K-fold cross-validation for model comparison

Different models of diffusion MRI can be compared based on their accuracy in fitting the diffusion signal. Here, we demonstrate this by comparing two models: the diffusion tensor model (DTI) and Constrained Spherical Deconvolution (CSD). These models differ from each other substantially. DTI approximates the diffusion pattern as a 3D Gaussian distribution, and has only 6 free parameters. CSD, on the other hand, fits many more parameters. The models aare also not nested, so they cannot be compared using the log-likelihood ratio.

A general way to perform model comparison is cross-validation [Hastie2008]. In this method, a model is fit to some of the data (a learning set) and the model is then used to predict a held-out set (a testing set). The model predictions can then be compared to estimate prediction error on the held out set. This method has been used for comparison of models such as DTI and CSD [Rokem2014], and has the advantage that it the comparison is imprevious to differences in the number of parameters in the model, and it can be used to compare models that are not nested.

In DIPY, we include an implementation of k-fold cross-validation. In this method, the data is divided into \(k\) different segments. In each iteration \(\frac{1}{k}th\) of the data is held out and the model is fit to the other \(\frac{k-1}{k}\) parts of the data. A prediction of the held out data is done and recorded. At the end of \(k\) iterations a prediction of all of the data will have been conducted, and this can be compared directly to all of the data.

First, we import that modules needed for this example. In particular, the reconst.cross_validation module implements k-fold cross-validation

import numpy as np
np.random.seed(2014)
import matplotlib.pyplot as plt

import dipy.data as dpd
import dipy.reconst.cross_validation as xval
import dipy.reconst.dti as dti
import dipy.reconst.csdeconv as csd
import scipy.stats as stats

We fetch some data and select a couple of voxels to perform comparisons on. One lies in the corpus callosum (cc), while the other is in the centrum semiovale (cso), a part of the brain known to contain multiple crossing white matter fiber populations.

dpd.fetch_stanford_hardi()
img, gtab = dpd.read_stanford_hardi()
data = img.get_data()

cc_vox = data[40, 70, 38]
cso_vox = data[30, 76, 38]

We initialize each kind of model:

dti_model = dti.TensorModel(gtab)
response, ratio = csd.auto_response(gtab, data, roi_radius=10, fa_thr=0.7)
csd_model = csd.ConstrainedSphericalDeconvModel(gtab, response)

Next, we perform cross-validation for each kind of model, comparing model predictions to the diffusion MRI data in each one of these voxels.

Note that we use 2-fold cross-validation, which means that in each iteration, the model will be fit to half of the data, and used to predict the other half.

dti_cc = xval.kfold_xval(dti_model, cc_vox, 2)
csd_cc = xval.kfold_xval(csd_model, cc_vox, 2, response)
dti_cso = xval.kfold_xval(dti_model, cso_vox, 2)
csd_cso = xval.kfold_xval(csd_model, cso_vox, 2, response)

We plot a scatter plot of the data with the model predictions in each of these voxels, focusing only on the diffusion-weighted measurements (each point corresponds to a different gradient direction). The two models are compared in each sub-plot (blue=DTI, red=CSD).

fig, ax = plt.subplots(1, 2)
fig.set_size_inches([12, 6])
ax[0].plot(cc_vox[~gtab.b0s_mask], dti_cc[~gtab.b0s_mask], 'o', color='b')
ax[0].plot(cc_vox[~gtab.b0s_mask], csd_cc[~gtab.b0s_mask], 'o', color='r')
ax[1].plot(cso_vox[~gtab.b0s_mask], dti_cso[~gtab.b0s_mask], 'o', color='b', label='DTI')
ax[1].plot(cso_vox[~gtab.b0s_mask], csd_cso[~gtab.b0s_mask], 'o', color='r', label='CSD')
plt.legend(loc='upper left')
for this_ax in ax:
    this_ax.set_xlabel('Data (relative to S0)')
    this_ax.set_ylabel('Model prediction (relative to S0)')
fig.savefig("model_predictions.png")
../_images/model_predictions.png

Model predictions.

We can also quantify the goodness of fit of the models by calculating an R-squared score:

cc_dti_r2 = stats.pearsonr(cc_vox[~gtab.b0s_mask], dti_cc[~gtab.b0s_mask])[0]**2
cc_csd_r2 = stats.pearsonr(cc_vox[~gtab.b0s_mask], csd_cc[~gtab.b0s_mask])[0]**2
cso_dti_r2 = stats.pearsonr(cso_vox[~gtab.b0s_mask], dti_cso[~gtab.b0s_mask])[0]**2
cso_csd_r2 = stats.pearsonr(cso_vox[~gtab.b0s_mask], csd_cso[~gtab.b0s_mask])[0]**2

print("Corpus callosum\n"
      "DTI R2 : %s\n"
      "CSD R2 : %s\n"
      "\n"
      "Centrum Semiovale\n"
      "DTI R2 : %s\n"
      "CSD R2 : %s\n" % (cc_dti_r2, cc_csd_r2, cso_dti_r2, cso_csd_r2))

This should look something like this:

Corpus callosum

DTI R2 : 0.782881752597

CSD R2 : 0.805764364116

Centrum Semiovale

DTI R2 : 0.431921832012

CSD R2 : 0.604806420501

As you can see, DTI is a pretty good model for describing the signal in the CC, while CSD is much better in describing the signal in regions of multiple crossing fibers.

References

[Hastie2008]Hastie, T., Tibshirani, R., Friedman, J. (2008). The Elements of Statistical Learning: Data Mining, Inference and Prediction. Springer-Verlag, Berlin
[Rokem2014]Rokem, A., Chan, K.L. Yeatman, J.D., Pestilli, F., Mezer, A., Wandell, B.A., 2014. Evaluating the accuracy of diffusion models at multiple b-values with cross-validation. ISMRM 2014.

Example source code

You can download the full source code of this example. This same script is also included in the dipy source distribution under the doc/examples/ directory.