Multi-GNN Recursive Kernel PCA Problem
You are tasked with designing a multi-layer trading strategy based on recursive Kernel PCA and interdependent ð‘-Layer GNNs to analyze ð‘›=28 currency pairs. Each GNN layer specializes in capturing different graph dynamics (e.g., volatility, profitability, momentum), while layers interact nonlinearly and recursively. The outputs from each GNN layer influence:
The adjacency matrices for the other layers.
The kernel matrices for volatility (ð¾ð‘‰) and profitability (ð¾Î ).
The node embeddings, which are combined and propagated across layers.
The challenge is to optimize the system by dynamically balancing these interactions while maximizing risk-adjusted profitability.
Enhanced Relationships Between ð‘-Layer GNNs Interdependent GNN Layers
GNN Layer Types:
GNN1 : Models volatility relationships.
GNN2 : Models profitability dynamics.
GNN3 : Models momentum and correlation patterns.
Cross-Layer Interactions:
Outputs from GNN1 dynamically update edge weights in GNN2 :
A_2^{(t+1)}[i][j] = f(A_2^{(t)}[i][j], h_i^{1,(t)}, h_j^{1,(t)}),
where â„Žð‘–1(ð‘¡) is the embedding of node ð‘– from GNN1.
Similarly, outputs from GNN2 influence GNN3 :
A_3^{(t+1)}[i][j] = g(A_3^{(t)}[i][j], h_i^{2,(t)}, h_j^{2,(t)}).
The final embeddings from GNN3 recursively update GNN1 :
A_1^{(t+1)}[i][j] = h(A_1^{(t)}[i][j], h_i^{3,(t)}, h_j^{3,(t)}).
Multi-Scale Aggregation:
Node embeddings are aggregated across all ð‘ layers:
h_i^{final} = Σ_{l=1}^N W_l \cdot h_i^{l,(t)}.
Step-by-Step Mathematical Representation
Step 1: Node and Edge Features
Node Features (ð‘¥ð‘–):
x_i = [\sigma_i(t), R_i(t), M_i(t), \rho_{i,j}(t)],
where:
ðœŽð‘–(ð‘¡): Volatility.
ð‘…ð‘–(ð‘¡): Return.
ð‘€ð‘–(ð‘¡): Momentum.
ðœŒð‘–,ð‘—(ð‘¡): Correlation.
Edge Features (ð‘’ð‘–ð‘—):
e_{ij} = [| \sigma_i(t) - \sigma_j(t) |, | R_i(t) - R_j(t) |, \rho_{i,j}(t)].
Adjacency Matrices:
For layer ð‘™:
A^{(l)}[i][j] = exp(-||x_i - x_j||^2 / (2 * \sigma^{(l)})).
Step 2: Multi-GNN Propagation Message Passing:
For each GNN layer ð‘™:
m_{ij}^{(l)} = A^{(l)}[i][j] * (W^{(l)} * h_j^{(l)}),
Node Update:
h_i^{(l+1)} = ReLU(Σ_{j ∈ N(i)} m_{ij}^{(l)} + b^{(l)}).
Cross-Layer Feedback:
Update edge weights in other layers:
A_{k}^{(t+1)}[i][j] = f(A_k^{(t)}[i][j], h_i^{l,(t)}, h_j^{l,(t)}),
where ð‘˜â‰ ð‘™.
Step 3: Kernel PCA
Kernel Updates:
Use embeddings â„Žð‘–ð‘“ð‘–ð‘›ð‘Žð‘™ to update kernel matrices:
K_V[i][j] = exp(-||h_i^{final} - h_j^{final}||^2 / (2 * \sigma_V^2)),
K_\Pi[i][j] = exp(-||h_i^{final} - h_j^{final}||^2 / (2 * \sigma_\Pi^2)).
Eigenvalue Decomposition:
K = Σ_{k=1}^m λ_k * (v_k ⊗ v_k),
where ðœ†ð‘˜ and ð‘£ð‘˜ are eigenvalues and eigenvectors.
Step 4: Recursive Feedback
Recursive Edge Updates:
A^{(t+1)} = f(A^{(t)}, K_V^{(t)}, K_\Pi^{(t)}).
Recursive Kernel Updates:
K_V^{(t+1)} = g(A^{(t+1)}, K_\Pi^{(t)}),
K_\Pi^{(t+1)} = h(A^{(t+1)}, K_V^{(t)}).
Step 5: Neural Networks
Threshold Predictions:
Θ_V(t) = NN_1(R_{V,i}(t)),
Θ_\Pi(t) = NN_2(R_{\Pi,i}(t)).
Meta Neural Network:
Θ_{final}(t) = MetaNN(Θ_V(t), Θ_\Pi(t)).
Step 6: Trade Execution
Signal Calculation:
S_{i,j}(t) = R_{V,i}(t) / sqrt( Σ_{k=1}^n R_{V,k}(t)^2 ).
Execution Condition:
| S_{i,j}(t) | > Θ_{final}(t).
Constraints
Variance Explained:
Σ_{k=1}^m λ_k / Σ_{k=1}^n λ_k ≥ 0.90.
Dynamic Kernel Widths:
σ_V(t) = σ_V(0) * (1 + κ * StdDev(V(t))),
σ_\Pi(t) = σ_\Pi(0) * (1 + κ * StdDev(Π(t))).
Feedback Convergence:
max(|K_V^{(t+1)} - K_V^{(t)}|, |K_\Pi^{(t+1)} - K_\Pi^{(t)}|) < ε.
Objective
Maximize risk-adjusted profitability by:
Extracting dominant volatility and profitability components with Kernel PCA.
Propagating and aggregating embeddings across ð‘-interdependent GNN layers.
Dynamically refining kernels and thresholds using recursive feedback.
#define PAIRS 28
string CurrencyPairs[PAIRS] = {
"EURUSD", "GBPUSD", "USDJPY", "GBPJPY", "USDCAD", "EURAUD", "EURJPY",
"AUDCAD", "AUDJPY", "AUDNZD", "AUDUSD", "CADJPY", "EURCAD", "EURCHF",
"EURGBP", "EURNZD", "GBPCAD", "GBPCHF", "NZDCAD", "NZDJPY", "NZDUSD",
"USDCHF", "CHFJPY", "AUDCHF", "GBPNZD", "NZDCHF", "CADCHF", "GBPAUD"
};
vars volatilities[PAIRS]; // Volatility for each pair
vars adjacencyMatrices[PAIRS][PAIRS][28]; // Adjacency matrices for 28 GNNs
vars nodeEmbeddings[PAIRS][5][28]; // Node embeddings for 28 GNNs
vars updatedEmbeddings[PAIRS][5]; // Final node embeddings after aggregation
vars kernelMatrix[PAIRS][PAIRS]; // Kernel matrix for PCA
vars eigenvalues[PAIRS]; // Eigenvalues from PCA
vars eigenvectors[PAIRS][PAIRS]; // Eigenvectors from PCA
vars reducedMatrix[PAIRS][3]; // Reduced components from PCA
vars thresholds[PAIRS]; // Thresholds for trade execution
vars signals[PAIRS]; // Final trading signals
int lookback = 50;
int NN; // Neural network handle for threshold prediction
int propagationDepth = 3; // GNN propagation depth
// Step 1: Calculate volatilities for all pairs
function calculateVolatilities() {
for (int i = 0; i < PAIRS; i++) {
volatilities[i] = StdDev(series(price(CurrencyPairs[i])), lookback);
}
}
// Step 2: Construct adjacency matrices for 28 GNNs
function constructAdjacencyMatrices() {
for (int l = 0; l < 28; l++) { // For each GNN layer
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
if (i != j) {
adjacencyMatrices[i][j][l] = exp(-pow(volatilities[i] - volatilities[j], 2) / (2 * 0.5^2)); // Gaussian kernel
} else {
adjacencyMatrices[i][j][l] = 0; // No self-loops
}
}
}
}
}
// Step 3: Propagate embeddings for each GNN
function propagateEmbeddings() {
for (int l = 0; l < 28; l++) { // For each GNN
for (int t = 0; t < propagationDepth; t++) { // Propagation steps
vars newEmbeddings[PAIRS][5];
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < 5; k++) {
newEmbeddings[i][k] = 0;
for (int j = 0; j < PAIRS; j++) {
newEmbeddings[i][k] += adjacencyMatrices[i][j][l] * nodeEmbeddings[j][k][l];
}
newEmbeddings[i][k] = ReLU(newEmbeddings[i][k]);
}
}
// Update embeddings
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < 5; k++) {
nodeEmbeddings[i][k][l] = newEmbeddings[i][k];
}
}
}
}
}
// Step 4: Aggregate embeddings across all GNNs
function aggregateEmbeddings() {
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < 5; k++) {
updatedEmbeddings[i][k] = 0;
for (int l = 0; l < 28; l++) {
updatedEmbeddings[i][k] += nodeEmbeddings[i][k][l];
}
updatedEmbeddings[i][k] /= 28; // Average across GNNs
}
}
}
// Step 5: Update kernel matrix using aggregated embeddings
function updateKernelMatrix() {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
kernelMatrix[i][j] = exp(-dotProduct(updatedEmbeddings[i], updatedEmbeddings[j]) / (2 * 0.5^2)); // Gaussian kernel
}
}
}
// Step 6: Perform Kernel PCA
function performKernelPCA() {
eigenDecomposition(kernelMatrix, eigenvalues, eigenvectors);
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < 3; j++) { // Top 3 components
reducedMatrix[i][j] = dotProduct(kernelMatrix[i], eigenvectors[j]);
}
}
}
// Step 7: Train neural network for threshold prediction
function trainNeuralNetwork() {
LayerPerceptron(10); // Hidden layer
LayerNormalize();
LayerPerceptron(1); // Output layer
NN = Train("ThresholdNN", reducedMatrix, thresholds, PAIRS);
}
// Step 8: Generate trade signals
function generateSignals() {
for (int i = 0; i < PAIRS; i++) {
signals[i] = Predict(NN, reducedMatrix[i]);
}
}
// Step 9: Execute trades
function executeTrades() {
for (int i = 0; i < PAIRS; i++) {
if (signals[i] > 0.5) {
enterLong(CurrencyPairs[i]);
} else if (signals[i] < -0.5) {
enterShort(CurrencyPairs[i]);
}
}
}
// Main function
function run() {
set(PLOTNOW);
// Step 1: Calculate volatilities
calculateVolatilities();
// Step 2: Construct adjacency matrices
constructAdjacencyMatrices();
// Step 3: Propagate embeddings for each GNN
propagateEmbeddings();
// Step 4: Aggregate embeddings
aggregateEmbeddings();
// Step 5: Update kernel matrix
updateKernelMatrix();
// Step 6: Perform Kernel PCA
performKernelPCA();
// Step 7: Train neural network for thresholds
trainNeuralNetwork();
// Step 8: Generate trade signals
generateSignals();
// Step 9: Execute trades
executeTrades();
}
