cnm | R Documentation |
Function cnm
can be used to compute the maximum
likelihood estimate of a nonparametric mixing distribution
(NPMLE) that has a one-dimensional mixing parameter, or simply
the mixing proportions with support points held fixed.
A finite mixture model has a density of the form
f(x; \pi, \theta, \beta) = \sum_{j=1}^k \pi_j f(x; \theta_j,
\beta).
where \pi_j \ge 0
and \sum_{j=1}^k \pi_j
=1
.
A nonparametric mixture model has a density of the form
f(x; G) = \int f(x; \theta) d G(\theta),
where G
is a mixing distribution
that is completely unspecified. The maximum likelihood estimate of
the nonparametric G
, or the NPMLE of G
, is known to be
a discrete distribution function.
Function cnm
implements the CNM algorithm that is proposed
in Wang (2007) and the hierarchical CNM algorithm of Wang and
Taylor (2013). The implementation is generic using S3
object-oriented programming, in the sense that it works for an
arbitrary family of mixture models defined by the user. The user,
however, needs to supply the implementations of the following
functions for their self-defined family of mixture models, as they
are needed internally by function cnm
:
initial(x, beta, mix, kmax)
valid(x, beta, theta)
logd(x, beta, pt, which)
gridpoints(x, beta, grid)
suppspace(x, beta)
length(x)
print(x, ...)
weight(x, ...)
While not needed by the algorithm for finding the solution, one may also implement
plot(x, mix, beta, ...)
so that the fitted model can be shown graphically in a
user-defined way. Inside cnm
, it is used when
plot="probability"
so that the convergence of the algorithm
can be graphically monitored.
For creating a new class, the user may consult the implementations
of these functions for the families of mixture models included in
the package, e.g., npnorm
and nppois
.
cnm(
x,
init = NULL,
model = c("npmle", "proportions"),
maxit = 100,
tol = 1e-06,
grid = 100,
plot = c("null", "gradient", "probability"),
verbose = 0
)
x |
a data object of some class that is fully defined by the user. The user needs to supply certain functions as described below. |
init |
list of user-provided initial values for the mixing
distribution |
model |
the type of model that is to estimated: the
non-parametric MLE (if |
maxit |
maximum number of iterations. |
tol |
a tolerance value needed to terminate an
algorithm. Specifically, the algorithm is terminated, if the
increase of the log-likelihood value after an iteration is less
than |
grid |
number of grid points that are used by the algorithm to
locate all the local maxima of the gradient function. A larger
number increases the chance of locating all local maxima, at the
expense of an increased computational cost. The locations of the
grid points are determined by the function |
plot |
whether a plot is produced at each iteration. Useful
for monitoring the convergence of the algorithm. If
|
verbose |
verbosity level for printing intermediate results in each iteration, including none (= 0), the log-likelihood value (= 1), the maximum gradient (= 2), the support points of the mixing distribution (= 3), the mixing proportions (= 4), and if available, the value of the structural parameter beta (= 5). |
family |
the name of the mixture family that is used to fit to the data. |
num.iterations |
number of iterations required by the algorithm |
max.gradient |
maximum value of the gradient function, evaluated at the beginning of the final iteration |
convergence |
convergence code. |
ll |
log-likelihood value at convergence |
mix |
MLE of the mixing distribution, being an object of the
class |
beta |
value of the structural parameter, that is held fixed throughout the computation. |
Yong Wang <yongwang@auckland.ac.nz>
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86
Wang, Y. and Taylor, S. M. (2013). Efficient computation of nonparametric survival functions via a hierarchical mixture formulation. Statistics and Computing, 23, 713-725.
nnls
, npnorm
,
nppois
, cnmms
.
## Simulated data
x = rnppois(200, disc(c(1,4), c(0.7,0.3))) # Poisson mixture
(r = cnm(x))
plot(r, x)
x = rnpnorm(200, disc(c(0,4), c(0.3,0.7)), sd=1) # Normal mixture
plot(cnm(x), x) # sd = 1
plot(cnm(x, init=list(beta=0.5)), x) # sd = 0.5
## Real-world data
data(thai)
plot(cnm(x <- nppois(thai)), x) # Poisson mixture
data(brca)
plot(cnm(x <- npnorm(brca)), x) # Normal mixture
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.