Estimation of probabilistic copulas from nonlinear correlations: Application to lagged teleconnections and monthly atmospheric forecasting.

Probabilistic lagged dependence (ranging from months to seasons) between atmospheric-oceanic variables, comes essentially from their linear and nonlinear statistical multivariate correlations. A new technique is presented to estimate the posterior conditional pdf of a scalar predictand y(t+lag) at lag tau, knowing a vector of predictor climatic indices X(t), taken at time t. For that, we apply a variant of the Kernel Canonical Correlation Analysis (KCCA) linking extended feature vectors f(Y) and g(X), filled with nonlinear and mixing functions (e.g. monomials, component products). The issued, leading canonical component pair (u,v) is then used to estimate the copula between X and Y, estimated as the Gaussian copula between Gaussian-anamorphed components ug, vg of u,v respectively. This copula works as a maximum-entropy copula, maximizing the Gaussian correlation Cor-g (Pearson correlation between ug, vg), captured by the feature vectors, and also maximizing the part  -0.5*log(1-cor-g^2) of the mutual information (MI) between X and Y. Moreover, Cor-g is much more outlier-resistant than the Pearson correlation. The above method is applied in two cases: 1) Y being a climatic index, (e.g. El-Niño index with lags tau in the range 0-48 months) and 2) Y being the local monthly temperature or precipitation for lags of 1-2 months. In both cases, X is taken as a set of climatic indices from the pool: El-Niño, NAO, AMO, PDO, IOD; QBO, TNA, TSA, SCAND, WE, EA-WR. The Gaussian-copula model improves the forecast of extreme situations, even beyond 1-2 standard deviations, providing a way of exploring probabilistic nonlinear forecasts and nonlinear lagged teleconnections. This work is supported by the Portuguese Fundação para a Ciência e Tecnologia, FCT, I.P./MCTES through national funds (PIDDAC): UID/50019/2025 and LA/P/0068/2020 https://doi.org/10.54499/LA/P/0068/2020).