I have added a demo file to the latest version of the fanplot package. It has lots of examples of different plotting styles to represent uncertainty in time series data. In the updated package I have added functionality to plot fan charts based on irregular time series objects from the zoo package, plus the use of alternative colour palettes from the RColorBrewer and colorspace packages. All plots are based on the th.mcmc object, the estimated posterior distributions of the volatility in daily returns from the Pound/Dollar exchange rate from 02/10/1981 to 28/6/1985. To run the demo file from your R console (ensure fanplot, zoo, tsbugs, RColorBrewer and colorspace packages are all installed beforehand);

# if you want plots in separate graphic devices
# do not run this first line...
par(mfrow = c(10,2))
# run demo
demo(sv_fan, package = "fanplot", ask = FALSE)

The demo script should output this set of plots:
If you wish, click on the image above and take a closer look in your browser. In R, you can save the PDF version of all the plots on one graphics device (which looks much better than what comes up in my R graphics device):

A couple of months ago, a paper I worked on with co-authors from the Centre of Population Change was published in Population and Environment. It summarised work we did as part of the UK Government Office for Science Foresight project on Migration and Global Environmental Change. Our aim was to build expert based forecasts of environmental immigrants to the UK. We conducted a Delphi survey of nearly 30 migration experts from academia, the civil service and non-governmental organisations to obtain estimates on the future levels of immigration to the UK in 2030 and 2060 with uncertainty. We also asked them what proportion of current and future immigration are/will be environmental migrants. The results were incorporated into a set of model averaged Bayesian time series models through prior distributions on the mean and variance terms.

The plots in the journal article got somewhat butchered during the publication process. Below is the non-butchered version for the future immigration to the UK alongside the past immigration data from the Office of National Statistics.
At first, I was a bit taken aback with this plot. A few experts thought there were going to be some very high levels of future immigration which cause the rather striking large upper tail. However, at a second glance, the central percentiles show a gentle decrease where these is only (approximately) a 30% chance of an increase in future migration from the 2010 level throughout the forecast period.

The expert based forecast for total immigration was combined with the responses to questions on the proportion of environmental migrants, to obtain an estimate on both the current level of environmental migration (which is not currently measured) and future levels:

As is the way with these things, we came across some problems in our project. The first, was with the definition of an environmental migrant, which is not completely nailed on in the migration literature. As a result the part of the uncertainty in the expert based forecasts are reflective of not only the future level but also of the measure itself. The second was with the elicitation of uncertainty. We used a Likert type scale, which caused some difficulties even during the later round of the Delphi survey. If I was to do over, then this I reckon problem could be much better addressed by getting experts to visualise their forecast fans in an interactive website, perhaps creating a shiny app with the fanplot package. Such an approach would result in smoother fans than those in the plots above, which were based on interpolations from expert answers at only two points of time in the future (2030 and 2060).

Over the next 50 years, the potential impact of environmental change on human livelihoods could be considerable, with one possible consequence being increased levels of human mobility. This paper explores how uncertainty about the level of immigration to the United Kingdom as a consequence of environmental factors elsewhere may be forecast using a methodology involving Bayesian models. The conceptual understanding of forecasting is advanced in three ways. First, the analysis is believed to be the first time that the Bayesian modelling approach has been attempted in relation to environmental mobility. Second, the paper considers the expediency of this approach by comparing the responses to a Delphi survey with conventional expectations about environmental mobility in the research literature. Finally, the values and assumptions of the expert evidence provided in the Delphi survey are interrogated to illustrate the limited set of conditions under which forecasts of environmental mobility, as set out in this paper, are likely to hold.

I have/will update this post as I expanded the fanplot package.

I managed to catch David Spiegelhalter’s Tails You Win on BBC iplayer last week. I missed it the first time round, only for my parents on my last visit home to tell me about a Statistician jumping out of a plane on TV. It was a great watch. Towards the end I spotted some fan charts used by the Bank of England to illustrate uncertainty in their forecasts, similar to this one:

They discussed how even in the tails of their GDP predictive distribution they missed the financial crisis by a long shot. This got me googling, trying and find you how they made the plots, something that (also) completely passed me by when I put together my fanplot package for R. As far as I could tell they did them in Excel, although (appropriately) I am not completely certain. There are also MATLAB files that can create fan charts. Anyhow, I thought I would have a go at replicating a Bank of England fan chart in R….

Split Normal (Two-Piece) Normal Distribution.

The Bank of England produce fan charts of forecasts for CPI and GDP in their quarterly Inflation Reports. They also provide data, in the form of mode, uncertainty and a skewness parameters of a split-normal distribution that underlie their fan charts (The Bank of England predominately refer to the equivalent, re-parametrised, two-piece normal distribution). The probability density of the split-normal distribution is given by Julio (2007) as

where represents the mode parameter, and the two standard deviations and can be derived given the overall uncertainty parameter, and skewness parameters, , as;

As no split normal distribution existed in R, I added routines for a density, distribution and quantile function, plus a random generator, to a new version (2.1) of the fanplot package. I used the formula in Julio (2007) to code each of the three functions, and checked the results against those from the fan chart MATLAB code.

Fan Chart Plots for CPI.

Once I had the qsplitnorm function working properly, producing the fan chart plot in R was pretty straight-forward. I added two data objects to the fanplot package to help readers reproduce my plots below. The first, cpi, is a time series object with past values of CPI index. The second, boe, is a data frame with historical details on the split normal parameters for CPI inflation between Q1 2004 to Q4 2013 forecasts by the Bank of England.

The 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 mode (), uncertainty () and skew () parameters:

Users can replicate past Bank of England fan charts for a particular period after creating a matrix object that contains values on the split-normal quantile function for a set of user defined probabilities. 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 the values evaluated from the qsplitnorm function in 6 rows and 13 columns, where rows represent the probabilities used in the calculation p and columns represent successive time periods.

The object cpival can then used to add a fan chart to the active R graphic device. In the code below, the area of the plot 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 the Bank of England style for plotting fan charts, the background for future values is set to 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
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 plotted, indicate that the data are a set of calculated values (as opposed to simulations) 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.

Default Fan Chart Plot.

By default, the fan function treats objects passed to the data argument as simulations from sequential distributions, rather than user-created values corresponding probabilities provided in the probs argument (as above). An alternative plot below, based on simulated data and default style settings in the fan function produces a fan chart with a greater array of coloured fans with labels and contour lines alongside selected percentiles of the future distribution. To illustrate we can simulate 10,000 values from the future split-normal distribution parameters from Q1 2013 in the boe0 data frame using the rsplitnorm function

#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 can then be added to the plot;

# truncate cpi series
cpi0 <- ts(cpi[time(cpi)<2013], start=start(cpi),
frequency=frequency(cpi) )
# past data
plot(cpi0, type = "l", lwd = 2,
xlim = c(y0 - 5, y0 + 3.25), ylim = c(-2, 7))
# add fan
fan(data = cpisim, start = y0, frequency = 4)

The fan 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 palate, providing a finer level of shading in the representation of future distributions. In addition, lines and labels are provided along each decile. The fan chart does not connect to the last observation as anchor = NULL by default.