1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | import numpy as np import matplotlib.pyplot as plt import scipy.stats from sklearn.linear_model import LogisticRegression from sklearn.metrics import ( accuracy_score, balanced_accuracy_score, confusion_matrix ) import pandas as pd import seaborn as sns from sklearn.linear_model import LinearRegression, RidgeCV from sklearn.metrics import r2_score from sklearn.model_selection import ShuffleSplit, KFold |
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 | def generate_classification_results(n=1000, seed=None, noise_sd=1, signal_sd=10, p_pos=0.5, shuffle=False, pred=None, clf=None): rng = np.random.RandomState(seed) X = rng.normal(size=n) * signal_sd noise = noise_sd * rng.randn(n) thresh = scipy.stats.scoreatpercentile(X, 100 * (1 - p_pos)) y = ((X + noise) > thresh).astype('int') if shuffle: rng.shuffle(y) if pred is not None: y_pred = pred * np.ones(n) else: clf = LogisticRegression() clf.fit(X[:, None], y) y_pred = clf.predict(X[:, None]) results = { 'acc': accuracy_score(y, y_pred), 'bacc': balanced_accuracy_score(y, y_pred), 'confusion': confusion_matrix(y, y_pred), 'n': n, 'seed': seed, 'noise_sd': noise_sd, 'signal_sd': signal_sd, 'p_pos': p_pos, 'shuffle': shuffle, 'thresh': thresh, 'clf': clf, 'X': X, 'y': y, 'pred': y_pred, } return(results) |
1 2 | r = generate_classification_results(p_pos=0.9) r |
{'acc': 0.984,
'bacc': np.float64(0.9481118290742605),
'confusion': array([[ 93, 10],
[ 6, 891]]),
'n': 1000,
'seed': None,
'noise_sd': 1,
'signal_sd': 10,
'p_pos': 0.9,
'shuffle': False,
'thresh': np.float64(-13.026289334267915),
'clf': LogisticRegression(),
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1 2 | r = generate_classification_results(p_pos=0.9, pred=1) r |
{'acc': 0.896,
'bacc': np.float64(0.5),
'confusion': array([[ 0, 104],
[ 0, 896]]),
'n': 1000,
'seed': None,
'noise_sd': 1,
'signal_sd': 10,
'p_pos': 0.9,
'shuffle': False,
'thresh': np.float64(-11.641292591387709),
'clf': None,
'X': array([-9.91180680e+00, -4.86661300e+00, 2.52586964e+01, 7.45769636e+00,
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1 2 3 4 5 | from sklearn.naive_bayes import GaussianNB clf = GaussianNB() r = generate_classification_results(p_pos=0.8, clf=clf, noise_sd=10) r |
{'acc': 0.805,
'bacc': np.float64(0.7218452237969413),
'confusion': array([[148, 128],
[ 67, 657]]),
'n': 1000,
'seed': None,
'noise_sd': 10,
'signal_sd': 10,
'p_pos': 0.8,
'shuffle': False,
'thresh': np.float64(-9.033025990136332),
'clf': LogisticRegression(),
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1.06236390e+01, -4.34598315e+00, 1.23199784e+00, 3.92188029e+00]),
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'pred': array([1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1,
1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1])}
1 2 3 4 | r_shuf = generate_classification_results(p_pos=0.8, clf=clf, noise_sd=10, shuffle=True) r_shuf['acc'] r_shuf['bacc'] |
np.float64(0.5)
1 2 3 4 | pred_proba = r['clf'].predict_proba(r['X'].reshape(-1, 1))[:,1] pred_df = pd.DataFrame({'X': r['X'], 'y': r['y'], 'pred_proba': pred_proba}) sns.displot(pred_df, x='pred_proba', hue='y', bins=100) plt.xlabel('Predicted probability') |
Text(0.5, 9.444444444444438, 'Predicted probability')
1 2 3 4 5 6 7 8 9 10 | from sklearn.metrics import roc_curve, roc_auc_score fpr, tpr, _ = roc_curve(r['y'], pred_proba) plt.plot(fpr, tpr) plt.xlabel('False positive rate') plt.ylabel('True positive rate') plt.title('ROC curve') plt.plot([0, 1], [0, 1], linestyle='--', color='gray') # fill area under curve plt.fill_between(fpr, tpr, color='lightgray') |
<matplotlib.collections.PolyCollection at 0x17fdcef60>
1 | roc_auc_score(r['y'], pred_proba) |
np.float64(0.8496426855632958)
1 | roc_auc_score(r_shuf['y'], r_shuf['pred']) |
np.float64(0.5)
Assessing regression peformance¶
1 2 3 4 5 6 7 8 9 | def generate_regression_data(n=50, nfeatures=40, noise_sd=1, seed=None): rng = np.random.RandomState(seed) X = rng.normal(size=(n, nfeatures)) beta = rng.normal(size=nfeatures) y = X @ beta + noise_sd * rng.normal(size=n) return(X, y, beta) X, y, beta = generate_regression_data(seed=1) |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | # first to standard linear regression lr = LinearRegression() lr.fit(X, y) y_pred = lr.predict(X) print(f'R2: {r2_score(y, y_pred):.03}') print(f'correlation: {np.corrcoef(y, y_pred)[0, 1]:.03}') # confirm that r2 is the square of the correlation assert np.allclose(r2_score(y, y_pred), np.corrcoef(y, y_pred)[0, 1]**2) print(f'R2 ({r2_score(y, y_pred):.03}) is ~equal to r^2 ({np.corrcoef(y, y_pred)[0, 1]**2:.03})') plt.plot(y, y_pred, 'o') plt.xlabel('True y') plt.ylabel('Predicted y') plt.title('Linear regression - in sample prediction') |
R2: 0.995 correlation: 0.998 R2 (0.995) is ~equal to r^2 (0.995)
Text(0.5, 1.0, 'Linear regression - in sample prediction')
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 | clf = LinearRegression kf = KFold(n_splits=8) pred_y = np.zeros_like(y) for split_idx, (train_idx, test_idx) in enumerate(kf.split(X)): train_X = X[train_idx, :] train_y = y[train_idx] test_X = X[test_idx, :] test_y = y[test_idx] # fit the model if clf is None: clf = RidgeCV model = clf() model.fit(train_X, train_y) pred_y[test_idx] = model.predict(test_X) print('R2: ', r2_score(y, pred_y)) print('correlation:', np.corrcoef(y, pred_y)[0, 1]) print(f'R2 ({r2_score(y, pred_y)}) is not equal to r^2 ({np.corrcoef(y, pred_y)[0, 1]**2})') plt.plot(y, pred_y, 'o') plt.xlabel('True y') plt.ylabel('Predicted y') plt.title('Linear regression - out of sample prediction') |
R2: 0.37077968542027107 correlation: 0.7354056951156511 R2 (0.37077968542027107) is not equal to r^2 (0.540821536408534)
Text(0.5, 1.0, 'Linear regression - out of sample prediction')
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 | # shuffled data y_shuf = y.copy() np.random.seed(1) np.random.shuffle(y_shuf) lr = LinearRegression() lr.fit(X, y_shuf) y_pred = lr.predict(X) print(f'R2 (in-sample): {r2_score(y_shuf, y_pred):.03}') print(f'correlation (in-sample): {np.corrcoef(y_shuf, y_pred)[0, 1]:.03}') clf = LinearRegression kf = KFold(n_splits=8) pred_y = np.zeros_like(y) for split_idx, (train_idx, test_idx) in enumerate(kf.split(X)): train_X = X[train_idx, :] train_y = y_shuf[train_idx] test_X = X[test_idx, :] test_y = y_shuf[test_idx] # fit the model if clf is None: clf = RidgeCV model = clf() model.fit(train_X, train_y) pred_y[test_idx] = model.predict(test_X) print(f'R2 (out-of-sample): {r2_score(y_shuf, pred_y):.03f}') print(f'correlation (out-of-sample): {np.corrcoef(y_shuf, pred_y)[0, 1]:.03f}') print('') print(f'R2 (mean prediction): {r2_score(y_shuf, np.mean(y_shuf) * np.ones_like(y_shuf)):.03f}') print(f'R2 ({r2_score(y_shuf, pred_y)}) is not equal to r^2 ({np.corrcoef(y_shuf, pred_y)[0, 1]**2})') plt.plot(y, pred_y, 'o') plt.xlabel('True y') plt.ylabel('Predicted y') plt.title('Linear regression (shuffled y)- out of sample prediction') |
R2 (in-sample): 0.889 correlation (in-sample): 0.943 R2 (out-of-sample): -13.970 correlation (out-of-sample): -0.121 R2 (mean prediction): 0.000 R2 (-13.970445357380074) is not equal to r^2 (0.01465717598485716)
Text(0.5, 1.0, 'Linear regression (shuffled y)- out of sample prediction')
1 |