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, …, b - step*.

Method `"lower"`

is the backward difference of the cdf *F*:

*p[x] = F(x) - F(x - h)*

for *x = a +
h, …, 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, …, 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] = (E[min(X, a)] - E[min(X, a + h)])/h + 1 - F(a)*

*
p[x] = (2 E[min(X, x)] - E[min(X, x - h)] - E[min(X, x + h)])/h, a < x < b*

*
p[b] = (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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