Common use of The Weighted Mean and Covariance Matrix Clause in Contracts

The Weighted Mean and Covariance Matrix. In some cases, one might want to compute the weighted arithmetic mean and a weighted sample covariance (matrix) where particular sample points should contribute more to the final mean (covariances) than others. For example, it could be reasonable to give larger weight to more recent data points if the statistics (e.g., mean) of the underlying distribution change over time (concept drifts). In such cases, we would want the estimator to track the data generating distribution changes and forget about older knowledge. But also other scenarios are possible, in which a weighting of the data points might be useful. In this section, we define the weighted arithmetic mean and weighted sample covariance, explore a few properties related to these weighted statistics, and then design an estimator for the sample mean and covariance exhibiting some forgetting. In general, the weighted arithmetic mean is defined quite straightforward as: Σ i=1 i x¯n = Σn w′xi wi′ where xi is the i-th data point (vector) and wi′ is the (unnormalized) weight assigned to the corresponding data point. If the weights are normalized to sum 1, then we get: Σ x¯n = wixi, (B.39) where the normalized weights are defined as: with the normalization factor w = wi′ wi′ Σ = n w′ , (B.40) Σ Wn = wi′.