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//! The output analysis module provides standard statistical analysis tools
//! for analyzing simulation outputs. Independent, identically-distributed
//! (IID) samples are analyzed with the `IndependentSample`. Time series
//! (including those with initialization bias and autocorrelation) can be
//! analyzed with `TerminatingSimulationOutput` or `SteadyStateOutput`.
use std::f64::INFINITY;
use num_traits::{Float, NumAssign};
use serde::{Deserialize, Serialize};
pub mod t_scores;
use crate::utils::errors::SimulationError;
use crate::utils::usize_sqrt;
fn sum<T: Float>(points: &[T]) -> T
where
f64: Into<T>,
{
points.iter().fold(0.0.into(), |sum, point| sum + *point)
}
/// This function calculates the sample mean from a set of points - a simple
/// arithmetic mean.
fn sample_mean<T: Float>(points: &[T]) -> Result<T, SimulationError>
where
f64: Into<T>,
{
Ok(sum(points) / usize_to_float(points.len())?)
}
/// This function calculates sample variance, given a set of points and the
/// sample mean.
fn sample_variance<T: Float>(points: &[T], mean: &T) -> Result<T, SimulationError>
where
f64: Into<T>,
{
Ok(points
.iter()
.fold(0.0.into(), |acc, point| acc + (*point - *mean).powi(2))
/ usize_to_float(points.len())?)
}
/// This function converts a usize to a Float, with an associated
/// `SimulationError` returned for failed conversions
fn usize_to_float<T: Float>(unconv: usize) -> Result<T, SimulationError> {
T::from(unconv).ok_or(SimulationError::FloatConvError)
}
/// The confidence interval provides an upper and lower estimate on a given
/// output, whether that output is an independent, identically-distributed
/// sample or time series data.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct ConfidenceInterval<T: Float> {
lower: T,
upper: T,
}
impl<T: Float> ConfidenceInterval<T>
where
f64: Into<T>,
{
pub fn lower(&self) -> T {
self.lower
}
pub fn upper(&self) -> T {
self.upper
}
pub fn half_width(&self) -> T {
(self.upper - self.lower) / 2.0.into()
}
}
/// The independent sample is for independent, identically-distributed (IID)
/// samples, or where treating the data as an IID sample is determined to be
/// reasonable. Typically, this will be non-time series data - no
/// autocorrelation. There are no additional requirements on the data beyond
/// being IID. For example, there are no normality assumptions. The
/// `TerminatingSimulationOutput` or `SteadyStateOutput` structs are
/// available for non-IID output analysis.
#[derive(Debug, Default, Clone, Serialize, Deserialize)]
pub struct IndependentSample<T> {
points: Vec<T>,
mean: T,
variance: T,
}
impl<T: Float> IndependentSample<T>
where
f64: Into<T>,
{
/// This constructor method creates an `IndependentSample` from a vector
/// of floating point values.
pub fn post(points: Vec<T>) -> Result<IndependentSample<T>, SimulationError> {
let mean = sample_mean(&points)?;
let variance = sample_variance(&points, &mean)?;
Ok(IndependentSample {
points,
mean,
variance,
})
}
/// Calculate the confidence interval of the mean, base on the provided
/// value of alpha.
pub fn confidence_interval_mean(
&self,
alpha: T,
) -> Result<ConfidenceInterval<T>, SimulationError> {
if self.points.len() == 1 {
return Ok(ConfidenceInterval {
lower: self.mean,
upper: self.mean,
});
}
let points_len: T = usize_to_float(self.points.len())?;
Ok(ConfidenceInterval {
lower: self.mean
- t_scores::t_score(alpha, self.points.len() - 1) * self.variance.sqrt()
/ points_len.sqrt(),
upper: self.mean
+ t_scores::t_score(alpha, self.points.len() - 1) * self.variance.sqrt()
/ points_len.sqrt(),
})
}
/// Return the sample mean.
pub fn point_estimate_mean(&self) -> T {
self.mean
}
/// Return the sample variance.
pub fn variance(&self) -> T {
self.variance
}
}
/// Terminating simulations are useful when the initial and final conditions
/// of a simulation are known, and set deliberately to match real world
/// conditions. For example, a simulation spanning a 9:00 to 17:00 work day
/// might use the terminating simulation approach to simulation experiments
/// and analysis. These initial and final conditions are known and of
/// interest.
#[derive(Debug, Default, Clone, Serialize, Deserialize)]
pub struct TerminatingSimulationOutput<T> {
time_series_replications: Vec<Vec<T>>,
replication_means: Vec<T>,
replications_mean: Option<T>,
replications_variance: Option<T>,
}
impl<T: Float> TerminatingSimulationOutput<T> {
/// This `TerminatingSimulationOutput` constructor method creates a new
/// terminating simulation output, with a single replication. The
/// `put_time_series` method is then used to load additional simulation
/// replications (time series).
pub fn post(time_series: Vec<T>) -> TerminatingSimulationOutput<T> {
TerminatingSimulationOutput {
time_series_replications: vec![time_series],
replication_means: Vec::new(),
replications_mean: None,
replications_variance: None,
}
}
/// This method loads a single simulation replication output into the
/// `TerminatingSimulationOutput` object. Typically, simulation analysis
/// will require many replications, and thus many `put_time_series`
/// calls.
pub fn put_time_series(&mut self, time_series: Vec<T>) {
self.time_series_replications.push(time_series);
}
}
/// Steady-state simulations are useful when the initial conditions and/or
/// final conditions of a simulation are not well-known or not of interest.
/// Steady-state simulation is interested in the long-run behavior of the
/// system, where initial condition effects are negligible. Steady-state
/// simulation analysis is primarily concerned with initialization bias (bias
/// caused by setting initial conditions of the simulation) and
/// auto-correlation (the tendency of a data point in a time series to show
/// correlation with the latest, previous values in that time series). When
/// the interest is a steady-state simulation output, standard simulation
/// design suggests the use of only a single simulation replication.
#[derive(Debug, Default, Clone, Serialize, Deserialize)]
pub struct SteadyStateOutput<T> {
time_series: Vec<T>,
/// Points are removed from the beginning of the sample for initialization
/// bias reduction.
deletion_point: Option<usize>,
/// Batching is used to combat autocorrelation in the time series.
batch_size: Option<usize>,
batch_count: Option<usize>,
batch_means: Vec<T>,
batches_mean: Option<T>,
batches_variance: Option<T>,
}
impl<T: Float + NumAssign> SteadyStateOutput<T>
where
f64: Into<T>,
{
/// This `SteadyStateOutput` constructor method takes the simulation
/// output time series, as a vector of floating point values.
pub fn post(time_series: Vec<T>) -> SteadyStateOutput<T> {
SteadyStateOutput {
time_series,
deletion_point: None,
batch_size: None,
batch_count: None,
batch_means: Vec::new(),
batches_mean: None,
batches_variance: None,
}
}
/// The steady-state output analysis in `set_to_fixed_budget` analyzes
/// the time series to determine the appropriate initialization data
/// deletion and batching strategies. Initialization data deletion and
/// batching reduce concerns around initialization bias and
/// autocorrelation, respectively. After this method determines the
/// strategy/configuration, the `calculate_batch_statistics` then
/// executes the processing.
fn set_to_fixed_budget(&mut self) -> Result<(), SimulationError> {
let mut s = 0.0.into();
let mut q = 0.0.into();
let mut d = self.time_series.len() - 2;
let mut mser = vec![0.0.into(); self.time_series.len() - 1];
let time_series_len: T = usize_to_float(self.time_series.len())?;
loop {
s += self.time_series[d + 1];
q += self.time_series[d + 1].powi(2);
mser[d] = q - s.powi(2) / (time_series_len - usize_to_float(d)?).powi(3);
if d == 0 {
// Find the minimum MSER in the first half of the time series
let min_mser = (0..(self.time_series.len() - 1) / 2)
.fold(INFINITY.into(), |min_mser, mser_index| {
min_mser.min(mser[mser_index])
});
// Use that point for deletion determination
self.deletion_point = mser.iter().position(|mser_value| *mser_value == min_mser);
break;
}
d -= 1;
}
// Schmeiser [1982] found that, for a fixed total sample size, there
// is little benefit from dividing it into more than k = 30 batches,
// even if we could do so and still retain independence between the
// batch means.
let deletion_point = self
.deletion_point
.ok_or(SimulationError::PrerequisiteCalcError)?;
let batch_count = usize::min(usize_sqrt(self.time_series.len() - deletion_point), 30);
self.batch_count = Some(batch_count);
let batch_size = (self.time_series.len() - deletion_point) / batch_count;
// if data are left over, eliminate from the beginning
self.deletion_point = Some(self.time_series.len() - batch_count * batch_size);
self.batch_size = Some(batch_size);
Ok(())
}
/// After the `set_to_fixed_budget` method analyzes the time series to
/// determine the appropriate initialization data deletion and batching
/// configuration, this method uses that configuration for calculation
/// and processing. This method stores the batch statistics in the
/// `SteadyStateOutput` struct, for later use in retrieving point and
/// confidence interval estimates.
fn calculate_batch_statistics(&mut self) -> Result<(), SimulationError> {
if self.batch_count.is_none() {
self.set_to_fixed_budget()?;
}
let deletion_point = self
.deletion_point
.ok_or(SimulationError::PrerequisiteCalcError)?;
let batch_size = self
.batch_size
.ok_or(SimulationError::PrerequisiteCalcError)?;
let batch_count = self
.batch_count
.ok_or(SimulationError::PrerequisiteCalcError)?;
let batch_means: Result<Vec<T>, SimulationError> = (0..batch_count)
.map(|batch_index| {
let batch_start_index = deletion_point + batch_size * batch_index;
let batch_end_index = deletion_point + batch_size * (batch_index + 1);
let points: Vec<T> = (batch_start_index..batch_end_index)
.map(|index| self.time_series[index])
.collect();
sample_mean(&points)
})
.collect();
self.batch_means = batch_means?;
let batches_mean = sample_mean(&self.batch_means)?;
self.batches_variance = Some(sample_variance(&self.batch_means, &batches_mean)?);
self.batches_mean = Some(batches_mean);
Ok(())
}
/// The method provides a confidence interval on the mean, for the
/// simuation output. If not already processed, the raw data will first
/// use standard approaches for initialization bias reduction and
/// autocorrelation management.
pub fn confidence_interval_mean(
&mut self,
alpha: T,
) -> Result<ConfidenceInterval<T>, SimulationError> {
if self.batches_mean.is_none() {
self.calculate_batch_statistics()?;
}
let batches_mean = self
.batches_mean
.ok_or(SimulationError::PrerequisiteCalcError)?;
let batch_count = self
.batch_count
.ok_or(SimulationError::PrerequisiteCalcError)?;
let f_batch_count: T = usize_to_float(batch_count)?;
let batches_variance = self
.batches_variance
.ok_or(SimulationError::PrerequisiteCalcError)?;
if batch_count == 1 {
return Ok(ConfidenceInterval {
lower: batches_mean,
upper: batches_mean,
});
}
Ok(ConfidenceInterval {
lower: batches_mean
- t_scores::t_score(alpha, batch_count) * batches_variance.sqrt()
/ f_batch_count.sqrt(),
upper: batches_mean
+ t_scores::t_score(alpha, batch_count - 1) * batches_variance.sqrt()
/ f_batch_count.sqrt(),
})
}
/// The method provides a point estimate on the mean, for the simulation
/// output. If not already processed, the raw data will first use
/// standard approaches for initialization bias reduction and
/// autocorrelation management.
pub fn point_estimate_mean(&mut self) -> Result<T, SimulationError> {
if self.batches_mean.is_none() {
self.calculate_batch_statistics()?;
}
self.batches_mean
.ok_or(SimulationError::PrerequisiteCalcError)
}
}
#[cfg(test)]
mod tests {
use super::*;
fn epsilon() -> f64 {
1.0e-12
}
#[test]
fn confidence_interval_mean() {
let sample = IndependentSample::post(vec![
1.02, 0.73, 3.20, 0.23, 1.76, 0.47, 1.89, 1.45, 0.44, 0.23,
]);
let confidence_interval = sample.unwrap().confidence_interval_mean(0.1).unwrap();
assert!((confidence_interval.lower - 0.7492630635369267).abs() < epsilon());
assert!((confidence_interval.upper - 1.534736936463073).abs() < epsilon());
}
}