Balance model complexity and cross-validated scoreΒΆ

This example balances model complexity and cross-validated score by finding a decent accuracy within 1 standard deviation of the best accuracy score while minimising the number of PCA components [1].

The figure shows the trade-off between cross-validated score and the number of PCA components. The balanced case is when n_components=6 and accuracy=0.80, which falls into the range within 1 standard deviation of the best accuracy score.

[1] Hastie, T., Tibshirani, R.,, Friedman, J. (2001). Model Assessment and Selection. The Elements of Statistical Learning (pp. 219-260). New York, NY, USA: Springer New York Inc..



The best_index_ is 2
The n_components selected is 6
The corresponding accuracy score is 0.80

# Author: Wenhao Zhang <>


import numpy as np
import matplotlib.pyplot as plt

from sklearn.datasets import load_digits
from sklearn.decomposition import PCA
from sklearn.model_selection import GridSearchCV
from sklearn.pipeline import Pipeline
from sklearn.svm import LinearSVC

def lower_bound(cv_results):
    Calculate the lower bound within 1 standard deviation
    of the best `mean_test_scores`.

    cv_results : dict of numpy(masked) ndarrays
        See attribute cv_results_ of `GridSearchCV`

        Lower bound within 1 standard deviation of the
        best `mean_test_score`.
    best_score_idx = np.argmax(cv_results['mean_test_score'])

    return (cv_results['mean_test_score'][best_score_idx]
            - cv_results['std_test_score'][best_score_idx])

def best_low_complexity(cv_results):
    Balance model complexity with cross-validated score.

    cv_results : dict of numpy(masked) ndarrays
        See attribute cv_results_ of `GridSearchCV`.

        Index of a model that has the fewest PCA components
        while has its test score within 1 standard deviation of the best
    threshold = lower_bound(cv_results)
    candidate_idx = np.flatnonzero(cv_results['mean_test_score'] >= threshold)
    best_idx = candidate_idx[cv_results['param_reduce_dim__n_components']
    return best_idx

pipe = Pipeline([
        ('reduce_dim', PCA(random_state=42)),
        ('classify', LinearSVC(random_state=42)),

param_grid = {
    'reduce_dim__n_components': [2, 4, 6, 8]

grid = GridSearchCV(pipe, cv=10, n_jobs=1, param_grid=param_grid,
                    scoring='accuracy', refit=best_low_complexity)
digits = load_digits(),

n_components = grid.cv_results_['param_reduce_dim__n_components']
test_scores = grid.cv_results_['mean_test_score']

plt.figure(), test_scores, width=1.3, color='b')

lower = lower_bound(grid.cv_results_)
plt.axhline(np.max(test_scores), linestyle='--', color='y',
            label='Best score')
plt.axhline(lower, linestyle='--', color='.5', label='Best score - 1 std')

plt.title("Balance model complexity and cross-validated score")
plt.xlabel('Number of PCA components used')
plt.ylabel('Digit classification accuracy')
plt.ylim((0, 1.0))
plt.legend(loc='upper left')

best_index_ = grid.best_index_

print("The best_index_ is %d" % best_index_)
print("The n_components selected is %d" % n_components[best_index_])
print("The corresponding accuracy score is %.2f"
      % grid.cv_results_['mean_test_score'][best_index_])

Total running time of the script: ( 0 minutes 17.695 seconds)

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