ALGORITHM 1 Prediction with manifold learning

Step 1: Perform eigen decomposition on kernel matrix and define projection matrix HKH = UΛUT, with U = [α1, α2, …αn] and Λ = diag(λ1, …, λn) Define projection matrix P = [V1, V2, …Vm], for each $V_k={\sum }_{i=1}^n\frac{{\alpha }_{k,i}}{\sqrt{{\lambda }_k}}\stackrel{\sim }{\varphi }({\mathbf{x}}_i)$

Step 2: Project training and testing data onto the constructed sub-manifold x̌i = PTϕ(xi)

Step 3: Calculate VAR coefficient matrix given T covariate-response pairs {(ši, y̌i)|i ∈ [1, T]}, for each ${\check{\mathbf{s}}}_i={[{\check{\mathbf{x}}}_{i-(p-1)}^T,\dots ,{\check{\mathbf{x}}}_i^T]}^T$ and y̌i = x̌i+τ Solve VAR coefficient matrix B̂ = YST(SST)−1

Step 4: Estimate testing response given testing covariates š $\check{\mathbf{y}}=\widehat{B}\left[\begin{array}{c}1\\ \check{\mathbf{s}}\end{array}\right]$

Step 5: Estimate pre-image through fixed-point iterations ${\widehat{\mathbf{x}}}_{t+1}^{\star }=\frac{{\sum }_{i=1}^n{\stackrel{\sim }{\gamma }}_{i}exp(-{\Vert {\widehat{\mathbf{x}}}_t^{\star }-{\mathbf{x}}_i\Vert }^2/c){\mathbf{x}}_i}{{\sum }_{i=1}^n{\stackrel{\sim }{\gamma }}_{i}exp(-{\Vert {\widehat{\mathbf{x}}}_t^{\star }-{\mathbf{x}}_i\Vert }^2/c)}$, Where ${\gamma }_i={\sum }_{k=1}^m{\check{y}}_k\frac{{\alpha }_{i,k}}{\sqrt{{\lambda }_k}}$, and ${\stackrel{\sim }{\gamma }}_i={\gamma }_i+\frac{1}{n}(1-{\sum }_{j=1}^n{\gamma }_j)$