discretize | R Documentation |
Compute a discrete probability mass function from a continuous cumulative distribution function (cdf) with various methods.
discretise
is an alias for discretize
.
discretize(cdf, from, to, step = 1,
method = c("upper", "lower", "rounding", "unbiased"),
lev, by = step, xlim = NULL)
discretise(cdf, from, to, step = 1,
method = c("upper", "lower", "rounding", "unbiased"),
lev, by = step, xlim = NULL)
cdf |
an expression written as a function of |
from, to |
the range over which the function will be discretized. |
step |
numeric; the discretization step (or span, or lag). |
method |
discretization method to use. |
lev |
an expression written as a function of |
by |
an alias for |
xlim |
numeric of length 2; if specified, it serves as default
for |
Usage is similar to curve
.
discretize
returns the probability mass function (pmf) of the
random variable obtained by discretization of the cdf specified in
cdf
.
Let F(x)
denote the cdf, E[\min(X, x)]
the
limited expected value at x
, h
the step, p_x
the probability mass at x
in the discretized distribution and
set a =
from
and b =
to
.
Method "upper"
is the forward difference of the cdf F
:
p_x = F(x + h) - F(x)
for x = a, a + h, \dots, b - step
.
Method "lower"
is the backward difference of the cdf F
:
p_x = F(x) - F(x - h)
for x = a +
h, \dots, b
and p_a = F(a)
.
Method "rounding"
has the true cdf pass through the
midpoints of the intervals [x - h/2, x + h/2)
:
p_x = F(x + h/2) - F(x - h/2)
for x = a + h, \dots, b - step
and p_a = F(a + h/2)
. The function assumes the cdf is continuous. Any
adjusment necessary for discrete distributions can be done via
cdf
.
Method "unbiased"
matches the first moment of the discretized
and the true distributions. The probabilities are as follows:
p_a = \frac{E[\min(X, a)] - E[\min(X, a + h)]}{h} + 1 - F(a)
p_x = \frac{2 E[\min(X, x)] - E[\min(X, x - h)] - E[\min(X, x +
h)]}{h}, \quad a < x < b
p_b = \frac{E[\min(X, b)] - E[\min(X, b - h)]}{h} - 1 + F(b),
A numeric vector of probabilities suitable for use in
aggregateDist
.
Vincent Goulet vincent.goulet@act.ulaval.ca
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.
aggregateDist
x <- seq(0, 5, 0.5)
op <- par(mfrow = c(1, 1), col = "black")
## Upper and lower discretization
fu <- discretize(pgamma(x, 1), method = "upper",
from = 0, to = 5, step = 0.5)
fl <- discretize(pgamma(x, 1), method = "lower",
from = 0, to = 5, step = 0.5)
curve(pgamma(x, 1), xlim = c(0, 5))
par(col = "blue")
plot(stepfun(head(x, -1), diffinv(fu)), pch = 19, add = TRUE)
par(col = "green")
plot(stepfun(x, diffinv(fl)), pch = 19, add = TRUE)
par(col = "black")
## Rounding (or midpoint) discretization
fr <- discretize(pgamma(x, 1), method = "rounding",
from = 0, to = 5, step = 0.5)
curve(pgamma(x, 1), xlim = c(0, 5))
par(col = "blue")
plot(stepfun(head(x, -1), diffinv(fr)), pch = 19, add = TRUE)
par(col = "black")
## First moment matching
fb <- discretize(pgamma(x, 1), method = "unbiased",
lev = levgamma(x, 1), from = 0, to = 5, step = 0.5)
curve(pgamma(x, 1), xlim = c(0, 5))
par(col = "blue")
plot(stepfun(x, diffinv(fb)), pch = 19, add = TRUE)
par(op)
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