Unit

`Scientific.Analysis`

Declared in Scientific.Analysis.pas

Provides numerical methods and elementary statistical transforms. The routines illustrate documentation ranging from short inline symbols such as $\mu$ and $\sigma$ to complete display equations.

Interface dependencies

Scientific.Core

Types

type-alias

`Scientific.Analysis.TDoubleArray`

Visibility publicSource Scientific.Analysis.pas:16:15


TDoubleArray = Array of Double

Owns a finite sequence $(x_1,\ldots,x_n)$ of real values.

class

`Scientific.Analysis.TGaussianFunction`

Visibility publicSource Scientific.Analysis.pas:25:20

Relationships:

Inherits from TRealFunction


TGaussianFunction = class(TRealFunction)
end

Represents a Gaussian density with mean $\mu$ and deviation $\sigma$ .

\mathcal{N}(x\mid\mu,\sigma^2)= \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left[-\frac{(x-\mu)^2}{2\sigma^2}\right].

Routines

routine

`Scientific.Analysis.ArithmeticMean`

Visibility publicSource Scientific.Analysis.pas:113:24


function ArithmeticMean(const Values: Array of Double): Double

Computes the arithmetic mean of $$n$$ observations.

\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i.

Parameters

Name	Description
`Values`	Observations $(x_1,\ldots,x_n)$ .

Returns

Their arithmetic mean $\bar{x}$ .

Raises

Exception	Condition
`Exception`	When the sequence is empty.

routine

`Scientific.Analysis.Entropy`

Visibility publicSource Scientific.Analysis.pas:151:17


function Entropy(const Probabilities: Array of Double): Double

Computes Shannon entropy in bits.

H(P)=-\sum_{i=1}^{n}p_i\log_2 p_i, \qquad 0\log_2 0:=0.

Parameters

Name	Description
`Probabilities`	Values $p_i\in[0,1]$ .

Returns

The information entropy $$H(P)$$ in bits.

Raises

Exception	Condition
`Exception`	When a probability lies outside $$[0,1]$$ .

routine

`Scientific.Analysis.Logistic`

Visibility publicSource Scientific.Analysis.pas:58:18


function Logistic(const X: Double): Double

Evaluates the logistic sigmoid.

\operatorname{sigmoid}(x)=\frac{1}{1+e^{-x}}.

Parameters

Name	Description
`X`	Real argument $$x$$ .

Returns

A value strictly between zero and one.

routine

`Scientific.Analysis.NewtonStep`

Visibility publicSource Scientific.Analysis.pas:85:20


function NewtonStep(
  const F: IRealFunction;
  const Derivative: IRealFunction;
  const X: Double
): Double

Performs one Newton-Raphson root update.

x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}.

Parameters

Name	Description
`F`	Function $$f$$ whose root is sought.
`Derivative`	Derivative function $$f'$$ .
`X`	Current approximation $$x_k$$ .

Returns

The next approximation $x_{k+1}$ .

Raises

Exception	Condition
`Exception`	When $$f'(x_k)$$ is numerically zero.

routine

`Scientific.Analysis.NormalPDF`

Visibility publicSource Scientific.Analysis.pas:72:19


function NormalPDF(
  const X: Double;
  const Mean: Double;
  const StandardDeviation: Double
): Double

Evaluates a normal probability density without constructing an object.

p(x)=\frac{e^{-z^2/2}}{\sigma\sqrt{2\pi}}, \qquad z=\frac{x-\mu}{\sigma}.

Parameters

Name	Description
`X`	Observation $$x$$ .
`Mean`	Distribution mean $\mu$ .
`StandardDeviation`	Positive standard deviation $\sigma$ .

Returns

The normal density $$p(x)$$ .

Raises

Exception	Condition
`Exception`	When $\sigma\leq 0$ .

routine

`Scientific.Analysis.PopulationVariance`

Visibility publicSource Scientific.Analysis.pas:124:28


function PopulationVariance(const Values: Array of Double): Double

Computes population variance around the arithmetic mean.

\sigma^2=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2.

Parameters

Name	Description
`Values`	Observations $(x_1,\ldots,x_n)$ .

Returns

Their population variance $\sigma^2$ .

Raises

Exception	Condition
`Exception`	When the sequence is empty.

routine

`Scientific.Analysis.SimpsonEstimate`

Visibility publicSource Scientific.Analysis.pas:101:25


function SimpsonEstimate(
  const F: IRealFunction;
  const A: Double;
  const B: Double
): Double

Estimates an integral with one Simpson panel.

\int_a^b f(x)\,dx \approx \frac{b-a}{6}\left[ f(a)+4f\!\left(\frac{a+b}{2}\right)+f(b) \right].

Parameters

Name	Description
`F`	Integrand $$f$$ .
`A`	Lower bound $$a$$ .
`B`	Upper bound $$b$$ .

Returns

The Simpson estimate of the definite integral.

routine

`Scientific.Analysis.Softmax`

Visibility publicSource Scientific.Analysis.pas:139:17


function Softmax(const Values: Array of Double): TDoubleArray

Converts arbitrary scores into a categorical probability distribution.

\operatorname{softmax}(z_i)= \frac{e^{z_i-m}}{\sum_{j=1}^{n}e^{z_j-m}}, \qquad m=\max_j z_j.

Subtracting $$m$$ leaves the result unchanged while improving numerical stability.

Parameters

Name	Description
`Values`	Scores $(z_1,\ldots,z_n)$ .

Returns

Probabilities whose sum is one, or an empty sequence.

Members

constructor

`Scientific.Analysis.TGaussianFunction.Create`

Visibility publicSource Scientific.Analysis.pas:35:23

Parent: Scientific.Analysis.TGaussianFunction


constructor Create(const AMean: Double; const AStandardDeviation: Double)

Creates $\mathcal{N}(\mu,\sigma^2)$ for a positive $\sigma$ .

Parameters

Name	Description
`AMean`	Distribution mean $\mu$ .
`AStandardDeviation`	Standard deviation $\sigma>0$ .

Raises

Exception	Condition
`Exception`	When $\sigma\leq 0$ .

method

`Scientific.Analysis.TGaussianFunction.Evaluate`

Visibility publicSource Scientific.Analysis.pas:41:22

Parent: Scientific.Analysis.TGaussianFunction


function Evaluate(const X: Double): Double; override

Evaluates the Gaussian probability density at $$x$$ .

Parameters

Name	Description
`X`	Observation $$x$$ .

Returns

The density $\mathcal{N}(x\mid\mu,\sigma^2)$ .

property

`Scientific.Analysis.TGaussianFunction.Mean`

Visibility publicSource Scientific.Analysis.pas:44:18

Parent: Scientific.Analysis.TGaussianFunction


Mean : Double

Distribution mean $\mu=\mathbb{E}[X]$ .

property

`Scientific.Analysis.TGaussianFunction.StandardDeviation`

Visibility publicSource Scientific.Analysis.pas:47:31

Parent: Scientific.Analysis.TGaussianFunction


StandardDeviation : Double

Standard deviation $\sigma=\sqrt{\operatorname{Var}(X)}$ .