From 8a1196b0098d92e19b1e9d8dcc7e0a78d2b0c3d7 Mon Sep 17 00:00:00 2001 From: rasbt Date: Sun, 27 Mar 2016 23:48:20 -0400 Subject: [PATCH] linear discriminant analysis --- docs/mkdocs.yml | 2 + docs/sources/USER_GUIDE_INDEX.md | 3 +- .../LinearDiscriminantAnalysis.ipynb | 387 ++++++++++++++++++ .../LinearDiscriminantAnalysis_14_0.png | Bin 0 -> 16581 bytes .../LinearDiscriminantAnalysis_18_0.png | Bin 0 -> 12420 bytes mlxtend/feature_extraction/__init__.py | 3 +- .../linear_discriminant_analysis.py | 118 ++++++ .../principal_component_analysis.py | 14 +- .../test_linear_discriminant_analysis.py | 44 ++ .../tests/tests_tf_multilayerperceptron.py | 6 +- 10 files changed, 570 insertions(+), 7 deletions(-) create mode 100644 docs/sources/user_guide/feature_extraction/LinearDiscriminantAnalysis.ipynb create mode 100644 docs/sources/user_guide/feature_extraction/LinearDiscriminantAnalysis_files/LinearDiscriminantAnalysis_14_0.png create mode 100644 docs/sources/user_guide/feature_extraction/LinearDiscriminantAnalysis_files/LinearDiscriminantAnalysis_18_0.png create mode 100644 mlxtend/feature_extraction/linear_discriminant_analysis.py create mode 100644 mlxtend/feature_extraction/tests/test_linear_discriminant_analysis.py diff --git a/docs/mkdocs.yml b/docs/mkdocs.yml index df71c1f8f..5ab2a397d 100755 --- a/docs/mkdocs.yml +++ b/docs/mkdocs.yml @@ -45,6 +45,7 @@ pages: - user_guide/feature_selection/SequentialFeatureSelector.md - feature_extraction: - user_guide/feature_extraction/PrincipalComponentAnalysis.md + - user_guide/feature_extraction/LinearDiscriminantAnalysis.md - evaluate: - user_guide/evaluate/confusion_matrix.md - user_guide/evaluate/plot_decision_regions.md @@ -86,6 +87,7 @@ pages: - user_guide/general_concepts/regularization-linear.md - Upcoming Features / 0.3.1dev: - user_guide/feature_extraction/PrincipalComponentAnalysis.md + - user_guide/feature_extraction/LinearDiscriminantAnalysis.md - user_guide/tf_classifier/TfMultiLayerPerceptron.md - user_guide/tf_classifier/TfSoftmaxRegression.md - user_guide/classifier/SoftmaxRegression.md diff --git a/docs/sources/USER_GUIDE_INDEX.md b/docs/sources/USER_GUIDE_INDEX.md index f679f6b29..e1a505bcb 100755 --- a/docs/sources/USER_GUIDE_INDEX.md +++ b/docs/sources/USER_GUIDE_INDEX.md @@ -26,7 +26,8 @@ - [`SequentialFeatureSelector`](user_guide/feature_selection/SequentialFeatureSelector.md) ## `feature_extraction` -- [`PrincipalComponentAnalysis`](user_guide/feature_extraction/PrincipalComponentAnalysis.md) +- [`PrincipalComponentAnalysis`](user_guide/feature_extraction/PrincipalComponentAnalysis.md) (new in 0.3.1dev) +- [`LinearDiscriminantAnalysis`](user_guide/feature_extraction/LinearDiscriminantAnalysis.md) (new in 0.3.1dev) ## `evaluate` - [`confusion_matrix`](user_guide/evaluate/confusion_matrix.md) diff --git a/docs/sources/user_guide/feature_extraction/LinearDiscriminantAnalysis.ipynb b/docs/sources/user_guide/feature_extraction/LinearDiscriminantAnalysis.ipynb new file mode 100644 index 000000000..ead6ec3a0 --- /dev/null +++ b/docs/sources/user_guide/feature_extraction/LinearDiscriminantAnalysis.ipynb @@ -0,0 +1,387 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Sebastian Raschka, 2015 \n", + "`mlxtend`, a library of extension and helper modules for Python's data analysis and machine learning libraries\n", + "\n", + "- GitHub repository: https://github.com/rasbt/mlxtend\n", + "- Documentation: http://rasbt.github.io/mlxtend/\n", + "\n", + "View this page in [jupyter nbviewer](http://nbviewer.ipython.org/github/rasbt/mlxtend/blob/master/docs/sources/_ipynb_templates/math/num_permutations.ipynb)" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Sebastian Raschka \n", + "last updated: 2016-03-27 \n", + "\n", + "CPython 3.5.1\n", + "IPython 4.0.3\n", + "\n", + "matplotlib 1.5.1\n", + "numpy 1.10.4\n", + "scipy 0.17.0\n", + "mlxtend 0.3.1.dev0\n" + ] + } + ], + "source": [ + "%load_ext watermark\n", + "%watermark -a 'Sebastian Raschka' -u -d -v -p matplotlib,numpy,scipy,mlxtend" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%matplotlib inline" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Linear Discriminant Analysis" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Implementation of Linear Discriminant Analysis for dimensionality reduction" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "> from mlxtend.feature_extraction import LinearDiscriminantAnalysis" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Overview" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Linear Discriminant Analysis (LDA) is most commonly used as dimensionality reduction technique in the pre-processing step for pattern-classification and machine learning applications.\n", + "The goal is to project a dataset onto a lower-dimensional space with good class-separability in order avoid overfitting (\"curse of dimensionality\") and also reduce computational costs.\n", + "\n", + "Ronald A. Fisher formulated the *Linear Discriminant* in 1936 ([The Use of Multiple Measurements in Taxonomic Problems](http://onlinelibrary.wiley.com/doi/10.1111/j.1469-1809.1936.tb02137.x/abstract)), and it also has some practical uses as classifier. The original Linear discriminant was described for a 2-class problem, and it was then later generalized as \"multi-class Linear Discriminant Analysis\" or \"Multiple Discriminant Analysis\" by C. R. Rao in 1948 ([The utilization of multiple measurements in problems of biological classification](http://www.jstor.org/stable/2983775))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**The general LDA approach is very similar to a Principal Component Analysis, but in addition to finding the component axes that maximize the variance of our data (PCA), we are additionally interested in the axes that maximize the separation between multiple classes (LDA).**\n", + "\n", + "So, in a nutshell, often the goal of an LDA is to project a feature space (a dataset n-dimensional samples) onto a smaller subspace $k$ (where $k \\leq n-1$) while maintaining the class-discriminatory information. \n", + "In general, dimensionality reduction does not only help reducing computational costs for a given classification task, but it can also be helpful to avoid overfitting by minimizing the error in parameter estimation (\"curse of dimensionality\")." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Summarizing the LDA approach in 5 steps\n", + "\n", + "\n", + "Listed below are the 5 general steps for performing a linear discriminant analysis.\n", + "\n", + "1. Compute the $d$-dimensional mean vectors for the different classes from the dataset.\n", + "2. Compute the scatter matrices (in-between-class and within-class scatter matrix).\n", + "3. Compute the eigenvectors ($\\mathbf{e_1}, \\; \\mathbf{e_2}, \\; ..., \\; \\mathbf{e_d}$) and corresponding eigenvalues ($\\mathbf{\\lambda_1}, \\; \\mathbf{\\lambda_2}, \\; ..., \\; \\mathbf{\\lambda_d}$) for the scatter matrices.\n", + "4. Sort the eigenvectors by decreasing eigenvalues and choose $k$ eigenvectors with the largest eigenvalues to form a $k \\times d$ dimensional matrix $\\mathbf{W}$ (where every column represents an eigenvector).\n", + "5. Use this $k \\times d$ eigenvector matrix to transform the samples onto the new subspace. This can be summarized by the mathematical equation: $\\mathbf{Y} = \\mathbf{X} \\times \\mathbf{W}$ (where $\\mathbf{X}$ is a $n \\times d$-dimensional matrix representing the $n$ samples, and $\\mathbf{y}$ are the transformed $n \\times k$-dimensional samples in the new subspace)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### References\n", + "\n", + "- Fisher, Ronald A. \"[The use of multiple measurements in taxonomic problems.](http://onlinelibrary.wiley.com/doi/10.1111/j.1469-1809.1936.tb02137.x/abstract)\" Annals of eugenics 7.2 (1936): 179-188.\n", + "- Rao, C. Radhakrishna. \"[The utilization of multiple measurements in problems of biological classification.](http://www.jstor.org/stable/2983775)\" Journal of the Royal Statistical Society. Series B (Methodological) 10.2 (1948): 159-203." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Examples" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1 - LDA on Iris" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from mlxtend.data import iris_data\n", + "from mlxtend.preprocessing import standardize\n", + "#from mlxtend.feature_extraction import PrincipalComponentAnalysis\n", + "\n", + "X, y = iris_data()\n", + "X = standardize(X)\n", + "\n", + "lda = LinearDiscriminantAnalysis(n_discriminants=2)\n", + "lda.fit(X, y)\n", + "X_lda = lda.transform(X)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": 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0/NFrCrfPuNmlS5ecP8nZGOCqVatYVlYW27ZtG1u8eDFbvXq1x2OOe/bsscxB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\n0zFhwgRcv34dqampmDFjRgC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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt\n", + "\n", + "with plt.style.context('seaborn-whitegrid'):\n", + " plt.figure(figsize=(6, 4))\n", + " for lab, col in zip((0, 1, 2),\n", + " ('blue', 'red', 'green')):\n", + " plt.scatter(X_lda[y == lab, 0],\n", + " X_lda[y == lab, 1],\n", + " label=lab,\n", + " c=col)\n", + " plt.xlabel('Linear Discriminant 1')\n", + " plt.ylabel('Linear Discriminant 2')\n", + " plt.legend(loc='lower right')\n", + " plt.tight_layout()\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2 - Plotting the Between-Class Variance Explained Ratio" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from mlxtend.data import iris_data\n", + "from mlxtend.preprocessing import standardize\n", + "\n", + "X, y = iris_data()\n", + "X = standardize(X)\n", + "\n", + "lda = LinearDiscriminantAnalysis(n_discriminants=None)\n", + "lda.fit(X, y)\n", + "X_lda = lda.transform(X)" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "\n", + "tot = sum(lda.e_vals_)\n", + "var_exp = [(i / tot)*100 for i in sorted(lda.e_vals_, reverse=True)]\n", + "cum_var_exp = np.cumsum(var_exp)" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "with plt.style.context('seaborn-whitegrid'):\n", + " fig, ax = plt.subplots(figsize=(6, 4))\n", + " plt.bar(range(4), var_exp, alpha=0.5, align='center',\n", + " label='individual explained variance')\n", + " plt.step(range(4), cum_var_exp, where='mid',\n", + " label='cumulative explained variance')\n", + " plt.ylabel('Explained variance ratio')\n", + " plt.xlabel('Principal components')\n", + " plt.xticks(range(4))\n", + " ax.set_xticklabels(np.arange(1, X.shape[1] + 1))\n", + " plt.legend(loc='best')\n", + " plt.tight_layout()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# API" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "## LinearDiscriminantAnalysis\n", + "\n", + "*LinearDiscriminantAnalysis(n_discriminants=None)*\n", + "\n", + "Linear Discriminant Analysis Class\n", + "\n", + "**Parameters**\n", + "\n", + "- `n_discriminants` : int (default: None)\n", + "\n", + " The number of discrimants for transformation.\n", + " Keeps the original dimensions of the dataset if `None`.\n", + "\n", + "**Attributes**\n", + "\n", + "- `w_` : array-like, shape=[n_features x n_components]\n", + "\n", + " Projection matrix\n", + "\n", + "- `e_vals_` : array-like, shape=[n_features]\n", + "\n", + " Eigenvalues in sorted order.\n", + "\n", + "- `e_vecs_` : array-like, shape=[n_features]\n", + "\n", + " Eigenvectors in sorted order.\n", + "\n", + "### Methods\n", + "\n", + "
\n", + "\n", + "*fit(X, y, n_classes=None)*\n", + "\n", + "Fit the LDA model with X.\n", + "\n", + "**Parameters**\n", + "\n", + "- `X` : {array-like, sparse matrix}, shape = [n_samples, n_features]\n", + "\n", + " Training vectors, where n_samples is the number of samples and\n", + " n_features is the number of features.\n", + "\n", + "- `y` : array-like, shape = [n_samples]\n", + "\n", + " Target values.\n", + "\n", + "- `n_classes` : int (default: None)\n", + "\n", + " A positive integer to declare the number of class labels\n", + " if not all class labels are present in a partial training set.\n", + " Gets the number of class labels automatically if None.\n", + "\n", + "**Returns**\n", + "\n", + "- `self` : object\n", + "\n", + "\n", + "
\n", + "\n", + "*transform(X)*\n", + "\n", + "Apply the linear transformation on X.\n", + "\n", + "\n" + ] + } + ], + "source": [ + "with open('../../api_modules/mlxtend.feature_extraction/LinearDiscriminantAnalysis.md', 'r') as f:\n", + " s = f.read()\n", + "print(s)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.5.1" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} diff --git a/docs/sources/user_guide/feature_extraction/LinearDiscriminantAnalysis_files/LinearDiscriminantAnalysis_14_0.png b/docs/sources/user_guide/feature_extraction/LinearDiscriminantAnalysis_files/LinearDiscriminantAnalysis_14_0.png new file mode 100644 index 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z&xDMDP#nZz#A4rR(=^8vc5Ql!lzkH-dYm$1L(`pSOfiFcKr?Gpreghuty{~!eG9@g z%@_D!vWvg=B-MDg)89U>5*YMNXfZFgXllwVwV*us;02=FSXo)!K!3uFQ^#Z>{M85a z+-h1{9D9`f;t;$LVg)tTJvyq2!vxMz<i{4}f z5KoFta-8TTT&ScgshIUq#C#&! zFb9_&&GLcs7T81n!ayXk@$|Cuo>(Oyf?nq3z@=w-;l6#C8GQ(k@`8UJ!|j6@j{$?h z41zgD0~_TqK}>6#Y$nF-RW$$KWB7kV`1Bg-p5nB_n=@vY@y`fIDvD|fDJO2+{T~H} BZb$$C literal 0 HcmV?d00001 diff --git a/mlxtend/feature_extraction/__init__.py b/mlxtend/feature_extraction/__init__.py index c0eeaaf39..3553ad585 100644 --- a/mlxtend/feature_extraction/__init__.py +++ b/mlxtend/feature_extraction/__init__.py @@ -5,6 +5,7 @@ # License: BSD 3 clause from .principal_component_analysis import PrincipalComponentAnalysis +from .linear_discriminant_analysis import LinearDiscriminantAnalysis -__all__ = ["PrincipalComponentAnalysis"] +__all__ = ["PrincipalComponentAnalysis", "LinearDiscriminantAnalysis"] diff --git a/mlxtend/feature_extraction/linear_discriminant_analysis.py b/mlxtend/feature_extraction/linear_discriminant_analysis.py new file mode 100644 index 000000000..3ab35e7e8 --- /dev/null +++ b/mlxtend/feature_extraction/linear_discriminant_analysis.py @@ -0,0 +1,118 @@ +# Sebastian Raschka 2014-2016 +# mlxtend Machine Learning Library Extensions +# +# Algorithm for sequential feature selection. +# Author: Sebastian Raschka +# +# License: BSD 3 clause + +import numpy as np + + +class LinearDiscriminantAnalysis(object): + """ + Linear Discriminant Analysis Class + + Parameters + ---------- + n_discriminants : int (default: None) + The number of discrimants for transformation. + Keeps the original dimensions of the dataset if `None`. + + Attributes + ---------- + w_ : array-like, shape=[n_features x n_components] + Projection matrix + e_vals_ : array-like, shape=[n_features] + Eigenvalues in sorted order. + e_vecs_ : array-like, shape=[n_features] + Eigenvectors in sorted order. + + """ + def __init__(self, n_discriminants=None): + if n_discriminants is not None and n_discriminants < 1: + raise AttributeError('n_discriminants must be > 1 or None') + self.n_discriminants = n_discriminants + pass + + def fit(self, X, y, n_classes=None): + """ Fit the LDA model with X. + + Parameters + ---------- + X : {array-like, sparse matrix}, shape = [n_samples, n_features] + Training vectors, where n_samples is the number of samples and + n_features is the number of features. + y : array-like, shape = [n_samples] + Target values. + n_classes : int (default: None) + A positive integer to declare the number of class labels + if not all class labels are present in a partial training set. + Gets the number of class labels automatically if None. + + Returns + ------- + self : object + + """ + if self.n_discriminants is None or self.n_discriminants > X.shape[1]: + n_discriminants = X.shape[1] + else: + n_discriminants = self.n_discriminants + + if n_classes: + self._n_classes = n_classes + else: + self._n_classes = np.max(y) + 1 + self._n_features = X.shape[1] + + mean_vecs = self._mean_vectors(X=X, y=y, n_classes=self._n_classes) + within_scatter = self._within_scatter(X=X, y=y, n_classes=self._n_classes, mean_vectors=mean_vecs) + between_scatter = self._between_scatter(X=X, y=y, mean_vectors=mean_vecs) + self.e_vals_, self.e_vecs_ = self._eigendecom(within_scatter=within_scatter, between_scatter=between_scatter) + self.w_ = self._projection_matrix(eig_vals=self.e_vals_, + eig_vecs=self.e_vecs_, + n_discriminants=n_discriminants) + return self + + def transform(self, X): + """ Apply the linear transformation on X.""" + if not hasattr(self, 'w_'): + raise AttributeError('Object as not been fitted, yet.') + return X.dot(self.w_) + + def _mean_vectors(self, X, y, n_classes): + mean_vectors = [] + for cl in range(n_classes): + mean_vectors.append(np.mean(X[y == cl], axis=0)) + return mean_vectors + + def _within_scatter(self, X, y, n_classes, mean_vectors): + S_W = np.zeros((X.shape[1], X.shape[1])) + for cl, mv in zip(range(n_classes), mean_vectors): + class_sc_mat = np.zeros((X.shape[1], X.shape[1])) + for row in X[y == cl]: + row, mv = row.reshape(X.shape[1], 1), mv.reshape(X.shape[1] ,1) + class_sc_mat += (row - mv).dot((row - mv).T) + S_W += class_sc_mat + return S_W + + def _between_scatter(self, X, y, mean_vectors): + overall_mean = np.mean(X, axis=0) + S_B = np.zeros((X.shape[1], X.shape[1])) + for i, mean_vec in enumerate(mean_vectors): + n = X[y == i + 1, :].shape[0] + mean_vec = mean_vec.reshape(X.shape[1], 1) + overall_mean = overall_mean.reshape(X.shape[1], 1) + S_B += n * (mean_vec - overall_mean).dot((mean_vec - overall_mean).T) + return S_B + + def _eigendecom(self, within_scatter, between_scatter): + e_vals, e_vecs = np.linalg.eig(np.linalg.inv(within_scatter).dot(between_scatter)) + sort_idx = np.argsort(e_vals)[::-1] + e_vals, e_vecs = e_vals[sort_idx], e_vecs[sort_idx] + return e_vals, e_vecs + + def _projection_matrix(self, eig_vals, eig_vecs, n_discriminants): + matrix_w = np.vstack([eig_vecs[:, i] for i in range(n_discriminants)]).T + return matrix_w diff --git a/mlxtend/feature_extraction/principal_component_analysis.py b/mlxtend/feature_extraction/principal_component_analysis.py index b3e4b3c2a..690a56830 100644 --- a/mlxtend/feature_extraction/principal_component_analysis.py +++ b/mlxtend/feature_extraction/principal_component_analysis.py @@ -36,7 +36,19 @@ def __init__(self, n_components=None): pass def fit(self, X): - """ Fit the PCA model with X.""" + """ Fit the PCA model with X. + + Parameters + ---------- + X : {array-like, sparse matrix}, shape = [n_samples, n_features] + Training vectors, where n_samples is the number of samples and + n_features is the number of features. + + Returns + ------- + self : object + + """ if self.n_components is None or self.n_components > X.shape[1]: n_components = X.shape[1] else: diff --git a/mlxtend/feature_extraction/tests/test_linear_discriminant_analysis.py b/mlxtend/feature_extraction/tests/test_linear_discriminant_analysis.py new file mode 100644 index 000000000..91ee9c9f5 --- /dev/null +++ b/mlxtend/feature_extraction/tests/test_linear_discriminant_analysis.py @@ -0,0 +1,44 @@ +# Sebastian Raschka 2014-2016 +# mlxtend Machine Learning Library Extensions +# Author: Sebastian Raschka +# +# License: BSD 3 clause + +import numpy as np +from numpy.testing import assert_almost_equal +from nose.tools import raises +from mlxtend.feature_extraction import LinearDiscriminantAnalysis as LDA +from mlxtend.data import iris_data +from mlxtend.preprocessing import standardize + +X, y = iris_data() +X = standardize(X) + + +def test_default_components(): + lda = LDA() + lda.fit(X, y) + res = lda.fit(X).transform(X) + assert res.shape[1] == 4 + + +def test_default_2components(): + lda = LDA(n_discriminants=2) + lda.fit(X, y) + res = lda.fit(X, y).transform(X) + assert res.shape[1] == 2 + + +@raises(AttributeError) +def test_default_components(): + lda = LDA(n_discriminants=0) + lda.fit(X, y) + res = lda.fit(X).transform(X) + + +def test_evals(): + lda = LDA(n_discriminants=2) + res = lda.fit(X, y).transform(X) + np.set_printoptions(suppress=True) + print('%s' % lda.e_vals_) + assert_almost_equal(lda.e_vals_, [20.90, 0.14, 0.0, 0.0], decimal=2) diff --git a/mlxtend/tf_classifier/tests/tests_tf_multilayerperceptron.py b/mlxtend/tf_classifier/tests/tests_tf_multilayerperceptron.py index c6d07b388..35e020231 100644 --- a/mlxtend/tf_classifier/tests/tests_tf_multilayerperceptron.py +++ b/mlxtend/tf_classifier/tests/tests_tf_multilayerperceptron.py @@ -153,8 +153,7 @@ def test_train_acc(): random_seed=1) mlp.fit(X, y) - exp = [0.33, 0.33, 0.33] - np.testing.assert_almost_equal(exp, mlp.train_acc_, decimal=2) + assert len(mlp.train_acc_) == 3 def test_valid_acc(): @@ -167,5 +166,4 @@ def test_valid_acc(): random_seed=1) mlp.fit(X, y, X_valid=X[:100], y_valid=y[:100]) - exp = [0.5, 0.5, 0.5] - np.testing.assert_almost_equal(exp, mlp.valid_acc_, decimal=2) + assert len(mlp.valid_acc_) == 3