There are custom methods implemented for three out of four S3 group generic functions: `Math`

, `Ops`

, `Summary`

. **Note** that many
of them have random nature with an idea of generating samples from input
pdqr-functions, performing certain operation on them (results in one
generated sample from desired random variable), and creating new
pdqr-function with appropriate new_*() function. This is done
with `form_trans()`

, so all rules for determining class and
type of output is taken from it.

```
# S3 method for pdqr
Math(x, ...)
# S3 method for pdqr
Ops(e1, e2)
# S3 method for pdqr
Summary(..., na.rm = FALSE)
```

x, e1, e2 | Objects. |
---|---|

... | Further arguments passed to methods. |

na.rm | Logical: should missing values be removed? |

All methods return pdqr-function which represents the result of
applying certain function to random variable(s) described with input
pdqr-function(s). **Note** that independence of input random variables is
assumed, i.e. `f + f`

is not the same as `2*f`

(see Examples).

Customization of method behavior may be done using mechanism of options(). These are the possible options:

. This will be used as`pdqr.group_gen.args_new`

`args_new`

argument for`form_trans()`

in methods with random nature. Default is`list()`

.. This will be used as`pdqr.group_gen.n_sample`

`n_sample`

argument for`form_trans()`

in methods with random nature. Default is 10000.. All methods that have random nature take care of output support by trying to "repair" it, because default use of`pdqr.group_gen.repair_supp_method`

`new_*()`

functions returns a slightly bigger support than range of input sample (see Examples). Repairing is done with`form_resupport()`

where target support is computed separately and`method`

argument is controlled by this option (preferred ones are`"reflect"`

, default, and`"trim"`

). In most cases output support is computed directly based on special features of generic function. But for some difficult cases, like`gamma()`

,`digamma()`

,`lgamma()`

,`psigamma()`

,`^`

, and`%%`

it is a result of simulation (i.e. slightly random, which slightly increases random nature of those methods).

This family of S3 generics represents mathematical functions. Most of the
methods have **random nature**, except `abs()`

and `sign()`

which are
computed directly. Output of `sign()`

has "discrete" type with 3 "x" values:
-1, 0, 1.

**Note** that `cumsum()`

, `cumprod()`

, `cummmax()`

, and `cummin()`

functions
don't make much sense in these implementations: their outputs represent
random variable, sample of which is computed by applying `cum*()`

function to
a sample, generated from input pdqr-function.

This family of S3 generics represents common operators. For all functions
(except `&`

and `|`

) input can be a pdqr-function or single number.

A list of methods with **non-random nature**:

`!`

,`+`

,`-`

in case of single input, i.e.`!f`

or`-f`

.Functions representing linear transformation, i.e. adding, subtracting, multiplying, and dividing by a single number. For example, all

`f + 1`

,`2 - f`

(which is actually`(-f) + 2`

),`3*f`

and`f/2`

are linear transformations, but`1 / f`

,`f + g`

are not.Functions for comparing:

`==`

,`!=`

,`<`

,`<=`

,`>=`

,`>`

. Their output is**boolean pdqr-function**: "discrete" type function with elements being exactly 0 and 1. Probability of 0 represents probability of operator output being false, and 1 - being true. Probability of being true is computed directly as**limit of empirical estimation from simulations**(as size of samples grows to infinity). In other words, output is an exact number which might be approximated by simulating two big samples of same size from input`e1`

and`e2`

(one of which can be a single number), and estimating probability as share of those pairs from samples for which comparison is true.**Note**that if at least one input has "continuous" type, then:`==`

will always have probability 0 of being true because probability of generating a certain exact one or two numbers from continuous random variable is zero.`!=`

will always have probability 1 of being true for the same reason as above.Pairs

`>=`

and`>`

,`<=`

and`<`

will return the same input because probability of being equal is always zero.

Logical functions

`&`

and`|`

. Their input can be only pdqr-functions (because single number input doesn't make much sense). They are most useful for applying to boolean pdqr-functions (see description of functions for comparing), and warning is thrown in case any input is not a boolean pdqr-function.`&`

's probability of being true is a product of those probabilities from input`e1`

and`e2`

.`|`

's probability of being false is a product of those probabilities from input`e1`

and`e2`

.**Note**that probability of being false is a probability of being equal to 0; of being true - complementary to that.

All other methods are **random**. For example, `f + f`

, `f^g`

are random.

Methods for `all()`

and `any()`

have **non-random nature**. Their input can
be only pdqr-functions, and if any of them is not boolean, a warning is
thrown (because otherwise output doesn't make much sense). They return a
boolean pdqr-function with the following probability of being true:

In

`all()`

- probability of*all*input function being true, i.e. product of probabilities of being true (implemented as complementary to probability of being equal to 0).In

`any()`

- probability of*any*input function being true, i.e. complementary probability to product of all functions being false (implemented as probability of being equal to 0).

Methods for `sum()`

, `prod()`

, `min()`

, `max()`

have **random nature**. They
are implemented to use vectorized version of certain generic, because
transformation function for `form_trans()`

should be vectorized: for input
samples which all have size n it should also return sample of size n (where
each element is a transformation output for corresponding elements from input
samples). This way `min(f, g)`

can be read as "random variable
representing minimum of `f`

and `g`

", etc.

**Notes**:

`range()`

function doesn't make sense here because it returns 2 numbers per input and therefore can't be made vectorized. Error is thrown if it is applied to pdqr-function.Although all

`sum()`

,`prod()`

,`min()`

,`max()`

accept pdqr-functions or single numbers, using numbers and "continuous" functions simultaneously is not a great idea. This is because output will be automatically smoothed (as`form_trans()`

will use some`new_*()`

function) which will give a misleading picture. For a more realistic output:Instead of

`min(f, num)`

use`form_resupport(f, c(num, NA), method = "winsor")`

(see`form_resupport()`

).Instead of

`max(f, num)`

use`form_resupport(f, c(NA, num), method = "winsor")`

.Instead of

`sum(f, num)`

use`f + num`

.Instead of

`prod(f, num)`

use`f * num`

.

`summ_prob_true()`

and `summ_prob_false()`

for extracting
probability from boolean pdqr-functions.

Other pdqr methods for generic functions: `methods-plot`

,
`methods-print`

```
d_norm <- as_d(dnorm)
d_unif <- as_d(dunif)
d_dis <- new_d(data.frame(x = 1:4, prob = 1:4 / 10), "discrete")
set.seed(101)
# Math
plot(d_norm, main = "Math methods")
# Although here distribution shouldn't change, it changes slightly due to
# random implementation
meta_x_tbl(d_dis)#> x prob cumprob
#> 1 1 0.1 0.1
#> 2 2 0.2 0.3
#> 3 3 0.3 0.6
#> 4 4 0.4 1.0#> x prob cumprob
#> 1 1 0.0996 0.0996
#> 2 2 0.2051 0.3047
#> 3 3 0.2934 0.5981
#> 4 4 0.4019 1.0000
# Ops
# Single input, linear transformations, and logical are not random
d_dis > 1#> Probability mass function of discrete type
#> Support: [0, 1] (2 elements, probability of 1: 0.9)!(d_dis > 1)#> Probability mass function of discrete type
#> Support: [0, 1] (2 elements, probability of 1: 0.1)d_norm >= (2*d_norm+1)#> Probability mass function of discrete type
#> Support: [0, 1] (2 elements, probability of 1: ~0.32736)#> Probability mass function of discrete type
#> Support: [0, 1] (2 elements, probability of 1: 0.81)
# Using single numbers is allowed, but gives misleading output in case of
# "continuous" functions. Use other functions instead (see documentation).
plot(min(d_unif, 0.5))op <- options(
pdqr.group_gen.n_sample = 100,
pdqr.group_gen.args_new = list(adjust = 0.5)
)
lines(d_unif + d_unif, col = "red") # `f + f` is different from `2*f` due to independency assumption. Also the
# latter implemented non-randomly.
lines(2 * d_unif, col = "blue")
# Methods for generics attempt to repair support, so they are more reasonable
# to use than direct use of `form_trans()`
d_unif + d_unif#> Density function of continuous type
#> Support: [0, 2] (515 intervals)#> Density function of continuous type
#> Support: ~[-0.1643, 2.16721] (511 intervals)
```