Density, distribution function, quantile function and random generation for the generally--altered, --inflated and --truncated 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. Altogether it can be abbreviated as GAAIIT--Pois(lambda.p)--Pois(lambda.a)--MLM--Pois(lambda.i)--MLM, and it is also known as the GAIT-Pois PNP combo where PNP stands for parametric and nonparametric.
dgaitpois(x, lambda.p, alt.mix = NULL, alt.mlm = NULL,
inf.mix = NULL, inf.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0,
pstr.mix = 0, pstr.mlm = 0, byrow.ai = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p,
deflation = FALSE, log = FALSE)
pgaitpois(q, lambda.p, alt.mix = NULL, alt.mlm = NULL,
inf.mix = NULL, inf.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0,
pstr.mix = 0, pstr.mlm = 0, byrow.ai = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p, lower.tail = TRUE)
qgaitpois(p, lambda.p, alt.mix = NULL, alt.mlm = NULL,
inf.mix = NULL, inf.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0,
pstr.mix = 0, pstr.mlm = 0, byrow.ai = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p)
rgaitpois(n, lambda.p, alt.mix = NULL, alt.mlm = NULL,
inf.mix = NULL, inf.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0,
pstr.mix = 0, pstr.mlm = 0, byrow.ai = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p)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 (inner) distribution.
The other two concern the parametric variant and
these outer distributions (usually spikes) may be
altered and/or inflated.
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 truncated values for the
parametric variant.
Each argument must have unique values only.
Assigning argument alt.mix
means that pobs.mix will be used.
Assigning argument inf.mix
means that pstr.mix will be used.
If alt.mix is of unit length
then the default probability mass function (PMF)
evaluated at alt.mix will be pobs.mix.
So having alt.mix = 0 corresponds to the
zero-inflated Poisson distribution (see Zipois).
Similar to the above, but for the nonparametric (MLM) variant.
Assigning argument alt.mlm
means that pobs.mlm will be used.
Assigning argument inf.mlm
means that pstr.mlm will be used.
Collectively, the above 6 arguments represent
5 disjoint sets of
special values and they are a proper subset of the support of the
distribution.
The first two arguments are coerced into a matrix of probabilities
using byrow.ai to determine the order of the elements
(similar to byrow in matrix, and
the .ai reinforces the behaviour that it applies to both
altered and inflated cases).
The first argument is recycled if necessary to become
n x length(alt.mlm).
The second argument becomes
n x length(inf.mlm).
Thus these arguments are not used unless
alt.mlm and inf.mlm are assigned.
If deflation = TRUE then pstr.mlm may be negative.
Vectors of probabilities that are recycled if necessary to
length \(n\).
The first argument is used when alt.mix is not NULL.
The second argument is used when inf.mix is not NULL.
Logical. If TRUE then pstr.mlm is allowed to have
negative values,
however, not too negative so that the final PMF becomes negative.
Of course, if the values are negative then they cannot be
interpreted as probabilities.
In theory, one could also allow pstr.mix to be negative,
but currently this is disallowed.
dgaitpois gives the density,
pgaitpois gives the distribution function,
qgaitpois gives the quantile function, and
rgaitpois generates random deviates.
The default values of the arguments correspond to ordinary
dpois,
ppois,
qpois,
rpois
respectively.
These functions allow any combination of 3 operator types:
truncation, alteration and inflation.
The precedence is
truncation, then alteration and lastly 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 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
because it is mixture of 3 Poisson distributions with
partitioned or nested support.
Any value of the support of the distribution that is
altered, inflated or truncated 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 five types of special values and they are represented by
alt.mix,
alt.mlm,
inf.mix,
inf.mlm,
truncate.
Jargonwise,
the outer distributions concern those special values which
are altered or inflated, and
the inner distribution concerns the remaining
support points that correspond directly to
the parent distribution.
These functions do what
Zapois,
Zipois,
Pospois
collectively did plus much more.
In the notation of Yee and Ma (2020)
these functions allow for the special cases:
(i) GAIT--Pois(lambda.p)--Pois(lambda.a,
alt.mix, pobs.mix)--Pois(lambda.i,
inf.mix, pstr.mix);
(ii) GAIT--Pois(lambda.p)--MLM(alt.mlm,
pobs.mlm)--MLM(inf.mlm, pstr.mlm).
Model (i) is totally parametric while model (ii) is the most
nonparametric possible.
Yee, T. W. and Ma, C. (2021). Generally--altered, --inflated and --truncated regression, with application to heaped and seeped counts. In preparation.
gaitpoisson,
multinomial,
specials,
spikeplot,
dgaitplot,
Zapois,
Zipois,
Pospois
Poisson;
Gaitbinom,
Gaitnbinom,
Gaitlog,
Gaitzeta.
# NOT RUN {
ivec <- c(6, 14); avec <- c(8, 11); lambda <- 10; xgrid <- 0:25
tvec <- 15; max.support <- 20; pobs.a <- 0.05; pstr.i <- 0.25
(ddd <- dgaitpois(xgrid, lambda, lambda.a = lambda + 5,
truncate = tvec, max.support = max.support, pobs.mix = pobs.a,
pobs.mlm = pobs.a, alt.mlm = avec,
pstr.mix = pstr.i, inf.mix = ivec))
# }
# NOT RUN {
plot(xgrid, ddd, type = "n", ylab = "Probability", xlab = "x",
main = "GAIT PNP Combo PMF---Poisson Parent")
mylwd <- 1
abline(v = avec, col = 'blue', lwd = mylwd)
abline(v = ivec, col = 'purple', lwd = mylwd)
abline(v = tvec, col = 'tan', lwd = mylwd)
abline(v = max.support, col = 'magenta', lwd = mylwd)
abline(h = c(pobs.a, pstr.i, 0:1), col = 'gray', lty = "dashed")
lines(xgrid, dpois(xgrid, lambda), col = 'gray', lty = "dashed") # f_{\pi}
lines(xgrid, ddd, type = "h", col = "pink", lwd = 7) # GAIT PNP combo PMF
points(xgrid[ddd == 0], ddd[ddd == 0], pch = 16, col = 'tan', cex = 2)
# }
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