Example of pattern recognition on simulated data#

This example simulates data according to a very simple sketch of brain imaging data and applies machine learning techniques to predict output values.

We use a very simple generating function to simulate data, as in Michel et al.[1], a linear model with a random design matrix X:

\mathbf{y} = \mathbf{X} \mathbf{w} + \mathbf{e}

  • w: the weights of the linear model correspond to the predictive brain regions. Here, in the simulations, they form a 3D image with 5, four of which in opposite corners and one in the middle, as plotted below.

  • X: the design matrix corresponds to the observed fMRI data. Here we simulate random normal variables and smooth them as in Gaussian fields.

  • e is random normal noise.

try:
    import matplotlib.pyplot as plt
except ImportError:
    raise RuntimeError("This script needs the matplotlib library")

from time import time

import nibabel
import numpy as np
from scipy import linalg
from scipy.ndimage import gaussian_filter
from sklearn import linear_model, svm
from sklearn.feature_selection import f_regression
from sklearn.model_selection import KFold
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import StandardScaler
from sklearn.utils import check_random_state

import nilearn.masking
from nilearn import decoding
from nilearn.plotting import show

A function to generate data#

def create_simulation_data(snr=0, n_samples=2 * 100, size=12, random_state=1):
    generator = check_random_state(random_state)
    roi_size = 2  # size / 3
    smooth_X = 1
    # Coefs
    w = np.zeros((size, size, size))
    w[0:roi_size, 0:roi_size, 0:roi_size] = -0.6
    w[-roi_size:, -roi_size:, 0:roi_size] = 0.5
    w[0:roi_size, -roi_size:, -roi_size:] = -0.6
    w[-roi_size:, 0:roi_size:, -roi_size:] = 0.5
    w[
        (size - roi_size) // 2 : (size + roi_size) // 2,
        (size - roi_size) // 2 : (size + roi_size) // 2,
        (size - roi_size) // 2 : (size + roi_size) // 2,
    ] = 0.5
    w = w.ravel()
    # Generate smooth background noise
    XX = generator.randn(n_samples, size, size, size)
    noise = []
    for i in range(n_samples):
        Xi = gaussian_filter(XX[i, :, :, :], smooth_X)
        Xi = Xi.ravel()
        noise.append(Xi)
    noise = np.array(noise)
    # Generate the signal y
    y = generator.randn(n_samples)
    X = np.dot(y[:, np.newaxis], w[np.newaxis])
    norm_noise = linalg.norm(X, 2) / np.exp(snr / 20.0)
    noise_coef = norm_noise / linalg.norm(noise, 2)
    noise *= noise_coef
    snr = 20 * np.log(linalg.norm(X, 2) / linalg.norm(noise, 2))
    print(f"SNR: {snr:.1f} dB")
    # Mixing of signal + noise and splitting into train/test
    X += noise
    X -= X.mean(axis=-1)[:, np.newaxis]
    X /= X.std(axis=-1)[:, np.newaxis]
    X_test = X[n_samples // 2 :, :]
    X_train = X[: n_samples // 2, :]
    y_test = y[n_samples // 2 :]
    y = y[: n_samples // 2]

    return X_train, X_test, y, y_test, snr, w, size

A simple function to plot slices#

def plot_slices(data, title=None):
    plt.figure(figsize=(5.5, 2.2))
    vmax = np.abs(data).max()
    for i in (0, 6, 11):
        plt.subplot(1, 3, i // 5 + 1)
        plt.imshow(
            data[:, :, i],
            vmin=-vmax,
            vmax=vmax,
            interpolation="nearest",
            cmap=plt.cm.RdBu_r,
        )
        plt.xticks(())
        plt.yticks(())
    plt.subplots_adjust(
        hspace=0.05, wspace=0.05, left=0.03, right=0.97, top=0.9
    )
    if title is not None:
        plt.suptitle(title)

Create data#

X_train, X_test, y_train, y_test, snr, coefs, size = create_simulation_data(
    snr=-10, n_samples=100, size=12
)

# Create masks for SearchLight. process_mask is the voxels where SearchLight
# computation is performed. It is a subset of the brain mask, just to reduce
# computation time.
mask = np.ones((size, size, size), dtype=bool)
mask_img = nibabel.Nifti1Image(mask.astype("uint8"), np.eye(4))
process_mask = np.zeros((size, size, size), dtype=bool)
process_mask[:, :, 0] = True
process_mask[:, :, 6] = True
process_mask[:, :, 11] = True
process_mask_img = nibabel.Nifti1Image(process_mask.astype("uint8"), np.eye(4))

coefs = np.reshape(coefs, [size, size, size])
plot_slices(coefs, title="Ground truth")
Ground truth
SNR: -10.0 dB

Run different estimators#

We can now run different estimators and look at their prediction score, as well as the feature maps that they recover. Namely, we will use

  • A support vector regression (SVM)

  • An elastic-net

  • A Bayesian ridge estimator, i.e. a ridge estimator that sets its parameter according to a metaprior

  • A ridge estimator that set its parameter by cross-validation

Note that the RidgeCV and the ElasticNetCV have names ending in CV that stands for cross-validation: in the list of possible alpha values that they are given, they choose the best by cross-validation.

bayesian_ridge = make_pipeline(StandardScaler(), linear_model.BayesianRidge())

estimators = [
    ("bayesian_ridge", bayesian_ridge),
    (
        "enet_cv",
        linear_model.ElasticNetCV(alphas=[5, 1, 0.5, 0.1], l1_ratio=0.05),
    ),
    ("ridge_cv", linear_model.RidgeCV(alphas=[100, 10, 1, 0.1], cv=5)),
    ("svr", svm.SVR(kernel="linear", C=0.001)),
    (
        "searchlight",
        decoding.SearchLight(
            mask_img,
            process_mask_img=process_mask_img,
            radius=2.7,
            scoring="r2",
            estimator=svm.SVR(kernel="linear"),
            cv=KFold(n_splits=4),
            verbose=1,
            n_jobs=2,
        ),
    ),
]

Run the estimators#

As the estimators expose a fairly consistent API, we can all fit them in a for loop: they all have a fit method for fitting the data, a score method to retrieve the prediction score, and because they are all linear models, a coef_ attribute that stores the coefficients w estimated

for name, estimator in estimators:
    t1 = time()
    if name != "searchlight":
        estimator.fit(X_train, y_train)
    else:
        X = nilearn.masking.unmask(X_train, mask_img)
        estimator.fit(X, y_train)
        del X
    elapsed_time = time() - t1

    if name != "searchlight":
        if name == "bayesian_ridge":
            coefs = estimator.named_steps["bayesianridge"].coef_
        else:
            coefs = estimator.coef_
        coefs = np.reshape(coefs, [size, size, size])
        score = estimator.score(X_test, y_test)
        title = (
            f"{name}: prediction score {score:.3f}, "
            f"training time: {elapsed_time:.2f}s"
        )

    else:  # Searchlight
        coefs = estimator.scores_
        title = (
            f"{estimator.__class__.__name__}: "
            f"training time: {elapsed_time:.2f}s"
        )

    # We use the plot_slices function provided in the example to
    # plot the results
    plot_slices(coefs, title=title)

    print(title)

_, p_values = f_regression(X_train, y_train)
p_values = np.reshape(p_values, (size, size, size))
p_values = -np.log10(p_values)
p_values[np.isnan(p_values)] = 0
p_values[p_values > 10] = 10
plot_slices(p_values, title="f_regress")

show()
  • bayesian_ridge: prediction score 0.114, training time: 0.17s
  • enet_cv: prediction score 0.528, training time: 0.24s
  • ridge_cv: prediction score 0.328, training time: 0.04s
  • svr: prediction score 0.345, training time: 0.00s
  • SearchLight: training time: 4.59s
  • f_regress
bayesian_ridge: prediction score 0.114, training time: 0.17s
enet_cv: prediction score 0.528, training time: 0.24s
ridge_cv: prediction score 0.328, training time: 0.04s
svr: prediction score 0.345, training time: 0.00s
[Parallel(n_jobs=2)]: Using backend LokyBackend with 2 concurrent workers.
[Parallel(n_jobs=2)]: Done   2 out of   2 | elapsed:    4.0s finished
SearchLight: training time: 4.59s

An exercise to go further#

As an exercice, you can use recursive feature elimination (RFE) with the SVM

Read the object’s documentation to find out how to use RFE.

Performance tip: increase the step parameter, or it will be very slow.

# from sklearn.feature_selection import RFE

References#

Total running time of the script: (0 minutes 7.176 seconds)

Estimated memory usage: 12 MB

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