Density, distribution function, quantile function and random generation for the generally--altered, --inflated, --truncated and --deflated Poisson distribution. Both parametric and nonparametric variants are supported; these are based on finite mixtures of the parent with itself and the multinomial logit model (MLM) respectively.
dgaitdpois(x, lambda.p, a.mix = NULL, a.mlm = NULL, i.mix = NULL,
i.mlm = NULL, d.mix = NULL, d.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0, pstr.mix = 0,
pstr.mlm = 0, pdip.mix = 0, pdip.mlm = 0, byrow.aid = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p,
lambda.d = lambda.p, log = FALSE)
pgaitdpois(q, lambda.p, a.mix = NULL, a.mlm = NULL, i.mix = NULL,
i.mlm = NULL, d.mix = NULL, d.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0, pstr.mix = 0,
pstr.mlm = 0, pdip.mix = 0, pdip.mlm = 0, byrow.aid = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p,
lambda.d = lambda.p, lower.tail = TRUE)
qgaitdpois(p, lambda.p, a.mix = NULL, a.mlm = NULL, i.mix = NULL,
i.mlm = NULL, d.mix = NULL, d.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0, pstr.mix = 0,
pstr.mlm = 0, pdip.mix = 0, pdip.mlm = 0, byrow.aid = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p, lambda.d = lambda.p)
rgaitdpois(n, lambda.p, a.mix = NULL, a.mlm = NULL, i.mix = NULL,
i.mlm = NULL, d.mix = NULL, d.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0, pstr.mix = 0,
pstr.mlm = 0, pdip.mix = 0, pdip.mlm = 0, byrow.aid = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p, lambda.d = lambda.p)
dgaitdpois
gives the density,
pgaitdpois
gives the distribution function,
qgaitdpois
gives the quantile function, and
rgaitdpois
generates random deviates.
The default values of the arguments correspond to ordinary
respectively.
Same meaning as in Poisson
.
Same meaning as in Poisson
.
Same meaning as in Poisson
,
i.e., for an ordinary Poisson distribution.
The first is for the main parent (or base) distribution.
The next two concern the parametric variant and
these distributions (usually spikes) may be
altered and/or inflated.
The last one concerns the deflated variant.
Short vectors are recycled.
numeric; these specify the set of truncated values.
The default value of NULL
means an empty set
for the former.
The latter is the
maximum support value so that any value larger
has been truncated (necessary because
truncate = (max.support + 1):Inf
is not allowed),
hence is needed for truncating the upper tail of the
distribution. Note that max(truncate) < max.support
must be satisfied otherwise an error message will be issued.
Vectors of nonnegative integers;
the altered, inflated and deflated values for the
parametric variant.
Each argument must have unique values only.
Assigning argument a.mix
means that pobs.mix
will be used.
Assigning i.mix
means that pstr.mix
will be used.
Assigning d.mix
means that pdip.mix
will be used.
If a.mix
is of unit length
then the default probability mass function (PMF)
evaluated at a.mix
will be pobs.mix
.
So having a.mix = 0
corresponds to the
zero-inflated Poisson distribution (see Zipois
).
Similar to the above, but for the nonparametric (MLM) variant.
For example, assigning a.mlm
means that pobs.mlm
will be used.
Collectively, the above 7 arguments represent
7 disjoint sets of
special values and they are a proper subset of the support
of the distribution.
The first three arguments are coerced into a matrix of
probabilities
using byrow.aid
to determine the order of the elements
(similar to byrow
in matrix
, and
the .aid
reinforces the behaviour that it applies to both
altered, inflated and deflated cases).
The first argument is recycled if necessary to become
n x length(a.mlm)
.
The second argument becomes
n x length(i.mlm)
.
The third argument becomes
n x length(d.mlm)
.
Thus these arguments are not used unless
a.mlm
, i.mlm
and d.mlm
are assigned.
For deflated models, pdip.mix
and pdip.mlm
are
positive-valued and VGAM will subtract these quantities;
the argument deflation
has been deprecated.
Vectors of probabilities that are recycled if necessary to
length \(n\).
The first argument is used when a.mix
is not NULL
.
The second argument is used when i.mix
is not NULL
.
The third argument is used when d.mix
is not NULL
.
T. W. Yee.
These functions allow any combination of 4 operator types:
truncation, alteration, inflation and deflation.
The precedence is
truncation, then alteration and lastly inflation and deflation.
Informally, deflation can be thought of as the
opposite of inflation.
This order minimizes the potential interference among the operators.
Loosely, a set of probabilities is set to 0 by truncation
and the remaining probabilities are scaled up.
Then a different set of probabilities are set to some
values pobs.mix
and/or pobs.mlm
and the remaining probabilities are rescaled up.
Then another different set of probabilities is inflated by
an amount pstr.mlm
and/or proportional
to pstr.mix
so that individual elements in this set have two sources.
Then another different set of probabilities is deflated by
an amount pdip.mlm
and/or proportional
to pdip.mix
.
Then all the probabilities are
rescaled so that they sum to unity.
Both parametric and nonparametric variants are implemented.
They usually have arguments with suffix
.mix
and .mlm
respectively.
The MLM is a loose coupling that effectively separates
the parent (or base) distribution from
the altered values.
Values inflated nonparametrically effectively have
their spikes shaved off.
The .mix
variant has associated with it
lambda.a
and lambda.i
and lambda.d
because it is mixture of 4 Poisson distributions with
partitioned or nested support.
Any value of the support of the distribution that is
altered, inflated, truncated or deflated
is called a special value.
A special value that is altered may mean that its probability
increases or decreases relative to the parent distribution.
An inflated special value means that its probability has
increased, provided alteration elsewhere has not made it decrease
in the first case.
There are seven types of special values and they are
represented by
a.mix
,
a.mlm
,
i.mix
,
i.mlm
,
d.mix
,
d.mlm
,
truncate
.
Terminology-wise, special values
are altered or inflated or truncated or deflated, and
the remaining support points that correspond directly to
the parent distribution are nonspecial or ordinary.
These functions do what
Zapois
,
Zipois
,
Pospois
collectively did plus much more.
In the notation of Yee and Ma (2022)
these functions allow for the special cases:
(i) GAIT--Pois(lambda.p
)--Pois(lambda.a
,
a.mix
, pobs.mix
)--Pois(lambda.i
,
i.mix
, pstr.mix
);
(ii) GAIT--Pois(lambda.p
)--MLM(a.mlm
,
pobs.mlm
)--MLM(i.mlm
, pstr.mlm
).
Model (i) is totally parametric while model (ii) is the most
nonparametric possible.
Yee, T. W. and Ma, C. (2022). Generally--altered, --inflated, --truncated and --deflated regression, with application to heaped and seeped data. In preparation.
gaitdpoisson
,
multinomial
,
specials
,
spikeplot
,
dgaitdplot
,
Zapois
,
Zipois
,
Pospois
Poisson
;
Gaitdbinom
,
Gaitdnbinom
,
Gaitdlog
,
Gaitdzeta
.
ivec <- c(6, 14); avec <- c(8, 11); lambda <- 10; xgrid <- 0:25
tvec <- 15; max.support <- 20; pobs.mix <- 0.05; pstr.i <- 0.25
dvec <- 13; pdip.mlm <- 0.05; pobs.mlm <- 0.05
(ddd <- dgaitdpois(xgrid, lambda, lambda.a = lambda + 5,
truncate = tvec, max.support = max.support, pobs.mix = pobs.mix,
pobs.mlm = pobs.mlm, a.mlm = avec,
pdip.mlm = pdip.mlm, d.mlm = dvec,
pstr.mix = pstr.i, i.mix = ivec))
if (FALSE) dgaitdplot(lambda, ylab = "Probability", xlab = "x",
truncate = tvec, max.support = max.support, pobs.mix = pobs.mix,
pobs.mlm = pobs.mlm, a.mlm = avec, all.lwd = 3,
pdip.mlm = pdip.mlm, d.mlm = dvec,
pstr.mix = pstr.i, i.mix = ivec, deflation = TRUE,
main = "GAITD Combo PMF---Poisson Parent")
Run the code above in your browser using DataLab