Problem: Recursive Kernel PCA with GNN for Profitability Optimization

You are tasked to design a trading strategy that integrates two Kernel PCAs (for volatility and profitability relationships) with a Graph Neural Network (GNN). The strategy dynamically updates node embeddings and kernel matrices to extract dominant patterns and optimize trading signals for a portfolio of 𝑛=28 currency pairs.

The goal is to maximize profitability Π, subject to market dynamics and constraints.

Step 1: Graph Construction
Graph Definition: Let the currency pairs be represented as a graph 𝐺=(𝑉,𝐸), where:

𝑉: Nodes, 𝑣𝑖 , represent currency pairs.
𝐸: Edges represent relationships between pairs.
Node Features: Each node 𝑣𝑖 is assigned a feature vector 𝑥𝑖, defined as:

x_i = [σ_i(t), R_i(t), M_i(t), ρ_{i,j}(t)]

where:

𝜎𝑖(𝑡): Volatility of 𝐶𝑖,
𝑅𝑖(𝑡): Return of 𝐶𝑖,𝑀𝑖(𝑡): Momentum of 𝐶𝑖,
𝜌𝑖,𝑗(𝑡): Correlation between 𝐶𝑖 and 𝐶𝑗.

Edge Weights: The weighted adjacency matrix 𝐴 is given by:

A[i][j] = exp(-||x_i - x_j||^2 / (2 * σ^2))

where 𝜎 is a kernel width parameter.

Step 2: Kernel Matrices

Volatility Kernel
𝐾𝑉 : Construct 𝐾𝑉K using a Gaussian kernel:

K_V[i][j] = exp(- (V_{i,j}(t) - V_{k,l}(t))^2 / (2 * σ_V^2))

where 𝑉𝑖,𝑗(𝑡)=∣𝜎𝑖(𝑡)−𝜎𝑗(𝑡)∣ is the volatility spread.

Profitability Kernel 𝐾Π​ : Construct 𝐾Π as:

K_Π[i][j] = exp(- (Π_{i,j}(t) - Π_{k,l}(t))^2 / (2 * σ_Π^2))

where Π𝑖,𝑗(𝑡)=∣𝑅𝑖(𝑡)−𝑅𝑗(𝑡)∣Π i,j​(t) is the profitability difference.

Eigenvalue Decomposition: Decompose 𝐾𝑉​ and 𝐾Π into 𝜆𝑘 (eigenvalues) and 𝑣𝑘 (eigenvectors):

K = Σ_{k=1}^m λ_k * (v_k ⊗ v_k)

where 𝑚 is the number of principal components.

Step 3: GNN Embedding Propagation
Message Passing: For each edge (𝑖,𝑗), compute the message 𝑚𝑖𝑗​:

m_{ij} = A[i][j] * (W * h_j^{(t)})

where 𝑊 is a weight matrix.

Node Updates: Update node embeddings ℎ𝑖(𝑡+1) using:

h_i^{(t+1)} = ReLU(W * Σ_{j∈N(i)} m_{ij} + b)

Recursive Feedback: Use updated embeddings ℎ𝑖(𝑡+1) to refine the kernel matrices:

K_V[i][j] = f(h_i^{(t+1)}, h_j^{(t+1)})
K_Π[i][j] = g(h_i^{(t+1)}, h_j^{(t+1)})

Step 4: Neural Networks
Threshold Prediction: Train two neural networks:

𝑁𝑁1 : Predicts volatility threshold Θ𝑉(𝑡):

Θ_V(t) = NN_1(R_{V,i}(t))

𝑁𝑁2 : Predicts profitability threshold ΘΠ(𝑡):

Θ_Π(t) = NN_2(R_{Π,i}(t))

Meta Neural Network: Combine Θ𝑉(𝑡) and ΘΠ(𝑡) using a meta network:

Θ_{final}(t) = MetaNN(Θ_V(t), Θ_Π(t))

Step 5: Trade Execution
Signal Calculation: Compute the normalized trade signal 𝑆𝑖,𝑗(𝑡):

S_{i,j}(t) = R_{V,i}(t) / sqrt(Σ_{k=1}^n R_{V,k}(t)^2)

Execution Criteria: Execute a trade if:

|S_{i,j}(t)| > Θ_{final}(t)

Constraints
Variance Explained: Select the top 𝑚 components such that:

Σ_{k=1}^m λ_k / Σ_{k=1}^n λ_k ≥ 0.90

Dynamic Kernel Widths: Update kernel widths dynamically:

σ_V(t) = σ_V(0) * (1 + κ * StdDev(V(t)))
σ_Π(t) = σ_Π(0) * (1 + κ * StdDev(Π(t)))

Feedback Convergence: Ensure feedback loops converge within
𝑡𝑚𝑎𝑥=10 iterations:

max(|K_V^{(t+1)} - K_V^{(t)}|, |K_Π^{(t+1)} - K_Π^{(t)}|) < ε

Goal Maximize cumulative profitability Π:

Π = Σ_{t=1}^T Σ_{i,j} (S_{i,j}(t) * Trade_{i,j}(t))

subject to constraints on variance explained, kernel widths, and feedback convergence.

Code
#define PAIRS 28
#define COMBINATIONS 378

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 profitability[COMBINATIONS];    // Profitability for each pair combination
vars adjacencyMatrix[PAIRS][PAIRS];  // GNN adjacency matrix
vars nodeEmbeddings[PAIRS][5];       // Node embeddings from GNN
vars kernelMatrix1[PAIRS][PAIRS];    // Kernel matrix for PCA-1 (volatility)
vars kernelMatrix2[PAIRS][PAIRS];    // Kernel matrix for PCA-2 (profitability)
vars ReducedMatrix1[PAIRS][PAIRS];   // Reduced components from PCA-1
vars ReducedMatrix2[PAIRS][PAIRS];   // Reduced components from PCA-2
vars Threshold1[PAIRS];              // Thresholds from NN-1
vars Threshold2[PAIRS];              // Thresholds from NN-2
vars MetaFeatures[PAIRS];            // Meta-features for MetaNN
vars FinalSignals[PAIRS];            // Final trade signals
int NN1, NN2, MetaNN;                // Neural network handles
int lookback = 50;                   // Lookback period

// Step 1: Calculate volatility and profitability
function calculateMetrics() {
    for (int i = 0; i < PAIRS; i++) {
        volatilities[i] = StdDev(series(price(CurrencyPairs[i])), lookback);
    }
    for (int k = 0; k < COMBINATIONS; k++) {
        int i = k / PAIRS;
        int j = k % PAIRS;
        profitability[k] = abs(priceClose(CurrencyPairs[i]) - priceClose(CurrencyPairs[j]));
    }
}

// Step 2: Construct adjacency matrix
function constructAdjacencyMatrix() {
    for (int i = 0; i < PAIRS; i++) {
        for (int j = 0; j < PAIRS; j++) {
            if (i != j) {
                adjacencyMatrix[i][j] = exp(-pow(volatilities[i] - volatilities[j], 2)); // Gaussian kernel
            } else {
                adjacencyMatrix[i][j] = 0;
            }
        }
    }
}

// Step 3: Propagate GNN embeddings
function propagateEmbeddings() {
    vars newEmbeddings[PAIRS][5];
    for (int t = 0; t < 3; t++) { // 3 propagation steps
        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] += adjacencyMatrix[i][j] * nodeEmbeddings[j][k];
                }
                newEmbeddings[i][k] = ReLU(newEmbeddings[i][k]);
            }
        }
        for (int i = 0; i < PAIRS; i++) {
            for (int k = 0; k < 5; k++) {
                nodeEmbeddings[i][k] = newEmbeddings[i][k];
            }
        }
    }
}

// Step 4: Update kernel matrices using GNN embeddings
function updateKernelMatrices() {
    for (int i = 0; i < PAIRS; i++) {
        for (int j = 0; j < PAIRS; j++) {
            kernelMatrix1[i][j] = exp(-dotProduct(nodeEmbeddings[i], nodeEmbeddings[j]) / 2); // Volatility kernel
            kernelMatrix2[i][j] = exp(-profitability[i * PAIRS + j] / 2);                     // Profitability kernel
        }
    }
}

// Step 5: Perform Kernel PCA
function performKernelPCA(vars kernelMatrix[PAIRS][PAIRS], vars ReducedMatrix[PAIRS][PAIRS]) {
    vars eigenvalues, eigenvectors;
    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 6: Train neural networks
function trainNeuralNetworks() {
    // NN-1: Thresholds from PCA-1
    LayerPerceptron(10);
    LayerNormalize();
    LayerPerceptron(1);
    NN1 = Train("NN1", ReducedMatrix1, Threshold1, PAIRS);

    // NN-2: Thresholds from PCA-2
    LayerPerceptron(10);
    LayerNormalize();
    LayerPerceptron(1);
    NN2 = Train("NN2", ReducedMatrix2, Threshold2, PAIRS);

    // Meta Neural Network
    LayerPerceptron(5);
    LayerNormalize();
    LayerPerceptron(1);
    MetaNN = Train("MetaNN", [Threshold1, Threshold2], FinalSignals, PAIRS);
}

// Step 7: Execute trades
function executeTrades() {
    for (int i = 0; i < PAIRS; i++) {
        if (FinalSignals[i] > 0.5) {
            enterLong(CurrencyPairs[i]);
        } else if (FinalSignals[i] < -0.5) {
            enterShort(CurrencyPairs[i]);
        }
    }
}

// Main function
function run() {
    set(PLOTNOW);

    // Step 1: Calculate metrics
    calculateMetrics();

    // Step 2: Construct adjacency matrix
    constructAdjacencyMatrix();

    // Step 3: Propagate embeddings
    propagateEmbeddings();

    // Step 4: Update kernel matrices
    updateKernelMatrices();

    // Step 5: Perform Kernel PCA
    performKernelPCA(kernelMatrix1, ReducedMatrix1);
    performKernelPCA(kernelMatrix2, ReducedMatrix2);

    // Step 6: Train neural networks
    trainNeuralNetworks();

    // Step 7: Execute trades
    executeTrades();
}

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