Probability Bounds Analysis S4 Library for R
Place this file on your computer and, from within R, select File/Source R code... from the main menu. Select this file and click the Open button to read it into R. You should see the completion message ":pbox> library loaded". Once the library has been loaded, you can define probability distributions
a = normal(5,1)
b = uniform(2,3)
and p-boxes
c = meanvariance(0,2)
d = lognormal(interval(6,7), interval(1,2))
e = mmms(0,10,3,1)
and perform mathematical operations on them, including the Frechet convolution such as
a %+% b
or a traditional convolution assuming independence
a %|+|% b
If you do not enclose the operator inside percent signs or vertical bars, the software tries to figure out how the arguments are related to one another. Expressions such as
a + b
a + log(a) * b
autoselect the convolution to use. If the software cannot tell what dependence the arguments have, it uses a Frechet convolution, which is conservative because it makes no assumption about what dependence the operands might have.
Variables containing probability distributions or p-boxes are assumed to be independent of one another unless one formally depends on the other, which happens if one was created as a function of the other. Assigning a variable containing a probability distribution or a p-box to another variable makes the two variables perfectly dependent. To make an independent copy of the distribution or p-box, use the samedistribution function, e.g., c = samedistribution(a).
By default, separately constructed distributions such as
a = normal(5,1)
b = uniform(2,3)
will be assumed to be independent (so their convolution a+b will be a precise distribution). You can acknowledge any dependencies between uncertain numbers by mentioning their dependence when you construct them with expressions like
b = uniform(2,3, depends=a)
or
b = pbox(uniform(2,3), depends=a)
You can also make an existing uncertain number dependent on another with an assignment like
b = pbox(b, depends=a)
You can acknowledge several dependencies at a time, as with
d = pbox(d, depends=c(a,b))
but you can't mention a variable in the 'depends' array before the value exists. Also, it only makes sense to specify such dependence among distributions and p-boxes. Using a scalar or an interval in the 'depends' array will precipitate an error.
As alternatives to independence and (unspecified) dependence, you can also specify that an uncertain number is perfectly or oppositely dependent on another.
d = beta(2,5, perfect=a)
or
d = beta(2,5, opposite=a)
These dependency specifications are automatically mutual, so it is not necessary to explicitly make the reciprocal assignment. Thus
a = N(5,1)
b = U(2,3, perfect=a)
c = N(15,2, perfect=b)
suffices to link c with a and vice versa. The assignments
automatically make a, b, and c mutually perfectly dependent.
Naturally, it is not possible to be perfectly (or oppositely)
dependent on more than one quantity unless they are also
mutually dependent in the same way. The indep, perfect,
opposite and depend functions check whether their two
arguments are independent, perfectly dependent, oppositely
dependent, or dependent, respectively. The depend function
returns an interval code that is zero if its arguments are
independent, +1 if they are perfect, [-1,+1] if they have
an unknown dependence, etc.
The defined mathematical infix operators include these tabled below.
| Auto | Frechet | Perfect | Opposite | Independent | |
|---|---|---|---|---|---|
| Addition | + | %+% | %/+/% | %o+o% | %|+|% |
| Subtraction | - | %-% | %/-/% | %o-o% | %|-|% |
| Product | * | %*% | %/*/% | %o*o% | %|*|% |
| Division | / | %/% | %///% | %o/o% | %|/|% |
| Minimum | %m% | %/m/% | %omo% | %|m|% | |
| Maximum | %M% | %/M/% | %oMo% | %|M|% | |
| Powers | ^ | %^% | %/^/% | %o^o% | %|^|% |
| Less than | < | %<% | %/</% | %o<o% | %|<|% |
| Greater than | > | %>% | %/>/% | %o>o% | %|>|% |
| Less or equal | <= | %<=% | %/<=/% | %o<=o% | %|<=|% |
| Greater/equal | >= | %>=% | %/>=/% | %o>=o% | %|>=|% |
| Conjunction | %&% | %|&|% | |||
| Disjunction | %|% | %|||% |
The basic operator symbols +, -, *, / and ^ have been overloaded so that they also work with uncertain numbers, i.e., probability distributions, p-boxes and intervals. Note that the operators %*% and %/% (which in R normally invoke matrix multiplication and integer division) have been reassigned (not overloaded, so they no longer do matrix multiplication and integer division). Also notice that there are no autoselected infix operators for minimum and maximum. The pmin and pmax functions will return the Frechet convolutions. Also notice that &, |, &&, || have not been overloaded for uncertain numbers because R has sealed those operators. You must use the operators with percent signs to compute conjunctions or disjunctions.
Alternatively, the various convolution operations can be accessed by calling functions:
autoselect(x,y,op)
frechetconv.pbox(x,y,op)
perfectconv.pbox(x,y,op)
oppositeconv.pbox(x,y,op)
conv.pbox(x,y,op)
positiveconv(x,y,op)
negativeconv(x,y,op)
where x and y are the operands and op denotes the operation, such as '+'.
In addition to these "in-fix" operators, several binary functions are also defined such as
env, imp, pmin, pmax, smin, smax, and, or, not
Note that the imp function gives the intersection of uncertain numbers. Several standard mathematical transformations have also been extended to handle p-boxes, including
exp, log, sqrt, atan, abs, negate, reciprocate, int
Use the output commands to see the resulting uncertain numbers, such as
show(c)
summary(d)
plot(a)
lines(b, col='blue')
There are a variety of standard functions you can use with distributions and p-boxes, such as
mean(a)
sd(b)
var(b)
median(c)
as well as some new functions such as
breadth(d)
leftside(c)
left(a)
prob(a, 3)
cut(a, 0.2)
This R library is under development. We would appreciate your comments, questions and suggestions.