67 lines
1.6 KiB
C
67 lines
1.6 KiB
C
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#include "common.h"
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#include <gsl/gsl_matrix.h>
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/* Builds the covariance matrix Σ
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* from the standard parameters (σ, ρ)
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* of a bivariate gaussian.
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*/
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gsl_matrix* normal_cov(struct par *p);
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/* Builds the mean vector of
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* a bivariate gaussian.
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*/
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gsl_vector* normal_mean(struct par *p);
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/* `fisher_proj(c1, c2)` computes the optimal
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* projection map, which maximises the separation
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* between the two classes.
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* The projection vector w is given by
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*
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* w = Sw⁻¹ (μ₂ - μ₁)
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*
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* where Sw = Σ₁ + Σ₂ is the so-called within-class
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* covariance matrix.
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*/
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gsl_vector* fisher_proj(sample_t *c1, sample_t *c2);
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/* `fisher_cut(ratio, w, c1, c2)` computes
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* the threshold (cut), on the line given by
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* `w`, to discriminates the classes `c1`, `c2`;
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* with `ratio` being the ratio of their prior
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* probabilities.
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*
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* The cut is fixed by the condition of
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* conditional probability being the
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* same for each class:
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*
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* P(c₁|x) p(x|c₁)⋅p(c₁)
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* ------- = --------------- = 1;
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* P(c₂|x) p(x|c₁)⋅p(c₂)
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*
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* where p(x|c) is the probability for point x
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* along the fisher projection line. If the classes
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* are bivariate gaussian then p(x|c) is simply
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* given by a normal distribution:
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*
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* Φ(μ=(w,μ), σ=(w,Σw))
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*
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* The solution is then
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*
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* t = (b/a) + √((b/a)² - c/a);
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*
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* where
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*
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* 1. a = S₁² - S₂²
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* 2. b = M₂S₁² - M₁S₂²
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* 3. c = M₂²S₁² - M₁²S₂² - 2S₁²S₂² log(α)
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* 4. α = p(c₁)/p(c₂)
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*
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*/
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double fisher_cut(
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double ratio,
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gsl_vector *w,
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sample_t *c1, sample_t *c2);
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