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Reports the credit-to-GDP gap for a country, which is the deviation of the credit ratio from its long-run trend and the quantity the Basel framework uses to trigger the countercyclical capital buffer. The thresholds applied are the regulatory ones — the buffer starts to build at a gap of two percentage points and is at maximum by ten — and they are returned with the result, because a phase judgeme

Returns harmonised consumer price inflation for several countries over a period, taken entirely from the European harmonised index so the numbers are actually comparable. That constraint is the product: national consumer price indices use different baskets, treat owner-occupied housing differently and rest on different base years, so placing them side by side produces a comparison that looks valid

Returns the outstanding cross-border positions of banks located in a reporting country against counterparties in another, from the BIS locational banking statistics, split into claims and liabilities and never netted silently. The separation matters: a jurisdiction showing large claims and large liabilities is intermediating, not lending, and reporting only a net figure hides the gross exposure th

Solves a bounded dense multi-output least-squares problem and returns the minimum-norm solution together with the diagnostics that say whether it can be trusted: numerical rank, the full singular value spectrum, the rank threshold actually applied, the residual, and the effective condition number. A rank-deficient system still gets an answer here, but it gets one that announces its own deficiency

Factors a matrix into rank-k left and right factors with their singular values, optionally centering the columns first, and states what the compression actually cost: the Frobenius reconstruction error against the input and the fraction of spectral energy retained. The full spectrum comes back too, so a caller can pick a different k without a second call. With centering enabled this is principal c

Produces the two transforms a stochastic model normally needs from a covariance: the whitening matrix that maps correlated data to unit-variance uncorrelated data, and the coloring matrix that goes back the other way. Both are returned so the round trip is available without a second call.

Applies the standard affine propagation of uncertainty: the mean goes through the transform and offset, the covariance through the transform on both sides, and the independent noise covariance is added. This is the prediction step of a Kalman filter and the workhorse of any linear error budget.

Computes the full eigendecomposition of a real symmetric operator and reports the signed eigenvalues, the orthonormal eigenvectors, and the inertia — how many eigenvalues are positive, negative and indistinguishable from zero at the stated tolerance. Two residuals are returned as evidence rather than as reassurance: how far the decomposition is from reproducing the input, and how far the eigenvect

Returns carbon dioxide emissions for one or more countries on three bases at once: per inhabitant, per unit of purchasing-power-adjusted output, and in absolute megatonnes, each with its own period and its trend over the requested window. Keeping the three apart is the point. Per capita answers how heavily a person there burdens the climate; per unit of output answers how efficiently value is prod