buchmann 2010) provides MATLAB
code to create fan charts for user supplied forecast distributions with
a limited amount of control for the plotted display.
In any of the fore-mentioned options users are restricted in either their ability to effectively adapt the properties of fan charts or create plots based on alternative models, values or simulated data. The aim of this article is to illustrate R code in the fanplot package to create fan charts of different styles and from a range of input data. These are demonstrated on data from sequential Monte Carlo Markov chain (MCMC) simulated distributions of parameters in a stochastic volatility model and expert based forecasts for the Consumer Price Index (CPI) of the Bank of England.
2 Fan charts for sequential simulated distributions
The fanplot package can used to display any form of sequential distributions along a plots x-axis. To illustrate, we use posterior density distributions of the estimated volatility of daily returns (\(y_t\)) from the Pound/Dollar exchange rate from 02/10/1981 to 28/6/1985. As Meyer and Yu (2002) show, posterior distributions for the volatility process can be estimated in WinBUGS by fitting the stochastic volatility model; \[y_t | \theta_t = \exp\left(\frac{1}{2}\theta_t\right)u_t \qquad u_t \sim N(0, 1) \qquad t=1,\ldots,n.\] The latent volatilities \(\theta_t\), which are unknown states in a state-space model terminology (Harvey 1990), are assumed to follow a Markovian transition over time given by the state equations: \[\theta_t | \theta_{t-1}, \mu, \phi, \tau^2 = \mu + \phi \log \sigma^2_{t-1} + v_t \qquad v_t \sim N(0, \tau^2) \qquad t=1,\ldots,n\] with \(\theta_0 \sim N(\mu, \tau^2)\).
A sample of the posterior distributions of \(\theta_t\) is contained in
the th.mcmc object of the fanplot package. It consists of (1000)
rows corresponding to MCMC simulations and (945) columns corresponding
to time points \(t\). Example code to replicate this object using
R2OpenBUGS
(Sturtz et al. 2005) is given in the help file for the th.mcmc object. It
is based on the BUGS model of Meyer and Yu (2002) replicated in the my1.R
file of the fanplot package. Time ordered simulated distributions,
such as th.mcmc, can be easily extracted from the sims.list element
of an R2OpenBUGS “bugs” object.
A fan chart of the evolution of the distribution of \(\theta_t\) in
Figure 1 can be plotted using either the fan0 or fan
function. The fan0 function, which we will first discuss, provides the
simplest representation;
library("fanplot")
fan0(data = th.mcmc) 
The plotting function calculates the values of 100 equally spaced
percentiles of each future distribution when the default
data.type = "simulations" is set. This allows 50 fans to be plotted
from the heat.colours colour palette, providing darker shadings for
the more probable percentiles. The axis limits are determined from the
data argument. By default, the y-axis limits to 85 percent of the
range of the MCMC distributions to reduce white space in the plot.
Similar plots of sequential distributions from alternative Bayesian
models can be easily plotted using the fan0 or fan functions. The
data argument accepts objects from a range of classes including “mcmc”
which is typically used to handle BUGS or JAGS results via the
read.coda or read.jags commands.
The data in th.mcmc are based on trading day observations only.
Irregular time series can be handled by passing a
zoo time series object
(Zeileis and Grothendieck 2005) to the data argument. The trading days are given in the
spvdx object of the
tsbugs package
(Abel et al. 2013).
library("zoo")
library("tsbugs")
# create irregular multiple time series object
th.mcmc2 <- zoo(th.mcmc, order.by = svpdx$date) #$
# plot
fan0(data = th.mcmc2, type = "interval", ln = c(0.5, 0.8, 0.95),
llab = TRUE, rcex = 0.6)
Basing the fan chart on a zoo time series allows the x-axis in
Figure 2 to use the corresponding trading days rather than
the observations index as in Figure 1. When argument
type = "interval" is set, the probs argument corresponds to
prediction intervals. Consequently, the default interval fan chart
comprises of three different shades, running from the darkest for the
50th prediction interval to the lightest for the 95th prediction
interval. Contour lines are are controlled by the ln argument, which
is set to NULL by default for fan0. In Figure 2, changes
in the default of ln overlays lines on the fan chart for the upper and
lower bounds of the 50th, 80th and 95th prediction intervals. A further
line is plotted along the median of \(\theta_t\) controlled by the
med.ln argument and shown when type = "interval". Labels on the
right hand side are by default added to correspond to the upper and
lower bounds of each plotted line. The text size of the labels are
controlled by the rcex argument. This is set to 0.6 to incorporate the
labels without extending the limits of the x-axis. The left labels are
added by setting llab = TRUE and take the same text properties as the
right labels. Users can customize many properties of the fan chart
shading, labels contour lines, labels and axis through a range of
arguments.
Spaghetti plots are a method of viewing data to visualize possible
values through a systems. They are commonly used on geographical data,
such as meteorological forecasts (Sanyal et al. 2010) to show possible or
realised paths, or over time, such in longitudinal data analysis
(Hedeker and Gibbons 2006). Spaghetti plots can also be used represent uncertainty
shown by a range of possible future trajectories or past estimates. For
example, using the th.mcmc2 object Figure 3 displays 20
random sets of \(\theta_t\) simulations plotted by setting the argument
style = "spaghetti";
# transparent fan with visible lines
fan0(th.mcmc2, ln = c(5, 20, 50, 80, 95), alpha = 0, ln.col = "darkorange", llab = TRUE)
# spaghetti lines
fan(th.mcmc2, style = "spaghetti", n.spag = 20, alpha = 0.3)
The initial fan chart is completely transparent from setting the
transparency argument alpha = 0. In order for the percentile lines to
be visible a non-transparent colour is used for the ln.col argument.
Lines are plotted according the user defined ln argument to provide
underlying uncertainty measures for the posterior probability
distribution. The spaghetti lines, which are semi-transparent, are based
on a random selection of simulations. They are superimposed on a fan
chart using the fan function, which operates in much the similar way
as fan0. The most important difference between the two is in the
default setting of the add argument, which controls whether to create
a new plot window for a fan chart or add it to an existing device. For
the fan function, add is set to TRUE and hence its is more
appropriately used to add a fan chart to an existing plotting device.
The fan function also adds lines and labels on select contours by
default as illustrated in the next section.
3 Bank of England fan charts
The Monetary Policy Committee (MPC) of the Bank of England produces fan charts of forecasts for Consumer Price Index (CPI) of inflation and Gross Domestic Product in their quarterly Inflation Reports. Alongside the fan charts, the Bank of England provides data, in the form of central location, uncertainty and skewness parameters of a split-normal distribution that underlie their fan charts.2 The probability density of the split-normal distribution is given by Julio (2007)3 as,
\[f(x; \mu, \sigma_1, \sigma_2) = \left\{ \begin{array}{ll} \frac{\sqrt 2}{\sqrt\pi (\sigma_1+\sigma_2)} e^{-\frac{1}{2\sigma_1^2}(x-\mu)^2} & \mbox{for } -\infty < x \leq \mu \\ \frac{\sqrt 2}{\sqrt\pi (\sigma_1+\sigma_2)} e^{-\frac{1}{2\sigma_2^2}(x-\mu)^2} & \mbox{for } \mu < x < \infty \\ \end{array}, \right.\]
where \(\mu\) represents the mode, and the two standard deviations \(\sigma_1\) and \(\sigma_2\) can be derived given the overall uncertainty parameter, \(\sigma\) and skewness parameters, \(\gamma\), as;
\[\sigma^2=\sigma^2_1(1+\gamma)=\sigma^2_2(1-\gamma).\]
Functions for the probability density, cumulative distribution, quantiles and random generation for the split-normal distribution can be found in the fanplot package.
The boe data frame provides historical details on the forecasts of the
MPC for CPI inflation between Q1 2004 to Q4 2013.
> head(boe)
time0 time mode uncertainty skew
1 2004 2004.00 1.34 0.2249 0
2 2004 2004.25 1.60 0.3149 0
3 2004 2004.50 1.60 0.3824 0
4 2004 2004.75 1.71 0.4274 0
5 2004 2005.00 1.77 0.4499 0
6 2004 2005.25 1.68 0.4761 0The first column time0 refers to the base year of forecast, the
second, time indexes future projections, whilst the remaining three
columns provide values for the corresponding projected central location
(\(\mu\)), uncertainty (\(\sigma\)) and skew (\(\gamma\)) parameters:
Bank of England style fan charts vary from quarter to quarter but follow
a similar theme throughout, which can be replicated in R using the
fanplot package. The input data given to a fan function to plot a
fan chart differs from the simulations of the previous section. Rather
than many simulations from distributions in each time period we can pass
a “matrix” object of time ordered values from the split-normal quantile
function.4 As is the case for passing simulated values to the fan
function, rows of the data object represent a set of user defined
probabilities and columns represent a set of time points. For example,
in the code below, a subset of the Bank of England future parameters of
CPI published in Q1 2013 are first selected. Then a vector of
probabilities related to the percentiles, that we ultimately would like
to plot different shaded fans for, are created. Finally, in a for
loop, the qsplitnorm function calculates the values for which the
time-specific (i) split-normal distribution will be less than or equal
to the probabilities of p.
# select relevant data
y0 <- 2013
boe0 <- subset(boe, time0 == y0)
k <- nrow(boe0)
# guess work to set percentiles the boe are plotting
p <- seq(0.05, 0.95, 0.05)
p <- c(0.01, p, 0.99)
# quantiles of split-normal distribution for each probability (row) at each future
# time point (column)
cpival <- matrix(NA, nrow = length(p), ncol = k)
for (i in 1:k)
cpival[, i] <- qsplitnorm(p, mode = boe0$mode[i],
sd = boe0$uncertainty[i],
skew = boe0$skew[i]) #$The new object cpival contains values evaluated from the qsplitnorm
function in 6 rows, for our selected probabilities used in the
calculation p, and 13 columns for successive time periods for which
the MPC provide future parameters.
The cpival object can be used to add a fan chart to an active R
graphic device. In the code below, the area of Figure 4 is
set up when plotting the past CPI data, contained in the time series
object cpi. The xlim arguments are set to ensure space on the right
hand side of the plotting area for the fan. Following as closely as
possible the Bank of England style for plotting fan charts for Q1
20135, the plotting area is set to near square, the background for
future values is a gray colour, y-axis are plotted on the right hand
side, a horizontal line are added for the CPI target and a vertical line
for the two-year ahead point.
# past data
plot(cpi, type = "l", col = "tomato", lwd = 2,
xlim = c(y0 - 5, y0 + 3), ylim = c(-2, 7),
xaxt = "n", yaxt = "n", ylab = "")
# background shading during forecast period
rect(y0 - 0.25, par("usr")[3] - 1, y0 + 3, par("usr")[4] + 1,
border = "gray90", col = "gray90")
# add fan
fan(data = cpival, data.type = "values", probs = p,
start = y0, frequency = 4, anchor = cpi[time(cpi) == y0 - 0.25],
fan.col = colorRampPalette(c("tomato", "gray90")), ln = NULL, rlab = NULL)
# boe aesthetics
axis(2, at = -2:7, las = 2, tcl = 0.5, labels = FALSE)
axis(4, at = -2:7, las = 2, tcl = 0.5)
axis(1, at = 2008:2016, tcl = 0.5)
axis(1, at = seq(2008, 2016, 0.25), labels = FALSE, tcl = 0.2)
abline(h = 2) # boe cpi target
abline(v = y0 + 1.75, lty = 2) # 2 year line
The fan chart itself is outputted from the fan function, where
arguments are set to ensure a close resemblance of the R plot to that
produced by the Bank of England. The first three arguments in the fan
function called in the above code, provide the cpival data to be
plotted, indicate that the data are a set of calculated values (as
opposed to simulations as in the previous examples) and provide the
probabilities that correspond to each row of cpival object. The next
two arguments define the start time and frequency of the data. These
operate in a similar fashion to those used when defining time series in
R with the ts function. The anchor argument is set to the value of
CPI before the start of the fan chart. This allows a join between the
value of the Q1 2013 observation and the fan chart. The fan.col
argument is set to a colour palette for shades between tomato and
gray90. The final two arguments are set to NULL to suppress the
plotting of contour lines at the boundary of each shaded fan and their
labels, as per the Bank of England style.
An alternative plot in Figure 5 is based on a regular time
series object of simulated data and some other style settings in the
fan function. These produce a fan chart with a greater array of
coloured fans with labels and contour lines alongside selected
percentiles of the future distribution. The input data is based upon
10,000 simulated values produced using the rsplitnorm function and the
future split-normal distribution parameters from Q1 2013 in the
truncated boe0 data frame;
# simulate future values
cpisim <- matrix(NA, nrow = 10000, ncol = k)
for (i in 1:k)
cpisim[, i] <- rsplitnorm(n = 10000, mode = boe0$mode[i],
sd = boe0$uncertainty[i],
skew = boe0$skew[i]) The fan chart based on the simulations in cpisim are then be added to
the truncated CPI data plot;
# truncate cpi series and plot
cpi0 <- ts(cpi[time(cpi) < 2013], start = start(cpi), frequency = frequency(cpi))
plot(cpi0, type = "l", lwd = 2, las = 1, ylab = "",
xlim = c(y0 - 5, y0 + 3.5), ylim = c(-2, 7))
# add fan
library("RColorBrewer")
fan(data = cpisim, type = "interval", probs = seq(0.01,0.99,0.01),
start = y0, frequency = 4, ln = c(50,80,95), med.ln = FALSE,
fan.col = colorRampPalette(colors = rev(brewer.pal(9, "Oranges"))))
This code shows how users can control multiple visual elements of the
fan chart, not previously illustrated. In Figure 5 the fan
is based on 50 shadings for 100 equally spaced percentiles of the future
distributions specified through the probs argument. The colour scheme
is based on the Oranges palette from the
RColorBrewer
package (Neuwirth 2014). Contour lines for the upper and lower intervals
of the 50th, 80th and 95th prediction intervals are imposed using the
ln argument. The median line, plotted by default when
type = "interval" is set, is removed using the med.ln argument.
Box plots (Tukey 1977) are commonly used as a simple descriptive
statistics to visualise data through their quartiles. The boxplot
function in R has many options including the display of multiple box
plots based on sequential distributions. However, when data is based on
a time series with multiple observations during a unit of time, such as
quarterly data, fixing the location of the plot on the x-axis can be
cumbersome. The fan function overcomes this problem when setting
style = "boxplot". In Figure 6 the simulated future CPI
data cpisim are passed to the data argument:
# plot past data
plot(cpi0, type = "l", xlim = c(y0-5, y0+3), ylim = c(-2, 7), lwd = 2)
# box plots
fan(cpisim, style = "boxplot", start = y0, frequency = 4, outline = FALSE)
The fan function allows users to easily the locate sequential
distributions of box plots on the x-axis using the start and
frequency arguments. Additional arguments in the fan function are
passed to boxplot. For example from the code above, outliers are
suppressed by setting outline = FALSE.
4 Summary
The fanplot package allows users to easily visualise uncertainty based
on either simulations from sequential distributions or values based on
pre-calculated quartiles of distribution. Interactive visualisations of
fan charts, as demonstrated in the net_elicit.R demo using the
shiny package
(RStudio and Inc. 2014), could potentially allow for an intuitive elicitation of
experts forecasts6. Data of various classes can be incorporated
including regular time series (“mts”), irregular time series (“zoo”) and
simulations from, say, MCMC via “matrix”, “data.frame” or “mcmc” type
objects. The fanplot package also has a range of options to adjust
colour shadings of fan charts, their lines and labels, and specify
whether to display prediction intervals or percentiles of
distributions.7
CRAN packages used
vars, forecast, fanplot, R2OpenBUGS, zoo, tsbugs, RColorBrewer, shiny
CRAN Task Views implied by cited packages
Econometrics, Environmetrics, Finance, GraphicalModels, MissingData, MixedModels, Spatial, TimeSeries, WebTechnologies
Note
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