Implementing Kalman Filter for Pair Trading Strategy Enhancement

kalman_filter.q

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+// Reference
+// https://www.quantstart.com/articles/State-Space-Models-and-the-Kalman-Filter/
+
+// Auxiliary functions
+// Creates a diagonal matrix
+eye:{(2#x)#1f,x#0f}
+// Creates a 0's symetric matrix
+zeros: {(x#0f)}
+// JUST WORKS FOR 1X2 MATRIXES -> transform a 1x2 matrix to 2x1
+transpose: {enlist each x}
+
+// This function implements the Kalman Filter algorithm to estimate the state of a linear dynamic system.
+// It takes the following parameters:
+/   x: The observation matrix (measurement) at time t.
+/   y: The state transition matrix (predicted state) at time t.
+/   delta: The observation covariance matrix, representing the uncertainty in the measurements.
+/   estimates: The estimated state mean vector at time t-1.
+/   covariances: The estimated state covariance matrix at time t-1.
+
+// The function returns:
+/   estimates: estimated updated state mean vector at time t
+/   covariances: estimated updated state covariance matrix at time t.
+
+kalmanFilter:{[x;y;delta;estimates; covariances]
+    // Initialize static values 
+    G: eye 2;
+    W: (delta % (1-delta)) * eye 2;
+    V: 1f;
+    m0: estimates;
+    c0: covariances;
+    Ft: x, 1f;
+
+    alphat: G mmu m0; // a_t = G_t * m_t-1 (EQ 1)
+    Rt: ((G mmu c0) mmu flip[G]) + W; // R_t = G_t * C_t-1 * T(G) + W_t (EQ 2)
+    ft: Ft mmu alphat; // f_t = T(F_t) * a_t (EQ 5)
+    et: y - ft; // e_t = y_t - f_t (EQ 3)
+    Qt: ((Ft mmu Rt) mmu transpose[Ft]) + V; // Q_t = T(F_t) * R_t * F_t + V_t (EQ 6)
+    At: ((Rt mmu transpose[Ft]) mmu 1%Qt); // A_t = R_t * F_t * inv(Q_t) (EQ 7)
+    mt: (At * et) + alphat; // m_t = a_t + A_t * e_t (EQ 4)
+    Ct:Rt-flip[enlist At] mmu enlist first[Qt] * At; // C_t = R_t - A_t * Q_t * T(A_t) (EQ 8)
+    (mt;Ct)}  // Return new updates
+