### Install Pragmastat

Source: https://pragmastat.dev/py

Install the library using pip.

```bash
pip install pragmastat==14.0.1
```

--------------------------------

### Install Pragmastat via Go

Source: https://pragmastat.dev/go

Use the go get command to add the specific version of the library to your project.

```bash
go get github.com/AndreyAkinshin/pragmastat/go/v14@v14.0.1
```

--------------------------------

### Sample Usage Examples

Source: https://pragmastat.dev/sample

Basic usage examples for selecting elements from a list and the identity case where selecting all elements returns the original order.

```text
Rng("demo-sample").Sample([1, 2, 3, 4, 5], 3)
```

```text
r.Sample(x, n) = x
```

--------------------------------

### Install Pragmastat from GitHub

Source: https://pragmastat.dev/r

Use the remotes package to install the Pragmastat library from the specified GitHub repository and subdirectory.

```R
install.packages("remotes") # If 'remotes' is not installed
remotes::install_github("AndreyAkinshin/pragmastat",
                        subdir = "r/pragmastat", ref = "v14.0.1")
library(pragmastat)
```

--------------------------------

### Pragmastat Library Usage Demonstration

Source: https://pragmastat.dev/go

A complete example demonstrating one-sample and two-sample statistics, confirmatory comparisons, random number generation, and distribution sampling. Helper functions are used to handle errors during measurement and sample creation.

```go
package main

import (
	"fmt"
	"log"

	pragmastat "github.com/AndreyAkinshin/pragmastat/go/v14"
)

func mustM(val pragmastat.Measurement, err error) pragmastat.Measurement {
	if err != nil {
		log.Fatal(err)
	}
	return val
}

func mustB(val pragmastat.Bounds, err error) pragmastat.Bounds {
	if err != nil {
		log.Fatal(err)
	}
	return val
}

func mustS(val *pragmastat.Sample, err error) *pragmastat.Sample {
	if err != nil {
		log.Fatal(err)
	}
	return val
}

func mustT(val *pragmastat.Threshold, err error) *pragmastat.Threshold {
	if err != nil {
		log.Fatal(err)
	}
	return val
}

func mustP(val []pragmastat.Projection, err error) []pragmastat.Projection {
	if err != nil {
		log.Fatal(err)
	}
	return val
}

func main() {
	// --- One-Sample ---

	xVals := make([]float64, 200)
	for i := range xVals {
		xVals[i] = float64(i + 1)
	}
	x := mustS(pragmastat.NewSample(xVals))

	fmt.Println(mustM(x.Center()).Value)                     // 100.5
	fmt.Println(mustB(x.CenterBounds(1e-3)))                 // [86;115]
	fmt.Println(mustM(x.Spread()).Value)                     // 59
	fmt.Println(mustB(x.SpreadBoundsWithSeed(1e-3, "demo"))) // [44;87]

	// --- Two-Sample ---

	sxVals := make([]float64, 200)
	for i := range sxVals {
		sxVals[i] = float64(i + 1)
	}
	syVals := make([]float64, 200)
	for i := range syVals {
		syVals[i] = float64(i + 101)
	}
	sx := mustS(pragmastat.NewSample(sxVals))
	sy := mustS(pragmastat.NewSample(syVals))

	fmt.Println(mustM(sx.Shift(sy)).Value)                           // -100
	fmt.Println(mustB(sx.ShiftBounds(sy, 1e-3)))                     // [-120;-80]
	fmt.Println(mustM(sx.Ratio(sy)).Value)                           // 0.5008354224706336
	fmt.Println(mustB(sx.RatioBounds(sy, 1e-3)))                     // [0.4066666666666668;0.5958333333333332]
	fmt.Println(mustM(sx.Disparity(sy)).Value)                       // -1.694915254237288
	fmt.Println(mustB(sx.DisparityBoundsWithSeed(sy, 1e-3, "demo"))) // [-3.1025641025641026;-0.8494623655913979]

	// --- Confirmatory Comparison ---

	th1 := mustT(pragmastat.NewThreshold(pragmastat.MetricCenter, pragmastat.NewNumberMeasurement(50), 1e-3))
	proj1 := mustP(pragmastat.Compare1(sx, []*pragmastat.Threshold{th1}))
	fmt.Println(proj1[0].Verdict) // greater

	th2 := mustT(pragmastat.NewThreshold(pragmastat.MetricShift, pragmastat.NewNumberMeasurement(0), 1e-3))
	proj2 := mustP(pragmastat.Compare2(sx, sy, []*pragmastat.Threshold{th2}))
	fmt.Println(proj2[0].Verdict) // less

	// --- Randomization ---

	rng := pragmastat.NewRngFromString("demo-uniform")
	fmt.Println(rng.UniformFloat64()) // 0.2640554428629759
	fmt.Println(rng.UniformFloat64()) // 0.9348534835582796

	rng = pragmastat.NewRngFromString("demo-uniform-int")
	fmt.Println(rng.UniformInt64(0, 100)) // 41

	rng = pragmastat.NewRngFromString("demo-sample")
	fmt.Println(pragmastat.RngSample(rng, []float64{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 3)) // [3 8 9]

	rng = pragmastat.NewRngFromString("demo-resample")
	fmt.Println(pragmastat.RngResample(rng, []float64{1, 2, 3, 4, 5}, 7)) // [3 1 3 2 4 1 2]

	rng = pragmastat.NewRngFromString("demo-shuffle")
	fmt.Println(pragmastat.RngShuffle(rng, []float64{1, 2, 3, 4, 5})) // [4 2 3 5 1]

	// --- Distributions ---

	rng = pragmastat.NewRngFromString("demo-dist-additive")
	fmt.Println(pragmastat.NewAdditive(0, 1).Sample(rng)) // 0.1741044867956819

	rng = pragmastat.NewRngFromString("demo-dist-multiplic")
	fmt.Println(pragmastat.NewMultiplic(0, 1).Sample(rng)) // 1.1273244602673853

	rng = pragmastat.NewRngFromString("demo-dist-exp")
	fmt.Println(pragmastat.NewExp(1).Sample(rng)) // 0.6589065267276553

	rng = pragmastat.NewRngFromString("demo-dist-power")
	fmt.Println(pragmastat.NewPower(1, 2).Sample(rng)) // 1.023677535537084

	rng = pragmastat.NewRngFromString("demo-dist-uniform")
	fmt.Println(pragmastat.NewUniform(0, 10).Sample(rng)) // 6.54043657816832
}
```

--------------------------------

### Install Pragmastat via Cargo

Source: https://pragmastat.dev/rs

Add the library to your project using the cargo command-line tool.

```bash
cargo add pragmastat@14.0.1
```

--------------------------------

### Install Pragmastat via npm

Source: https://pragmastat.dev/ts

Use this command to add the library to your project dependencies.

```bash
npm i pragmastat@14.0.1
```

--------------------------------

### Pragmastat Library Demonstration

Source: https://pragmastat.dev/rs

A comprehensive example covering one-sample and two-sample statistics, confirmatory comparisons, random number generation, probability distributions, and unit system management.

```rust
use pragmastat::distributions::{Additive, Distribution, Exp, Multiplic, Power, Uniform};
use pragmastat::*;

fn main() {
    // --- One-Sample (Sample-based API) ---

    let x = Sample::new((1..=200).map(|i| i as f64).collect()).unwrap();

    println!("{}", center(&x).unwrap().value); // 100.5
    let bounds = center_bounds(&x, 1e-3).unwrap();
    println!("{{lower: {}, upper: {}}}", bounds.lower, bounds.upper); // {lower: 86, upper: 115}
    println!("{}", spread(&x).unwrap().value); // 59
    let bounds = spread_bounds_with_seed(&x, 1e-3, "demo").unwrap();
    println!("{{lower: {}, upper: {}}}", bounds.lower, bounds.upper); // {lower: 44, upper: 87}

    // --- Two-Sample (Sample-based API) ---

    let x = Sample::new((1..=200).map(|i| i as f64).collect()).unwrap();
    let y = Sample::new((101..=300).map(|i| i as f64).collect()).unwrap();

    println!("{}", shift(&x, &y).unwrap().value); // -100
    let bounds = shift_bounds(&x, &y, 1e-3).unwrap();
    println!("{{lower: {}, upper: {}}}", bounds.lower, bounds.upper); // {lower: -120, upper: -80}
    println!("{}", ratio(&x, &y).unwrap().value); // 0.5008354224706334
    let bounds = ratio_bounds(&x, &y, 1e-3).unwrap();
    println!("{{lower: {}, upper: {}}}", bounds.lower, bounds.upper); // {lower: 0.4066666666666668, upper: 0.5958333333333332}
    println!("{}", disparity(&x, &y).unwrap().value); // -1.694915254237288
    let bounds = disparity_bounds_with_seed(&x, &y, 1e-3, "demo").unwrap();
    println!("{{lower: {}, upper: {}}}", bounds.lower, bounds.upper); // {lower: -3.1025641025641026, upper: -0.8494623655913979}

    // --- Confirmatory Comparison ---

    let th = Threshold::new(Metric::Center, Measurement::number(50.0), 1e-3).unwrap();
    println!("{}", compare1(&x, &[th]).unwrap()[0].verdict().as_str()); // greater

    let th = Threshold::new(Metric::Shift, Measurement::number(0.0), 1e-3).unwrap();
    println!("{}", compare2(&x, &y, &[th]).unwrap()[0].verdict().as_str()); // less

    // --- Randomization ---

    let mut rng = Rng::from_string("demo-uniform");
    println!("{}", rng.uniform_f64()); // 0.2640554428629759
    println!("{}", rng.uniform_f64()); // 0.9348534835582796

    let mut rng = Rng::from_string("demo-uniform-int");
    println!("{}", rng.uniform_i64(0, 100)); // 41

    let mut rng = Rng::from_string("demo-sample");
    println!(
        "{:?}",
        rng.sample(&[0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0], 3)
    ); // [3.0, 8.0, 9.0]

    let mut rng = Rng::from_string("demo-resample");
    println!("{:?}", rng.resample(&[1.0, 2.0, 3.0, 4.0, 5.0], 7)); // [3.0, 1.0, 3.0, 2.0, 4.0, 1.0, 2.0]

    let mut rng = Rng::from_string("demo-shuffle");
    println!("{:?}", rng.shuffle(&[1.0, 2.0, 3.0, 4.0, 5.0])); // [4.0, 2.0, 3.0, 5.0, 1.0]

    // --- Distributions ---

    let mut rng = Rng::from_string("demo-dist-additive");
    println!("{}", Additive::new(0.0, 1.0).sample(&mut rng)); // 0.17410448679568188

    let mut rng = Rng::from_string("demo-dist-multiplic");
    println!("{}", Multiplic::new(0.0, 1.0).sample(&mut rng)); // 1.1273244602673853

    let mut rng = Rng::from_string("demo-dist-exp");
    println!("{}", Exp::new(1.0).sample(&mut rng)); // 0.6589065267276553

    let mut rng = Rng::from_string("demo-dist-power");
    println!("{}", Power::new(1.0, 2.0).sample(&mut rng)); // 1.023677535537084

    let mut rng = Rng::from_string("demo-dist-uniform");
    println!("{}", Uniform::new(0.0, 10.0).sample(&mut rng)); // 6.54043657816832

    // --- Unit system ---

    let ms = MeasurementUnit::new("ms", "Time", "ms", "Millisecond", 1_000_000);
    let ns = MeasurementUnit::new("ns", "Time", "ns", "Nanosecond", 1);

    let s = Sample::with_unit(vec![1.0, 2.0, 3.0], ms).unwrap();
    let converted = s.convert_to(&ns).unwrap();
    println!("Converted: {:?}", converted.values()); // [1000000.0, 2000000.0, 3000000.0]

    let registry = UnitRegistry::standard();
    let unit = registry.resolve("number").unwrap();
    println!("Resolved: {}", unit.full_name()); // Number
}
```

--------------------------------

### Pragmastat Library Usage Demo

Source: https://pragmastat.dev/kt

Comprehensive examples covering one-sample and two-sample statistics, confirmatory comparisons, custom measurement units, randomization, and probability distributions.

```kotlin
package dev.pragmastat.demo

import dev.pragmastat.*
import dev.pragmastat.distributions.*

fun main() {
    // --- One-Sample ---

    val x = Sample.of((1..200).map { it.toDouble() })
    println(center(x)) // 100.5
    println(spread(x)) // 59.0
    println(centerBounds(x, Probability(1e-3))) // Bounds(lower=86.0, upper=115.0, unit=...)
    println(spreadBounds(x, Probability(1e-3), "demo")) // Bounds(lower=44.0, upper=87.0, unit=...)

    // --- Two-Sample ---

    val sx = Sample.of((1..200).map { it.toDouble() })
    val sy = Sample.of((101..300).map { it.toDouble() })
    println(shift(sx, sy)) // -100.0
    println(shiftBounds(sx, sy, Probability(1e-3))) // Bounds(lower=-120.0, upper=-80.0, unit=...)
    println(ratio(sx, sy)) // 0.5008354224706336
    println(ratioBounds(sx, sy, Probability(1e-3))) // Bounds(lower=0.4066..., upper=0.5958..., unit=...)
    println(disparity(sx, sy)) // -1.694915254237288
    println(disparityBounds(sx, sy, Probability(1e-3), "demo")) // Bounds(lower=-3.1025..., upper=-0.8494..., unit=...)

    // --- Confirmatory Comparison ---

    println(compare1(sx, listOf(Threshold(Metric.Center, Measurement(50.0), Probability(1e-3))))[0].verdict.toId()) // greater
    println(compare2(sx, sy, listOf(Threshold(Metric.Shift, Measurement(0.0), Probability(1e-3))))[0].verdict.toId()) // less

    // --- Custom units ---

    val ns = MeasurementUnit("ns", "Time", "ns", "Nanosecond", 1)
    val us = MeasurementUnit("us", "Time", "us", "Microsecond", 1000)
    val ms = MeasurementUnit("ms", "Time", "ms", "Millisecond", 1_000_000)

    val registry = UnitRegistry.standard()
    registry.register(ns)
    registry.register(us)
    registry.register(ms)

    val timeSample = Sample.of(listOf(100.0, 200.0, 300.0, 400.0, 500.0), ns)
    println(center(timeSample)) // 300 ns
    println(center(timeSample.convertTo(us))) // 0.3 us

    // --- Randomization ---

    var rng = Rng("demo-uniform")
    println(rng.uniformDouble()) // 0.2640554428629759
    println(rng.uniformDouble()) // 0.9348534835582796

    rng = Rng("demo-uniform-int")
    println(rng.uniformInt(0, 100)) // 41

    rng = Rng("demo-sample")
    println(rng.sample(listOf(0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0), 3)) // [3.0, 8.0, 9.0]

    rng = Rng("demo-resample")
    println(rng.resample(listOf(1.0, 2.0, 3.0, 4.0, 5.0), 7)) // [3.0, 1.0, 3.0, 2.0, 4.0, 1.0, 2.0]

    rng = Rng("demo-shuffle")
    println(rng.shuffle(listOf(1.0, 2.0, 3.0, 4.0, 5.0))) // [4.0, 2.0, 3.0, 5.0, 1.0]

    // --- Distributions ---

    rng = Rng("demo-dist-additive")
    println(Additive(0.0, 1.0).sample(rng)) // 0.17410448679568188

    rng = Rng("demo-dist-multiplic")
    println(Multiplic(0.0, 1.0).sample(rng)) // 1.1273244602673853

    rng = Rng("demo-dist-exp")
    println(Exp(1.0).sample(rng)) // 0.6589065267276553

    rng = Rng("demo-dist-power")
    println(Power(1.0, 2.0).sample(rng)) // 1.023677535537084

    rng = Rng("demo-dist-uniform")
    println(Uniform(0.0, 10.0).sample(rng)) // 6.54043657816832
}
```

--------------------------------

### Compare1 Usage Examples

Source: https://pragmastat.dev/compare1

Demonstrates comparing a sample against Center and Spread metrics with defined thresholds.

```text
Compare1([1..10], [(Center, 20, 1e-3)])
```

```text
Compare1([1..10], [(Spread, 0.1, 1e-3)])
```

--------------------------------

### Pragmastat Library Usage Demo

Source: https://pragmastat.dev/r

Comprehensive examples demonstrating one-sample and two-sample metrics, confirmatory comparisons, sample-based interfaces, randomization, and distribution sampling.

```R
library(pragmastat)

# --- One-Sample (legacy vector interface) ---

x <- 1:200

print(center(x)) # 100.5
bounds <- center_bounds(x, 1e-3)
print(paste("[", bounds$lower, ", ", bounds$upper, "]", sep = "")) # [86, 115]
print(spread(x)) # 59
bounds <- spread_bounds(x, 1e-3, seed = "demo")
print(paste("[", bounds$lower, ", ", bounds$upper, "]", sep = "")) # [44, 87]

# --- Two-Sample (legacy vector interface) ---

x <- 1:200
y <- 101:300

print(shift(x, y)) # -100
bounds <- shift_bounds(x, y, 1e-3)
print(paste("[", bounds$lower, ", ", bounds$upper, "]", sep = "")) # [-120, -80]
print(ratio(x, y)) # 0.500835
bounds <- ratio_bounds(x, y, 1e-3)
print(paste("[", bounds$lower, ", ", bounds$upper, "]", sep = "")) # [0.406666666666667, 0.595833333333333]
print(disparity(x, y)) # -1.694915
bounds <- disparity_bounds(x, y, 1e-3, seed = "demo")
print(paste("[", bounds$lower, ", ", bounds$upper, "]", sep = "")) # [-3.1025641025641, -0.849462365591398]

# --- Confirmatory Comparison ---

sx <- Sample$new(x)
t1 <- list(list(metric = METRIC_CENTER, value = Measurement$new(50), misrate = 1e-3))
print(compare1(sx, t1)[[1]]$verdict) # "greater"

sy <- Sample$new(y)
t2 <- list(list(metric = METRIC_SHIFT, value = Measurement$new(0), misrate = 1e-3))
print(compare2(sx, sy, t2)[[1]]$verdict) # "less"

# --- Sample-based interface ---

sx <- Sample$new(1:200)
m <- center(sx)
print(paste("center:", m$value, "unit:", m$unit$id))
b <- center_bounds(sx, 1e-3)
print(paste("center_bounds: [", b$lower, ", ", b$upper, "] unit:", b$unit$id, sep = ""))

sx <- Sample$new(1:200)
sy <- Sample$new(101:300)
m <- shift(sx, sy)
print(paste("shift:", m$value, "unit:", m$unit$id))
m <- ratio(sx, sy)
print(paste("ratio:", m$value, "unit:", m$unit$id))

# --- Randomization ---

r <- rng("demo-uniform")
print(r$uniform_float()) # 0.2640554
print(r$uniform_float()) # 0.9348535

r <- rng("demo-uniform-int")
print(r$uniform_int(0, 100)) # 41

r <- rng("demo-sample")
print(r$sample(0:9, 3)) # 3 8 9

r <- rng("demo-resample")
print(r$resample(c(1, 2, 3, 4, 5), 7)) # 3 1 3 2 4 1 2

r <- rng("demo-shuffle")
print(r$shuffle(c(1, 2, 3, 4, 5))) # 4 2 3 5 1

# --- Distributions ---

r <- rng("demo-dist-additive")
dist <- dist_additive(0, 1)
print(dist$sample(r)) # 0.1741045

r <- rng("demo-dist-multiplic")
dist <- dist_multiplic(0, 1)
print(dist$sample(r)) # 1.127324

r <- rng("demo-dist-exp")
dist <- dist_exp(1)
print(dist$sample(r)) # 0.6589065

r <- rng("demo-dist-power")
dist <- dist_power(1, 2)
print(dist$sample(r)) # 1.023678

r <- rng("demo-dist-uniform")
dist <- dist_uniform(0, 10)
print(dist$sample(r)) # 6.540437
```

--------------------------------

### Install Pragmastat via Build Systems

Source: https://pragmastat.dev/kt

Configuration snippets for adding Pragmastat as a dependency in Maven and Gradle projects.

```xml
<dependency>
    <groupId>dev.pragmastat</groupId>
    <artifactId>pragmastat</artifactId>
    <version>14.0.1</version>
</dependency>
```

```groovy
implementation 'dev.pragmastat:pragmastat:14.0.1'
```

```kotlin
implementation("dev.pragmastat:pragmastat:14.0.1")
```

--------------------------------

### Shuffle Usage Examples

Source: https://pragmastat.dev/shuffle

Basic usage of the Shuffle method for generating a shuffled copy of an array.

```text
Rng("demo-shuffle").Shuffle([1, 2, 3, 4, 5])
```

```text
r.Shuffle(x)
```

--------------------------------

### Center Estimator Namespace Setup

Source: https://pragmastat.dev/center

Required imports and namespace declaration for the Pragmastat Center estimator implementation.

```csharp
using Pragmastat.Internal;
using Pragmastat.Randomization;

namespace Pragmastat.Algorithms;
```

--------------------------------

### Violation Schema Examples

Source: https://pragmastat.dev/assumptions

Standardized JSON format for reporting assumption violations.

```json
{
  "id": "positivity",
  "subject": "x"
}
```

```json
{
  "id": "domain",
  "subject": "misrate"
}
```

--------------------------------

### Reproducible Shuffle with Seed

Source: https://pragmastat.dev/shuffle

Example of passing a seeded generator to ensure reproducible results.

```text
Shuffle(data, Rng("experiment-1"))
```

--------------------------------

### Comparison Entry Points

Source: https://pragmastat.dev/compare1

Public methods for executing comparisons, enforcing metric compatibility constraints.

```csharp
  public static IReadOnlyList<Projection> Compare1(Sample x, IReadOnlyList<Threshold> thresholds, string? seed)
  {
    Assertion.NonWeighted("x", x);
    Assertion.NotNullOrEmpty("thresholds", thresholds);
    Assertion.ItemNotNull("thresholds", thresholds);

    foreach (var threshold in thresholds)
    {
      if (threshold.Metric is Metric.Shift or Metric.Ratio or Metric.Disparity)
        throw new ArgumentException(
          $"Metric {threshold.Metric} is not supported by Compare1. Use Compare2 instead.",
          nameof(thresholds));
    }

    foreach (var threshold in thresholds)
    {
      if (!threshold.Value.NominalValue.IsFinite())
        throw new ArgumentOutOfRangeException(nameof(thresholds), "threshold.Value must be finite");
    }

    var normalizedValues = new Measurement[thresholds.Count];
    for (int i = 0; i < thresholds.Count; i++)
    {
      var spec = GetSpec(Compare1Specs, thresholds[i].Metric);
      normalizedValues[i] = spec.ValidateAndNormalize(thresholds[i], x, null);
    }

    return Execute(Compare1Specs, x, null, thresholds, normalizedValues, seed);
  }

  public static IReadOnlyList<Projection> Compare2(
    Sample x, Sample y, IReadOnlyList<Threshold> thresholds, string? seed)
  {
    Assertion.NonWeighted("x", x);
    Assertion.NonWeighted("y", y);
    Assertion.CompatibleUnits(x, y);
    Assertion.NotNullOrEmpty("thresholds", thresholds);
    Assertion.ItemNotNull("thresholds", thresholds);

    foreach (var threshold in thresholds)
    {
      if (threshold.Metric is Metric.Center or Metric.Spread)
        throw new ArgumentException(
          $"Metric {threshold.Metric} is not supported by Compare2. Use Compare1 instead.",
          nameof(thresholds));
    }
```

--------------------------------

### Compare2 Algorithm Implementation

Source: https://pragmastat.dev/compare2

Core logic for validating inputs, normalizing metrics, and executing the comparison projection.

```csharp
  public static IReadOnlyList<Projection> Compare2(
    Sample x, Sample y, IReadOnlyList<Threshold> thresholds, string? seed)
  {
    Assertion.NonWeighted("x", x);
    Assertion.NonWeighted("y", y);
    Assertion.CompatibleUnits(x, y);
    Assertion.NotNullOrEmpty("thresholds", thresholds);
    Assertion.ItemNotNull("thresholds", thresholds);

    foreach (var threshold in thresholds)
    {
      if (threshold.Metric is Metric.Center or Metric.Spread)
        throw new ArgumentException(
          $"Metric {threshold.Metric} is not supported by Compare2. Use Compare1 instead.",
          nameof(thresholds));
    }

    foreach (var threshold in thresholds)
    {
      if (!threshold.Value.NominalValue.IsFinite())
        throw new ArgumentOutOfRangeException(nameof(thresholds), "threshold.Value must be finite");
    }

    var normalizedValues = new Measurement[thresholds.Count];
    for (int i = 0; i < thresholds.Count; i++)
    {
      var spec = GetSpec(Compare2Specs, thresholds[i].Metric);
      normalizedValues[i] = spec.ValidateAndNormalize(thresholds[i], x, y);
    }

    return Execute(Compare2Specs, x, y, thresholds, normalizedValues, seed);
  }

  private static MetricSpec GetSpec(MetricSpec[] specs, Metric metric)
  {
    foreach (var spec in specs)
      if (spec.Metric == metric) return spec;
    throw new ArgumentException($"No spec found for metric {metric}");
  }

  private static IReadOnlyList<Projection> Execute(
    MetricSpec[] canonicalSpecs,
    Sample x,
    Sample? y,
    IReadOnlyList<Threshold> thresholds,
    Measurement[] normalizedValues,
    string? seed)
  {
    var results = new Projection[thresholds.Count];

    var byMetric = thresholds
      .Select((t, i) => (t, i, normalizedValues[i]))
      .GroupBy(item => item.t.Metric)
      .ToDictionary(g => g.Key, g => g.ToList());

    foreach (var spec in canonicalSpecs)
    {
      if (!byMetric.TryGetValue(spec.Metric, out var entries)) continue;
      var estimate = spec.Estimate(x, y);
      foreach (var (threshold, inputIndex, normalizedValue) in entries)
      {
        var bounds = (seed != null && spec.SeededBounds != null)
          ? spec.SeededBounds(x, y, threshold.Misrate, seed)
          : spec.Bounds(x, y, threshold.Misrate);
        var verdict = ComputeVerdict(bounds, normalizedValue);
        results[inputIndex] = new Projection(threshold, estimate, bounds, verdict);
      }
    }

    return results;
  }

  private static ComparisonVerdict ComputeVerdict(Bounds bounds, Measurement normalizedThreshold)
  {
    double t = normalizedThreshold.NominalValue;
    if (bounds.Lower > t) return ComparisonVerdict.Greater;
    if (bounds.Upper < t) return ComparisonVerdict.Less;
    return ComparisonVerdict.Inconclusive;
  }
}
```

--------------------------------

### Input Preparation Helpers

Source: https://pragmastat.dev/spread

Utility methods for copying and validating input data, including an optional sorting step.

```csharp
private static double[] CopyAndSort(IReadOnlyList<double> values)
  {
    var a = new double[values.Count];
    for (int i = 0; i < a.Length; i++)
    {
      double v = values[i];
      if (double.IsNaN(v)) throw new ArgumentException("NaN not allowed.", nameof(values));
      a[i] = v;
    }

    Array.Sort(a);
    return a;
  }

  private static double[] CopyTrusted(IReadOnlyList<double> values)
  {
    // Trust caller that values are sorted; still copy to array for fast indexed access.
    var a = new double[values.Count];
    for (int i = 0; i < a.Length; i++)
    {
      double v = values[i];
      if (double.IsNaN(v)) throw new ArgumentException("NaN not allowed.", nameof(values));
      a[i] = v;
    }

    return a;
  }
```

--------------------------------

### Sample Algorithm Implementation

Source: https://pragmastat.dev/sample

Pseudocode for the selection sampling algorithm that performs a single pass over the data.

```text
seen = 0, selected = 0
for each element x at position i:
   if uniform() < (k - selected) / (n - seen):
       output x
       selected += 1
   seen += 1
```

--------------------------------

### Implement Ratio Estimation in C#

Source: https://pragmastat.dev/ratio

Computes quantiles of pairwise ratios by delegating to the Shift algorithm in log-space. Requires input samples to be strictly positive.

```csharp
namespace Pragmastat.Algorithms;

using System;
using System.Collections.Generic;
using System.Linq;
using Pragmastat.Exceptions;
using Pragmastat.Internal;

/// <summary>
/// Computes quantiles of pairwise ratios via log-transformation and ShiftImpl delegation.
/// Ratio(x, y) = exp(Shift(log(x), log(y)))
/// </summary>
internal static class RatioImpl
{
  /// <summary>
  /// Computes quantiles of all pairwise ratios { x_i / y_j }.
  /// Time: O((m + n) * log(precision)) per quantile. Space: O(m + n).
  /// </summary>
  /// <remarks>
  /// Log-transformation preserves sort order for positive values.
  /// </remarks>
  /// <param name="x">First sample (must be strictly positive).</param>
  /// <param name="y">Second sample (must be strictly positive).</param>
  /// <param name="p">Probabilities in [0, 1].</param>
  /// <param name="assumeSorted">If true, assumes x and y are already sorted in ascending order.</param>
  /// <returns>Quantile values for each probability in p.</returns>
  public static double[] Estimate(IReadOnlyList<double> x, IReadOnlyList<double> y, double[] p, bool assumeSorted = false)
  {
    // Log-transform both samples (includes positivity check)
    var logX = MathExtensions.Log(x, Subject.X);
    var logY = MathExtensions.Log(y, Subject.Y);

    // Delegate to ShiftImpl in log-space
    var logResult = ShiftImpl.Estimate(logX, logY, p, assumeSorted);

    // Exp-transform back to ratio-space
    return logResult.Select(v => Math.Exp(v)).ToArray();
  }
}
```

--------------------------------

### Compute Pairwise Quantiles and Bounds in C#

Source: https://pragmastat.dev/center-bounds

Provides methods to find the k-th smallest pairwise average and calculate lower and upper bounds for a sorted input list.

```csharp
using System.Diagnostics;

namespace Pragmastat.Algorithms;

/// <summary>
/// Efficiently computes quantiles from all pairwise averages (x[i] + x[j]) / 2 for i ≤ j.
/// Uses binary search with counting function to avoid materializing all N(N+1)/2 pairs.
/// </summary>
internal static class CenterQuantilesImpl
{
  private static bool IsSorted(IReadOnlyList<double> list)
  {
    for (int i = 1; i < list.Count; i++)
      if (list[i] < list[i - 1])
        return false;
    return true;
  }
  /// <summary>
  /// Relative epsilon for floating-point comparisons in binary search convergence.
  /// </summary>
  private const double RelativeEpsilon = 1e-14;
  /// <summary>
  /// Compute specified quantile from pairwise averages.
  /// </summary>
  /// <param name="sorted">Sorted input array.</param>
  /// <param name="k">1-based rank of the desired quantile.</param>
  /// <returns>The k-th smallest pairwise average.</returns>
  public static double Quantile(IReadOnlyList<double> sorted, long k)
  {
    Debug.Assert(
      IsSorted(sorted),
      "CenterQuantilesImpl.Quantile: input must be sorted");
    int n = sorted.Count;
    if (n == 0)
      throw new ArgumentException("Input cannot be empty", nameof(sorted));

    long totalPairs = (long)n * (n + 1) / 2;
    if (k < 1 || k > totalPairs)
      throw new ArgumentOutOfRangeException(nameof(k), $"k must be in range [1, {totalPairs}]");

    if (n == 1)
      return sorted[0];

    return FindExactQuantile(sorted, k);
  }

  /// <summary>
  /// Compute both lower and upper bounds from pairwise averages.
  /// </summary>
  /// <param name="sorted">Sorted input array.</param>
  /// <param name="marginLo">Rank of lower bound (1-based).</param>
  /// <param name="marginHi">Rank of upper bound (1-based).</param>
  /// <returns>Lower and upper quantiles.</returns>
  public static (double Lo, double Hi) Bounds(IReadOnlyList<double> sorted, long marginLo, long marginHi)
  {
    Debug.Assert(
      IsSorted(sorted),
      "CenterQuantilesImpl.Bounds: input must be sorted");
    int n = sorted.Count;
    if (n == 0)
      throw new ArgumentException("Input cannot be empty", nameof(sorted));

    long totalPairs = (long)n * (n + 1) / 2;

    marginLo = Max(1, Min(marginLo, totalPairs));
    marginHi = Max(1, Min(marginHi, totalPairs));

    double lo = FindExactQuantile(sorted, marginLo);
    double hi = FindExactQuantile(sorted, marginHi);

    return (Min(lo, hi), Max(lo, hi));
  }

  /// <summary>
  /// Count pairwise averages ≤ target value.
  /// </summary>
  private static long CountPairsLessOrEqual(IReadOnlyList<double> sorted, double target)
  {
    int n = sorted.Count;
    long count = 0;
    // j is not reset: as i increases, threshold decreases monotonically
    int j = n - 1;

    for (int i = 0; i < n; i++)
    {
      double threshold = 2 * target - sorted[i];

      while (j >= 0 && sorted[j] > threshold)
        j--;

      if (j >= i)
        count += j - i + 1;
    }

    return count;
  }

  /// <summary>
  /// Find exact k-th pairwise average using selection algorithm.
  /// </summary>
  private static double FindExactQuantile(IReadOnlyList<double> sorted, long k)
  {
    int n = sorted.Count;
    long totalPairs = (long)n * (n + 1) / 2;

    if (n == 1)
      return sorted[0];

    if (k == 1)
      return sorted[0];

    if (k == totalPairs)
      return sorted[n - 1];

    double lo = sorted[0];
    double hi = sorted[n - 1];
    const double eps = RelativeEpsilon;

    var candidates = new List<double>(n);

    while (hi - lo > eps * Max(1.0, Max(Abs(lo), Abs(hi))))
    {
      double mid = 0.5 * lo + 0.5 * hi;
      long countLessOrEqual = CountPairsLessOrEqual(sorted, mid);

      if (countLessOrEqual >= k)
        hi = mid;
      else
        lo = mid;
    }

    double target = 0.5 * lo + 0.5 * hi;

    for (int i = 0; i < n; i++)
    {
      double threshold = 2 * target - sorted[i];

      int left = i;
      int right = n;

      while (left < right)
      {
        int m = (left + right) / 2;
        if (sorted[m] < threshold - eps)
          left = m + 1;
        else
          right = m;
      }

      if (left < n && left >= i && Abs(sorted[left] - threshold) < eps * Max(1.0, Abs(threshold)))
        candidates.Add(0.5 * sorted[i] + 0.5 * sorted[left]);

      if (left > i)
      {
        double avgBefore = 0.5 * sorted[i] + 0.5 * sorted[left - 1];
        if (avgBefore <= target + eps)
          candidates.Add(avgBefore);
      }
    }

    if (candidates.Count == 0)
      return target;

    candidates.Sort();

    foreach (double candidate in candidates)
    {
      long countAtCandidate = CountPairsLessOrEqual(sorted, candidate);
      if (countAtCandidate >= k)
        return candidate;
    }

    return target;
  }
}
```

--------------------------------

### Perform Statistical Analysis and Randomization

Source: https://pragmastat.dev/py

Demonstrates one-sample and two-sample metrics, confirmatory comparisons, random number generation, and sampling from various distributions.

```python
from pragmastat import (
    Metric,
    Rng,
    Sample,
    Threshold,
    center,
    center_bounds,
    compare1,
    compare2,
    disparity,
    disparity_bounds,
    ratio,
    ratio_bounds,
    shift,
    shift_bounds,
    spread,
    spread_bounds,
)
from pragmastat.distributions import Additive, Exp, Multiplic, Power, Uniform


def main():
    # --- One-Sample ---

    x = Sample(list(range(1, 201)))

    result = center(x)
    print(result.value)  # 100.5
    bounds = center_bounds(x, 1e-3)
    print(f"Bounds(lower={bounds.lower}, upper={bounds.upper})")  # Bounds(lower=86.0, upper=115.0)
    print(spread(x).value)  # 59.0
    bounds = spread_bounds(x, 1e-3, seed="demo")
    print(f"Bounds(lower={bounds.lower}, upper={bounds.upper})")  # Bounds(lower=44.0, upper=87.0)

    # --- Two-Sample ---

    x = Sample(list(range(1, 201)))
    y = Sample(list(range(101, 301)))

    print(shift(x, y).value)  # -100.0
    bounds = shift_bounds(x, y, 1e-3)
    print(f"Bounds(lower={bounds.lower}, upper={bounds.upper})")  # Bounds(lower=-120.0, upper=-80.0)
    print(ratio(x, y).value)  # 0.5008354224706334
    bounds = ratio_bounds(x, y, 1e-3)
    print(
        f"Bounds(lower={bounds.lower}, upper={bounds.upper})"
    )  # Bounds(lower=0.4066666666666668, upper=0.5958333333333332)
    print(disparity(x, y).value)  # -1.694915254237288
    bounds = disparity_bounds(x, y, 1e-3, seed="demo")
    print(
        f"Bounds(lower={bounds.lower}, upper={bounds.upper})"
    )  # Bounds(lower=-3.1025641025641026, upper=-0.8494623655913979)

    # --- Confirmatory Comparison ---

    x = Sample(list(range(1, 201)))
    projection = compare1(x, [Threshold(Metric.CENTER, 50.0, 1e-3)])[0]
    print(projection.verdict.value)  # greater

    x = Sample(list(range(1, 201)))
    y = Sample(list(range(101, 301)))
    projection = compare2(x, y, [Threshold(Metric.SHIFT, 0.0, 1e-3)])[0]
    print(projection.verdict.value)  # less

    # --- Randomization ---

    rng = Rng("demo-uniform")
    print(rng.uniform_float())  # 0.2640554428629759
    print(rng.uniform_float())  # 0.9348534835582796

    rng = Rng("demo-uniform-int")
    print(rng.uniform_int(0, 100))  # 41

    rng = Rng("demo-sample")
    print(rng.sample([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 3))  # [3, 8, 9]

    rng = Rng("demo-resample")
    print(rng.resample([1, 2, 3, 4, 5], 7))  # [3, 1, 3, 2, 4, 1, 2]

    rng = Rng("demo-shuffle")
    print(rng.shuffle([1, 2, 3, 4, 5]))  # [4, 2, 3, 5, 1]

    # --- Distributions ---

    rng = Rng("demo-dist-additive")
    print(Additive(0, 1).sample(rng))  # 0.17410448679568188

    rng = Rng("demo-dist-multiplic")
    print(Multiplic(0, 1).sample(rng))  # 1.1273244602673853

    rng = Rng("demo-dist-exp")
    print(Exp(1).sample(rng))  # 0.6589065267276553

    rng = Rng("demo-dist-power")
    print(Power(1, 2).sample(rng))  # 1.023677535537084

    rng = Rng("demo-dist-uniform")
    print(Uniform(0, 10).sample(rng))  # 6.54043657816832


if __name__ == "__main__":
    main()
```