% \VignetteIndexEntry{Fitting the Nelson--Siegel--Svensson model with Differential Evolution}
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% \VignetteKeyword{Differential Evolution}
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\begin{document}
{\raggedright{\LARGE Fitting the Nelson--Siegel--Svensson model\\[0.2ex]%
with Differential Evolution}}\medskip

\noindent Enrico Schumann\\
\noindent \texttt{es@enricoschumann.net}\\
\bigskip


\section{Introduction}

\noindent In this tutorial we look into fitting the
Nelson--Siegel--Svensson (NSS) model to data; for more details, please
see \citep{Gilli2011b}.  Further information can be found in
\citet{Gilli2010h} and \citet{Gilli2010i}.

We start by attaching the package.  Since we will use a stochastic
technique for optimisation, we should be running several restarts (see
\citealp[Chapter~12]{Gilli2011b}, for a discussion).  The variable
\texttt{nRuns} sets the number of restarts for the examples to come.
We set it to only two here to keep the build-time of the package
acceptable; increase it to check the stochastics of the solutions.  We
set a seed to make the computations reproducible.
<<>>=
require("NMOF")
nRuns <- 2L
set.seed(112233)
@

\section{Fitting the NS model to given zero rates}
\subsection*{The NS model}

We create a `true' yield curve \texttt{yM} with given parameters
\texttt{betaTRUE}.  The times-to-payment, measured in years, are
collected in the vector \texttt{tm}.
<<fig = TRUE, height = 3>>=
tm <- c(c(1, 3, 6, 9)/12, 1:10)
betaTRUE <- c(6, 3, 8, 1)
yM <- NS(betaTRUE, tm)
par(ps = 11, bty = "n", las = 1, tck = 0.01,
    mgp = c(3, 0.2, 0), mar = c(4, 4, 1, 1))
plot(tm, yM, xlab = "maturities in years", ylab = "yields in %")
@

The aim is to fit a smooth curve through these points. Since we have
used the model to create the points, we should be able to obtain a
perfect fit. We start with the objective function \texttt{OF}. It
takes two arguments: \texttt{param}, which is a candidate solution (a
numeric vector), and the list \texttt{data}, which holds all other
variables. It returns the maximum absolute difference between
a vector of observed  (`market') yields \texttt{yM}, and the model's
yields for parameters \texttt{param}.
<<>>=
OF <- function(param, data) {
    y <- data$model(param, data$tm)
    maxdiff <- y - data$yM
    maxdiff <- max(abs(maxdiff))
    if (is.na(maxdiff))
        maxdiff <- 1e10
    maxdiff
}
@

We have a added a crude but effective safeguard against `strange'
parameter values that lead to \texttt{NA} values: the objective
function returns a large positive value.  We minimise, and hence
parameters that produce \texttt{NA} values are marked as bad.

In this first example, we set up \texttt{data} as follows:
<<>>=
data <- list(yM = yM,
             tm = tm,
          model = NS,
             ww = 0.1,
            min = c( 0,-15,-30, 0),
            max = c(15, 30, 30,10))
@

We add a \texttt{model} (a function; in this case \texttt{NS}) that
describes the mapping from parameters to a yield curve, and vectors
\texttt{min} and \texttt{max} that we will later use as
constraints. \texttt{ww} is a penalty weight, explained below.

\texttt{OF} will take a candidate solution \texttt{param}, transform
this solution via \texttt{data\$model} into yields, and compare these
yields with \texttt{yM}, which here means to compute the maximum
absolute difference.
<<>>=
param1 <- betaTRUE         ## the solution...
OF(param1, data)           ## ...gives 0
param2 <- c(5.7, 3, 8, 2)  ## anything else
OF(param2, data)           ## ... gives a postive number
@

We can also compare the solutions in terms of yield curves.
<<fig = TRUE, height = 3>>=
par(ps = 11, bty = "n", las = 1, tck = 0.01,
    mgp = c(3, 0.2, 0), mar = c(4, 4, 1, 1))
plot(tm, yM, xlab = "maturities in years", ylab = "yields in %")
lines(tm, NS(param1, tm), col = "blue")
lines(tm, NS(param2, tm), col = "red")
legend(x = "topright",
       legend = c("true yields", "param1", "param2"),
       col = c("black", "blue", "red"),
       pch = c(1, NA, NA), lty = c(0, 1, 1))
@

We generally want to obtain parameters such that certain constraints
are met. We include these through a penalty function.
<<>>=
penalty <- function(mP, data) {
    minV <- data$min
    maxV <- data$max
    ww <- data$ww
    ## if larger than maxV, element in A is positiv
    A <- mP - as.vector(maxV)
    A <- A + abs(A)
    ## if smaller than minV, element in B is positiv
    B <- as.vector(minV) - mP
    B <- B + abs(B)
    ## beta 1 + beta2 > 0
    C <- ww*((mP[1L, ] + mP[2L, ]) - abs(mP[1L, ] + mP[2L, ]))
    A <- ww * colSums(A + B) - C
    A
}
@

We already have \texttt{data}, so let us see what the function does
to solutions that violate a constraint.  Suppose we have a population
\texttt{mP} of three solutions (the \texttt{m} in \texttt{mP} is to
remind us that we deal with a matrix).
<<>>=
param1 <- c( 6, 3, 8, -1)
param2 <- c( 6, 3, 8,  1)
param3 <- c(-1, 3, 8,  1)

mP <- cbind(param1,param2,param3)
rownames(mP) <- c("b1","b2","b3","lambda")
mP
@

The first and the third solution violate the constraints. In the
first solution, $\lambda$ is negative; in the third solution,
$\beta_1$ is negative.
<<>>=
penalty(mP,data)
@

The parameter \texttt{ww} controls how heavily we penalise.
<<>>=
data$ww <- 0.5
penalty(mP,data)
@

For valid solutions, the penalty should be zero.
<<>>=
param1 <- c( 6, 3, 8, 1)
param2 <- c( 6, 3, 8, 1)
param3 <- c( 2, 3, 8, 1)
mP <- cbind(param1, param2, param3)
rownames(mP) <- c("b1","b2","b3","lambda")
penalty(mP, data)
@
Note that \texttt{penalty} works on the complete population at once;
there is no need to loop over the solutions.

So we can run a test. We start by defining the parameters of DE. Note
in particular that we pass the penalty function, and that we set
\texttt{loopPen} to \texttt{FALSE}.

<<>>=
algo <- list(nP = 100L,   ## population size
             nG = 500L,   ## number of generations
              F = 0.50,   ## step size
             CR = 0.99,   ## prob. of crossover
            min = c( 0,-15,-30, 0),
            max = c(15, 30, 30,10),
            pen = penalty,
         repair = NULL,
         loopOF = TRUE,   ## loop over popuation? yes
        loopPen = FALSE,  ## loop over popuation? no
     loopRepair = TRUE,   ## loop over popuation? yes
       printBar = FALSE)
@

\texttt{DEopt} is then called with the objective function \texttt{OF},
the list \texttt{data}, and the list \texttt{algo}.
<<>>=
sol <- DEopt(OF = OF, algo = algo, data = data)
@

To check whether the objective function works properly, we
compare the maximum error with the returned objective function value
-- they should be the same.
<<>>=
max( abs(data$model(sol$xbest, tm) - data$model(betaTRUE, tm)) )
sol$OFvalue
@

As a benchmark, we run the function \texttt{nlminb} from the
\textsf{stats} package. This is not a fair test: \texttt{nlminb} is
not appropriate for such problems. (But then, if we found that it
performed better than DE, we would have a strong indication that
something is wrong with our implementation of DE.)  We use a random
starting value \texttt{s0}.
<<>>=
s0 <- algo$min + (algo$max - algo$min) * runif(length(algo$min))
sol2 <- nlminb(s0, OF, data = data,
                           lower = data$min,
                           upper = data$max,
                           control = list(eval.max = 50000L,
                                          iter.max = 50000L))
@

Again, we compare the returned objective function value and the
maximum error.
<<>>=
max( abs(data$model(sol2$par, tm) - data$model(betaTRUE,tm)) )
sol2$objective
@

To compare our two solutions (DE and \texttt{nlminb}), we can plot
them together with the true yields curve. But it is important to
stress that the results of both algorithms are stochastic: in the case
of DE because it deliberately uses randomness; in the case of
\texttt{nlminb} because we set the starting value randomly.  To get
more meaningful results we should run both algorithms several times.
To keep the build-time of the vignette down, we only run both methods
once. But increase \texttt{nRuns} for more restarts.

<<fig = TRUE, height = 3>>=
par(ps = 11, bty = "n", las = 1, tck = 0.01,
    mgp = c(3, 0.2, 0), mar = c(4, 4, 1, 1))
plot(tm, yM, xlab = "maturities in years",
             ylab = "yields in %")
algo$printDetail <- FALSE
for (i in seq_len(nRuns)) {
    sol <- DEopt(OF = OF, algo = algo, data = data)
    lines(tm, data$model(sol$xbest,tm), col = "blue")
    s0 <- algo$min + (algo$max-algo$min) * runif(length(algo$min))
    sol2 <- nlminb(s0, OF, data = data,
                           lower = data$min,
                           upper = data$max,
                           control = list(eval.max = 50000L,
                                          iter.max = 50000L))

    lines(tm,data$model(sol2$par,tm), col = "darkgreen", lty = 2)
}

legend(x = "topright", legend = c("true yields", "DE", "nlminb"),
       col = c("black","blue","darkgreen"),
       pch = c(1, NA, NA), lty = c(0, 1, 2))
@

It is no error that there tpyically appears to be only one curve for
DE: there are, in fact, \texttt{nRuns} lines, but they are printed on
top of each other.


\subsection*{Other constraints}
The parameter constraints on the NS (and NSS) model are to make sure
that the resulting zero rates are nonnegative. But in fact, they do
not guarantee positive rates.
<<fig = TRUE, height = 3>>=
tm <- seq(1, 10, length.out = 100)   ## 1 to 10 years
betaTRUE <- c(3, -2, -8, 1.5)        ## 'true' parameters
yM <- NS(betaTRUE, tm)
par(ps = 11, bty = "n", las = 1, tck = 0.01,
    mgp = c(3, 0.2, 0), mar = c(4, 4, 1, 1))
plot(tm, yM, xlab = "maturities in years", ylab = "yields in %")
abline(h = 0)
@

This is really a made-up example, but nevertheless we may want to
include safeguards against such parameter vectors: we could include
just one constraint that all rates are greater than zero. This can be
done, again, with a penalty function.
<<>>=
penalty2 <- function(param, data) {
    y <- data$model(param, data$tm)
    maxdiff <- abs(y - abs(y))
    sum(maxdiff) * data$ww
}
@
Check:
<<>>=
penalty2(c(3, -2, -8, 1.5),data)
@

This penalty function only works for a single solution, so it is
actually simplest to write it directly into the objective function.
<<>>=
OFa <- function(param,data) {
    y <- data$model(param,data$tm)
    aux <- y - data$yM
    res <- max(abs(aux))
    ## compute the penalty
    aux <- y - abs(y) ## aux == zero for nonnegative y
    aux <- -sum(aux) * data$ww
    res <- res + aux
    if (is.na(res)) res <- 1e10
    res
}
@
So just as a numerical test: suppose the above parameters were true,
and interest rates were negative.

<<fig = TRUE, height = 3>>=
algo$pen <- NULL; data$yM <- yM; data$tm <- tm
par(ps = 11, bty = "n", las = 1, tck = 0.01,
    mgp = c(3, 0.2, 0), mar = c(4, 4, 1, 1))
plot(tm, yM, xlab = "maturities in years", ylab = "yields in %")
abline(h = 0)
sol <- DEopt(OF = OFa, algo = algo, data = data)
lines(tm,data$model(sol$xbest,tm), col = "blue")
legend(x = "topleft", legend = c("true yields", "DE (constrained)"),
       col = c("black", "blue"),
       pch = c(1, NA, NA), lty = c(0, 1, 2))
@


\section{Fitting the NSS model to given zero rates}
There is little that we need to change if we want to use the NSS model
instead. We just have to pass a different \texttt{model} to the
objective function (and change the
\texttt{min}/\texttt{max}-vectors). An example follows. Again, we fix
true parameters and try to recover them.
<<>>=
tm <- c(c(1, 3, 6, 9)/12, 1:10)
betaTRUE <- c(5,-2,5,-5,1,6)
yM <- NSS(betaTRUE, tm)
@

The lists \texttt{data} and \texttt{algo} are almost the same as
before; the objective function stays exactly the same.
<<>>=
data <- list(yM = yM,
             tm = tm,
          model = NSS,
            min = c( 0,-15,-30,-30,  0,5),
            max = c(15, 30, 30, 30,  5,  10),
             ww = 1)

algo <- list(nP = 100L,
             nG = 500L,
              F = 0.50,
             CR = 0.99,
            min = c( 0,-15,-30,-30,  0,5),
            max = c(15, 30, 30, 30,  5,  10),
            pen = penalty,
         repair = NULL,
         loopOF = TRUE,
        loopPen = FALSE,
     loopRepair = TRUE,
       printBar = FALSE,
    printDetail = FALSE)
@

It remains to run the algorithm. (Again, we check the returned
objective function value.)
<<>>=
sol <- DEopt(OF = OF, algo = algo, data = data)
max( abs(data$model(sol$xbest, tm) - data$model(betaTRUE, tm)) )
sol$OFvalue
@

We compare the results with \texttt{nlminb}.
<<>>=
s0 <- algo$min + (algo$max - algo$min) * runif(length(algo$min))
sol2 <- nlminb(s0,OF,data = data,
                           lower = data$min,
                           upper = data$max,
                         control = list(eval.max = 50000L,
                                        iter.max = 50000L))
max( abs(data$model(sol2$par, tm) - data$model(betaTRUE, tm)) )
sol2$objective
@

Finally, we compare the yield curves resulting from several
runs. (Recall that the number of runs is controlled by \texttt{nRuns},
which we have set initially.)
<<fig = TRUE, height = 3>>=
par(ps = 11, bty = "n", las = 1, tck = 0.01,
    mgp = c(3, 0.2, 0), mar = c(4, 4, 1, 1))
plot(tm, yM, xlab = "maturities in years", ylab = "yields in %")
for (i in seq_len(nRuns)) {
    sol <- DEopt(OF = OF, algo = algo, data = data)
    lines(tm, data$model(sol$xbest,tm), col = "blue")
    s0 <- algo$min + (algo$max - algo$min) * runif(length(algo$min))
    sol2 <- nlminb(s0, OF, data = data,
                           lower = data$min,
                           upper = data$max,
                           control = list(eval.max = 50000L,
                                          iter.max = 50000L))

    lines(tm, data$model(sol2$par,tm), col = "darkgreen", lty = 2)
}

legend(x = "topright", legend = c("true yields", "DE", "nlminb"),
       col = c("black","blue","darkgreen"),
       pch = c(1,NA,NA), lty = c(0,1,2), bg = "white")
@


\section{Fitting the NSS model to given bond prices}
\begin{framed}
  \noindent The section was removed to reduce the build-time of the
  package. The examples were moved to the `NMOF manual' (see
  \url{http://enricoschumann.net/NMOF.htm}). The code is in the
  subdirectory \texttt{NMOFex}; to show the code in \textsf{R}, you can use
  the function \texttt{system.file}.
<<eval = FALSE>>=
whereToLook <- system.file("NMOFex/NMOFman.R", package = "NMOF")
file.show(whereToLook, title = "NMOF examples")
@
\end{framed}



\section{Fitting the NSS model to given yields-to-maturity}
\begin{framed}
  \noindent The section was removed to reduce the build-time of the
  package. The examples were moved to the  `NMOF manual' (see
  \url{http://enricoschumann.net/NMOF.htm}). The code is in the
  subdirectory \texttt{NMOFex}; to show the code in \textsf{R}, you can use
  the function \texttt{system.file}.
<<eval = FALSE>>=
whereToLook <- system.file("NMOFex/NMOFman.R", package = "NMOF")
file.show(whereToLook, title = "NMOF examples")
@
\end{framed}

\bibliographystyle{plainnat}
\bibliography{NMOF}

\end{document}
