"""
Cross-validation analysis of diffusion models.
"""
import numpy as np
import dipy.core.gradients as gt
from dipy.testing.decorators import warning_for_keywords
[docs]
@warning_for_keywords()
def coeff_of_determination(data, model, *, axis=-1):
r"""Calculate the coefficient of determination for a model prediction,
relative to data.
Parameters
----------
data : ndarray
The data
model : ndarray
The predictions of a model for this data. Same shape as the data.
axis: int, optional
The axis along which different samples are laid out.
Returns
-------
COD : ndarray
The coefficient of determination. This has shape `data.shape[:-1]`
Notes
-----
See: https://en.wikipedia.org/wiki/Coefficient_of_determination
The coefficient of determination is calculated as:
.. math::
R^2 = 100 * (1 - \frac{SSE}{SSD})
where SSE is the sum of the squared error between the model and the data
(sum of the squared residuals) and SSD is the sum of the squares of the
deviations of the data from the mean of the data (variance * N).
"""
residuals = data - model
ss_err = np.sum(residuals**2, axis=axis)
demeaned_data = data - np.mean(data, axis=axis)[..., np.newaxis]
ss_tot = np.sum(demeaned_data**2, axis=axis)
# Don't divide by 0:
if np.all(ss_tot == 0.0):
return np.nan
return 100 * (1 - (ss_err / ss_tot))
[docs]
def kfold_xval(model, data, folds, *model_args, **model_kwargs):
"""Perform k-fold cross-validation.
It generates out-of-sample predictions for each measurement.
See :footcite:p:`Rokem2014` for further details about the method.
Parameters
----------
model : Model class instance
The type of the model to use for prediction. The corresponding Fit
object must have a `predict` function implemented One of the following:
`reconst.dti.TensorModel` or
`reconst.csdeconv.ConstrainedSphericalDeconvModel`.
data : ndarray
Diffusion MRI data acquired with the GradientTable of the model. Shape
will typically be `(x, y, z, b)` where `xyz` are spatial dimensions and
b is the number of bvals/bvecs in the GradientTable.
folds : int
The number of divisions to apply to the data
model_args : list
Additional arguments to the model initialization
model_kwargs : dict
Additional key-word arguments to the model initialization. If contains
the kwarg `mask`, this will be used as a key-word argument to the `fit`
method of the model object, rather than being used in the
initialization of the model object
Notes
-----
This function assumes that a prediction API is implemented in the Model
class for which prediction is conducted. That is, the Fit object that gets
generated upon fitting the model needs to have a `predict` method, which
receives a GradientTable class instance as input and produces a predicted
signal as output.
It also assumes that the model object has `bval` and `bvec` attributes
holding b-values and corresponding unit vectors.
References
----------
.. footbibliography::
"""
_ = model_kwargs.pop("rng", np.random.default_rng())
# This should always be there, if the model inherits from
# dipy.reconst.base.ReconstModel:
gtab = model.gtab
data_b = data[..., ~gtab.b0s_mask]
div_by_folds = np.mod(data_b.shape[-1], folds)
# Make sure that an equal number of samples get left out in each fold:
if div_by_folds != 0:
msg = "The number of folds must divide the diffusion-weighted "
msg += "data equally, but "
msg = f"np.mod({data_b.shape[-1]}, {folds}) is {div_by_folds}"
raise ValueError(msg)
data_0 = data[..., gtab.b0s_mask]
S0 = np.mean(data_0, -1)
n_in_fold = data_b.shape[-1] / folds
prediction = np.zeros(data.shape)
# We are going to leave out some randomly chosen samples in each iteration:
order = np.random.permutation(data_b.shape[-1])
nz_bval = gtab.bvals[~gtab.b0s_mask]
nz_bvec = gtab.bvecs[~gtab.b0s_mask]
# Pop the mask, if there is one, out here for use in every fold:
mask = model_kwargs.pop("mask", None)
gtgt = gt.gradient_table # Shorthand
for k in range(folds):
fold_mask = np.ones(data_b.shape[-1], dtype=bool)
fold_idx = order[int(k * n_in_fold) : int((k + 1) * n_in_fold)]
fold_mask[fold_idx] = False
this_data = np.concatenate([data_0, data_b[..., fold_mask]], -1)
this_gtab = gtgt(
np.hstack([gtab.bvals[gtab.b0s_mask], nz_bval[fold_mask]]),
bvecs=np.concatenate([gtab.bvecs[gtab.b0s_mask], nz_bvec[fold_mask]]),
)
left_out_gtab = gtgt(
np.hstack([gtab.bvals[gtab.b0s_mask], nz_bval[~fold_mask]]),
bvecs=np.concatenate([gtab.bvecs[gtab.b0s_mask], nz_bvec[~fold_mask]]),
)
this_model = model.__class__(this_gtab, *model_args, **model_kwargs)
this_fit = this_model.fit(this_data, mask=mask)
if not hasattr(this_fit, "predict"):
err_str = f"Models of type: {this_model.__class__} "
err_str += "do not have an implementation of model prediction"
err_str += " and do not support cross-validation"
raise ValueError(err_str)
this_predict = S0[..., None] * this_fit.predict(gtab=left_out_gtab, S0=1)
idx_to_assign = np.where(~gtab.b0s_mask)[0][~fold_mask]
prediction[..., idx_to_assign] = this_predict[..., np.sum(gtab.b0s_mask) :]
# For the b0 measurements
prediction[..., gtab.b0s_mask] = S0[..., None]
return prediction