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, checkd = FALSE)
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.
Logical.
If TRUE then the density is computed
at floor(q) with the same parameters.
This can help detect whether the PMF is invalid.
If so, then NaNs are returned.
See Example 2 below.
It is possible that the GAITD PMF is invalid because
of too much inflation and/or deflation.
This would result in some probabilities exceeding
unity or being negative.
Hence x should ideally contain these types
of special values so that this can be detected.
If so then a NaN is returned and
a warning is issued, e.g.,
same as dnorm(0, 0, sd = -1).
To help checking,
pgaitdpois(q, ...) calls
dgaitdpois(floor(q), ...) if checkd
is TRUE.
That is, given the parameters, it is impractical to determine whether the PMF is valid. To do this, one would have to compute the PMF at all values of its support and check that they are nonnegative and sum to unity. Hence one must be careful to input values from the parameter space, especially for inflation and deflation. See Example 2 below.
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 (2023)
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. (2024). Generally altered, inflated, truncated and deflated regression. Statistical Science, 39 (in press).
gaitdpoisson,
multinomial,
specials,
spikeplot,
dgaitdplot,
Zapois,
Zipois,
Pospois
Poisson;
Gaitdbinom,
Gaitdnbinom,
Gaitdlog,
Gaitdzeta.
# Example 1
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")
# Example 2: detection of an invalid PMF
xgrid <- 1:3 # Does not cover the special values purposely
(ddd <- dgaitdpois(xgrid, 1, pdip.mlm = 0.1, d.mlm = 5,
pstr.mix = 0.95, i.mix = 0)) # Undetected
xgrid <- 0:13 # Wider range so this detects the problem
(ddd <- dgaitdpois(xgrid, 1, pdip.mlm = 0.1, d.mlm = 5,
pstr.mix = 0.95, i.mix = 0)) # Detected
sum(ddd, na.rm = TRUE) # Something gone awry
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