From eaded72f8f17ab585e1530f476c406cdfd7ef062 Mon Sep 17 00:00:00 2001 From: PRAKUL HIREMATH <175131562+prakulhiremath@users.noreply.github.com> Date: Fri, 10 Apr 2026 17:47:05 +0530 Subject: [PATCH] Add v2 of Latent Micro-Regimes in Limit Order Books This version introduces key changes including the identification of pre-stress build-up and crisis regimes, a new method for detection via posterior entropy, and improvements in signal processing to reduce flooding. --- Experiments/v2 | 541 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 541 insertions(+) create mode 100644 Experiments/v2 diff --git a/Experiments/v2 b/Experiments/v2 new file mode 100644 index 0000000..98f6025 --- /dev/null +++ b/Experiments/v2 @@ -0,0 +1,541 @@ +# ============================================================ +# Latent Micro-Regimes in Limit Order Books: +# Identification and Early Detection — v2 +# ───────────────────────────────────────── +# Key changes over v1: +# 1. Regime 1 = pre-stress build-up, Regime 2 = crisis +# → HMM entry into regime 1 is the early-warning signal +# 2. Detection via posterior ENTROPY rise, not hard-switch +# 3. Minimum-gap deduplication (no signal flooding) +# 4. Posterior probability smoothing for cleaner signals +# 5. Baselines use identical deduplication +# ============================================================ +# !pip install hmmlearn scikit-learn scipy numpy pandas matplotlib + +import warnings +warnings.filterwarnings("ignore") + +import numpy as np +import pandas as pd +import matplotlib.pyplot as plt +from scipy import stats +from scipy.stats import gaussian_kde +from sklearn.preprocessing import StandardScaler +from hmmlearn.hmm import GaussianHMM + +# ───────────────────────────────────────── +# 0. Global Configuration +# ───────────────────────────────────────── +SEED = 42 +T = 12_000 +N_REGIMES = 3 +MAX_LAG = 60 # evaluation window (timesteps) +FW_WINDOW = 20 # forward window for stress label +STRESS_PCT = 95 # percentile for stress threshold +N_BOOT = 2_000 +MIN_GAP = 15 # minimum timesteps between two signals (dedup) +PENALTY = -MAX_LAG + +np.random.seed(SEED) + +# ───────────────────────────────────────── +# 1. Structured Latent Regime Generator +# Regime 0 = Normal (baseline, calm) +# Regime 1 = Pre-stress build-up ← the key early-warning state +# Regime 2 = Crisis / stress peak +# +# Causal chain: 0 → 1 → 2 → (0 or 1) +# The HMM must learn that regime 1 is a harbinger of regime 2. +# We encode this by: +# • Making spread/depth in regime 1 intermediate +# • Making transitions 0→1 more likely than 0→2 directly +# • Stress events defined on FUTURE spread (no leakage) +# ───────────────────────────────────────── + +REGIME_PARAMS = { + # spread: mu, sigma (log-normal) + # depth: mu, sigma (AR-1) + # imb: mu, sigma (truncated normal) + # vol_base: scaling factor + 0: dict(sp_mu=1.5, sp_sig=0.25, dp_mu=120, dp_sig=10, ib_mu=0.00, ib_sig=0.07, vb=0.35), + 1: dict(sp_mu=3.2, sp_sig=0.55, dp_mu= 88, dp_sig=16, ib_mu=0.22, ib_sig=0.13, vb=1.10), + 2: dict(sp_mu=7.5, sp_sig=1.20, dp_mu= 42, dp_sig=22, ib_mu=0.48, ib_sig=0.22, vb=2.90), +} + +def build_causal_transition_matrix() -> np.ndarray: + """ + Transition matrix with causal structure: + - 0 → 1 much more likely than 0 → 2 (stress builds gradually) + - 1 → 2 is the crisis trigger + - 2 → 0 or 1 (recovery) + """ + P = np.array([ + [0.970, 0.028, 0.002], # Normal → mostly stays + [0.060, 0.900, 0.040], # Pre-stress → can escalate + [0.100, 0.150, 0.750], # Crisis → decays back + ]) + # Normalise rows (safety) + return P / P.sum(axis=1, keepdims=True) + + +def simulate_latent_chain(T, P, pi0, rng): + K = P.shape[0] + Z = np.empty(T, dtype=int) + Z[0] = rng.choice(K, p=pi0) + for t in range(1, T): + Z[t] = rng.choice(K, p=P[Z[t - 1]]) + return Z + + +def generate_lob_data(T, rng): + """ + Simulate LOB features under a causal Markov-switching model. + + Hawkes-like self-excitation on spread ensures bursts cluster + without trivially revealing regime via a single feature. + """ + P = build_causal_transition_matrix() + pi0 = np.array([0.85, 0.12, 0.03]) + Z = simulate_latent_chain(T, P, pi0, rng) + + spread = np.zeros(T) + depth = np.zeros(T) + imbalance = np.zeros(T) + + # Spread: log-normal + Hawkes excitation + hawkes = 0.0 + decay = 0.88 + for t in range(T): + p = REGIME_PARAMS[Z[t]] + hawkes = hawkes * decay + eps = rng.normal(0, p['sp_sig']) + spread[t] = np.exp(np.log(p['sp_mu']) + 0.12 * hawkes + eps) + if spread[t] > np.exp(np.log(p['sp_mu']) + p['sp_sig']): + hawkes += 0.25 + + # Depth: mean-reverting AR(1) + phi = 0.93 + depth[0] = REGIME_PARAMS[Z[0]]['dp_mu'] + for t in range(1, T): + p = REGIME_PARAMS[Z[t]] + depth[t] = phi * depth[t-1] + (1-phi) * p['dp_mu'] + rng.normal(0, p['dp_sig']) + depth = np.clip(depth, 5, None) + + # Imbalance: truncated normal + for t in range(T): + p = REGIME_PARAMS[Z[t]] + imbalance[t] = np.clip(rng.normal(p['ib_mu'], p['ib_sig']), -1, 1) + + # Rolling spread vol (20-step) + roll_vol = pd.Series(spread).pct_change().rolling(20).std().fillna(0).values + + # OFI proxy + ofi = imbalance * np.abs(np.diff(spread, prepend=spread[0])) + + X = np.column_stack([spread, depth, imbalance, roll_vol, ofi]) + return X, Z + + +# ───────────────────────────────────────── +# 2. Feature Engineering & Normalisation +# ───────────────────────────────────────── + +def engineer_features(X_raw): + spread = X_raw[:, 0] + depth = X_raw[:, 1] + imbalance = X_raw[:, 2] + roll_vol = X_raw[:, 3] + ofi = X_raw[:, 4] + + sd_ratio = spread / (depth + 1e-6) + abs_imb = np.abs(imbalance) + cum_ofi = pd.Series(ofi).rolling(50, min_periods=1).mean().values + roll_depth = pd.Series(depth).rolling(20).mean().fillna(method='bfill').values + + X_full = np.column_stack([ + spread, depth, imbalance, roll_vol, ofi, + sd_ratio, abs_imb, cum_ofi, roll_depth + ]) + scaler = StandardScaler() + X_scaled = scaler.fit_transform(X_full) + return X_scaled, scaler + + +# ───────────────────────────────────────── +# 3. HMM Fitting +# ───────────────────────────────────────── + +def fit_hmm(X, n_components=N_REGIMES, n_restarts=10, rng_seed=SEED): + best_score, best_model = -np.inf, None + for k in range(n_restarts): + model = GaussianHMM( + n_components = n_components, + covariance_type = "full", + n_iter = 300, + tol = 1e-6, + random_state = rng_seed + k, + init_params = "stmc", + params = "stmc", + ) + try: + model.fit(X) + score = model.score(X) + if score > best_score: + best_score, best_model = score, model + except Exception: + continue + if best_model is None: + raise RuntimeError("HMM fitting failed.") + return best_model + + +# ───────────────────────────────────────── +# 4. Stress Event Definition (no leakage) +# ───────────────────────────────────────── + +def define_stress_events(X_raw, fw=FW_WINDOW, pct=STRESS_PCT): + spread = X_raw[:, 0] + threshold = np.percentile(spread, pct) + sigma = [t for t in range(len(spread) - fw) + if np.mean(spread[t+1:t+fw+1]) > threshold] + return np.array(sigma, dtype=int) + + +# ───────────────────────────────────────── +# 5. Signal Computation (the core upgrade) +# ───────────────────────────────────────── + +def state_entropy(posterior): + """Shannon entropy of the posterior state distribution at each step.""" + eps = 1e-12 + return -np.sum(posterior * np.log(posterior + eps), axis=1) + + +def deduplicate(indices, min_gap=MIN_GAP): + """ + Keep only the FIRST index in each cluster of indices + that are within min_gap of each other. + This prevents signal flooding. + """ + if len(indices) == 0: + return indices + out = [indices[0]] + for idx in indices[1:]: + if idx - out[-1] >= min_gap: + out.append(idx) + return np.array(out, dtype=int) + + +def model_signals(model, X_scaled, Z_hat, + entropy_pct=80, smooth_window=5, min_gap=MIN_GAP): + """ + Uncertainty-based early-warning signal. + + Strategy (three complementary triggers, unioned then deduped): + A) Entropy spike: posterior uncertainty rises sharply + → captures "the HMM is unsure", a known pre-transition marker + B) Pre-stress state entry: Z_hat transitions INTO the intermediate + state (state with 2nd-highest mean spread) + → aligns detection with causal regime structure + C) Entropy trend: rolling entropy slope turns positive + → catches gradual build-up before a hard switch + + All triggers are deduplicated so N(τ) stays controlled. + """ + posterior = model.predict_proba(X_scaled) # (T, K) + + # ── Smooth posterior to reduce HMM jitter ── + smooth_post = pd.DataFrame(posterior).rolling( + smooth_window, min_periods=1, center=True).mean().values + + entropy = state_entropy(smooth_post) + + # ── Identify the "pre-stress" HMM state ── + # We rank states by their mean spread (feature 0 in raw space) + means_raw = model.means_[:, 0] # spread dim of HMM means + state_rank = np.argsort(means_raw) # [low, mid, high] + prestress_state = state_rank[1] # intermediate spread state + + # ── Trigger A: entropy spike ── + ent_thresh = np.percentile(entropy, entropy_pct) + sig_A = np.where(entropy > ent_thresh)[0] + + # ── Trigger B: entry into pre-stress state ── + enters_prestress = (Z_hat[1:] == prestress_state) & (Z_hat[:-1] != prestress_state) + sig_B = np.where(enters_prestress)[0] + + # ── Trigger C: entropy slope turns positive ── + ent_slope = pd.Series(entropy).diff(5).fillna(0).values + slope_thresh = np.percentile(ent_slope[ent_slope > 0], 70) + sig_C = np.where(ent_slope > slope_thresh)[0] + + # ── Union + deduplicate ── + all_signals = np.unique(np.concatenate([sig_A, sig_B, sig_C])) + tau = deduplicate(all_signals, min_gap=min_gap) + return tau + + +def imbalance_baseline(X_raw, pct=90, min_gap=MIN_GAP): + imb = np.abs(X_raw[:, 2]) + threshold = np.percentile(imb, pct) + raw = np.where(imb > threshold)[0] + return deduplicate(raw, min_gap=min_gap) + + +def volatility_baseline(X_raw, pct=90, min_gap=MIN_GAP): + roll_vol = X_raw[:, 3] + threshold = np.percentile(roll_vol, pct) + raw = np.where(roll_vol > threshold)[0] + return deduplicate(raw, min_gap=min_gap) + + +# ───────────────────────────────────────── +# 6. Lead-Time Evaluation +# ───────────────────────────────────────── + +def compute_lead_times(tau, sigma, max_lag=MAX_LAG): + deltas = np.empty(len(tau), dtype=float) + for i, t in enumerate(tau): + cands = sigma[(sigma > t) & (sigma <= t + max_lag)] + deltas[i] = (cands[0] - t) if len(cands) > 0 else PENALTY + return deltas + + +def evaluation_metrics(deltas): + valid = deltas > 0 + return dict( + mean_delta = float(np.mean(deltas)), + pct_early = float(np.mean(valid)), + mean_early = float(np.mean(deltas[valid])) if valid.any() else 0.0, + std_delta = float(np.std(deltas)), + n_tau = int(len(deltas)), + n_early = int(valid.sum()), + ) + + +# ───────────────────────────────────────── +# 7. Bootstrap CI + Mann–Whitney +# ───────────────────────────────────────── + +def bootstrap_ci(deltas, stat_fn=np.mean, n_boot=N_BOOT, alpha=0.05, seed=SEED): + rng = np.random.default_rng(seed) + boot = np.array([stat_fn(rng.choice(deltas, size=len(deltas), replace=True)) + for _ in range(n_boot)]) + return float(np.percentile(boot, 100*alpha/2)), float(np.percentile(boot, 100*(1-alpha/2))) + + +def mannwhitney_test(a, b): + return stats.mannwhitneyu(a, b, alternative="two-sided") + + +# ───────────────────────────────────────── +# 8. Publication-Quality Visualisation +# ───────────────────────────────────────── + +PALETTE = {"Model": "#2C6FAC", "Imbalance": "#D94F3D", "Volatility": "#5AAE61"} + +plt.rcParams.update({ + "font.family" : "serif", + "font.size" : 11, + "axes.spines.top" : False, + "axes.spines.right": False, + "axes.linewidth" : 0.8, + "figure.dpi" : 150, +}) + + +def plot_entropy_and_regimes(X_raw, Z_true, Z_hat, entropy, tau_model, + sigma, n_show=2500): + """ + Four-panel diagnostic: + 1. Spread with true regime shading + 2. Posterior entropy (with signal thresholds and triggers marked) + 3. Inferred HMM state + 4. Depth with stress events + """ + t_end = min(n_show, len(Z_true)) + t_ax = np.arange(t_end) + spread = X_raw[:t_end, 0] + depth = X_raw[:t_end, 1] + + tau_vis = tau_model[tau_model < t_end] + sigma_vis = sigma[sigma < t_end] + + regime_colors = {0: "#DDEEFF", 1: "#FFF3CD", 2: "#FFDDDD"} + + fig, axes = plt.subplots(4, 1, figsize=(12, 9), sharex=True, + gridspec_kw={"height_ratios": [2, 2, 1, 2]}) + + # Panel 1 — Spread + true shading + ax = axes[0] + for k, c in regime_colors.items(): + ax.fill_between(t_ax, 0, spread.max()*1.1, + where=Z_true[:t_end]==k, color=c, alpha=0.55, + label=f"State {k}") + ax.plot(t_ax, spread, lw=0.7, color="#1A1A2E") + ax.set_ylabel("Bid-Ask Spread") + ax.legend(loc="upper right", fontsize=8, frameon=False, ncol=3) + ax.set_title("True Regimes · Posterior Entropy · Inferred States · Market Depth", + fontsize=12, pad=8) + + # Panel 2 — Entropy + model triggers + ax = axes[1] + ent_vis = entropy[:t_end] + ax.plot(t_ax, ent_vis, lw=0.9, color="#555555", alpha=0.85, label="Entropy") + ent_thresh = np.percentile(entropy, 80) + ax.axhline(ent_thresh, color="#FF8800", lw=1.0, ls="--", label="80th pct threshold") + ax.vlines(tau_vis, ent_vis.min(), ent_vis.max(), + color=PALETTE["Model"], lw=0.7, alpha=0.6, label="Model signal τ") + ax.set_ylabel("Posterior Entropy") + ax.legend(loc="upper right", fontsize=8, frameon=False) + + # Panel 3 — Inferred HMM state + ax = axes[2] + ax.step(t_ax, Z_hat[:t_end], lw=0.9, color=PALETTE["Model"]) + ax.set_ylabel("HMM State") + ax.set_yticks([0, 1, 2]) + + # Panel 4 — Depth + stress events + ax = axes[3] + ax.plot(t_ax, depth, lw=0.7, color="#2D6A4F") + ax.vlines(sigma_vis, depth.min(), depth.max(), + color=PALETTE["Imbalance"], lw=0.7, alpha=0.5, label="Stress event σ") + ax.set_ylabel("Market Depth") + ax.set_xlabel("Timestep") + ax.legend(loc="upper right", fontsize=8, frameon=False) + + fig.tight_layout() + plt.show() + + +def plot_lead_time_densities(delta_dict, max_lag=MAX_LAG): + fig, ax = plt.subplots(figsize=(8, 4.5)) + x_grid = np.linspace(-max_lag-5, max_lag+5, 600) + + for i, (name, deltas) in enumerate(delta_dict.items()): + color = PALETTE[name] + valid = deltas[deltas > PENALTY] + if len(valid) > 5: + kde = gaussian_kde(valid, bw_method="scott") + ax.plot(x_grid, kde(x_grid), lw=2.2, color=color, label=name) + ax.fill_between(x_grid, kde(x_grid), alpha=0.13, color=color) + pf = np.mean(deltas <= PENALTY) + ax.annotate(f"missed={pf:.1%}", xy=(-max_lag+1, 0.004*(i+1)), + color=color, fontsize=8.5) + + ax.axvline(0, color="gray", lw=1.0, ls="--", label="Zero lead-time") + ax.set_xlabel("Lead time Δ (timesteps)", labelpad=8) + ax.set_ylabel("Density", labelpad=8) + ax.set_title("Lead-Time Distribution: Model vs Baselines", fontsize=13, pad=10) + ax.legend(frameon=False, fontsize=10) + ax.set_xlim(-max_lag-2, max_lag+2) + fig.tight_layout() + plt.show() + + +def plot_results_table(results_df): + fig, ax = plt.subplots(figsize=(11, 2.2)) + ax.axis("off") + tbl = ax.table(cellText=results_df.values, colLabels=results_df.columns, + cellLoc="center", loc="center") + tbl.auto_set_font_size(False) + tbl.set_fontsize(9.5) + tbl.scale(1.2, 1.6) + for j in range(len(results_df.columns)): + tbl[0, j].set_facecolor("#2C6FAC") + tbl[0, j].set_text_props(color="white", fontweight="bold") + fig.tight_layout() + plt.show() + + +# ───────────────────────────────────────── +# 9. Main Pipeline +# ───────────────────────────────────────── + +def run_experiment(): + rng = np.random.default_rng(SEED) + + print("=" * 62) + print(" Step 1 / 6 — Generating causal LOB data …") + X_raw, Z_true = generate_lob_data(T, rng) + dist = " | ".join([f"State {k}: {(Z_true==k).mean():.1%}" for k in range(N_REGIMES)]) + print(f" {T:,} timesteps | {dist}") + + print("\n Step 2 / 6 — Feature engineering …") + X_scaled, scaler = engineer_features(X_raw) + print(f" Feature matrix: {X_scaled.shape}") + + print("\n Step 3 / 6 — Fitting HMM (10 restarts) …") + model = fit_hmm(X_scaled) + Z_hat = model.predict(X_scaled) + ll = model.score(X_scaled) + print(f" Best log-likelihood: {ll:,.2f} | Converged: {model.monitor_.converged}") + + # Identify which HMM state corresponds to each true regime + # (sorted by mean spread value in the HMM) + means_sp = model.means_[:, 0] + state_rank = np.argsort(means_sp) + print(f" HMM state ranking by spread: {state_rank.tolist()} " + f"(low→high spread)") + + print("\n Step 4 / 6 — Stress event definition …") + sigma = define_stress_events(X_raw) + print(f" Stress events: {len(sigma):,} ({len(sigma)/T:.1%} of timesteps)") + + print("\n Step 5 / 6 — Computing signals …") + # Model signals: uncertainty + pre-stress entry + posterior = model.predict_proba(X_scaled) + smooth_post = pd.DataFrame(posterior).rolling(5, min_periods=1, center=True).mean().values + entropy = state_entropy(smooth_post) + + tau_model = model_signals(model, X_scaled, Z_hat) + tau_imb = imbalance_baseline(X_raw) + tau_vol = volatility_baseline(X_raw) + + print(f" Signals — Model: {len(tau_model)} | " + f"Imbalance: {len(tau_imb)} | Volatility: {len(tau_vol)}") + + delta_model = compute_lead_times(tau_model, sigma) + delta_imb = compute_lead_times(tau_imb, sigma) + delta_vol = compute_lead_times(tau_vol, sigma) + + delta_dict = {"Model": delta_model, "Imbalance": delta_imb, "Volatility": delta_vol} + + print("\n Step 6 / 6 — Statistical validation …") + rows = [] + for name, deltas in delta_dict.items(): + m = evaluation_metrics(deltas) + lo, hi = bootstrap_ci(deltas) + rows.append({ + "Detector" : name, + "Mean Δ" : f"{m['mean_delta']:+.2f}", + "95% CI" : f"[{lo:+.2f}, {hi:+.2f}]", + "% Early" : f"{m['pct_early']:.1%}", + "Mean Δ | early" : f"{m['mean_early']:+.2f}", + "Std Δ" : f"{m['std_delta']:.2f}", + "N(τ)" : m['n_tau'], + "N(early)" : m['n_early'], + }) + + results_df = pd.DataFrame(rows) + print("\n" + results_df.to_string(index=False)) + + print("\n Pairwise Mann–Whitney U tests (two-sided):") + pairs = [("Model","Imbalance"),("Model","Volatility"),("Imbalance","Volatility")] + for a, b in pairs: + u, p = mannwhitney_test(delta_dict[a], delta_dict[b]) + sig = "***" if p<0.001 else ("**" if p<0.01 else ("*" if p<0.05 else "ns")) + print(f" {a:12s} vs {b:12s}: U={u:,.0f} p={p:.4f} {sig}") + + print("\n Rendering figures …") + plot_entropy_and_regimes(X_raw, Z_true, Z_hat, entropy, tau_model, sigma) + plot_lead_time_densities(delta_dict) + plot_results_table(results_df) + + print("\n Experiment complete.") + return results_df, delta_dict, model, X_raw, Z_true, Z_hat, sigma, entropy + + +if __name__ == "__main__": + (results_df, delta_dict, model, + X_raw, Z_true, Z_hat, sigma, entropy) = run_experiment()