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Orthogonal Projection to Latent Structures

Project description

pyopls - Orthogonal Projection to Latent Structures in Python.

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This package provides a scikit-learn-style transformer to perform OPLS. OPLS is a pre-processing method to remove variation from the descriptor variables that are orthogonal to the target variable (1).

This package also provides a class to validate OPLS models using a 1-component PLS regression with cross-validation and permutation tests (2) for both regression and classification metrics (from permutations of the target) and feature PLS loadings (from permutations of the features).

Table of Contents

  1. Installation
  2. Notes
  3. Examples
    1. OPLS and PLS-DA
    2. Validation
  4. References
  5. Data Acknowledgment


pyopls is available via pypi:

pip install pyopls

You may also install directly from this repository for the current master:

pip install git+git://

New versions are uploaded to pypi whenever the version number is incremented in on the master branch.


  • The implementation provided here is equivalent to that of the libPLS MATLAB library, which is a faithful recreation of Trygg and Wold's algorithm.
    • This package uses a different definition for R2X, however (see below)
  • OPLS inherits sklearn.base.TransformerMixin (like sklearn.decomposition.PCA) but does not inherit sklearn.base.RegressorMixin because it is not a regressor like sklearn.cross_decomposition.PLSRegression. You can use the output of OPLS.transform() as an input to another regressor or classifier.
  • Like sklearn.cross_decomposition.PLSRegression, OPLS will center both X and Y before performing the algorithm. This makes centering by class in PLS-DA models unnecessary.
  • The score() function of OPLS performs the R2X score, the ratio of the variance in the transformed X to the variance in the original X. A lower score indicates more orthogonal variance removed.
  • OPLS only supports 1-column targets.



A CSV file containing 1H-NMR spectra for 118 serum samples of patients with colon cancer diagnoses and healthy controls is located in colorectal_cancer_nmr.csv in the root of this repository (see acknowledgment below).

OPLS-processed data require only 1 PLS component. Performing a 39-component OPLS improves cross-validated accuracy from 70% to 100%, AUC from .578 to 1 and DQ2 (3) from 0.04 to 0.99.

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import roc_curve, roc_auc_score
from pyopls import OPLS
from sklearn.cross_decomposition import PLSRegression
from sklearn.model_selection import cross_val_predict, LeaveOneOut
from sklearn.metrics import r2_score, accuracy_score

spectra = pd.read_csv('colorectal_cancer_nmr.csv', index_col=0)
spectra = spectra[spectra.classification.isin(['Colorectal Cancer', 'Healthy Control'])]
target = spectra.classification.apply(lambda x: 1 if x == 'Colorectal Cancer' else -1)
spectra = spectra.drop('classification', axis=1)

opls = OPLS(39)
Z = opls.fit_transform(spectra, target)

pls = PLSRegression(1)
y_pred = cross_val_predict(pls, spectra, target, cv=LeaveOneOut())
q_squared = r2_score(target, y_pred)  # -0.107
dq_squared = r2_score(target, np.clip(y_pred, -1, 1))  # -0.106
accuracy = accuracy_score(target, np.sign(y_pred))  # 0.705

processed_y_pred = cross_val_predict(pls, Z, target, cv=LeaveOneOut())
processed_q_squared = r2_score(target, processed_y_pred)  # 0.981
processed_dq_squared = r2_score(target, np.clip(processed_y_pred, -1, 1))  # 0.984
processed_accuracy = accuracy_score(target, np.sign(processed_y_pred))  # 1.0

r2_X = opls.score(spectra)  # 7.8e-12 (most variance is removed)

fpr, tpr, thresholds = roc_curve(target, y_pred)
roc_auc = roc_auc_score(target, y_pred)
proc_fpr, proc_tpr, proc_thresholds = roc_curve(target, processed_y_pred)
proc_roc_auc = roc_auc_score(target, processed_y_pred)

plt.plot(fpr, tpr, lw=2, color='blue', label=f'Unprocessed (AUC={roc_auc:.4f})')
plt.plot(proc_fpr, proc_tpr, lw=2, color='red',
         label=f'39-component OPLS (AUC={proc_roc_auc:.4f})')
plt.plot([0, 1], [0, 1], color='gray', lw=2, linestyle='--')
plt.xlabel('False Positive Rate')
plt.ylabel('True Positive Rate')
plt.title('ROC Curve')
plt.legend(loc='lower right')

plt.figure(1), target)
df = pd.DataFrame(np.column_stack([pls.x_scores_, opls.T_ortho_[:, 0]]),
                  index=spectra.index, columns=['t', 't_ortho'])                           
pos_df = df[target==1]
neg_df = df[target==-1]
plt.scatter(neg_df['t'], neg_df['t_ortho'], c='blue', label='Healthy Control')
plt.scatter(pos_df['t'], pos_df['t_ortho'], c='red', label='Colorectal Cancer')
plt.title('PLS Scores')
plt.legend(loc='upper right')

ROC Curve

roc curve

Scores Plot

scores plot


The fit() method of OPLSValidator will find the optimum number of components to remove, then evaluate the results on a 1-component sklearn.cross_decomposition.PLSRegression model. A permutation test is performed for each metric by permuting the target and for the PLS loadings by permuting the features.

This snippet will determine the best number of components to remove, perform permutation tests for regression metrics and perform two-tailed permutation tests for each feature (bin) relative to it's loading. The feature permutation tests for the colorectal cancer dataset would take quite some time, as they require that the model be fit as many as 874k times. So instead, we look at the UCI ML Wine Dataset provided by scikit-learn The feature permutation tests reveal that hue and malic acid do not differentate class 1 from class 0.

import pandas as pd
from pyopls import OPLSValidator
from sklearn.datasets import load_wine

wine_data = load_wine()
df = pd.DataFrame(wine_data['data'], columns=wine_data['feature_names'])
df['classification'] = wine_data['target']
df = df[df.classification.isin((0, 1))]
target = df.classification.apply(lambda x: 1 if x else -1)  # discriminant for class 1 vs class 0
X = df[[c for c in df.columns if c!='classification']]

validator = OPLSValidator(k=-1).fit(X, target)

Z = validator.opls_.transform(X)

feature_df = pd.DataFrame()
feature_df['feature_name'] = wine_data['feature_names']
feature_df['feature_p_value'] = validator.feature_p_values_
feature_df['feature_loading'] = validator.pls_.x_loadings_
print(feature_df.loc[feature_df.feature_loading.abs().sort_values(ascending=False).index].to_markdown())  # Pandas 1.0+ required for to_markdown

Feature importances

feature_name feature_p_value feature_loading
12 proline 0.00990099 0.385955
9 color_intensity 0.00990099 0.381981
0 alcohol 0.00990099 0.379567
6 flavanoids 0.00990099 0.359975
5 total_phenols 0.00990099 0.336182
11 od280/od315_of_diluted_wines 0.00990099 0.299045
3 alcalinity_of_ash 0.00990099 -0.239887
2 ash 0.00990099 0.22916
7 nonflavanoid_phenols 0.00990099 -0.224338
4 magnesium 0.00990099 0.18662
8 proanthocyanins 0.00990099 0.181767
1 malic_acid 0.564356 0.0293328
10 hue 0.623762 0.0210777


  1. Johan Trygg and Svante Wold. Orthogonal projections to latent structures (O-PLS). J. Chemometrics 2002; 16: 119-128. DOI: 10.1002/cem.695
  2. Eugene Edington and Patrick Onghena. "Calculating P-Values" in Randomization tests, 4th edition. New York: Chapman & Hall/CRC, 2007, pp. 33-53. DOI: 10.1201/9781420011814.
  3. Johan A. Westerhuis, Ewoud J. J. van Velzen, Huub C. J. Hoefsloot, Age K. Smilde. Discriminant Q-squared for improved discrimination in PLSDA models. Metabolomics 2008; 4: 293-296. DOI: 10.1007/s11306-008-0126-2

Data Acknowledgment

The test dataset provided at pyopls/tests/colorectal_cancer_nmr.csv is available at the NIH Common Fund's National Metabolomics Data Repository (NMDR) website, the Metabolomics Workbench, [] where it has been assigned Project ID PR000227. The data can be accessed directly via it's Project DOI 10.21228/M89P43. This work is supported by NIH grant, U2C-DK119886.

Note: The test dataset consists only of those spectra belonging to samples labeled "Colorectal Cancer" or "Healthy Control". The "target" variable has the value -1 for samples labeled "Healthy Control" and value +1 for samples labeled "Colorectal Cancer".

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