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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 judgement built on invented cut-offs is an opinion while one built on the supervisory thresholds can be…

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 because everything is expressed in percent and is not.

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 that actually fails in a crisis.

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.

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 instead of a plausible vector with no warning attached.

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.

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 component analysis, and the column means are returned as part of the model.

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 eigenvectors are from being orthonormal.

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 produced; absolute answers what actually reaches the atmosphere.