nsys-math-utils 0.2.0

Math types and traits
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195

//! Data analysis routines

use crate::approx;
use crate::num_traits as num;
use crate::algebra;
use crate::traits::*;
use crate::types::*;

/// Compute the covariance matrix of a list of 2D points
pub fn covariance_2 <S> (points : &[Point2 <S>]) -> Matrix2 <S> where
  S : Field + num::NumCast + MaybeSerDes
{
  let npoints      = S::from (points.len()).unwrap();
  let mean         = points.iter().map (|p| p.0).sum::<Vector2 <_>>() / npoints;
  let centered     = points.iter().map (|p| p.0 - mean);
  let [xx, xy, yy] = centered.map (|p| [p.x * p.x, p.x * p.y, p.y * p.y ])
    .fold ([S::zero(); 3], |acc, item| std::array::from_fn (|i| acc[i] + item[i]));
  let div = S::one() / (npoints - S::one());
  let cov_xx = xx * div;
  let cov_xy = xy * div;
  let cov_yx = cov_xy;
  let cov_yy = yy * div;
  Matrix2::from_col_arrays ([
    [cov_xx, cov_xy],
    [cov_yx, cov_yy]
  ])
}

/// Compute the covariance matrix of a list of 3D points
pub fn covariance_3 <S> (points : &[Point3 <S>]) -> Matrix3 <S> where
  S : Field + num::NumCast + MaybeSerDes
{
  let npoints      = S::from (points.len()).unwrap();
  let mean         = points.iter().map (|p| p.0).sum::<Vector3 <_>>() / npoints;
  let centered     = points.iter().map (|p| p.0 - mean);
  let [xx, xy, xz, yy, yz, zz] = centered.map (|p| [
    p.x * p.x,
    p.x * p.y,
    p.x * p.z,
    p.y * p.y,
    p.y * p.z,
    p.z * p.z
  ]).fold ([S::zero(); 6], |acc, item| std::array::from_fn (|i| acc[i] + item[i]));
  let div = S::one() / (npoints - S::one());
  Matrix3::from_col_arrays ([
    [xx, xy, xz],
    [xy, yy, yz],
    [xz, yz, zz]
  ]) * div
}

/// 2D principal component analysis
pub fn pca_2 <S> (points : &[Point2 <S>]) -> Rotation2 <S> where
  S : Real + num::NumCast + approx::RelativeEq <Epsilon=S> + std::fmt::Debug
    + MaybeSerDes,
  Vector2 <S> : NormedVectorSpace <S>
{
  let covariance = covariance_2 (points);
  // covariance matrix is symmetric, therefore eigenvectors should be orthogonal and
  // thus point in the direction of the principal components
  let (eigenvalues, mut eigenvectors) : (Vec<_>, Vec<_>) =
    algebra::linear::eigensystem_2 (covariance).iter().copied().map (Option::unwrap)
      .unzip();
  let eigenvectors = {
    // normalize eigenvectors and create array
    eigenvectors.iter_mut().for_each (|v| *v = NormedVectorSpace::normalize (*v));
    <[Vector2 <S>; 2]>::try_from (eigenvectors.as_slice()).unwrap()
      .map (Vector2::into_array)
  };
  // the principal component with the most variance corresponds to the largest
  // eigenvalue, we take this to be the first principal component
  debug_assert!(eigenvalues[0] >= eigenvalues[1]);
  let mut mat = Matrix2::from_col_arrays (eigenvectors);
  // ensure determinant == 1
  if mat.determinant().is_negative() {
    mat.cols[0] *= -S::one();
  }
  Rotation2::new_approx (mat).unwrap()
}

/// 3D principal component analysis
pub fn pca_3 <S> (points : &[Point3 <S>]) -> Rotation3 <S> where
  S : Real + num::NumCast + approx::RelativeEq <Epsilon=S> + std::fmt::Debug
    + MaybeSerDes,
  Vector2 <S> : NormedVectorSpace <S>
{
  let covariance = covariance_3 (points);
  // covariance matrix is symmetric, therefore eigenvectors should be orthogonal and
  // thus point in the direction of the principal components
  let (eigenvalues, mut eigenvectors) : (Vec<_>, Vec<_>) =
    algebra::linear::eigensystem_3 (covariance).iter().flatten().copied().unzip();
  let eigenvectors = {
    // normalize eigenvectors and create array
    eigenvectors.iter_mut().for_each (|v| *v = NormedVectorSpace::normalize (*v));
    <[Vector3 <S>; 3]>::try_from (eigenvectors.as_slice()).unwrap()
      .map (Vector3::into_array)
  };
  // the principal component with the most variance corresponds to the largest
  // eigenvalue, we take this to be the first principal component
  debug_assert!(eigenvalues[0] >= eigenvalues[1]);
  debug_assert!(eigenvalues[1] >= eigenvalues[2]);
  let mut mat = Matrix3::from_col_arrays (eigenvectors);
  // ensure determinant == 1
  if mat.determinant().is_negative() {
    mat.cols[0] *= -S::one();
  }
  Rotation3::new_approx (mat).unwrap()
}

#[cfg(test)]
mod tests {
  #![expect(clippy::unreadable_literal)]

  use super::*;
  #[test]
  fn covariance_2d() {
    let points = [
      [ 1.0, 2.0],
      [ 1.0, 3.0],
      [10.0, 2.0]
    ].map (Point2::<f32>::from);
    let covariance = covariance_2 (points.as_slice());
    assert_eq!(covariance.into_col_arrays(),
      [ [27.0, -1.5],
        [-1.5, 0.3333333] ]);
  }

  #[test]
  fn pca_2d() {
    let points = [
      [-1.0, -1.0],
      [-1.0,  1.0],
      [ 1.0, -1.0],
      [ 1.0,  1.0]
    ].map (Point2::<f32>::from);
    let components = pca_2 (points.as_slice()).into_col_arrays();
    assert_eq!(components[0], [1.0, 0.0]);
    assert_eq!(components[1], [0.0, 1.0]);
  }

  #[test]
  fn pca_3d() {
    use std::f32::consts::{FRAC_1_SQRT_2, FRAC_PI_4};
    let points = [
      [ 1.0, -1.0, -1.0],
      [ 1.0, -1.0,  1.0],
      [ 1.0,  1.0, -1.0],
      [ 1.0,  1.0,  1.0],
      [-1.0, -1.0, -1.0],
      [-1.0, -1.0,  1.0],
      [-1.0,  1.0, -1.0],
      [-1.0,  1.0,  1.0]
    ].map (Point3::<f32>::from);
    let components = pca_3 (points.as_slice()).into_col_arrays();
    assert_eq!(components[0], [1.0, 0.0, 0.0]);
    assert_eq!(components[1], [0.0, 1.0, 0.0]);
    assert_eq!(components[2], [0.0, 0.0, 1.0]);

    // this case returns axis-aligned components, this agrees when numpy output (except
    // that numpy returns the identity matrix)
    let rotation = Rotation3::from_angle_y (FRAC_PI_4.into());
    let points = [
      [ 1.0, -1.0, -1.0],
      [ 1.0, -1.0,  1.0],
      [ 1.0,  1.0, -1.0],
      [ 1.0,  1.0,  1.0],
      [-1.0, -1.0, -1.0],
      [-1.0, -1.0,  1.0],
      [-1.0,  1.0, -1.0],
      [-1.0,  1.0,  1.0]
    ].map (Point3::<f32>::from).map (|p| rotation.rotate (p));
    let components = pca_3 (points.as_slice()).into_col_arrays();
    assert_eq!(components[0], [0.0, -1.0, 0.0]);
    assert_eq!(components[1], [1.0,  0.0, 0.0]);
    assert_eq!(components[2], [0.0,  0.0, 1.0]);

    let rotation = Rotation3::from_angle_y (FRAC_PI_4.into());
    let points = [
      [ 1.0, -1.0, -4.0],
      [ 1.0, -1.0,  4.0],
      [ 1.0,  1.0, -4.0],
      [ 1.0,  1.0,  4.0],
      [-1.0, -1.0, -4.0],
      [-1.0, -1.0,  4.0],
      [-1.0,  1.0, -4.0],
      [-1.0,  1.0,  4.0]
    ].map (Point3::<f32>::from).map (|p| rotation.rotate (p));
    let components = pca_3 (points.as_slice());
    approx::assert_relative_eq!(components.mat(), Matrix3::from_col_arrays ([
      [FRAC_1_SQRT_2, 0.0, FRAC_1_SQRT_2],
      [0.0, 1.0, 0.0],
      [-FRAC_1_SQRT_2, 0.0, FRAC_1_SQRT_2]
    ]));
  }
}