require(fitdistrplus) require(knitr) #for kable() function set.seed(12345) options(digits = 3)
We present very quickly the main optimization methods. Please refer to Numerical Optimization (Nocedal \& Wright, 2006) or Numerical Optimization: theoretical and practical aspects (Bonnans, Gilbert, Lemarechal \& Sagastizabal, 2006) for a good introduction. We consider the following problem $\min_x f(x)$ for $x\in\mathbb{R}^n$.
The Nelder-Mead method is one of the most well known derivative-free methods that use only values of $f$ to search for the minimum. It consists in building a simplex of $n+1$ points and moving/shrinking this simplex into the good direction.
Reflection:
Expansion:
The Nelder-Mead method is available in optim
.
By default, in optim
, $\alpha=1$, $\beta=1/2$, $\gamma=2$ and $\sigma=1/2$.
For smooth non-linear function, the following method is generally used: a local method combined with line search work on the scheme $x_{k+1} =x_k + t_k d_{k}$, where the local method will specify the direction $d_k$ and the line search will specify the step size $t_k \in \mathbb{R}$.
A desirable property for $d_k$ is that $d_k$ ensures a descent $f(x_{k+1}) < f(x_{k})$. Newton methods are such that $d_k$ minimizes a local quadratic approximation of $f$ based on a Taylor expansion, that is $q_f(d) = f(x_k) + g(x_k)^Td +\frac{1}{2} d^T H(x_k) d$ where $g$ denotes the gradient and $H$ denotes the Hessian.
The \emph{exact Newton} consists in using the exact solution of local minimization problem $d_k = - H(x_k)^{-1} g(x_k)$.
In practice, other methods are preferred (at least to ensure positive definiteness).
The \emph{quasi-Newton} method approximates the Hessian by a matrix $H_k$ as a function of $H_{k-1}$, $x_k$, $f(x_k)$ and then $d_k$ solves the system $H_k d = - g(x_k)$.
Some implementation may also directly approximate the inverse of the Hessian $W_k$ in order to compute $d_k = -W_k g(x_k)$. Using the Sherman-Morrison-Woodbury formula, we can switch between $W_k$ and $H_k$.
To determine $W_k$, first it must verify the secant equation $H_k y_k =s_k$ or $y_k=W_k s_k$ where $y_k = g_{k+1}-g_k$ and $s_k=x_{k+1}-x_k$. To define the $n(n-1)$ terms, we generally impose a symmetry and a minimum distance conditions. We say we have a rank 2 update if $H_k = H_{k-1} + a u u^T + b v v^T$ and a rank 1 update if $H_k = H_{k-1} + a u u^T $. Rank $n$ update is justified by the spectral decomposition theorem.
There are two rank-2 updates which are symmetric and preserve positive definiteness
In R
, the so-called BFGS scheme is implemented in optim
.
Another possible method (which is initially arised from quadratic problems) is the nonlinear conjugate gradients. This consists in computing directions $(d_0, \dots, d_k)$ that are conjugate with respect to a matrix close to the true Hessian $H(x_k)$. Directions are computed iteratively by $d_k = -g(x_k) + \beta_k d_{k-1}$ for $k>1$, once initiated by $d_1 = -g(x_1)$. $\beta_k$ are updated according a scheme:
There exists also three-term formula for computing direction
$d_k = -g(x_k) + \beta_k d_{k-1}+\gamma_{k} d_t$ for $toptim
.
Let $\phi_k(t) = f(x_k + t d_k)$ for a given direction/iterate $(d_k, x_k)$. We need to find conditions to find a satisfactory stepsize $t_k$. In literature, we consider the descent condition: $\phi_k'(0) < 0$ and the Armijo condition: $\phi_k(t) \leq \phi_k(0) + t c_1 \phi_k'(0)$ ensures a decrease of $f$. Nocedal \& Wright (2006) presents a backtracking (or geometric) approach satisfying the Armijo condition and minimal condition, i.e. Goldstein and Price condition.
This backtracking linesearch is available in optim
.
To simplify the benchmark of optimization methods, we create a fitbench
function that computes
the desired estimation method for all optimization methods.
This function is currently not exported in the package.
fitbench <- function(data, distr, method, grad=NULL, control=list(trace=0, REPORT=1, maxit=1000), lower=-Inf, upper=+Inf, ...)
fitbench <- fitdistrplus:::fitbench
The density of the beta distribution is given by $$ f(x; \delta_1,\delta_2) = \frac{x^{\delta_1-1}(1-x)^{\delta_2-1}}{\beta(\delta_1,\delta_2)}, $$ where $\beta$ denotes the beta function, see the NIST Handbook of mathematical functions https://dlmf.nist.gov/. We recall that $\beta(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$. There the log-likelihood for a set of observations $(x_1,\dots,x_n)$ is $$ \log L(\delta_1,\delta_2) = (\delta_1-1)\sum_{i=1}^n\log(x_i)+ (\delta_2-1)\sum_{i=1}^n\log(1-x_i)+ n \log(\beta(\delta_1,\delta_2)) $$ The gradient with respect to $a$ and $b$ is $$ \nabla \log L(\delta_1,\delta_2) = \left(\begin{matrix} \sum\limits_{i=1}^n\ln(x_i) - n\psi(\delta_1)+n\psi( \delta_1+\delta_2) \ \sum\limits_{i=1}^n\ln(1-x_i)- n\psi(\delta_2)+n\psi( \delta_1+\delta_2) \end{matrix}\right), $$ where $\psi(x)=\Gamma'(x)/\Gamma(x)$ is the digamma function, see the NIST Handbook of mathematical functions https://dlmf.nist.gov/.
R
implementationAs in the fitdistrplus
package, we minimize the opposite of the log-likelihood:
we implement the opposite of the gradient in grlnL
. Both the log-likelihood and its gradient
are not exported.
lnL <- function(par, fix.arg, obs, ddistnam) fitdistrplus:::loglikelihood(par, fix.arg, obs, ddistnam) grlnlbeta <- fitdistrplus:::grlnlbeta
#(1) beta distribution n <- 200 x <- rbeta(n, 3, 3/4) grlnlbeta(c(3, 4), x) #test hist(x, prob=TRUE) lines(density(x), col="red") curve(dbeta(x, 3, 3/4), col="green", add=TRUE) legend("topleft", lty=1, col=c("red","green"), leg=c("empirical", "theoretical"))
Define control parameters.
ctr <- list(trace=0, REPORT=1, maxit=1000)
Call mledist
with the default optimization function
(optim
implemented in stats
package)
with and without the gradient for the different optimization methods.
unconstropt <- fitbench(x, "beta", "mle", grad=grlnlbeta, lower=0)
In the case of constrained optimization, mledist
permits the direct use
of constrOptim
function (still implemented in stats
package)
that allow linear inequality constraints by using a logarithmic barrier.
Use a exp/log transformation of the shape parameters $\delta_1$ and $\delta_2$ to ensure that the shape parameters are strictly positive.
dbeta2 <- function(x, shape1, shape2, log) dbeta(x, exp(shape1), exp(shape2), log=log) #take the log of the starting values startarg <- lapply(fitdistrplus:::start.arg.default(x, "beta"), log) #redefine the gradient for the new parametrization grbetaexp <- function(par, obs, ...) grlnlbeta(exp(par), obs) * exp(par) expopt <- fitbench(x, distr="beta2", method="mle", grad=grbetaexp, start=startarg) #get back to original parametrization expopt[c("fitted shape1", "fitted shape2"), ] <- exp(expopt[c("fitted shape1", "fitted shape2"), ])
Then we extract the values of the fitted parameters, the value of the corresponding log-likelihood and the number of counts to the function to minimize and its gradient (whether it is the theoretical gradient or the numerically approximated one).
Results are displayed in the following tables:
(1) the original parametrization without specifying the gradient (-B
stands for bounded version),
(2) the original parametrization with the (true) gradient (-B
stands for bounded version and -G
for gradient),
(3) the log-transformed parametrization without specifying the gradient,
(4) the log-transformed parametrization with the (true) gradient (-G
stands for gradient).
kable(unconstropt[, grep("G-", colnames(unconstropt), invert=TRUE)], digits=3)
kable(unconstropt[, grep("G-", colnames(unconstropt))], digits=3)
kable(expopt[, grep("G-", colnames(expopt), invert=TRUE)], digits=3)
kable(expopt[, grep("G-", colnames(expopt))], digits=3)
Using llsurface
, we plot the log-likehood surface around the true value (green) and the fitted parameters (red).
llsurface(min.arg=c(0.1, 0.1), max.arg=c(7, 3), plot.arg=c("shape1", "shape2"), nlev=25, plot.np=50, data=x, distr="beta", back.col = FALSE) points(unconstropt[1,"BFGS"], unconstropt[2,"BFGS"], pch="+", col="red") points(3, 3/4, pch="x", col="green")
We can simulate bootstrap replicates using the bootdist
function.
b1 <- bootdist(fitdist(x, "beta", method="mle", optim.method="BFGS"), niter=100, parallel="snow", ncpus=2) summary(b1) plot(b1) abline(v=3, h=3/4, col="red", lwd=1.5)
The p.m.f. of the Negative binomial distribution is given by $$ f(x; m,p) = \frac{\Gamma(x+m)}{\Gamma(m)x!} p^m (1-p)^x, $$ where $\Gamma$ denotes the beta function, see the NIST Handbook of mathematical functions https://dlmf.nist.gov/. There exists an alternative representation where $\mu=m (1-p)/p$ or equivalently $p=m/(m+\mu)$. Thus, the log-likelihood for a set of observations $(x_1,\dots,x_n)$ is $$ \log L(m,p) = \sum_{i=1}^{n} \log\Gamma(x_i+m) -n\log\Gamma(m) -\sum_{i=1}^{n} \log(x_i!) + mn\log(p) +\sum_{i=1}^{n} {x_i}\log(1-p) $$ The gradient with respect to $m$ and $p$ is $$ \nabla \log L(m,p) = \left(\begin{matrix} \sum_{i=1}^{n} \psi(x_i+m) -n \psi(m) + n\log(p) \ mn/p -\sum_{i=1}^{n} {x_i}/(1-p) \end{matrix}\right), $$ where $\psi(x)=\Gamma'(x)/\Gamma(x)$ is the digamma function, see the NIST Handbook of mathematical functions https://dlmf.nist.gov/.
R
implementationAs in the fitdistrplus
package, we minimize the opposite of the log-likelihood: we implement the opposite of the gradient in grlnL
.
grlnlNB <- function(x, obs, ...) { m <- x[1] p <- x[2] n <- length(obs) c(sum(psigamma(obs+m)) - n*psigamma(m) + n*log(p), m*n/p - sum(obs)/(1-p)) }
#(1) beta distribution n <- 200 trueval <- c("size"=10, "prob"=3/4, "mu"=10/3) x <- rnbinom(n, trueval["size"], trueval["prob"]) hist(x, prob=TRUE, ylim=c(0, .3)) lines(density(x), col="red") points(min(x):max(x), dnbinom(min(x):max(x), trueval["size"], trueval["prob"]), col="green") legend("topleft", lty=1, col=c("red","green"), leg=c("empirical", "theoretical"))
Define control parameters and make the benchmark.
ctr <- list(trace=0, REPORT=1, maxit=1000) unconstropt <- fitbench(x, "nbinom", "mle", grad=grlnlNB, lower=0) unconstropt <- rbind(unconstropt, "fitted prob"=unconstropt["fitted mu",] / (1+unconstropt["fitted mu",]))
In the case of constrained optimization, mledist
permits the direct use
of constrOptim
function (still implemented in stats
package)
that allow linear inequality constraints by using a logarithmic barrier.
Use a exp/log transformation of the shape parameters $\delta_1$ and $\delta_2$ to ensure that the shape parameters are strictly positive.
dnbinom2 <- function(x, size, prob, log) dnbinom(x, exp(size), 1/(1+exp(-prob)), log=log) #transform starting values startarg <- fitdistrplus:::start.arg.default(x, "nbinom") startarg$mu <- startarg$size / (startarg$size+startarg$mu) startarg <- list(size=log(startarg[[1]]), prob=log(startarg[[2]]/(1-startarg[[2]]))) #redefine the gradient for the new parametrization Trans <- function(x) c(exp(x[1]), plogis(x[2])) grNBexp <- function(par, obs, ...) grlnlNB(Trans(par), obs) * c(exp(par[1]), plogis(x[2])*(1-plogis(x[2]))) expopt <- fitbench(x, distr="nbinom2", method="mle", grad=grNBexp, start=startarg) #get back to original parametrization expopt[c("fitted size", "fitted prob"), ] <- apply(expopt[c("fitted size", "fitted prob"), ], 2, Trans)
Then we extract the values of the fitted parameters, the value of the corresponding log-likelihood and the number of counts to the function to minimize and its gradient (whether it is the theoretical gradient or the numerically approximated one).
Results are displayed in the following tables:
(1) the original parametrization without specifying the gradient (-B
stands for bounded version),
(2) the original parametrization with the (true) gradient (-B
stands for bounded version and -G
for gradient),
(3) the log-transformed parametrization without specifying the gradient,
(4) the log-transformed parametrization with the (true) gradient (-G
stands for gradient).
kable(unconstropt[, grep("G-", colnames(unconstropt), invert=TRUE)], digits=3)
kable(unconstropt[, grep("G-", colnames(unconstropt))], digits=3)
kable(expopt[, grep("G-", colnames(expopt), invert=TRUE)], digits=3)
kable(expopt[, grep("G-", colnames(expopt))], digits=3)
Using llsurface
, we plot the log-likehood surface around the true value (green) and the fitted parameters (red).
llsurface(min.arg=c(5, 0.3), max.arg=c(15, 1), plot.arg=c("size", "prob"), nlev=25, plot.np=50, data=x, distr="nbinom", back.col = FALSE) points(unconstropt["fitted size","BFGS"], unconstropt["fitted prob","BFGS"], pch="+", col="red") points(trueval["size"], trueval["prob"], pch="x", col="green")
We can simulate bootstrap replicates using the bootdist
function.
b1 <- bootdist(fitdist(x, "nbinom", method="mle", optim.method="BFGS"), niter=100, parallel="snow", ncpus=2) summary(b1) plot(b1) abline(v=trueval["size"], h=trueval["mu"], col="red", lwd=1.5)
Based on the two previous examples, we observe that all methods converge to the same
point. This is rassuring.
However, the number of function evaluations (and the gradient evaluations) is
very different from a method to another.
Furthermore, specifying the true gradient of the log-likelihood does not
help at all the fitting procedure and generally slows down the convergence.
Generally, the best method is the standard BFGS method or the BFGS method
with the exponential transformation of the parameters.
Since the exponential function is differentiable, the asymptotic properties are
still preserved (by the Delta method) but for finite-sample this may produce a small bias.
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