7. Inference in high dimension#

7.1. Naive inference in high dimension is ill-posed#

In some cases, data represent high-dimensional measurements of some phenomenon of interest (e.g. imaging or genotyping). The common characteristic of these problems is to be very high-dimensional and lead to correlated features. Both aspects are clearly detrimental to conditional inference, making it both expensive and powerless:

  • Expensive: most learers are quadratic or cubic in the number of features. Moreover per-feature inference generally entails a loop over features

  • powerless: As dimensionality and correlation increase, it becomes harder and harder to isolate the contribution of each variable, meaning that conditional inference is ill-posed.

This is illustrated in the above example, where the Desparsified Lasso (hidimstat.DesparsifiedLasso) struggles to identify relevant features. We need some data to start:

>>> n_samples = 100
>>> shape = (40, 40)
>>> roi_size = 4  # size of the edge of the four predictive regions

# generating the data
>>> from hidimstat._utils.scenario import multivariate_simulation_spatial
>>> X_init, y, beta, epsilon = multivariate_simulation_spatial(
>>>     n_samples, shape, roi_size, signal_noise_ratio=10., smooth_X=1)

Then we perform inference on this data using the Desparsified Lasso:

>>> from hidimstat.desparsified_lasso import DesparsifiedLasso
>>> dlasso = DesparsifiedLasso().fit(X_init, y)
>>> dlasso.importance(X_init, y) # compute importance score and associated
    corrected p-values

# compute estimated support
>>> import numpy as np
>>> alpha = .05 # alpha is the significance level for the statistical test
>>> selected_dl = dlasso.pvalues_ < alpha / (shape[0] * shape[1])
>>> true_support = beta > 0
>>> print(f'Desparsified Lasso selected {np.sum(selected_dl * true_support)}
    features among {np.sum(true_support)} ')
Desparsified Lasso selected 19 features among 64

7.2. Feature Grouping and its shortcomings#

As discussed earlier, feature grouping is a meaningful solution to deal with such cases: it reduces the number of features to condition on, and generally also decreases the level of correlation between features.

See also

As hinted in Nicolai Meinshausen and Buhlmann[1] an efficient way to deal with such configuration is to take the per-group average of the features: this leads to a reduced design. After inference, all the feature in a given group obtain the p-value of the group representative. When the inference engine is Desparsified Lasso, the resulting method is called Clustered Desparsified lasso, or hidimstat.CluDL.

Using the same example as previously, we start by defining a clustering method that will perform the grouping. For image data, Ward clustering is a good default model, because it takes into account the neighboring structure among pixels, which avoids creating overly messy clusters:

>>> from sklearn.feature_extraction import image
>>> from sklearn.cluster import FeatureAgglomeration
>>> n_clusters = 200
>>> connectivity = image.grid_to_graph(n_x=shape[0], n_y=shape[1])
>>> ward = FeatureAgglomeration(
>>>     n_clusters=n_clusters, connectivity=connectivity, linkage="ward")

Equipped with this, we can use CluDL:

>>> from hidimstat import CluDL
>>> from sklearn.linear_model import LassoCV
>>> cludl = CluDL(
    clustering=ward,
    desparsified_lasso=DesparsifiedLasso(estimator=LassoCV()))
>>> cludl.fit_importance(X_init, y)
# compute estimated support
>>> selected_cdl = cludl.fwer_selection(alpha, n_tests=n_clusters)
>>> print(f'Clustered Desparsified Lasso selected
    {np.sum(selected_cdl *  true_support)} features among
    {np.sum(true_support)}')
Clustered Desparsified Lasso selected 51 features among 64

Note that inference is also way faster on the compressed representation.

The issue is that very-high-dimensional data (biological, images, etc.) do not have any canonical grouping structure. Hence, they rely on grouping obtained from the data, typically with clustering technique. However, the resulting clusters bring some undesirable randomness. Think that imputing slightly different data would lead to different clusters. Since there is no globally optimal clustering, the wiser solution is to average the results across clusterings. Since it may not be a good idea to average p-values, an alternative ensembling or aggregation strategy is used instead. When the inference engine is Desparsified Lasso, the resulting method is called Ensemble of Clustered Desparsified lasso, or hidimstat.EnCluDL.

The behavior is illustrated here:

>>> from hidimstat import EnCluDL

# ensemble of clustered desparsified lasso (EnCluDL)
>>> encludl = EnCluDL(
>>>     clustering=ward,
        desparsified_lasso=DesparsifiedLasso(estimator=LassoCV()),
        n_bootstraps=20,
        random_state=0)
>>> encludl.fit_importance(X_init, y)
>>> selected_ecdl = encludl.fwer_selection(alpha, n_tests=n_clusters)
>>> print(f'Ensemble of Clustered Desparsified Lasso selected
    {np.sum(selected_ecdl *  true_support)} features among
    {np.sum(true_support)} ')
Ensemble of Clustered Desparsified Lasso selected 57 features among 64

7.3. What type of Control does this Ensemble of CLustered inference come with ?#

Ensemble of Clustered Inference is not a local method, so control cannot be maintained at each brain site in isolation. The notion of a false positive must be mitigated by the non-local characteristic of the inference performed. Thus, we introduce the concept of a \(\delta\)-false positive: A detection is a delta-false positive if it is at a distance greater than \(\delta\) from the support, which is the set of true positives. Thus, what is controlled is the \(\delta\)-FWER, i.e., the probability of reporting a single \(\delta\)-false positive. In other words, EnCluDL will likely only report detections at a distance less than \(\delta\) from the true support.

What is \(\delta\) ? It is the diameter of the clusters used in the CluDL procedure.

The details of the method and the underlying guarantees are described in Chevalier et al.[2]

7.4. References#