Concept of summary functions is to take one or more pdqr-function(s) and return a summary value (which shouldn't necessarily be a number). Argument `method`

is used to choose function-specific algorithm of computation.

**Note** that some summary functions can accumulate pdqr approximation error (like `summ_moment()`

for example). For better precision increase number intervals for piecewise-linear density using either `n`

argument for `density()`

in `new_*()`

or `n_grid`

argument in `as_*()`

.

We will use the following distributions throughout this vignette:

```
my_beta <- as_d(dbeta, shape1 = 2, shape2 = 5)
my_norm <- as_d(dnorm, mean = 0.5)
my_beta_mix <- form_mix(list(my_beta, my_beta + 1))
```

Although they both are continuous, discrete distributions are also fully supported.

```
# Usage of `summ_center()`
summ_center(my_beta, method = "mean")
#> [1] 0.2857136
summ_center(my_beta, method = "median")
#> [1] 0.2644498
summ_center(my_beta, method = "mode")
#> [1] 0.2000014
# Usage of wrappers
summ_mean(my_beta)
#> [1] 0.2857136
summ_median(my_beta)
#> [1] 0.2644498
summ_mode(my_beta)
#> [1] 0.2000014
# `summ_mode()` can compute local modes instead of default global
summ_mode(my_beta_mix, method = "local")
#> [1] 0.2000014 1.2000014
```

```
# Usage of `summ_spread()`
summ_spread(my_beta, method = "sd")
#> [1] 0.1597178
summ_spread(my_beta, method = "var")
#> [1] 0.02550977
summ_spread(my_beta, method = "iqr")
#> [1] 0.2283162
summ_spread(my_beta, method = "mad")
#> [1] 0.1122417
summ_spread(my_beta, method = "range")
#> [1] 0.955573
# Usage of wrappers
summ_sd(my_beta)
#> [1] 0.1597178
summ_var(my_beta)
#> [1] 0.02550977
summ_iqr(my_beta)
#> [1] 0.2283162
summ_mad(my_beta)
#> [1] 0.1122417
summ_range(my_beta)
#> [1] 0.955573
```

`summ_moment()`

has extra arguments for controlling the nature of moment (which can be combined):

```
summ_moment(my_beta, order = 3)
#> [1] 0.0476182
summ_moment(my_beta, order = 3, central = TRUE)
#> [1] 0.002429287
summ_moment(my_beta, order = 3, standard = TRUE)
#> [1] 11.68727
summ_moment(my_beta, order = 3, absolute = TRUE)
#> [1] 0.0476182
```

There are wrappers for most common moments: skewness and kurtosis:

```
summ_skewness(my_beta)
#> [1] 0.596237
# This by default computes excess kurtosis
summ_kurtosis(my_beta)
#> [1] -0.1202127
# Use `excess = FALSE` to compute non-excess kurtotsis
summ_kurtosis(my_beta, excess = FALSE)
#> [1] 2.879787
```

`summ_quantile(f, probs)`

is essentially a more strict version of `as_q(f)(probs)`

:

```
summ_quantile(my_beta, probs = seq(0, 1, by = 0.25))
#> [1] 0.0000000 0.1611628 0.2644498 0.3894790 0.9555730
```

`summ_entropy()`

computes differential entropy (which can be negative) for “continuous” type pdqr-functions, and information entropy for “discrete”:

```
summ_entropy(my_beta)
#> [1] -0.4845421
summ_entropy(new_d(1:10, type = "discrete"))
#> [1] 2.302585
```

`summ_entropy2()`

computes entropy based summary of relation between a pair of distributions. There are two methods: default “relative” (for relative entropy which is Kullback-Leibler divergence) and “cross” (for cross-entropy). It handles different supports by using `clip`

(default `exp(-20)`

) value instead of 0 during `log()`

computation. Order of input does matter: `summ_entropy2()`

uses support of the first pdqr-function as integration/summation reference.

```
summ_entropy2(my_beta, my_norm)
#> [1] 1.439193
summ_entropy2(my_norm, my_beta)
#> [1] 11.61849
summ_entropy2(my_norm, my_beta, clip = exp(-10))
#> [1] 5.289639
summ_entropy2(my_beta, my_norm, method = "cross")
#> [1] 0.9546508
```

Distributions can be summarized with regions: union of closed intervals. Region is represented as data frame with rows representing intervals and two columns “left” and “right” with left and right interval edges respectively.

`summ_interval()`

summarizes input pdqr-function with single interval based on the desired coverage level supplied in argument `level`

. It has three methods:

- Default “minwidth”: interval with total probability of
`level`

that has minimum width. - “percentile”:
`0.5*(1-level)`

and`1 - 0.5*(1-level)`

quantiles. - “sigma”: interval centered at the mean of distribution. Left and right edges are distant from center by the amount of standard deviation multiplied by
`level`

's critical value (computed from normal distribution). Corresponds to classical confidence interval of sample based on assumption of normality.

```
summ_interval(my_beta, level = 0.9, method = "minwidth")
#> left right
#> 1 0.03015543 0.5252921
summ_interval(my_beta, level = 0.9, method = "percentile")
#> left right
#> 1 0.06284986 0.5818016
summ_interval(my_beta, level = 0.9, method = "sigma")
#> left right
#> 1 0.02300124 0.548426
```

`summ_hdr()`

computes highest density region (HDR) of a distribution: set of intervals with the lowest total width among all sets with total probability not less than an input `level`

. With unimodal distribution it is essentially the same as `summ_interval()`

with “minwidth” method.

```
# Unimodal distribution
summ_hdr(my_beta, level = 0.9)
#> left right
#> 1 0.03013169 0.5253741
# Multimodal distribution
summ_hdr(my_beta_mix, level = 0.9)
#> left right
#> 1 0.03015315 0.5252785
#> 2 1.03015315 1.5252785
# Compare this to single interval of minimum width
summ_interval(my_beta_mix, level = 0.9, method = "minwidth")
#> left right
#> 1 0.05125316 1.448297
```

There is a `region_*()`

family of functions which helps working with them:

```
beta_mix_hdr <- summ_hdr(my_beta_mix, level = 0.9)
beta_mix_interval <- summ_interval(my_beta_mix, level = 0.9)
# Test if points are inside region
region_is_in(beta_mix_hdr, x = seq(0, 2, by = 0.5))
#> [1] FALSE TRUE FALSE TRUE FALSE
# Compute total probability of a region
region_prob(beta_mix_hdr, f = my_beta_mix)
#> [1] 0.899991
# Pdqr-function doesn't need to be the same as used for computing region
region_prob(beta_mix_hdr, f = my_norm)
#> [1] 0.336239
# Compute height of region: minimum value of d-function inside region
region_height(beta_mix_hdr, f = my_beta_mix)
#> [1] 0.400163
# Compute width of region: sum of interval widths
region_width(beta_mix_hdr)
#> [1] 0.9902507
# Compare widths with single interval
region_width(beta_mix_interval)
#> [1] 1.397043
# Draw region on existing plot
plot(my_beta_mix, main = "90% highest density region")
region_draw(beta_mix_hdr)
```

Function `summ_distance()`

takes two pdqr-functions and returns a distance between two distributions they represent. Many methods of computation are available. This might be useful for doing comparison statistical inference.

```
# Kolmogorov-Smirnov distance
summ_distance(my_beta, my_norm, method = "KS")
#> [1] 0.419766
# Total variation distance
summ_distance(my_beta, my_norm, method = "totvar")
#> [1] 0.730451
# Probability of one distribution being bigger than other, normalized to [0;1]
summ_distance(my_beta, my_norm, method = "compare")
#> [1] 0.1678761
# Wassertein distance: "average path density point should travel while
# transforming from one into another"
summ_distance(my_beta, my_norm, method = "wass")
#> [1] 0.6952109
# Cramer distance: integral of squared difference of p-functions
summ_distance(my_beta, my_norm, method = "cramer")
#> [1] 0.1719884
# "Align" distance: path length for which one of distribution should be "moved"
# towards the other so that they become "aligned" (probability of one being
# greater than the other is 0.5)
summ_distance(my_beta, my_norm, method = "align")
#> [1] 0.2147014
# "Entropy" distance: `KL(f, g) + KL(g, f)`, where `KL()` is Kullback-Leibler
# divergence. Usually should be used for distributions with same support, but
# works even if they are different (with big numerical penalty).
summ_distance(my_beta, my_norm, method = "entropy")
#> [1] 13.05768
```

Function `summ_separation()`

computes a threshold that optimally separates distributions represented by pair of input pdqr-functions. In other words, `summ_separation()`

solves a binary classification problem with one-dimensional linear classifier: values not more than some threshold are classified as one class, and more than threshold - as another. Order of input functions doesn't matter.

```
summ_separation(my_beta, my_norm, method = "KS")
#> [1] 0.6175981
summ_separation(my_beta, my_norm, method = "F1")
#> [1] 0.007545242
```

Functions `summ_classmetric()`

and `summ_classmetric_df()`

compute metric(s) of classification setup, similar to one used in `summ_separation()`

. Here classifier threshold should be supplied and order of input matters. Classification is assumed to be done as follows: any x value not more than threshold value is classified as “negative”; if more - “positive”. Classification metrics are computed based on two pdqr-functions: `f`

, which represents the distribution of values which should be classified as “negative” (“true negative”), and `g`

- the same for “positive” (“true positive”).

```
# Many threshold values can be supplied
thres_vec <- seq(0, 1, by = 0.2)
summ_classmetric(f = my_beta, g = my_norm, threshold = thres_vec, method = "F1")
#> [1] 0.5138193 0.5436321 0.6089061 0.6131005 0.5522756 0.4715757
# In `summ_classmetric_df()` many methods can be supplied
summ_classmetric_df(
f = my_beta, g = my_norm, threshold = thres_vec, method = c("GM", "F1", "MCC")
)
#> threshold GM F1 MCC
#> 1 0.0 0.0000000 0.5138193 -0.42709306
#> 2 0.2 0.4614729 0.5436321 -0.03892981
#> 3 0.4 0.6433485 0.6089061 0.31475763
#> 4 0.6 0.6643221 0.6131005 0.48370132
#> 5 0.8 0.6176386 0.5522756 0.48316011
#> 6 1.0 0.5554612 0.4715757 0.42709306
```

With `summ_roc()`

and `summ_rocauc()`

one can compute data frame of ROC curve points and ROC AUC value respectively. There is also a `roc_plot()`

function for predefined plotting of ROC curve.

```
my_roc <- summ_roc(my_beta, my_norm)
head(my_roc)
#> threshold fpr tpr
#> 1 5.253425 0 0.000000e+00
#> 2 5.243918 0 4.811646e-08
#> 3 5.234412 0 9.845779e-08
#> 4 5.224905 0 1.511167e-07
#> 5 5.215398 0 2.061949e-07
#> 6 5.205891 0 2.637984e-07
summ_rocauc(my_beta, my_norm)
#> [1] 0.583938
roc_plot(my_roc)
```

'pdqr' has functions that can order set of distributions. They are `summ_order()`

, `summ_sort()`

, and `summ_rank()`

, which are analogues of `order()`

, `sort()`

, and `rank()`

respectively. They take a list of pdqr-functions as input, establish their ordering based on specified method, and return the desired output.

There are two sets of methods:

- Method “compare” uses the following ordering relation: pdqr-function
`f`

is greater than`g`

if and only if`P(f >= g) > 0.5`

, or in 'pdqr' code`summ_prob_true(f >= g) > 0.5`

. This method orders input based on this relation and`order()`

function.**Notes**:- This relation doesn't strictly define ordering because it is not transitive. It is solved by first preordering input list based on method “mean” and then calling
`order()`

. - Because comparing two pdqr-functions can be time consuming, this method becomes rather slow as number of distributions grows. To increase computation speed (sacrificing a little bit of approximation precision), use less intervals in piecewise-linear approximation of density for “continuous” types of pdqr-functions.

- This relation doesn't strictly define ordering because it is not transitive. It is solved by first preordering input list based on method “mean” and then calling
- Methods “mean”, “median”, and “mode” are based on
`summ_center()`

: ordering of distributions is defined as ordering of corresponding measures of distribution's center.

```
# Here the only clear "correct" ordering is that `a <= b`.
f_list <- list(a = my_beta, b = my_beta + 1, c = my_norm)
# Returns an integer vector representing a permutation which rearranges f_list
# in desired order
summ_order(f_list, method = "compare")
#> [1] 1 3 2
# In this particular case of `f_list` all orderings agree with each other, but
# generally this is not the case: for any pair of methods there is a case
# when they disagree with each other
summ_order(f_list, method = "mean")
#> [1] 1 3 2
summ_order(f_list, method = "median")
#> [1] 1 3 2
summ_order(f_list, method = "mode")
#> [1] 1 3 2
# Use `decreasing = TRUE` to sort decreasingly
summ_order(f_list, method = "compare", decreasing = TRUE)
#> [1] 2 3 1
# Sort list
summ_sort(f_list)
#> $a
#> Density function of continuous type
#> Support: ~[0, 0.95557] (10000 intervals)
#>
#> $c
#> Density function of continuous type
#> Support: ~[-4.25342, 5.25342] (10000 intervals)
#>
#> $b
#> Density function of continuous type
#> Support: ~[1, 1.95557] (10000 intervals)
summ_sort(f_list, decreasing = TRUE)
#> $b
#> Density function of continuous type
#> Support: ~[1, 1.95557] (10000 intervals)
#>
#> $c
#> Density function of continuous type
#> Support: ~[-4.25342, 5.25342] (10000 intervals)
#>
#> $a
#> Density function of continuous type
#> Support: ~[0, 0.95557] (10000 intervals)
# Rank elements: 1 indicates "the smallest", `length(f_list)` - "the biggest"
summ_rank(f_list)
#> a b c
#> 1 3 2
```

Functions `summ_prob_true()`

and `summ_prob_false()`

should be used to extract probabilities from boolean pdqr-functions: outputs of comparing basic operators (like `>=`

, `==`

, etc.):

```
summ_prob_true(my_beta >= my_norm)
#> [1] 0.416062
summ_prob_false(my_beta >= 2*my_norm)
#> [1] 0.6391
```

`summ_pval()`

computes p-value(s) of observed statistic(s) based on the distribution. You can compute left, right, or two-sided p-values with methods “left”, “right”, and “both” respectively. By default multiple input values are adjusted for multiple comparisons (using stats::p.adjust()):

```
# By default two-sided p-value is computed
summ_pval(my_beta, obs = 0.7)
#> [1] 0.02186803
summ_pval(my_beta, obs = 0.7, method = "left")
#> [1] 0.989066
summ_pval(my_beta, obs = 0.7, method = "right")
#> [1] 0.01093401
# Multiple values are adjusted with `p.adjust()` with "holm" method by default
obs_vec <- seq(0, 1, by = 0.1)
summ_pval(my_beta, obs = obs_vec)
#> [1] 0.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000
#> [6] 1.0000000000 0.4915085377 0.1530761780 0.0255840348 0.0009720023
#> [11] 0.0000000000
# Use `adjust = "none"` to not adjust
summ_pval(my_beta, obs = obs_vec, adjust = "none")
#> [1] 0.0000000000 0.2285302047 0.6892806594 0.8403488674 0.4665584871
#> [6] 0.2187482323 0.0819180896 0.0218680254 0.0031980044 0.0001080003
#> [11] 0.0000000000
```