Passing Additional / Different Arguments to
nloptr()
The nloptr
package is extremely flexible, as it provides numerous arguments to
users for tailoring the optimization to the problem at hand. In the
nloptr function, most of these arguments are specified via
the opts argument, a convention we partially borrow for
brcal(). However, we allow users to directly specify a few
arguments that are typically relegated to opts for easier
adjustments. All other arguments, except those related to the objective
and constraint functions, are available to users by passing a list of
these arguments opts. We do not allow users to change the
objective or constraint functions, or their Jacobian functions, as these
are the backbone of this method and the sole purpose for using the
brcal function. For those who wish to optimize a different
function or use a different constraint, we recommend using the
nloptr function directly.
Start Values
While the algorithm used by optim() was not very
sensitive to the starting location of \(\delta\) and \(\gamma\), we’ve found the nature of
boldness-recalibration makes nloptr() more sensitive to
starting location. By default, brcal() initializes the
optimization at the MLEs for \(\delta\)
and \(\gamma\). While we’ve found this
to be the most stable approach to setting a starting location, users can
specify their own starting location by specifying x0 and
setting start_at_MLEs=FALSE.
To show the sensitivity of this approach to the starting values, try
set x0 to \(\delta = 10\),
\(\gamma = -5\) like we did with
optim(). The line of code below will run
brcal() with these starting values, however we do not run
it automatically in this vignette as it causes an error. In this case,
the algorithm cannot converge and ends up crashing due to trying extreme
values of the parameters. We encourage readers to run this code
themselves to verify this result.
try(brcal(hockey$x, hockey$y, x0 = c(10, -5), start_at_MLEs = FALSE))
> iteration: 1
> x = (10.000000, -5.000000)
> f(x) = -0.264782
> g(x) = 0.950000
> iteration: 2
> x = (10.000000, -5.000000)
> f(x) = -0.264782
> g(x) = 0.950000
> iteration: 3
> x = (9.994848, -5.044965)
> f(x) = -0.266818
> g(x) = 0.950000
...
> iteration: 27
> x = (3.550802, -324.404849)
> f(x) = -0.476697
> g(x) = 0.950000
> iteration: 28
> x = (4.390822, -445.720424)
> f(x) = -0.477751
> g(x) = 0.950000
> iteration: 29
> x = (5.542112, -615.971882)
> f(x) = nan
> Error in optim(par = par, fn = llo_lik, ..., x = x, y = y, neg = TRUE, :
> function cannot be evaluated at initial parameters
Algorithm
By default, the brcal function uses the Augmented
Lagrangian Algorithm (AUGLAG) (Conn, Gould, and
Toint 1991; Birgin and Martínez 2008), which requires an inner
optimization routine. We use Sequential Least-Squares Quadratic
Programming (SLSQP) (Kraft 1988, 1994) as
the inner optimization routine, by default. For a complete list of inner
and outer optimization algorithms, see the NLopt website. Note
that not all algorithms can be used with a non-linear constraint (which
is necessary for boldness-recalibration). Also note that other
algorithms may be more sensitive to starting conditions or may need
different stopping criteria to converge.
All changes to the inner algorithm are made via passing a sub-list to
local_opts, which itself an entry of the list passed to
opts. Instead of SLSQP as the inner algorithm, let’s try
the method of moving asymptotes (MMA) (Svanberg
2002).
br90_inMMA <- brcal(hockey$x, hockey$y, t=0.9,
opts=list(local_opts=list(algorithm="NLOPT_LD_MMA")))
> iteration: 1
> x = (0.945397, 1.400573)
> f(x) = -0.123926
> g(x) = -0.098849
> iteration: 2
> x = (0.945397, 1.400573)
> f(x) = -0.123926
> g(x) = -0.098849
...
> iteration: 104
> x = (0.866252, 2.011971)
> f(x) = -0.169010
> g(x) = -0.000001
> iteration: 105
> x = (0.866252, 2.011971)
> f(x) = -0.169010
> g(x) = -0.000001
br90_inMMA$nloptr
>
> Call:
>
> nloptr::nloptr(x0 = c(0.945397, 1.400573), eval_f = obj_f, eval_grad_f = obj_grad_f,
> eval_g_ineq = constr_g, eval_jac_g_ineq = constr_grad_g,
> opts = list(local_opts = list(algorithm = "NLOPT_LD_MMA",
> eval_f = obj_f, eval_grad_f = obj_grad_f, eval_g_ineq = constr_g,
> eval_jac_g_ineq = constr_grad_g, xtol_rel = 1e-06), algorithm = "NLOPT_LD_AUGLAG",
> maxeval = 500, xtol_rel = 1e-06, print_level = 3), t = 0.9,
> tau = FALSE, Pmc = 0.5, epsilon = 2.22044604925031e-16)
>
>
> Minimization using NLopt version 2.7.1
>
> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because
> xtol_rel or xtol_abs (above) was reached. )
>
> Number of Iterations....: 105
> Termination conditions: maxeval: 500 xtol_rel: 1e-06
> Number of inequality constraints: 1
> Number of equality constraints: 0
> Optimal value of objective function: -0.169010270051815
> Optimal value of controls: 0.8662522 2.011971
Now, let’s try changing the outer algorithm to MMA. This is a great
example of an algorithm that is more sensitive to both starting
conditions and stopping criteria. In order to get convergence here,
we’ll opt to start the optimizer at c(1,2) and relax the
relative tolerance. Stopping criteria is discussed in a later
section.
br90_outMMA <- brcal(hockey$x, hockey$y, t=0.9, x0=c(1, 2), start_at_MLEs = FALSE,
xtol_rel_outer = 1e-05,
opts=list(algorithm="NLOPT_LD_MMA"))
> iteration: 1
> x = (1.000000, 2.000000)
> f(x) = -0.166367
> g(x) = 0.229068
> iteration: 2
> x = (0.938217, 2.023699)
> f(x) = -0.168894
> g(x) = 0.075137
...
> iteration: 15
> x = (0.866256, 2.011972)
> f(x) = -0.169010
> g(x) = 0.000000
> iteration: 16
> x = (0.866238, 2.011969)
> f(x) = -0.169010
> g(x) = -0.000000
br90_outMMA$nloptr
>
> Call:
> nloptr::nloptr(x0 = c(1, 2), eval_f = obj_f, eval_grad_f = obj_grad_f,
> eval_g_ineq = constr_g, eval_jac_g_ineq = constr_grad_g,
> opts = list(algorithm = "NLOPT_LD_MMA", maxeval = 500, xtol_rel = 1e-05,
> print_level = 3, local_opts = list(algorithm = "NLOPT_LD_SLSQP",
> eval_grad_f = obj_grad_f, eval_jac_g_ineq = constr_grad_g,
> xtol_rel = 1e-06, eval_f = obj_f, eval_g_ineq = constr_g)),
> t = 0.9, tau = FALSE, Pmc = 0.5, epsilon = 2.22044604925031e-16)
>
>
> Minimization using NLopt version 2.7.1
>
> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because
> xtol_rel or xtol_abs (above) was reached. )
>
> Number of Iterations....: 16
> Termination conditions: maxeval: 500 xtol_rel: 1e-05
> Number of inequality constraints: 1
> Number of equality constraints: 0
> Optimal value of objective function: -0.169010299710146
> Optimal value of controls: 0.8662376 2.011969
Note that many of the algorithms do not use an inner algorithm. In
most of these cases, local_opts will be ignored, otherwise
nloptr() may throw a warning or error.
Bounds
Following the convention of nloptr(), arguments
lb and ub allow users to specify the lower and
upper bounds of optimization, respectively. The bounds set using
lb and ub are used for both the inner and
outer algorithms. By default, we only place a lower bound on \(\delta\) requiring \(\delta > 0\). Similar to setting the
bounds of optimization in optim(), setting the lower bound
to -Inf and the upper bound to Inf will leave
the corresponding parameter unbounded.
For example, the code below shows how to bound \(0.25 < \delta < 5\) and leave \(\gamma\) unbounded.
br90_bound <- brcal(hockey$x, hockey$y, t=0.9, lb=c(0.25, -Inf), ub=c(5, Inf))
> iteration: 1
> x = (0.945397, 1.400573)
> f(x) = -0.123926
> g(x) = -0.098849
> iteration: 2
> x = (0.945397, 1.400573)
> f(x) = -0.123926
> g(x) = -0.098849
...
> iteration: 175
> x = (0.866252, 2.011972)
> f(x) = -0.169010
> g(x) = 0.000000
> iteration: 176
> x = (0.866252, 2.011972)
> f(x) = -0.169010
> g(x) = 0.000000
br90_bound$nloptr
>
> Call:
>
> nloptr::nloptr(x0 = c(0.945397, 1.400573), eval_f = obj_f, eval_grad_f = obj_grad_f,
> eval_g_ineq = constr_g, eval_jac_g_ineq = constr_grad_g,
> opts = list(algorithm = "NLOPT_LD_AUGLAG", maxeval = 500,
> maxtime = -1, xtol_rel = 1e-06, print_level = 3, local_opts = list(algorithm = "NLOPT_LD_SLSQP",
> eval_grad_f = obj_grad_f, eval_jac_g_ineq = constr_grad_g,
> xtol_rel = 1e-06)), t = 0.9, tau = FALSE, Pmc = 0.5,
> epsilon = 2.22044604925031e-16)
>
>
> Minimization using NLopt version 2.7.1
>
> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because
> xtol_rel or xtol_abs (above) was reached. )
>
> Number of Iterations....: 176
> Termination conditions: maxeval: 500 xtol_rel: 1e-06
> Number of inequality constraints: 1
> Number of equality constraints: 0
> Optimal value of objective function: -0.169010301103791
> Optimal value of controls: 0.8662523 2.011972
Stopping criteria
Since there are quite a few options for stopping criteria that can be
set via opts in the nloptr function, we select
just a few to specify directly in the brcal function. As
with optim(), it is important to check convergence and
adjust the stopping criteria accordingly. Keep in mind that stopping
criteria also work together so several adjustments may be necessary.
Argument maxeval sets the maximum number of iterations
(approximately) allowed by the outer optimization routine. Argument
maxtime sets the maximum number of seconds (approximately)
the outer algorithm will be allowed to run. Argument
xtol_rel_outer and xtol_rel_inner set the
relative tolerance for the change in \(\delta\) and \(\gamma\) between iterations of the outer
and inner algorithms, respectively. Again, nloptr allows
several other options for stopping criteria that can be specified via
opts.
The example below shows how to set the maximum number of evaluations
to 100, the maximum time to 30 seconds, and the relative tolerance of
both the inner and outer algorithms to 0.001. These are not necessarily
recommended settings, rather they are intended for demonstration
purposes only.
br90_stop <- brcal(hockey$x, hockey$y, t=0.9, maxeval=100, maxtime = 30,
xtol_rel_outer = 0.001, xtol_rel_inner = 0.001)
> iteration: 1
> x = (0.945397, 1.400573)
> f(x) = -0.123926
> g(x) = -0.098849
> iteration: 2
> x = (0.945397, 1.400573)
> f(x) = -0.123926
> g(x) = -0.098849
...
> iteration: 82
> x = (0.866252, 2.011972)
> f(x) = -0.169010
> g(x) = -0.000000
> iteration: 83
> x = (0.866252, 2.011972)
> f(x) = -0.169010
> g(x) = -0.000000
br90_stop$nloptr
>
> Call:
>
> nloptr::nloptr(x0 = c(0.945397, 1.400573), eval_f = obj_f, eval_grad_f = obj_grad_f,
> eval_g_ineq = constr_g, eval_jac_g_ineq = constr_grad_g,
> opts = list(algorithm = "NLOPT_LD_AUGLAG", maxeval = 100,
> maxtime = 30, xtol_rel = 0.001, print_level = 3, local_opts = list(algorithm = "NLOPT_LD_SLSQP",
> eval_grad_f = obj_grad_f, eval_jac_g_ineq = constr_grad_g,
> xtol_rel = 0.001)), t = 0.9, tau = FALSE, Pmc = 0.5,
> epsilon = 2.22044604925031e-16)
>
>
> Minimization using NLopt version 2.7.1
>
> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because
> xtol_rel or xtol_abs (above) was reached. )
>
> Number of Iterations....: 83
> Termination conditions: maxeval: 100 maxtime: 30 xtol_rel: 0.001
> Number of inequality constraints: 1
> Number of equality constraints: 0
> Optimal value of objective function: -0.169010294069083
> Optimal value of controls: 0.8662524 2.011972
Suppressing output from nloptr()
To minimize the amount of output printed at each iteration of the
optimizer, we leverage print_level from
nloptr(). By default, we choose to print the maximum amount
of information by setting print_level=3. To simply reduce
the output, try print_level=1…
br90 <- brcal(hockey$x, hockey$y, t=0.9, print_level=1)
> iteration: 1
> f(x) = -0.123926
> iteration: 2
> f(x) = -0.123926
> iteration: 3
> f(x) = -0.130028
...
> iteration: 174
> f(x) = -0.169010
> iteration: 175
> f(x) = -0.169010
> iteration: 176
> f(x) = -0.169010
…or print_level=2.
br90 <- brcal(hockey$x, hockey$y, t=0.9, print_level=2)
> iteration: 1
> f(x) = -0.123926
> g(x) = -0.098849
> iteration: 2
> f(x) = -0.123926
> g(x) = -0.098849
> iteration: 3
> f(x) = -0.130028
> g(x) = -0.098758
...
> iteration: 174
> f(x) = -0.169010
> g(x) = 0.000000
> iteration: 175
> f(x) = -0.169010
> g(x) = 0.000000
> iteration: 176
> f(x) = -0.169010
> g(x) = 0.000000
Or to fully suppress, use print_level=0. Notice nothing
prints from the following call.
br90 <- brcal(hockey$x, hockey$y, t=0.9, print_level=0)
Passing Additional / Different Arguments to
optim()
While optim() is not used for the non-linear constrained
optimization for finding the boldness-recalibration parameters, it is
used throughout this package wherever the MLEs are needed, including in
the calculation for the posterior model posterior
(bayes_ms(), llo_lrt(),
mle_recal, brcal(), line_plot(),
and plot_params()).
While we’ve found that optim() converges in most cases
we’ve tried using the default settings in this package, it is important
to double check convergence. It is for this reason that we allow users
to pass additional arguments to optim and include the
returned list from optim() in several of our own functions
so users can see the convergence information. If the algorithm did not
converge, which can be checked via list elements
optim_details$convergence and
optim_details$message from bayes_ms(),
llo_lrt(), or mle_recal(), users should adjust
the arguments passed to optim() using ... to
achieve convergence before trusting results. Below are a few examples of
how to adjust the call to optim(). See the documentation
for optim() for more information.
Conventions for Passing Arguments to optim()
The BRcal package uses two different conventions for passing
arguments to optim() depending on the situation. For the
functions where optim() is the only function to which we
are passing additional arguments (bayes_ms(),
llo_lrt(), and mle_recal()), we pass arguments
to optim() using the .... The next few
sections demonstrate how to pass arguments in this way. While the
examples in this section use bayes_ms(), the same
conventions apply to llo_lrt() and
mle_recal().
For the functions where we pass additional arguments to multiple
functions (brcal(), plot_params(), and
lineplot()), we cannot leverage the flexibility of
.... Instead, users can pass arguments to
optim() in the form of a list via
optim_options. Demonstration of passing arguments to
optim() in this fashion can be found in Passing Arguments via a List.
Start Values
In some cases, the optimization routine may sensitive to the starting
location of the optimizer. By default, all calls to optim()
in this package use users can change the starting values of \(\delta\) and \(\gamma\) by passing them as a vector via
argument par.
In the following example, we will start the optimizer at \(\delta = 10\), \(\gamma = -5\).
bayes_ms(hockey$x, hockey$y, par = c(10, -5))
> $Pmc
> [1] 0.5
>
> $BIC_Mc
> [1] 1148.637
>
> $BIC_Mu
> [1] 1157.902
>
> $BF
> [1] 0.009730526
>
> $posterior_model_prob
> [1] 0.9903632
>
> $MLEs
> [1] 0.9455187 1.4004799
>
> $optim_details
> $optim_details$par
> [1] 0.9455187 1.4004799
>
> $optim_details$value
> [1] 572.1846
>
> $optim_details$counts
> function gradient
> 61 NA
>
> $optim_details$convergence
> [1] 0
>
> $optim_details$message
> NULL
In this case, the optimizer does not seem overly sensitive to start
values, as \(\delta = 10\), \(\gamma = -5\) are not close to the solution
(the MLEs) at \(\delta = 0.9455\),
\(\gamma = 1.4005\). The algorithm
converged and we get the same solution as when we started at \(\delta = 0.5\), \(\gamma = 0.5\).
Stopping Criteria
Sometimes it’s useful to adjust the optimization stopping criteria to
improve convergence or refine the solution. We use the default settings
in optim(). These can be adjust via the argument
control which takes a list with an array of possible
components. We defer to the documentation for optim() for
further explanation of how to use control. Below is a
simple example where we set the relative convergence tolerance
(reltol) to 1e-10 instead of the default of
sqrt(.Machine$double.eps).
bayes_ms(hockey$x, hockey$y, control=list(reltol=1e-10))
> $Pmc
> [1] 0.5
>
> $BIC_Mc
> [1] 1148.637
>
> $BIC_Mu
> [1] 1157.902
>
> $BF
> [1] 0.009730632
>
> $posterior_model_prob
> [1] 0.9903631
>
> $MLEs
> [1] 0.945395 1.401402
>
> $optim_details
> $optim_details$par
> [1] 0.945395 1.401402
>
> $optim_details$value
> [1] 572.1846
>
> $optim_details$counts
> function gradient
> 83 NA
>
> $optim_details$convergence
> [1] 0
>
> $optim_details$message
> NULL
Notice how we arrive at a slightly different solution and there were
more function calls than when using the default.
Algorithm and Bounds
It’s not uncommon that some optimization algorithms work better than
others depending on the nature of the problem. We use the default
algorithm "Nelder-Mead" (Nelder and
Mead 1965) which we’ve found to reliable in most cases we’ve
tried. This algorithm does not accept bounds on the parameters. To
circumvent the fact that \(\delta >
0\), meaning one of our parameters of interest is bounded, we
perform the optimization in terms of \(log(\delta)\). Should users prefer to use a
bounded algorithm, the algorithm can be specified using
method and the bounds can be specified using
lower and upper. For example, let’s say we
want to use "L-BFGS-B" where we bound \(0 < \delta < 10\) and \(-1 < \gamma < 25\):
bayes_ms(hockey$x, hockey$y, method = "L-BFGS-B", lower = c(0, -1), upper = c(10, 25))
> $Pmc
> [1] 0.5
>
> $BIC_Mc
> [1] 1148.637
>
> $BIC_Mu
> [1] 1157.902
>
> $BF
> [1] 0.009730632
>
> $posterior_model_prob
> [1] 0.9903631
>
> $MLEs
> [1] 0.9453906 1.4013926
>
> $optim_details
> $optim_details$par
> [1] 0.9453906 1.4013926
>
> $optim_details$value
> [1] 572.1846
>
> $optim_details$counts
> function gradient
> 8 8
>
> $optim_details$convergence
> [1] 0
>
> $optim_details$message
> [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
Or, if we’d rather only impose the lower bound on \(\delta\) and leave \(\gamma\) unbounded, we could use the
following snippet:
bayes_ms(hockey$x, hockey$y, method = "L-BFGS-B",
lower = c(0, -Inf), upper = c(Inf, Inf))
> $Pmc
> [1] 0.5
>
> $BIC_Mc
> [1] 1148.637
>
> $BIC_Mu
> [1] 1157.902
>
> $BF
> [1] 0.009730632
>
> $posterior_model_prob
> [1] 0.9903631
>
> $MLEs
> [1] 0.9453906 1.4013926
>
> $optim_details
> $optim_details$par
> [1] 0.9453906 1.4013926
>
> $optim_details$value
> [1] 572.1846
>
> $optim_details$counts
> function gradient
> 8 8
>
> $optim_details$convergence
> [1] 0
>
> $optim_details$message
> [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
Suppressing the output from optim()
Once you are confident that the algorithm converges via the arguments
you’ve specified, you can set optim_details=FALSE to reduce
the output if desired. As long as you continue to use the same settings,
the convergence should not change. Below is what the output looks like
using the hockey data when
optim_details=FALSE.
bayes_ms(hockey$x, hockey$y, optim_details=FALSE)
> $Pmc
> [1] 0.5
>
> $BIC_Mc
> [1] 1148.637
>
> $BIC_Mu
> [1] 1157.902
>
> $BF
> [1] 0.009730538
>
> $posterior_model_prob
> [1] 0.9903632
>
> $MLEs
> [1] 0.9453966 1.4005730
Passing Arguments via a List
As previously mentioned, the brcal,
plot_params, and lineplot functions use
optim() under the hood but we choose not to leverage the
flexibility of ... since they also pass additional
arguments to other functions. For this reason, we instead pass
additional arguments to optim() in the form of a list via
the parameter optim_options. While we only demonstrate this
using brcal() below, the same convention applies to
plot_params() and lineplot().
The example below shows how to use optim_options to
change the algorithm to "L-BFGS-B", set bounds on \(\delta\) and \(\gamma\), and set the max number of
iterations to 200 in the brcal function. Note we specify
print_level = 0 and only print the list returned from
nloptr() to reduce output.
br <- brcal(hockey$x, hockey$y, t=0.9, print_level = 0,
optim_options=list(method="L-BFGS-B",
lower=c(0.0001, 10),
upper=c(0.0001, 10),
control=list(maxit=200)))
br$nloptr
>
> Call:
>
> nloptr::nloptr(x0 = c(0.945397, 1.400573), eval_f = obj_f, eval_grad_f = obj_grad_f,
> eval_g_ineq = constr_g, eval_jac_g_ineq = constr_grad_g,
> opts = list(algorithm = "NLOPT_LD_AUGLAG", maxeval = 500,
> maxtime = -1, xtol_rel = 1e-06, print_level = 0, local_opts = list(algorithm = "NLOPT_LD_SLSQP",
> eval_grad_f = obj_grad_f, eval_jac_g_ineq = constr_grad_g,
> xtol_rel = 1e-06)), t = 0.9, tau = FALSE, Pmc = 0.5,
> epsilon = 2.22044604925031e-16)
>
>
> Minimization using NLopt version 2.7.1
>
> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because
> xtol_rel or xtol_abs (above) was reached. )
>
> Number of Iterations....: 82
> Termination conditions: maxeval: 500 xtol_rel: 1e-06
> Number of inequality constraints: 1
> Number of equality constraints: 0
> Optimal value of objective function: -0.169010282889692
> Optimal value of controls: 0.8662175 2.011966
While not demonstrated here, the options in the sections above that
are passed via ... can also be specified in a list passed
to optim_options.
Linear in Log Odds (LLO) function LLO()
The LLO function allows you to apply a linear in log
odds (LLO) adjustment to predicted probabilities in x.
Specifically it shifts the probabilities (on the log odds scale) by
delta and scales them by gamma. Note that the
value supplied to delta must be greater than 0 and
gamma can be any real number, though very extreme values
may cause instabilities (which we’ll see later).
By default, this function is used under the hood of all other
functions in this package, specifically when making any adjustment to
probabilities (i.e. MLE recalibration, boldness-recalibration). While
most users of this package probably will not use the LLO()
function directly, there are some cases where users may want to do so.
One example of this could be when you have estimates for \(\delta\) and \(\gamma\) based on historical data and want
to LLO-adjust unlabeled out of sample predictions. A user could use the
LLO function to LLO-adjust the new set of predictions by
passing them to function argument x, and the parameter
estimates to delta and gamma. The following
sections provide examples of how to use the LLO
function.
Using LLO() with MLEs or Boldness-Recalibration
Estimates
Another example of when users may want to use LLO()
directly is in getting the MLE recalibrated set. We have already
demonstrated how to go about this using mle_recal().
Alternatively, you can get the MLE recalibrated set manually by first
getting the MLEs from another function (such as bayes_ms()
or llo_lrt()) and then passing those as delta
and gamma in LLO(). This may be useful in
cases where you have large data and finding the MLEs is slower than
desired. If you already have the MLEs from a previous function call to
bayes_ms() or llo_lrt() then you don’t need to
take the time to re-solve for the MLEs. Additionally, this approach
serves as a good alternative to using mle_recal() with
probs_only=TRUE.
In the example below, we’ll use the MLEs for FiveThirtyEight’s
predictions we saved from bayes_ms() earlier on and pass
those to LLO(). Notice the probabilities below are
identical to those we got from mle_recal().
hockey$x_mle <- LLO(x=hockey$x, delta=bt538$MLEs[1], gamma=bt538$MLEs[2])
hockey$x_mle
> [1] 0.5633646 0.6814612 0.5628440 0.5117077 0.5675526 0.4223768 0.3802169
> [8] 0.4748424 0.5231574 0.3000678 0.5266953 0.5292678 0.4957091 0.5647970
> [15] 0.6538569 0.4845104 0.4090460 0.5792843 0.6937447 0.6000152 0.3961722
...
> [855] 0.7210362 0.6593455 0.4586214 0.7750603 0.5841900 0.4826292 0.4080026
> [862] 0.6701504 0.6561462 0.4814185 0.7421488 0.6786381 0.3255808 0.4814569
Let’s visualize what this LLO-adjustment looks like by plotting the
original probabilities (hockey$x) on the x-axis and the MLE
recalibrated probabilities (hockey$x_mle) on the y-axis.
Note that the points below are the actual observations and the curve
represents how any prediction along the 0-1 range would behave under
delta=0.9453966 and gamma=1.400573.
# plot original vs LLO-adjusted via MLEs
ggplot2::ggplot(data=hockey, mapping=aes(x=x, y=x_mle)) +
stat_function(fun=LLO,
args=list(delta=bt538$MLEs[1],
gamma=bt538$MLEs[2]),
geom="line",
linewidth=1,
color="black",
xlim=c(0, 1)) +
geom_point(aes(color=winner), alpha=0.75, size=2) +
lims(x=c(0,1), y=c(0,1)) +
labs(x="Original", y="LLO(x, 2, 2)", color="Winner") +
theme_bw()
Similarly, say you have already obtained boldness-recalibration
estimates but did not save the vector of boldness-recalibrated
probabilities. You can use LLO() to boldness-recalibrate
with your estimates without taking the time to let the optimizer in
brcal() re-solve for these estimates.
In the example below, we’ll use the 90% boldness-recalibration
estimates for FiveThirtyEight’s predictions we saved from
brcal() earlier on and pass those to LLO().
This essentially achieves the same probabilities in probs
returned by brcal() by LLO-adjusting via the returned
boldness-recalibration parameters in BR_params returned by
brcal().
LLO(x=hockey$x, br90$BR_params[1], br90$BR_params[2])
> [1] 0.57521324 0.73683075 0.57447030 0.50109237 0.58118451 0.37458745
> [7] 0.31759475 0.44828613 0.51755392 0.21761392 0.52264063 0.52633875
> [13] 0.47812060 0.57725659 0.70072899 0.46208535 0.35630619 0.59785587
...
> [859] 0.60479909 0.45939663 0.35488466 0.72219848 0.70377197 0.45766716
> [865] 0.81088037 0.73320058 0.24804606 0.45772207
Extreme Value of delta or gamma
As previously mentioned, using extreme values of delta
or gamma can cause instabilities. For example, let’s see
what happens when when we set delta=2,
gamma=1000. In the plot below, we show the original
probabilities on the x-axis and the LLO-adjusted probabilities via
delta=2, gamma=1000 on the y-axis. Notice that
nearly all predictions were pushed to approximately 0 or 1.
# LLO-adjust using delta=2, gamma=1000
hockey$x2 <- LLO(hockey$x, delta=2, gamma=1000)
# plot original vs LLO-adjusted via 2,2
ggplot2::ggplot(data=hockey, mapping=aes(x=x, y=x2)) +
stat_function(fun=LLO,
args=list(delta=2, gamma=1000),
geom="line",
linewidth=1,
color="black",
xlim=c(0, 1)) +
geom_point(aes(color=winner), alpha=0.75, size=2) +
lims(x=c(0,1), y=c(0,1)) +
labs(x="Original", y="LLO(x, 2, 1000)", color="Winner") +
theme_bw()
While this didn’t cause problems plotting the LLO curve, there are
cases in this package where probabilities very close to 0 or 1 will
cause instability and possibly even break the code. With this in mind,
we do have safeguard in place to keep predictions from getting to close
(numerically speaking) to 0 or 1, which is discussed further in Specifying epsilon.
Specifying epsilon
When probability predictions that are “too close” to 0 or 1,
calculations using the log likelihood become unstable. This is due to
the fact that the logit function is not defined at 0 and 1. To mitigate
this instability, we use epsilon to specify how close is
“too close”. For values in x that are less than
epsilon (i.e. too close to zero), these are replaced with
epsilon. For values in x greater than
1 - epsilon (i.e. too close to 1), these are replaced with
1 - epsilon. Naturally, epsilon must take on a
value between 0 and 1, but it is recommended that this be a very small
number. By default, we use .Machine$double.eps.
In this example with the hockey data, there are no
values close to 0 or 1 in x or rand, so
epsilon is not used at all.
Specifying event
Throughout this package, the functions assume by default that
y is a vector of 0s and 1s where a 1 represents a “event”
and 0 represents a “non-event”. Additionally, these functions expect
that the probabilities in x are the probabilities of
“event” at each observation. While switching the labels of “event” and
“non-event” will not change the fundamental conclusions, it is good
practice to make sure event is defined properly. However,
there may be cases where you want to pass a binary vector of outcomes
that are not defined as 0s and 1s. To define which of the two values in
y should be considered a “success” or the event of
interest, users may set argument event to that value.
For example, let’s use the column winner from the
hockey dataset as our vector of binary outcomes instead of
y in bayes_ms(). We can do this by specifying
event = "home". We’ll continue to set
optim_details=FALSE for easier reading of the output.
bayes_ms(hockey$x, hockey$winner, event="home", optim_details=FALSE)
> $Pmc
> [1] 0.5
>
> $BIC_Mc
> [1] 1148.637
>
> $BIC_Mu
> [1] 1157.902
>
> $BF
> [1] 0.009730538
>
> $posterior_model_prob
> [1] 0.9903632
>
> $MLEs
> [1] 0.9453966 1.4005730
As expected, the results are identical to our previous call.
Maybe you want to approach this problem from the perspective of an
event being a home team loss. To do so, you need to pass the predicted
probabilities of a home team loss as x = 1 - hockey$x, and
specify which value in y is an event via
event=0 for y=hockey$y…
bayes_ms(1-hockey$x, hockey$y, event=0, optim_details=FALSE)
> $Pmc
> [1] 0.5
>
> $BIC_Mc
> [1] 1148.637
>
> $BIC_Mu
> [1] 1157.902
>
> $BF
> [1] 0.009730605
>
> $posterior_model_prob
> [1] 0.9903632
>
> $MLEs
> [1] 1.057957 1.401514
… or event="away" for hockey$winner.
bayes_ms(1-hockey$x, hockey$winner, event="away", optim_details=FALSE)
> $Pmc
> [1] 0.5
>
> $BIC_Mc
> [1] 1148.637
>
> $BIC_Mu
> [1] 1157.902
>
> $BF
> [1] 0.009730605
>
> $posterior_model_prob
> [1] 0.9903632
>
> $MLEs
> [1] 1.057957 1.401514
Note that we get the same results using either specification.
Additionally, the posterior model probability is the same as when we
considered event to be a home team win, as one might expect with binary
event predictions.
Visualizing Posterior Model Probability as a Function of \(\delta\) and \(\gamma\) via
plot_params()
The goal of the plot_params function is to visualize how
the posterior model probability changes under different recalibration
parameters, as this is used in boldness-recalibration. Let’s see what
the default version of this plot looks like with FiveThirtyEight’s
data.
plot_params(hockey$x, hockey$y)
> Warning in get_zmat(x = x, y = y, Pmc = Pmc, len.out = k, lower = c(dlim[1], :
> There may be innaccuracies near gamma=0
You might immediately notice the imagine is grainy and unnecessarily
small. In the following sections, we will demonstrate how to fix these
issues and make further adjustments via arguments in
plot_params().
Setting bounds and grid density
To adjust the grid of \(\delta\) and
\(\gamma\) values, we can use
dlim and glim to “zoom in” on the area of
non-zero posterior model probabilities. Let’s reduce the range of \(\delta\) and \(\gamma\) so we can better visualize the
area of non-zero posterior model probability of calibration.
plot_params(hockey$x, hockey$y, dlim=c(0.5, 1.5), glim=c(0.25, 2.75))
Next, we can use k to form a denser grid (and thus a
smoother plotting surface). However, it’s important to note that the
larger k values used, the slower this plot will
generate.
plot_params(hockey$x, hockey$y, k=200, dlim=c(0.5, 1.5), glim=c(0.25, 2.75))
Saving and Reusing z
Once you’ve settled on the bounds for \(\delta\) and \(\gamma\) you want to plot and the density
of the grid via k, you can save the underlying matrix of
posterior model probabilities to reuse while making minor plot
adjustments. This will save you the hassle of having to recalculate
these values, which is where the bottleneck on time occurs. To save this
matrix, set return_z=TRUE and store the returned list from
plot_params().
zmat_list <- plot_params(hockey$x, hockey$y, k=200, dlim=c(0.5, 1.5), glim=c(0.25, 2.75),
return_z = TRUE)
The returned list is printed below. Note that the matrix is the first
entry in the list named z. Additionally, the bounds passed
via dlim and glim, and k are
returned as a reminder of what values were used to construct
z.
zmat_list
> $z
> 0.25 0.262562814070352 0.275125628140704
> 0.5 7.232601e-35 1.480850e-34 3.016613e-34
> 0.505025125628141 3.785684e-34 7.721926e-34 1.567085e-33
...
> $dlim
> [1] 0.5 1.5
>
> $glim
> [1] 0.25 2.75
>
> $k
> [1] 200
While it’s a little hard to see what z looks like in the
output above, below shows that z is a 200 by 200 matrix
(where 200 is the grid size set by k). The rows and columns
are named based on the value of \(\delta\) and \(\gamma\) for which the posterior model
probability was calculated.
class(zmat_list$z)
> [1] "matrix" "array"
dim(zmat_list$z)
> [1] 200 200
Now we can reuse the matrix to adjust the plot by passing
zmat_list$z to z in
plot_params(). However, this trick is only useful so long
as you do not expand the bounds over which you want to plot. Below shows
the same z matrix plotted on (1) the same range, (2) a
smaller range, and (3) a larger range than over which is was calculated.
The white that appears in the third frame reflects the fact that the
posterior model probability was not calculated for those grid cells
since they are outside of the original bounds.
par(mfrow=c(1,3))
plot_params(z=zmat_list$z, k=200, dlim=c(0.5, 1.5), glim=c(0.25, 2.75), main="Same range as z")
plot_params(z=zmat_list$z, k=200, dlim=c(0.7, 1.3), glim=c(0.5, 2.5), main="Smaller range than z")
plot_params(z=zmat_list$z, k=200, dlim=c(1e-04, 5), glim=c(1e-04, 5), main="Larger range than z")
We’ll continue to use this trick throughout to reduce plotting
time.
Adding contours and using countors_only=TRUE
To add contours at specific levels of the posterior model probability
of calibration, users can specify these levels in a vector via
t_levels. Below, we add contours at t = 0.95, 0.9, and 0.8.
See (Passing Additional / Different Arguments to
image.plot() and contour())[#contour_opts] for
guidance on adjusting the contours (i.e. changing color, line width,
labels, etc).
plot_params(z=zmat_list$z, t_levels=c(0.95, 0.9, 0.8),
k=200, dlim=c(0.5, 1.5), glim=c(0.25, 2.75))
To plot the contour surface without the color background, users
should specify contours_only=TRUE as shown below. We’ll
also add a few more contour levels.
plot_params(z=zmat_list$z, t_levels=c(0.99, 0.95, 0.9, 0.8, 0.7),
k=200, dlim=c(0.5, 1.5), glim=c(0.25, 2.75), contours_only=TRUE)
Passing Additional / Different Arguments to
image.plot() and contour()
While plot_params() obscures the call to
image.plot() and contour(), we allow users to
leverage their full capabilities via arguments
imgplt_options and contour_options. To pass
additional arguments to image.plot() and
contour(), users can create a list whose entries match the
names of the arguments in these functions and pass them to
imgplt_options and contour_options,
respectively.
For example, below we can move the legend to appear horizontally
below the plot using argument horizontal = TRUE in
image.plot(). Additionally, we’ll reduce the number of
colors used for plotting via nlevel=10 and add a legend
label via legend.lab. Notice how these are passed as a list
to imgplt_options.
plot_params(z=zmat_list$z, k=200, dlim=c(0.5, 1.5), glim=c(0.25, 2.75),
imgplt_options=list(horizontal = TRUE, nlevel=10, legend.lab="Posterior Model Prob"))
We can also pass additional arguments to contour() via
contour_options. Below, we’ll draw contours at t = 0.99 and
0.1. To make them dotted lines instead of solid, we’ll use
lty="dotted", we’ll change the contour color to hot pink,
and we’ll increase the line width using lwd=2.
plot_params(z=zmat_list$z, k=200, dlim=c(0.5, 1.5), glim=c(0.25, 2.75), t_levels = c(0.99, 0.1),
contour_options=list(lty = "dotted", col="hotpink", lwd=2))
Visualizing Recalibrated Probability Forecasts via
lineplot()
The goal of the lineplot function is to visualize how
predicted probabilities change under different recalibration parameters.
By default this function shows changes in the original probabilities to
the MLE recalibrated probabilities. Let’s see what this looks like with
the FiveThirtyEight hockey data.
lineplot(x=hockey$x, y=hockey$y)
Note that this function leverages the flexibility of the
ggplot2 package, so cosmetic modification and how the plot
is returned is a little different than plot_params. We will
demonstrate this throughout the following sections.
Specifying Levels of Boldness-Recalibration via
t_levels
Similar to specifying contour levels in plot_params,
users can specify levels of boldness-recalibration using
t_levels in lineplot(). Below we add 95%, 90%,
and 80% boldness-recalibration to the plot. Notice in the lineplot for
FiveThirtyEight below that the posterior model probabilities on the
x-axis are not in ascending or descending order. Instead, they simply
follow the ordering of how one might use the BRcal package:
first looking at the original predictions, then maximizing calibration,
then examining how far they can spread out predictions while maintaining
calibration with boldness-recalibration.
lineplot(x=hockey$x, y=hockey$y, t_levels = c(0.95, 0.9, 0.8))
Saving and reusing df
While the time expense of lineplot isn’t as high as
plot_params, there is still a call to optim
for MLE recalibration and a call to nloptr for each level
of boldness-recalibration. Similar to returning the z
matrix in plot_params, users of lineplot can
specify return_df to return the underlying dataframe used
to construct the plot.
By default, return_df=FALSE and lineplot()
only returns the ggplot object of the lineplot. When
return_df=TRUE, a list is returned with two entries: (1)
plot which is the ggplot object of the
lineplot and (2) df which is the underlying dataframe. The
code below shows how to specify return_df=TRUE and saved
the returned list.
lp <- lineplot(x=hockey$x, y=hockey$y, return_df = TRUE, t_levels = c(0.95, 0.9, 0.8))
To extract the plot, we can use the following code.
We can also extract the dataframe using lp$df. The first
and last few rows of the dataframe plus a summary of the dataframe are
printed below. Notice there are 6 columns, here are their brief
descriptions (more details below):
probs: the values of each predicted probability under
each setoutcome: the corresponding outcome for each predicted
probability of calibrationpost: the posterior model probability of the set as a
wholeid: the id of each probability used for mapping
observations between setsset: the set with which the probability belongs tolabel: the label used for the x-axis in the
lineplot
lp$df
> probs outcome post id set label
> 1 0.55528238 1 0.9903632 1 Original Original \n(0.99036)
> 2 0.64177575 1 0.9903632 2 Original Original \n(0.99036)
> 3 0.55490924 1 0.9903632 3 Original Original \n(0.99036)
> 4 0.51837525 0 0.9903632 4 Original Original \n(0.99036)
> 5 0.55828545 0 0.9903632 5 Original Original \n(0.99036)
> 6 0.45427667 0 0.9903632 6 Original Original \n(0.99036)
...
> 4336 0.45558625 1 0.8000000 864 80% B-R 80% B-R\n(0.8)
> 4337 0.81615909 1 0.8000000 865 80% B-R 80% B-R\n(0.8)
> 4338 0.73767174 1 0.8000000 866 80% B-R 80% B-R\n(0.8)
> 4339 0.24187950 0 0.8000000 867 80% B-R 80% B-R\n(0.8)
> 4340 0.45564257 0 0.8000000 868 80% B-R 80% B-R\n(0.8)
summary(lp$df)
> probs outcome post id set
> Min. :0.09135 0:2025 Min. :0.8000 Min. : 1.0 80% B-R :868
> 1st Qu.:0.43415 1:2315 1st Qu.:0.9000 1st Qu.:217.8 90% B-R :868
> Median :0.53261 Median :0.9500 Median :434.5 95% B-R :868
> Mean :0.53226 Mean :0.9278 Mean :434.5 MLE Recal:868
> 3rd Qu.:0.63269 3rd Qu.:0.9904 3rd Qu.:651.2 Original :868
> Max. :0.91671 Max. :0.9989 Max. :868.0
> label
> Original \n(0.99036) :868
> MLE Recal. \n(0.99885):868
> 95% B-R\n(0.95) :868
> 90% B-R\n(0.9) :868
> 80% B-R\n(0.8) :868
>
The probs column contains each set of predicted
probabilities that are plotted (i.e. the original, MLE recalibrated, and
each set of boldness-recalibrated probabilities). The
outcome column contains the corresponding outcome for each
value in probs. Each set of probabilities and outcomes are
essentially “stacked” on top of each other. The id column
contains a unique id for each observation to map how observations change
between sets. This essentially tells the plotting function how to
connect (with line) the same observation as is changes from the original
set to MLE- or boldness-recalibration. The set column
contains a factor for which set that observation belongs to. The
post column contains the value of the posterior model
probability for the set as a whole, so this value will be the same for
all observations in the same set. Lastly, the label column
is a string that is used to label the x-axis for each set in the
lineplot and is also the same for all observations in a set.
Since this dataframe has each set of probabilities stacked, the
number of rows in the dataframe will be \(\text{# of sets } \times n\), where \(n\) is the number of observations per set.
For example, in the data frame above, there are 5 sets (original, MLE,
95%, 90%, and 80% boldness-recalibration) with 868 observations total.
So the length of the dataframe is 5 * 868 = 4340. This is confirmed
below.
Now with this dataframe saved, we can pass it back via
df without needing to specify x or
y and we get the same plot as before, but much more
quickly. This functionality makes it easier to make cosmetic adjustments
to your plot without needing to wait for computations under the
hood.
lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8))
Users can also choose to omit specific boldness-recalibration sets
even if they are included in df. In the code below, we omit
the 90% and 80% boldness-recalibrated sets by only setting
t_levels=0.95.
lineplot(df=lp$df, t_levels=c(0.95))
Omitting the Original or MLE-Recalibrated sets from plotting via
plot_original and plot_MLE
To omit either the original set or the MLE-recalibrated set from
plotting, users can specify plot_original=FALSE or
plot_MLE=FALSE, respectively. Below is an example of
omitting the original set by setting plot_original=FALSE
while plotting the MLE-recalibrated and boldness-recalibration sets
returned in df above.
lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8), plot_original=FALSE)
Cosmetic Adjustments
A key difference in how this plot is created compared to
plot_params() is that it uses ggplot2 instead
of base R graphics. Users of lineplot() have
quite a few options for making cosmetic changes to these plots. The
first set of options we’ll discuss are those related to the points and
lines already present on the lineplot. To modify the appearance of the
points, users can pass options to geom_point() via a list
to ggpoint_options. For example, the code below shows how
to change the shape and size of the points.
lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8), ggpoint_options=list(size=3, shape=1))
lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8), ggline_options=list(linewidth=2, alpha=0.1))
An important thing to note about these two options is that their
purpose is solely for adjustments that are not part of the
aes statement in geom_point() or
geom_line(). Should users specify aes options
via ggpoint_options or ggline_options, this
will cause ggplot() to create a new layer of points or
lines on top of those following the default settings we use under the
hood.
Another option for making cosmetic adjustments is to use the
convention of ggplot2 and add additional
ggplot2 functions to the returned lineplot using
+. For example, the code below shows how we could add the
scale_color_manual() function to our plot to change the
colors of the points and lines. Notice that ggplot2 throws
a warning about another color scale already being present. This is
because under the hood we have already specified the default color
scheme to be red and blue. A similar warning may appear in any case
where a user adds a function that is already present in our original
construction of the lineplot.
lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8)) +
scale_color_manual(values = c("orchid", "darkgreen"))
> Scale for colour is already present.
> Adding another scale for colour, which will replace the existing scale.
Thinning
Another strategy to save time when plotting is to reduce, or “thin”,
the amount of data plotted. When sample sizes are large, the plot can
become overcrowded and slow to plot. We provide three options for
thinning:
thin_to: the observations in (x,y) are randomly sampled
without replacement to form a set of size thin_tothin_prop: the observations in (x,y) are randomly
sampled without replacement to form a set that is thin_prop
* 100% of the original size of (x,y)thin_by: the observations in (x,y) are thinned by
selecting every thin_by observation
By default, all three of these settings are set to NULL,
meaning no thinning is performed. Users can only specify one thinning
strategy at a time. Care should be taken in selecting a thinning
approach based on the nature of your data and problem. Note that MLE
recalibration and boldness-recalibration will be done using the full
set, so the posterior model probabilities are those of the full set.
Also note that if a thinning strategy is used with
return_df=TRUE, the returned data frame will only
contain the reduced set (i.e. the data after
thinning).
Below is an example of each of these strategies. Notice that each
plot looks a little different since the observations selected are a
little different under each strategy.
lp1 <- lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8), thin_to=500)
lp2 <- lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8), thin_prop=0.5)
lp3 <- lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8), thin_by=2)
gridExtra::grid.arrange(lp1, lp2, lp3, ncol=3)
By default, there is a seed set so that the observations selected
stay the same through successive calls using the same thinning strategy.
To change the seed, users can specify their own via seed.
Notice in the example below that we’re using the same strategy as panel
one in the above set of plots but the points are slightly different.
lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8), thin_to=500, seed = 47)
Alternatively, if users would prefer no seed be set, they can set
seed=NULL. In the plot below, we again have a different set
of points and lines. In fact, each compilation of this vignette will
have a different set.
lineplot(df=lp$df, t_levels=c(0.95, 0.9, 0.8), thin_to=500, seed=NULL)
Passing Additional Arguments to optim and
nloptr
Similar to the plot_params function, this function
allows users to specify additional arguments to optim via
optim_options. Additionally, we also allow users to pass
arguments to nloptr via nloptr_options in a
similar fashion. In this case, the list passed via
nloptr_options is passed to opts in the
brcal function. For more information on how to structure
this list, see Passing Additional / Different
Arguments to nloptr().
