API Reference: Hidden Markov Model (HMM)

The HMM class provides a complete implementation of Hidden Markov Models with Gaussian emissions for regime detection, time series modeling, and state inference.

Quick Start

from optimizr import HMM
import numpy as np

# Create model with 2 hidden states (e.g., bull/bear market)
model = HMM(n_states=2)

# Train on returns data
returns = np.random.randn(1000, 1)  # Should be 2D: (n_samples, n_features)
model.fit(returns, n_iterations=100, tolerance=1e-6)

# Decode most likely state sequence (Viterbi)
states = model.predict(returns)

# Compute log-likelihood (for model comparison)
logp = model.score(returns)

print(f"Log-likelihood: {logp:.2f}")
print(f"Decoded states: {states[:10]}")

Constructor

HMM(n_states: int)

Creates a new Hidden Markov Model with Gaussian emissions.

Parameters:

  • n_states (int): Number of hidden states/regimes. Common choices:

    • n_states=2: Binary regime (e.g., bull/bear, high/low volatility)

    • n_states=3: Three-regime model (e.g., bull/sideways/bear)

    • n_states>3: Fine-grained regime detection (requires more data)

Returns:

  • HMM object with random initialization

Initialization:

  • Transition matrix \(A\): Uniform with slight self-transition bias

  • Initial state distribution \(\pi\): Uniform

  • Emission parameters (means \(\mu_i\), covariances \(\Sigma_i\)): From K-means clustering

Example:

# Binary regime model
hmm_2 = HMM(n_states=2)

# Three-regime model for more nuanced detection
hmm_3 = HMM(n_states=3)

When to use:

  • n_states=2: Most common, sufficient for many applications

  • Higher n_states: When you have strong prior belief in multiple regimes and sufficient data (>1000 samples per state)

Methods

fit(X, n_iterations=100, tolerance=1e-6, n_init=1, random_state=None)

Trains the HMM on observed data using the Baum-Welch (Expectation-Maximization) algorithm.

Parameters:

  • X (np.ndarray): Training data of shape (n_samples, n_features)

    • For univariate time series: reshape to (n, 1) with X.reshape(-1, 1)

    • For multivariate: pass directly as (n, d) where d is feature dimension

  • n_iterations (int, default=100): Maximum number of EM iterations

    • Typical range: 50-200

    • More iterations → better convergence but slower training

  • tolerance (float, default=1e-6): Convergence threshold

    • Algorithm stops when log-likelihood improvement < tolerance

    • Typical range: 1e-8 to 1e-4

    • Smaller values → tighter convergence but more iterations

  • n_init (int, default=1): Number of random initializations

    • The best model (highest log-likelihood) is kept

    • Recommended: 5-10 for production models (helps avoid local minima)

  • random_state (int, optional): Random seed for reproducibility

Returns:

  • self: The fitted HMM object (for method chaining)

Algorithm: Baum-Welch (EM for HMMs)

The Baum-Welch algorithm iteratively refines model parameters:

  1. E-step: Compute state occupation probabilities

    • Forward pass: \(\alpha_t(i) = P(O_1, \ldots, O_t, S_t = i \mid \lambda)\)

    • Backward pass: \(\beta_t(i) = P(O_{t+1}, \ldots, O_T \mid S_t = i, \lambda)\)

    • State probabilities: \(\gamma_t(i) = \frac{\alpha_t(i)\beta_t(i)}{\sum_j \alpha_t(j)\beta_t(j)}\)

    • Transition probabilities: \(\xi_t(i,j) = \frac{\alpha_t(i)a_{ij}b_j(O_{t+1})\beta_{t+1}(j)}{\sum_{i,j}\alpha_t(i)a_{ij}b_j(O_{t+1})\beta_{t+1}(j)}\)

  2. M-step: Update model parameters

    • Initial probabilities: \(\pi_i = \gamma_1(i)\)

    • Transition matrix: \(a_{ij} = \frac{\sum_{t=1}^{T-1}\xi_t(i,j)}{\sum_{t=1}^{T-1}\gamma_t(i)}\)

    • Emission means: \(\mu_i = \frac{\sum_{t=1}^T \gamma_t(i) O_t}{\sum_{t=1}^T \gamma_t(i)}\)

    • Emission covariances: \(\Sigma_i = \frac{\sum_{t=1}^T \gamma_t(i)(O_t - \mu_i)(O_t - \mu_i)^T}{\sum_{t=1}^T \gamma_t(i)}\)

  3. Convergence: Repeat until log-likelihood change < tolerance

Example:

import numpy as np
from optimizr import HMM

# Simulate two-regime data
np.random.seed(42)
n = 2000

# Regime 1: low volatility (first 1000 samples)
regime1 = np.random.normal(0.0, 0.5, 1000)
# Regime 2: high volatility (last 1000 samples)
regime2 = np.random.normal(0.0, 2.0, 1000)
data = np.concatenate([regime1, regime2]).reshape(-1, 1)

# Train HMM
hmm = HMM(n_states=2)
hmm.fit(data, n_iterations=200, tolerance=1e-6, n_init=5)

print("Training complete")

Convergence diagnostics:

# Plot log-likelihood over iterations (requires storing history)
# Check if converged before max_iter
# Verify parameters make sense (e.g., distinct means for each state)

Typical training time:

  • 1000 samples, 2 states, 100 iterations: ~50-100ms

  • 10000 samples, 3 states, 200 iterations: ~500ms-1s

predict(X)

Decodes the most likely sequence of hidden states using the Viterbi algorithm.

Parameters:

  • X (np.ndarray): Observation sequence of shape (n_samples, n_features)

    • Must match feature dimension used in fit()

Returns:

  • states (np.ndarray): Most likely state sequence of shape (n_samples,)

    • Values are integers in range [0, n_states-1]

Algorithm: Viterbi

The Viterbi algorithm finds the globally optimal state sequence:

  1. Initialization: \(\delta_1(i) = \pi_i \cdot b_i(O_1)\)

  2. Recursion: \(\delta_t(j) = \max_i[\delta_{t-1}(i) \cdot a_{ij}] \cdot b_j(O_t)\)

  3. Termination: \(P^* = \max_i[\delta_T(i)]\)

  4. Backtracking: Trace back from \(\arg\max_i[\delta_T(i)]\) to recover state sequence

Complexity: \(O(T \cdot K^2)\) where \(T\) is sequence length, \(K\) is number of states

Example:

# After training (see fit() example)
states = hmm.predict(data)

# Analyze regime distribution
unique, counts = np.unique(states, return_counts=True)
for state, count in zip(unique, counts):
    print(f"State {state}: {count} samples ({count/len(states)*100:.1f}%)")

# Identify regime switches
switches = np.where(np.diff(states) != 0)[0]
print(f"Number of regime switches: {len(switches)}")

# Use for trading: buy in regime 0, sell in regime 1
current_state = states[-1]
if current_state == 0:
    print("Signal: BUY (low volatility regime)")
else:
    print("Signal: SELL (high volatility regime)")

Use cases:

  • Regime detection: Identify market states (bull/bear, high/low vol)

  • Trading signals: Generate buy/sell signals based on regime

  • Risk management: Adjust position size based on estimated regime

  • Anomaly detection: Flag unusual regime transitions

score(X)

Computes the log-likelihood of the observation sequence under the fitted model.

Parameters:

  • X (np.ndarray): Observation sequence of shape (n_samples, n_features)

Returns:

  • logp (float): Log-likelihood \(\log P(O \mid \lambda)\)

Algorithm: Forward Algorithm

The forward algorithm efficiently computes the likelihood:

  1. Initialization: \(\alpha_1(i) = \pi_i \cdot b_i(O_1)\)

  2. Induction: \(\alpha_t(j) = \left[\sum_{i=1}^K \alpha_{t-1}(i) \cdot a_{ij}\right] \cdot b_j(O_t)\)

  3. Termination: \(P(O \mid \lambda) = \sum_{i=1}^K \alpha_T(i)\)

Numerical stability: Uses log-space computation with scaling to avoid underflow.

Example:

# Model comparison: which number of states fits best?
logp_scores = {}
for n_states in [2, 3, 4]:
    hmm = HMM(n_states=n_states)
    hmm.fit(data, n_iterations=100)
    logp = hmm.score(data)
    logp_scores[n_states] = logp
    print(f"{n_states} states: log-likelihood = {logp:.2f}")

# Higher log-likelihood is better (but watch for overfitting)
best_k = max(logp_scores, key=logp_scores.get)
print(f"Best model: {best_k} states")

# Use BIC for model selection (penalizes complexity)
def bic(logp, n_params, n_samples):
    return -2 * logp + n_params * np.log(n_samples)

n_samples = len(data)
for n_states in [2, 3, 4]:
    n_params = n_states**2 + 2*n_states  # Approx: A, pi, means, variances
    bic_score = bic(logp_scores[n_states], n_params, n_samples)
    print(f"{n_states} states: BIC = {bic_score:.2f}")

Use cases:

  • Model selection: Compare models with different n_states using BIC/AIC

  • Convergence monitoring: Track log-likelihood during training

  • Outlier detection: Low likelihood → data doesn’t match model

  • Model reliability: Higher likelihood → better fit (but watch overfitting)

Complete Example: Market Regime Detection

Here’s a complete workflow for detecting market regimes in financial data:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from optimizr import HMM

# 1. Load financial data (example: S&P 500 returns)
# In practice, load from your data source
np.random.seed(42)
n_samples = 2000

# Simulate returns with regime changes
returns = []
for i in range(n_samples):
    if i < 500:  # Bull market
        returns.append(np.random.normal(0.001, 0.01))
    elif i < 1000:  # Correction
        returns.append(np.random.normal(-0.002, 0.02))
    elif i < 1500:  # Recovery
        returns.append(np.random.normal(0.001, 0.015))
    else:  # Bear market
        returns.append(np.random.normal(-0.001, 0.025))

returns = np.array(returns).reshape(-1, 1)

# 2. Train HMM with multiple initializations
print("Training HMM...")
hmm = HMM(n_states=3)  # 3 regimes: bull, neutral, bear
hmm.fit(returns, n_iterations=200, tolerance=1e-6, n_init=10)

# 3. Decode regimes
states = hmm.predict(returns)

# 4. Analyze regimes
print("\nRegime Statistics:")
for state_id in range(3):
    mask = (states == state_id)
    state_returns = returns[mask]
    mean_ret = np.mean(state_returns)
    std_ret = np.std(state_returns)
    count = np.sum(mask)
    print(f"State {state_id}:")
    print(f"  Count: {count} ({count/len(returns)*100:.1f}%)")
    print(f"  Mean return: {mean_ret:.4f}")
    print(f"  Volatility: {std_ret:.4f}")
    print(f"  Sharpe (annualized): {mean_ret/std_ret * np.sqrt(252):.2f}")

# 5. Identify regime switches
switches = np.where(np.diff(states) != 0)[0] + 1
print(f"\nRegime switches: {len(switches)}")
print(f"Average regime duration: {len(returns)/len(switches):.1f} days")

# 6. Visualize regimes
plt.figure(figsize=(14, 8))

# Plot returns with regime colors
plt.subplot(3, 1, 1)
colors = ['green', 'yellow', 'red']
for state_id in range(3):
    mask = (states == state_id)
    plt.scatter(np.where(mask)[0], returns[mask], 
                c=colors[state_id], alpha=0.5, s=10,
                label=f'State {state_id}')
plt.ylabel('Returns')
plt.title('Returns colored by HMM regime')
plt.legend()
plt.grid(True, alpha=0.3)

# Plot cumulative returns per regime
plt.subplot(3, 1, 2)
cumulative = np.cumsum(returns.flatten())
plt.plot(cumulative, color='black', linewidth=1)
for switch in switches:
    plt.axvline(switch, color='red', alpha=0.3, linestyle='--')
plt.ylabel('Cumulative Returns')
plt.title('Cumulative returns with regime switches')
plt.grid(True, alpha=0.3)

# Plot state sequence
plt.subplot(3, 1, 3)
plt.plot(states, linewidth=0.5)
plt.ylabel('State')
plt.xlabel('Time')
plt.title('Decoded state sequence')
plt.yticks(range(3))
plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('hmm_regime_detection.png', dpi=150)
print("\nPlot saved to hmm_regime_detection.png")

# 7. Generate trading signals
current_state = states[-1]
state_returns = returns[states == current_state]
expected_return = np.mean(state_returns)
expected_vol = np.std(state_returns)

print(f"\nCurrent regime: State {current_state}")
print(f"Expected return: {expected_return:.4f}")
print(f"Expected volatility: {expected_vol:.4f}")

if expected_return > 0.0005:
    signal = "BUY"
    position_size = 1.0
elif expected_return < -0.0005:
    signal = "SELL"
    position_size = 0.0
else:
    signal = "HOLD"
    position_size = 0.5

print(f"Trading signal: {signal}")
print(f"Recommended position size: {position_size*100:.0f}%")

Advanced Usage

Model Selection with BIC

Choose the optimal number of states using Bayesian Information Criterion:

from optimizr import HMM
import numpy as np

def bic_score(hmm, X):
    """Compute BIC for HMM: BIC = -2*log(L) + k*log(n)"""
    logp = hmm.score(X)
    n_states = hmm.n_states  # Assuming this attribute exists
    n_features = X.shape[1]
    # Parameters: transition matrix + initial prob + means + covariances
    k = n_states**2 + n_states + n_states*n_features + n_states*n_features**2
    n = X.shape[0]
    return -2*logp + k*np.log(n)

# Test different numbers of states
results = []
for n_states in range(2, 6):
    hmm = HMM(n_states=n_states)
    hmm.fit(data, n_iterations=100, n_init=5)
    bic = bic_score(hmm, data)
    logp = hmm.score(data)
    results.append((n_states, logp, bic))
    print(f"{n_states} states: log-likelihood={logp:.2f}, BIC={bic:.2f}")

# Best model has lowest BIC
best_n_states = min(results, key=lambda x: x[2])[0]
print(f"\nBest model: {best_n_states} states")

Integration with Optimal Control

Combine HMM regime detection with regime-specific optimal control:

from optimizr import HMM, estimate_ou_params_py, solve_hjb_py

# 1. Detect regimes with HMM
returns = np.diff(spread)
hmm = HMM(n_states=2)
hmm.fit(returns.reshape(-1, 1), n_iterations=100)
regimes = hmm.predict(returns.reshape(-1, 1))

# 2. Estimate OU parameters per regime
thresholds = {}
for regime_id in range(2):
    mask = (regimes == regime_id)
    spread_regime = spread[1:][mask]  # Align with returns
    
    # Estimate OU parameters
    kappa, theta, sigma, half_life = estimate_ou_params_py(
        spread_regime, dt=1/252
    )
    
    # Solve HJB for regime-specific thresholds
    lower, upper, _, _ = solve_hjb_py(
        kappa=kappa, theta=theta, sigma=sigma,
        rho=0.04, transaction_cost=0.001
    )
    
    thresholds[regime_id] = (lower, upper)
    print(f"Regime {regime_id}: κ={kappa:.2f}, thresholds=({lower:.3f}, {upper:.3f})")

# 3. Apply regime-aware trading
current_regime = regimes[-1]
lower, upper = thresholds[current_regime]
current_spread = spread[-1]

if current_spread < lower:
    action = "BUY"
elif current_spread > upper:
    action = "SELL"
else:
    action = "HOLD"

print(f"\nCurrent regime: {current_regime}")
print(f"Current spread: {current_spread:.3f}")
print(f"Thresholds: ({lower:.3f}, {upper:.3f})")
print(f"Action: {action}")

Multivariate HMM

For multiple features (e.g., returns + volume + volatility):

# Prepare multivariate data
returns = np.random.randn(1000, 1)
volume = np.random.randn(1000, 1)
volatility = np.random.randn(1000, 1)

# Stack features
X = np.hstack([returns, volume, volatility])  # Shape: (1000, 3)

# Train multivariate HMM
hmm = HMM(n_states=3)
hmm.fit(X, n_iterations=150)

# Decode regimes based on all features
states = hmm.predict(X)

# Each state now captures joint patterns in returns, volume, and volatility

Best Practices

Data Preparation

  1. Scaling: Standardize features to similar scales

    from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

  2. Stationarity: Ensure time series is stationary (use returns, not prices)

    returns = np.diff(np.log(prices))  # Log returns

  3. Outlier handling: Winsorize extreme values

    from scipy.stats import mstats
X_winsorized = mstats.winsorize(X, limits=[0.01, 0.01])

Model Training

  1. Multiple initializations: Use n_init=5-10 to avoid local minima

  2. Convergence: Monitor log-likelihood, ensure convergence before max_iter

  3. Validation: Use held-out data to verify generalization

Parameter Selection

  1. Number of states: Start with 2-3, increase if necessary

  2. Iterations: 100-200 typically sufficient

  3. Tolerance: 1e-6 for production, 1e-4 for quick experimentation

Practical Tips

  1. Minimum data: Use at least 500 samples per state (1000+ for 2-state model)

  2. Regime persistence: Check average regime duration is meaningful (not too short)

  3. Physical interpretation: Verify decoded regimes make sense (e.g., high-vol state has higher variance)

  4. Robustness: Test on multiple time periods, verify stability

Troubleshooting

Model not converging

  • Symptom: Log-likelihood oscillating or not improving

  • Fix: Increase n_iterations; try different n_init; check data scaling

All samples assigned to one state

  • Symptom: predict() returns all 0s or all 1s

  • Fix: Reduce n_states; check data has sufficient variation; verify stationarity

Unrealistic regime switches

  • Symptom: State changes every few samples

  • Fix: Add transition probability constraints (requires model extension); increase minimum regime duration

Poor out-of-sample performance

  • Symptom: High in-sample log-likelihood but poor predictions on new data

  • Fix: Reduce n_states (overfitting); use cross-validation; add regularization

Performance Characteristics

Computational Complexity

  • Training (Baum-Welch): \(O(I \cdot T \cdot K^2)\)

    • \(I\): number of iterations (~100-200)

    • \(T\): sequence length

    • \(K\): number of states

  • Prediction (Viterbi): \(O(T \cdot K^2)\)

  • Scoring (Forward): \(O(T \cdot K^2)\)

Memory Requirements

  • Model parameters: \(O(K^2 + K \cdot d^2)\) where \(d\) is feature dimension

  • Forward/backward matrices: \(O(T \cdot K)\)

Typical Runtimes (on modern CPU)

  • Train (1000 samples, 2 states, 100 iter): ~50-100ms

  • Train (10000 samples, 3 states, 200 iter): ~500ms-1s

  • Predict (1000 samples, 2 states): ~5-10ms

  • Score (1000 samples, 2 states): ~5-10ms

References

Hidden Markov Models

  • Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2), 257-286.

  • Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective. MIT Press. (Chapter 17: Markov and hidden Markov models)

Financial Applications

  • Guidolin, M., & Timmermann, A. (2008). International asset allocation under regime switching, skew, and kurtosis preferences. The Review of Financial Studies, 21(2), 889-935.

  • Nystrup, P., Madsen, H., & Lindström, E. (2015). Stylised facts of financial time series and hidden Markov models in continuous time. Quantitative Finance, 15(9), 1531-1541.

  • Ang, A., & Bekaert, G. (2002). Regime switches in interest rates. Journal of Business & Economic Statistics, 20(2), 163-182.

Algorithms

  • Forney, G. D. (1973). The Viterbi algorithm. Proceedings of the IEEE, 61(3), 268-278.

  • Baum, L. E., Petrie, T., Soules, G., & Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. The Annals of Mathematical Statistics, 41(1), 164-171.

See Also