Linear combination of weak estimators over fuzzy classifiers? Having:


a set of soft fuzzy classifiers (classification onto overlapping sets) $C_i(x) \to [0,1]$;
a corresponding set of weak estimators $R_i(z)$ of the form $R_i(z) = \mathit{EX}(y\mid z)$.


The estimators $R_i$ are just some kind of regression, Kalman or particle filters. The classifiers $C_i$ are fixed and static. How to make a strong estimator out of a weighted combination of the form:

$$L(x, z) = \sum_{i}C_i(x)R_i(z)Q_i$$

In other words how to choose the weights $Q_i$? Is there some kind of online approach to this problem? 

Here is brief description of a practical application. When an event $E$ is registered, multiple measurements are made. Based on these measurements, the classifiers $C_i$ make a soft assignment of the event to multiple overlapping categories. What we get is fit ratios for the soft clusters.

Now there is some chance that event $E$ may trigger a subsequent event $D$, depending on another variable $z$ -- independent from the event $E$. We know that all the soft cluster "memberships" may influence the probability of event $D$ being triggered.

We want to estimate the probability that $E$ triggers $D$, given the $C_i$ fitness ratios and value of $z$.
You might find a solution for this by checking out Viola &amp; Jones face detection algorithm (and object detection in general) http://www.cs.ubc.ca/~lowe/425/slides/13-ViolaJones.pdf. Particularly the AdaBoost algorithm for building a strong classifier from weak classifiers. https://www.cs.princeton.edu/~schapire/papers/explaining-adaboost.pdf

In this algorithm this is used particularly for feature selection, where it is wanted to select just the features that grouped together classify better, discarding the other as noisy features.

The approach to obtain this strong classifier is having a set of examples X, a set of weights W and a set of expected results (classifications) Y. For a two-class example (e.g. face o no face image) the first weak classifier is selected and then the classification error is found between this classifier and the expected output. For the samples that were misclassified their weights are incremented, so the next weak classifier to find is more biased to neglect this wrong classification.

The algorithm (AdaBoost) converges when the sign of the sum of classifications for each sample outputs the correct classification.

For example:

Y = +1 -1 +1 +1

WC1=+1 +1 +1 +1
WC2=-1 -1 -1 -1
WC3=+1 -1 +1 +1


So the strong classifier is WC1+WC2+WC3:

SC=+1 -1 +1 +1 == Y


Hope this solves your question.