Machine Learning#

Matrices #

mlstatpy.ml.matrices.gram_schmidt(mat, change=False)[source][source]#

Applies the Gram-Schmidt process. Due to performance, every row is considered as a vector.

@param mat matrix @param change returns the matrix to change the basis @return new matrix or (new matrix, change matrix)

The function assumes the matrix mat is horizontal: it has more columns than rows.

Note

The implementation could be improved by directly using BLAS function.

<<<

import numpy
from mlstatpy.ml.matrices import gram_schmidt

X = numpy.array([[1.0, 2.0, 3.0, 4.0], [5.0, 6.0, 6.0, 6.0], [5.0, 6.0, 7.0, 8.0]])
T, P = gram_schmidt(X, change=True)
print(T)
print(P)

>>>

    [[ 0.183  0.365  0.548  0.73 ]
     [ 0.736  0.502  0.024 -0.453]
     [ 0.651 -0.67  -0.181  0.308]]
    [[ 0.183  0.     0.   ]
     [-0.477  0.243  0.   ]
     [-1.814 -1.81   2.303]]

mlstatpy.ml.matrices.linear_regression(X, y, algo=None)[source][source]#

Solves the linear regression problem, find \(\beta\) which minimizes \(\norme{y - X\beta}\), based on the algorithm Arbre de décision optimisé pour les régressions linéaires.

Paramètres:

X – features
y – targets
algo – None to use the standard algorithm \(\beta = (X'X)^{-1} X'y\), “gram”, “qr”

Renvoie:

beta

<<<

import numpy
from mlstatpy.ml.matrices import linear_regression

X = numpy.array([[1.0, 2.0, 3.0, 4.0], [5.0, 6.0, 6.0, 6.0], [5.0, 6.0, 7.0, 8.0]]).T
y = numpy.array([0.1, 0.2, 0.19, 0.29])
beta = linear_regression(X, y, algo="gram")
print(beta)

>>>

    [ 0.077  0.037 -0.032]

algo=None computes \(\beta = (X'X)^{-1} X'y\). algo='qr' uses a QR decomposition and calls function dtrtri to invert an upper triangular matrix. algo='gram' uses gram_schmidt and then computes the solution of the linear regression (see above for a link to the algorithm).

mlstatpy.ml.matrices.streaming_gram_schmidt_update(Xk, Pk)[source][source]#

Updates matrix \(P_k\) to produce \(P_{k+1}\) which is the matrix P in algorithm Streaming Linear Regression. The function modifies the matrix Pk given as an input.

@param Xk kth row @param Pk matrix P at iteration k-1

mlstatpy.ml.matrices.streaming_gram_schmidt(mat, start=None)[source][source]#

Solves the linear regression problem, find \(\beta\) which minimizes \(\norme{y - X\beta}\), based on algorithm Streaming Gram-Schmidt.

@param mat matrix @param start first row to start iteration, X.shape[1] by default @return iterator on

The function assumes the matrix mat is horizontal: it has more columns than rows.

<<<

import numpy
from mlstatpy.ml.matrices import streaming_gram_schmidt

X = numpy.array(
    [[1, 0.5, 10.0, 5.0, -2.0], [0, 0.4, 20, 4.0, 2.0], [0, 0.7, 20, 4.0, 2.0]],
    dtype=float,
).T

for i, p in enumerate(streaming_gram_schmidt(X.T)):
    print("iteration", i, "\n", p)
    t = X[: i + 3] @ p.T
    print(t.T @ t)

>>>

    iteration 0 
     [[ 0.099  0.     0.   ]
     [-0.953  0.482  0.   ]
     [-0.287 -3.338  3.481]]
    [[ 1.000e+00 -1.310e-15 -2.238e-15]
     [-1.310e-15  1.000e+00  1.390e-14]
     [-2.238e-15  1.390e-14  1.000e+00]]
    iteration 1 
     [[ 0.089  0.     0.   ]
     [-0.308  0.177  0.   ]
     [-0.03  -3.334  3.348]]
    [[ 1.000e+00 -3.570e-16 -1.808e-15]
     [-3.570e-16  1.000e+00  2.423e-15]
     [-1.808e-15  2.423e-15  1.000e+00]]
    iteration 2 
     [[ 0.088  0.     0.   ]
     [-0.212  0.128  0.   ]
     [-0.016 -3.335  3.342]]
    [[ 1.000e+00  1.756e-17 -4.660e-15]
     [ 1.756e-17  1.000e+00  9.833e-16]
     [-4.660e-15  9.833e-16  1.000e+00]]

mlstatpy.ml.matrices.streaming_linear_regression_update(Xk, yk, XkXk, bk)[source][source]#

Updates coefficients \(\beta_k\) to produce \(\beta_{k+1}\) in Streaming Linear Regression. The function modifies the matrix Pk given as an input.

Paramètres:

Xk – kth row
yk – kth target
XkXk – matrix \(X_{1..k}'X_{1..k}\), updated by the function
bk – current coefficient (updated by the function)

mlstatpy.ml.matrices.streaming_linear_regression(mat, y, start=None)[source][source]#

Streaming algorithm to solve a linear regression. See Streaming Linear Regression.

@param mat features @param y expected target @return iterator on coefficients

<<<

import numpy
from mlstatpy.ml.matrices import streaming_linear_regression, linear_regression

X = numpy.array(
    [[1, 0.5, 10.0, 5.0, -2.0], [0, 0.4, 20, 4.0, 2.0], [0, 0.7, 20, 4.0, 3.0]],
    dtype=float,
).T
y = numpy.array([1.0, 0.3, 10, 5.1, -3.0])

for i, bk in enumerate(streaming_linear_regression(X, y)):
    bk0 = linear_regression(X[: i + 3], y[: i + 3])
    print("iteration", i, bk, bk0)

>>>

    iteration 0 [ 1.     0.667 -0.667] [ 1.     0.667 -0.667]
    iteration 1 [ 1.03   0.682 -0.697] [ 1.03   0.682 -0.697]
    iteration 2 [ 1.036  0.857 -0.875] [ 1.036  0.857 -0.875]

mlstatpy.ml.matrices.streaming_linear_regression_gram_schmidt_update(Xk, yk, Xkyk, Pk, bk)[source][source]#

Updates coefficients \(\beta_k\) to produce \(\beta_{k+1}\) in Streaming Linear Regression. The function modifies the matrix Pk given as an input.

Paramètres:

Xk – kth row
yk – kth target
Xkyk – matrix \(X_{1..k}' y_{1..k}\) (updated by the function)
Pk – Gram-Schmidt matrix produced by the streaming algorithm (updated by the function)

Renvoie:

bk current coefficient (updated by the function)

mlstatpy.ml.matrices.streaming_linear_regression_gram_schmidt(mat, y, start=None)[source][source]#

Streaming algorithm to solve a linear regression with Gram-Schmidt algorithm. See Streaming Linear Regression version Gram-Schmidt.

@param mat features @param y expected target @return iterator on coefficients

<<<

import numpy
from mlstatpy.ml.matrices import streaming_linear_regression, linear_regression

X = numpy.array(
    [[1, 0.5, 10.0, 5.0, -2.0], [0, 0.4, 20, 4.0, 2.0], [0, 0.7, 20, 4.0, 3.0]],
    dtype=float,
).T
y = numpy.array([1.0, 0.3, 10, 5.1, -3.0])

for i, bk in enumerate(streaming_linear_regression(X, y)):
    bk0 = linear_regression(X[: i + 3], y[: i + 3])
    print("iteration", i, bk, bk0)

>>>

    iteration 0 [ 1.     0.667 -0.667] [ 1.     0.667 -0.667]
    iteration 1 [ 1.03   0.682 -0.697] [ 1.03   0.682 -0.697]
    iteration 2 [ 1.036  0.857 -0.875] [ 1.036  0.857 -0.875]

Métriques #

class mlstatpy.ml.roc.ROC(y_true=None, y_score=None, sample_weight=None, df=None)[source][source]

Helper to draw a ROC curve.

Initialisation with a dataframe and two or three columns:

column 1: score (y_score)
column 2: expected answer (boolean) (y_true)
column 3: weight (optional) (sample_weight)

Paramètres:

y_true – if df is None, y_true, y_score, sample_weight must be filled, y_true is whether or None the answer is true. y_true means the prediction is right.
y_score – score prediction
sample_weight – weights
df – dataframe or array or list, it must contains 2 or 3 columns always in the same order

mlstatpy.ml.voronoi.voronoi_estimation_from_lr(L, B, C=None, D=None, cl=0, qr=True, max_iter=None, verbose=False)[source][source]#

Determines a Voronoi diagram close to a convex partition defined by a logistic regression in n classes. \(M \in \mathbb{M}_{nd}\) a row matrix \((L_1, ..., L_n)\). Every border between two classes i and j is defined by: \(\scal{L_i}{X} + B = \scal{L_j}{X} + B\).

The function looks for a set of points from which the Voronoi diagram can be inferred. It is done through a linear regression with norm L1. See Régression logistique, diagramme de Voronoï, k-Means.

@param L matrix @param B vector @param C additional conditions (see below) @param D addition condictions (see below) @param cl class on which the additional conditions applies @param qr use quantile regression @param max_iter number of condition to remove until convergence @param verbose display information while training @return matrix \(P \in \mathbb{M}_{nd}\)

The function solves the linear system:

\[\begin{split}\begin{array}{rcl} & \Longrightarrow & \left\{\begin{array}{l} \scal{\frac{L_i-L_j}{\norm{L_i-L_j}}}{P_i + P_j} + 2 \frac{B_i - B_j}{\norm{L_i-L_j}} = 0 \\ \scal{P_i- P_j}{u_{ij}} - \scal{P_i - P_j}{\frac{L_i-L_j} {\norm{L_i-L_j}}} \scal{\frac{L_i-L_j}{\norm{L_i-L_j}}}{u_{ij}}=0 \end{array} \right. \end{array}\end{split}\]

If the number of dimension is big and the number of classes small, the system has multiple solution. Addition condition must be added such as \(CP_i=D\) where i=cl, \(P_i\) is the Voronoï point attached to class cl. Quantile regression is not implemented in scikit-learn. We use QuantileLinearRegression.

After the first iteration, the function determines the furthest pair of points and removes it from the list of equations. If max_iter is None, the system goes until the number of equations is equal to the number of points * 2, otherwise it stops after max_iter removals. This is not the optimal pair to remove as they could still be neighbors but it should be a good heuristic.

Plus proches voisins #

class mlstatpy.ml.kppv.NuagePoints[source][source]: Définit une classe de nuage de points. On suppose qu’ils sont définis par une matrice, chaque ligne est un élément.

class mlstatpy.ml.kppv_laesa.NuagePointsLaesa(nb_pivots)[source][source]: Implémente l’algorithme des plus proches voisins, version LAESA.

Tree and neural networks #

class mlstatpy.ml._neural_tree_node.NeuralTreeNode(weights, bias=None, activation='sigmoid', nodeid=-1, tag=None)[source][source]

One node in a neural network.

Paramètres:

weights – weights
bias – bias, if None, draws a random number
activation – activation function
nodeid – node id
tag – unused but to add information on how this node was created

class mlstatpy.ml.neural_tree.NeuralTreeNet(dim, empty=True)[source][source]

Node ensemble.

Paramètres:

dim – space dimension
empty – empty network, other adds an identity node

<<<

import numpy
from mlstatpy.ml.neural_tree import NeuralTreeNode, NeuralTreeNet

w1 = numpy.array([-0.5, 0.8, -0.6])

neu = NeuralTreeNode(w1[1:], bias=w1[0], activation="sigmoid")
net = NeuralTreeNet(2, empty=True)
net.append(neu, numpy.arange(2))

ide = NeuralTreeNode(numpy.array([1.0]), bias=numpy.array([0.0]), activation="identity")

net.append(ide, numpy.arange(2, 3))

X = numpy.abs(numpy.random.randn(10, 2))
pred = net.predict(X)
print(pred)

>>>

    /home/xadupre/github/mlstatpy/mlstatpy/ml/_neural_tree_node.py:184: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
      self.coef[0] = bias
    [[0.978 0.905 0.435 0.435]
     [0.894 0.123 0.535 0.535]
     [0.089 0.577 0.315 0.315]
     [1.633 1.054 0.543 0.543]
     [0.288 0.415 0.373 0.373]
     [0.157 1.061 0.267 0.267]
     [0.181 1.277 0.246 0.246]
     [0.546 0.604 0.395 0.395]
     [0.924 0.263 0.52  0.52 ]
     [1.691 1.358 0.509 0.509]]

class mlstatpy.ml.neural_tree.BaseNeuralTreeNet(estimator, optimizer=None, max_iter=100, early_th=None, verbose=False, lr=None, lr_schedule=None, l1=0.0, l2=0.0, momentum=0.9)[source][source]

Classifier or regressor following scikit-learn API.

Paramètres:

estimator – instance of NeuralTreeNet.
X – training set
y – training labels
optimizer – optimizer, by default, it is SGDOptimizer.
max_iter – number maximum of iterations
early_th – early stopping threshold
verbose – more verbose
lr – to overwrite learning_rate_init if optimizer is None (unused otherwise)
lr_schedule – to overwrite lr_schedule if optimizer is None (unused otherwise)
l1 – L1 regularization if optimizer is None (unused otherwise)
l2 – L2 regularization if optimizer is None (unused otherwise)
momentum – used if optimizer is None

class mlstatpy.ml.neural_tree.NeuralTreeNetClassifier(estimator, optimizer=None, max_iter=100, early_th=None, verbose=False, lr=None, lr_schedule=None, l1=0.0, l2=0.0, momentum=0.9)[source][source]

Classifier following scikit-learn API.

Paramètres:

estimator – instance of NeuralTreeNet.
optimizer – optimizer, by default, it is SGDOptimizer.
max_iter – number maximum of iterations
early_th – early stopping threshold
verbose – more verbose
lr – to overwrite learning_rate_init if optimizer is None (unused otherwise)
lr_schedule – to overwrite lr_schedule if optimizer is None (unused otherwise)
l1 – L1 regularization if optimizer is None (unused otherwise)
l2 – L2 regularization if optimizer is None (unused otherwise)
momentum – used if optimizer is None

class mlstatpy.ml.neural_tree.NeuralTreeNetRegressor(estimator, optimizer=None, max_iter=100, early_th=None, verbose=False, lr=None, lr_schedule=None, l1=0.0, l2=0.0, momentum=0.9)[source][source]

Regressor following scikit-learn API.

Paramètres:

estimator – instance of NeuralTreeNet.
optimizer – optimizer, by default, it is SGDOptimizer.
max_iter – number maximum of iterations
early_th – early stopping threshold
verbose – more verbose
lr – to overwrite learning_rate_init if optimizer is None (unused otherwise)
lr_schedule – to overwrite lr_schedule if optimizer is None (unused otherwise)
l1 – L1 regularization if optimizer is None (unused otherwise)
l2 – L2 regularization if optimizer is None (unused otherwise)
momentum – used if optimizer is None

Machine Learning#

Matrices#

Métriques#

Plus proches voisins#

Tree and neural networks#

Matrices #

Métriques #

Plus proches voisins #

Tree and neural networks #