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Computes the unbiased sample covariance between predictions and labels.

The streaming_covariance function creates four local variables, comoment, mean_prediction, mean_label, and count, which are used to compute the sample covariance between predictions and labels across multiple batches of data. The covariance is ultimately returned as an idempotent operation that simply divides comoment by count - 1. We use count - 1 in order to get an unbiased estimate.

The algorithm used for this online computation is described in Specifically, the formula used to combine two sample comoments is C_AB = C_A + C_B + (E[x_A] - E[x_B]) * (E[y_A] - E[y_B]) * n_A * n_B / n_AB The comoment for a single batch of data is simply sum((x - E[x]) * (y - E[y])), optionally weighted.

If weights is not None, then it is used to compute weighted comoments, means, and count. NOTE: these weights are treated as "frequency weights", as opposed to "reliability weights". See discussion of the difference on

To facilitate the computation of covariance across multiple batches of data, the function creates an update_op operation, which updates underlying variables and returns the updated covariance.

predictions A Tensor of arbitrary size.
labels A Tensor of the same size as predictions.
weights Optional Tensor indicating the frequency with which an example is sampled. Rank must be 0, or the same rank as labels, and must be broadcastable to labels (i.e., all dimensions must be either 1, or the same as the corresponding labels dimension).
metrics_collections An optional list of collections that the metric value variable should be added to.
updates_collections An optional list of collections that the metric update ops should be added to.
name An optional variable_scope name.

covariance A Tensor representing the current unbiased sample covariance, comoment / (count - 1).
update_op An operation that updates the local variables appropriately.

ValueError If labels and predictions are of different sizes or if either metrics_collections or updates_collections are not a list or tuple.