Fig. 1

Graphical representation of the method. Starting from stock price time series (a), a rolling window is applied to compute the correlation among each pair of stocks (b). At each time step a distance matrix is created. Once the rolling window has produced \( \mathrm{Z} \) distance matrices, those matrices are embedded into a 3D-tensor (c). When the steps of the moving window exceed \( \mathrm{Z} \), the tensor is allowed to move in time with each new step, as new data are available (c - solid line vs. dashed line). This temporal shift of the tensor permits to compare the forecasts produced by two consecutive decompositions and generates a signal whose sign indicates the future direction of the stock prices. For graphical purposes, the decomposition of two consecutive tensors and the resulting links prediction are drawn using solid and dashed lines. The two consecutive tensors are approximated as the linear combination of three vectors (d) representing spatial (\( \mathbf{v} \)) and temporal (\( \mathbf{u} \)) relationships between stocks; \( \mathcal{D}\cong \upbeta \mathbf{v}\circ \mathbf{v}\circ \mathbf{u} \). The exponential smoothing (d - green lines) applied to \( \mathbf{u} \) extracts a scalar \( \uptau \) representing the forecast of temporal profile for the next period (e - red squares). The forecast of the future distance matrix is obtained as a linear combination of the two spatial dissimilarity vectors, the parameter \( \upbeta \) and of the forecast \( \uptau \) of the temporal profile vector; \( \widehat{\mathbf{D}}=\uptau \upbeta \mathbf{v}{\mathbf{v}}^{\mathrm{T}} \) (e and f - red squares). Finally, the connection intensity of each asset is quantified via a centrality score by summing the distances that link each asset to the others (g). The signal for each stock is obtained as the difference between the centrality values obtained in two predictions (h)