Bayesian STS - Bayesian Structural Time Series

Python: NimbusSTS | Julia: Coming Soon
Mathematical Model: State-Space Model with Extended Kalman Filter (EKF) Bayesian STS is a stateful Bayesian classification model designed for non-stationary BCI data. It combines feature-based classification with latent state dynamics, allowing it to adapt to temporal drift, electrode changes, and long-session fatigue.

Available in Python SDK:

Python SDK: NimbusSTS class (sklearn-compatible with state management)
Julia SDK: Coming in future release

This is the only model in the SDK that explicitly handles temporal dynamics and non-stationary distributions.

Overview

Bayesian STS extends beyond traditional static classifiers by modeling latent state evolution over time:

✅ Temporal state dynamics with Extended Kalman Filter
✅ Drift adaptation for non-stationary data
✅ State management API for explicit time propagation
✅ Online learning with delayed feedback support
✅ Cross-session transfer with state persistence
✅ Uncertainty quantification for predictions and states
✅ Fast inference (~20-30ms per trial)

Quick Start

Python

from nimbus_bci import NimbusSTS
import numpy as np

# Create and fit classifier
clf = NimbusSTS(transition_cov=0.01, num_steps=100)
clf.fit(X_train, y_train)

# Stateful prediction with time propagation
for x_new in streaming_data:
    clf.propagate_state()  # Advance time
    prediction = clf.predict(x_new.reshape(1, -1))
    
    # After feedback arrives
    clf.partial_fit(x_new.reshape(1, -1), [true_label])

# Inspect and save state for next session
z_mean, z_cov = clf.get_latent_state()

When to Use Bayesian STS

Bayesian STS is ideal for:

Non-stationary data with temporal drift
Long BCI sessions (>30 minutes) with fatigue effects
Cross-day experiments with electrode position changes
Adaptive BCI systems with delayed feedback
Online learning scenarios with continuous adaptation
Environments with changing noise characteristics

Use Bayesian LDA or Bayesian GMM instead if:

Data is stationary (class distributions don’t change over time)
Sessions are short (<10 minutes)
You need the absolute fastest inference (<15ms)
Complexity of state management is not warranted

Model Architecture

Mathematical Foundation (State-Space Model)

Bayesian STS implements a state-space model with Extended Kalman Filter inference: Latent Dynamics:

z_t = A @ z_{t-1} + w_t,  where w_t ~ N(0, Q)

Observation Model:

logits = W @ x_t + H @ z_t + b
p(y=k | x_t, z_t) = softmax(logits)_k

Where:

z_t = latent state at time t (captures temporal patterns like class prior drift)
A = state transition matrix (default: identity for random walk)
Q = process noise covariance (controls drift speed)
W = feature weight matrix
H = state-to-logit projection matrix
x_t = observed features

Key Innovation: The latent state z captures temporal patterns that persist across samples, such as gradual shifts in class priors due to fatigue or electrode drift.

Inference with Extended Kalman Filter

During training, the EKF updates the latent state using observed labels (measurement update). During inference:

propagate_state(): Advance the prior without labels (time update only)
partial_fit(): Update state with new label (measurement update)
predict_proba(): Never mutates state (consistent with sklearn API)

Hyperparameters

Bayesian STS supports configurable hyperparameters for optimal performance: Available Hyperparameters:

Parameter	Type	Default	Range	Description
`state_dim`	int or None	None	[1, n_classes]	Dimension of latent state (default: n_classes - 1)
`w_loc`	float	0.0	[-10, 10]	Prior mean for feature weights
`w_scale`	float	1.0	[0.1, 10]	Prior scale for feature weights
`transition_cov`	float or None	None	[0.001, 1.0]	Process noise covariance Q (drift speed)
`observation_cov`	float	1.0	[0.1, 10]	Observation noise covariance R
`transition_matrix`	ndarray or None	None	(state_dim, state_dim)	State transition matrix A (default: identity)
`learning_rate`	float	0.1	[0.01, 1.0]	Step size for parameter updates
`num_steps`	int	50	[20, 200]	Number of learning iterations
`rng_seed`	int	0	any	Random seed for reproducibility

Critical Parameter: transition_cov (Q) This controls how fast the latent state can drift:

0.001: Very slow drift (multi-day stability)
- Use for: Short sessions, stable recording conditions
- Example: 10-minute calibration sessions
0.01: Moderate drift (within-session adaptation)
- Use for: Standard BCI sessions (30-60 minutes)
- Example: Motor imagery with gradual fatigue
0.1: Fast drift (rapid environmental changes)
- Use for: Highly non-stationary environments
- Example: Mobile BCI, changing electrode impedance

Rule of thumb: Set to 1% of expected signal variance. If None, auto-estimated from data.

Model Structure

The NimbusSTS classifier maintains:

Feature weights W (learned during training)
State projection H (learned during training)
Current state mean z_mean (updated online)
Current state covariance z_cov (updated online)
Initial state z_mean_init, z_cov_init (for reset)

Usage

1. Basic Training and Prediction

Python

from nimbus_bci import NimbusSTS
import numpy as np

# Generate sample data (in practice, use real EEG features)
np.random.seed(42)
n_samples, n_features = 100, 16
X_train = np.random.randn(n_samples, n_features)
y_train = np.random.randint(0, 4, n_samples)

# Train with moderate drift tracking
clf = NimbusSTS(
    transition_cov=0.05,  # Moderate drift
    num_steps=100,
    verbose=True
)
clf.fit(X_train, y_train)

# Standard prediction (each sample treated independently)
predictions = clf.predict(X_test)
probabilities = clf.predict_proba(X_test)

print(f"Predictions: {predictions}")
print(f"Probabilities shape: {probabilities.shape}")

Important: predict() and predict_proba() never mutate the state (sklearn API compatibility). For time-ordered evaluation, use propagate_state() explicitly.

2. Stateful Prediction with Time Propagation

For time-ordered streaming data, explicitly propagate state between samples:

Python

from nimbus_bci import NimbusSTS
import numpy as np

# Train model
clf = NimbusSTS(transition_cov=0.05)
clf.fit(X_train, y_train)

# Streaming prediction with state propagation
predictions = []
for x_t in X_stream:  # Time-ordered samples
    # Advance state one step (EKF time update)
    clf.propagate_state(n_steps=1)
    
    # Predict using current state
    pred = clf.predict(x_t.reshape(1, -1))[0]
    predictions.append(pred)
    
    print(f"Prediction: {pred}")

# Compare with vs without state propagation
acc_with_state = np.mean(predictions == y_stream)
acc_without_state = clf.score(X_stream, y_stream)  # Treats samples independently

print(f"Accuracy with state propagation: {acc_with_state:.2%}")
print(f"Accuracy without state propagation: {acc_without_state:.2%}")

3. Online Learning with Delayed Feedback

The canonical BCI paradigm: predict → user acts → receive feedback → update

Python

from nimbus_bci import NimbusSTS

# Initial training (calibration phase)
clf = NimbusSTS(transition_cov=0.05, learning_rate=0.1)
clf.fit(X_calibration, y_calibration)

# Online session with delayed feedback
for trial in online_session:
    # 1. Advance time (no measurement)
    clf.propagate_state()
    
    # 2. Predict using current state
    prediction = clf.predict(trial.features.reshape(1, -1))[0]
    
    # 3. User performs action based on prediction
    execute_action(prediction)
    
    # 4. Get feedback (true label) after user completes action
    true_label = wait_for_feedback()
    
    # 5. Update state with measurement (EKF update)
    clf.partial_fit(trial.features.reshape(1, -1), [true_label])
    
    print(f"Trial: pred={prediction}, true={true_label}")

Why this matters: In real BCI, labels aren’t available at prediction time. The model must predict using only the prior, then update when feedback arrives.

4. State Inspection and Transfer

Save and restore latent state across sessions:

Python

from nimbus_bci import NimbusSTS
import pickle

# Day 1: Train and save state
clf_day1 = NimbusSTS()
clf_day1.fit(X_day1, y_day1)

# Get final state
z_final, P_final = clf_day1.get_latent_state()
print(f"Final state mean: {z_final}")
print(f"Final state uncertainty: {np.diag(P_final)}")

# Save model and state
with open("model_day1.pkl", "wb") as f:
    pickle.dump({'model': clf_day1, 'state': (z_final, P_final)}, f)

# Day 2: Transfer state with increased uncertainty
with open("model_day1.pkl", "rb") as f:
    saved = pickle.load(f)

clf_day2 = saved['model']

# Quick calibration (just a few trials)
clf_day2.fit(X_day2_calib, y_day2_calib)

# Transfer previous state (with decay)
z_prior, P_prior = saved['state']
clf_day2.set_latent_state(
    z_mean=z_prior * 0.5,  # Decay mean toward 0
    z_cov=P_prior * 2.0    # Increase uncertainty
)

# Use model with transferred state
predictions = clf_day2.predict(X_day2_test)

Use case: Cross-day transfer learning. Start with informed prior from previous session, but increase uncertainty to allow adaptation.

5. State Reset and Management

Reset state to initial values from training:

Python

from nimbus_bci import NimbusSTS

clf = NimbusSTS()
clf.fit(X_train, y_train)

# Run one session
for _ in range(100):
    clf.propagate_state()
    # ... predictions ...

# Reset for new session (clean slate)
clf.reset_state()

# State is now back to initial values
z_reset, P_reset = clf.get_latent_state()
print(f"Reset state mean: {z_reset}")  # Near zero
print(f"Reset state cov diagonal: {np.diag(P_reset)}")  # Identity

6. Batch Inference

Standard sklearn-compatible batch inference:

Python

from nimbus_bci import NimbusSTS
import numpy as np

clf = NimbusSTS()
clf.fit(X_train, y_train)

# Batch inference (samples treated as independent)
predictions = clf.predict(X_test)
probabilities = clf.predict_proba(X_test)
confidences = np.max(probabilities, axis=1)

# Calculate accuracy
accuracy = np.mean(predictions == y_test)
print(f"Accuracy: {accuracy * 100:.1f}%")
print(f"Mean confidence: {np.mean(confidences):.3f}")

7. Streaming Inference with StreamingSessionSTS

Real-time chunk-by-chunk processing with state management:

For detailed streaming examples, see Python SDK Streaming Inference.

Python

from nimbus_bci import NimbusSTS
from nimbus_bci.inference import StreamingSessionSTS
from nimbus_bci.data import BCIMetadata

# Train model
clf = NimbusSTS(transition_cov=0.05)
clf.fit(X_train, y_train)

# Setup streaming session
metadata = BCIMetadata(
    sampling_rate=250.0,
    paradigm="motor_imagery",
    feature_type="csp",
    n_features=16,
    n_classes=4,
    chunk_size=125,  # 500ms chunks
    temporal_aggregation="logvar"
)

session = StreamingSessionSTS(clf, metadata)

# Process chunks with automatic state propagation
for chunk in eeg_stream:  # (n_features, chunk_size)
    result = session.process_chunk(chunk)
    print(f"Chunk: pred={result.prediction}, conf={result.confidence:.3f}")

# Finalize trial
final_result = session.finalize_trial()
print(f"Final: pred={final_result.prediction}")

# Reset for next trial
session.reset()

Key feature: StreamingSessionSTS automatically calls propagate_state() between chunks, properly handling temporal dynamics.

Hyperparameter Tuning

Fine-tune NimbusSTS for your specific drift characteristics.

When to Tune Hyperparameters

Consider tuning when:

Default performance is unsatisfactory on non-stationary data
You observe significant drift or fatigue effects
You need to balance adaptation speed vs stability
Cross-session performance is poor

Tuning `transition_cov` (Critical Parameter)

The process noise covariance controls drift speed:

For Stable, Short Sessions

from nimbus_bci import NimbusSTS

# Minimal drift (short sessions, stable conditions)
clf = NimbusSTS(
    transition_cov=0.001,  # Very slow drift
    num_steps=50
)
clf.fit(X_train, y_train)

Use when:

Session duration < 15 minutes
Excellent electrode stability
Controlled lab environment
Minimal user fatigue

For Standard Sessions with Gradual Drift

from nimbus_bci import NimbusSTS

# Moderate drift (typical BCI session)
clf = NimbusSTS(
    transition_cov=0.01,  # Moderate drift (default behavior)
    num_steps=100,
    learning_rate=0.1
)
clf.fit(X_train, y_train)

Use when:

Session duration 30-60 minutes
Standard BCI recording conditions
Gradual fatigue or attention changes
This is the recommended starting point

For Highly Non-Stationary Environments

from nimbus_bci import NimbusSTS

# Fast drift (rapid environmental changes)
clf = NimbusSTS(
    transition_cov=0.1,  # Fast drift
    observation_cov=2.0,  # Increase measurement noise tolerance
    num_steps=100,
    learning_rate=0.2  # Faster adaptation
)
clf.fit(X_train, y_train)

Use when:

Mobile BCI or changing environments
Electrode impedance changes during session
Rapid user state changes
Real-world deployment scenarios

Auto-Estimation

If unsure, let the model estimate transition_cov from data:

from nimbus_bci import NimbusSTS

# Auto-estimate process noise
clf = NimbusSTS(
    transition_cov=None,  # Auto-estimate
    num_steps=100
)
clf.fit(X_train, y_train)

# Check estimated value
print(f"Estimated transition_cov: {clf.model_.params['Q'][0, 0]:.4f}")

Hyperparameter Search Example

Systematically search for optimal hyperparameters:

from sklearn.model_selection import GridSearchCV
from nimbus_bci import NimbusSTS

# Define parameter grid
param_grid = {
    'transition_cov': [0.001, 0.01, 0.05, 0.1],
    'learning_rate': [0.05, 0.1, 0.2],
    'num_steps': [50, 100]
}

# Grid search
grid = GridSearchCV(
    NimbusSTS(),
    param_grid,
    cv=3,  # Use fewer folds (STS is stateful)
    scoring='accuracy',
    n_jobs=-1,
    verbose=1
)

grid.fit(X_train, y_train)

print(f"\nBest hyperparameters:")
print(f"  transition_cov: {grid.best_params_['transition_cov']}")
print(f"  learning_rate: {grid.best_params_['learning_rate']}")
print(f"  num_steps: {grid.best_params_['num_steps']}")
print(f"  CV accuracy: {grid.best_score_*100:.1f}%")

# Use best model
best_clf = grid.best_estimator_

Quick Tuning Guidelines

Scenario	`transition_cov`	`learning_rate`	`num_steps`	Notes
Short, stable sessions	0.001	0.1	50	Minimal drift
Standard BCI (30-60 min)	0.01-0.05	0.1	100	Recommended default
Long sessions (>1 hour)	0.05-0.1	0.1-0.2	100	Allow more adaptation
Mobile/real-world	0.1-0.5	0.2	100-200	Fast adaptation
Cross-day transfer	0.01	0.05	50	Start conservative

Pro Tip: Start with transition_cov=0.01 and num_steps=100. If you observe drift (accuracy degrades over time), increase transition_cov. If predictions are too noisy, decrease it.

Training Requirements

Data Requirements

Minimum: 40 trials per class
Recommended: 80+ trials per class for stable initialization
For fine-tuning: 10-20 trials with partial_fit()

NimbusSTS requires at least 2 samples to initialize state statistics. Training will raise an error if any class has fewer than 2 samples.

Feature Normalization

Critical for STS models!Normalization is even more important for NimbusSTS than static models, as drift can amplify scale differences.

from sklearn.preprocessing import StandardScaler
import pickle

# Estimate normalization from training data
scaler = StandardScaler()
X_train_norm = scaler.fit_transform(X_train)

# Train on normalized features
clf = NimbusSTS()
clf.fit(X_train_norm, y_train)

# Save scaler with model
with open("model_with_scaler.pkl", "wb") as f:
    pickle.dump({'model': clf, 'scaler': scaler}, f)

# Always apply same normalization
X_test_norm = scaler.transform(X_test)
predictions = clf.predict(X_test_norm)

See Feature Normalization for complete details.

Feature Requirements

NimbusSTS expects preprocessed features, not raw EEG: ✅ Required preprocessing:

Bandpass filtering (paradigm-specific)
Artifact removal (ICA recommended)
Spatial filtering (CSP for motor imagery)
Feature extraction (log-variance, bandpower, etc.)
Temporal aggregation (for batch training)

❌ NOT accepted:

Raw EEG channels
Unfiltered data

See Preprocessing Requirements.

Performance Characteristics

Computational Performance

Operation	Latency	Notes
Training	15-40 seconds	100 iterations, 100 trials per class
Batch Inference	20-30ms per trial	Slightly slower than static models
propagate_state()	<1ms	Fast time update
partial_fit()	5-15ms	Online EKF update
Streaming Chunk	20-30ms	Includes state propagation

All measurements on standard CPU (no GPU required).

Classification Accuracy

Scenario	Typical Accuracy	Comparison to Static Models
Stationary data (short sessions)	70-85%	Similar to NimbusLDA/GMM
Non-stationary data (long sessions)	75-90%	+5-15% vs static models
Cross-day with state transfer	65-80%	+10-20% vs training from scratch
Online adaptation	Improves over time	Adapts to user state changes

Key Insight: NimbusSTS shines on non-stationary data where static models degrade over time. For short, stationary sessions, use NimbusLDA for faster inference.

Latency Trade-offs

NimbusLDA:     ~10-15ms  ✓ Fastest
NimbusGMM:     ~15-25ms  ✓ Fast
NimbusSTS:     ~20-30ms  ✓ Still real-time capable
NimbusSoftmax: ~15-25ms  ✓ Fast

The ~5-10ms overhead of NimbusSTS is worth it for non-stationary scenarios where static models would degrade.

Model Inspection

View Current State

import numpy as np

# Get latent state
z_mean, z_cov = clf.get_latent_state()

print("Latent state:")
print(f"  Mean: {z_mean}")
print(f"  Covariance diagonal: {np.diag(z_cov)}")
print(f"  Uncertainty (trace): {np.trace(z_cov):.3f}")

Monitor State Evolution

from nimbus_bci import NimbusSTS
import matplotlib.pyplot as plt

clf = NimbusSTS(transition_cov=0.05)
clf.fit(X_train, y_train)

# Track state over time
z_history = []
uncertainty_history = []

for x_t in X_stream:
    clf.propagate_state()
    z_mean, z_cov = clf.get_latent_state()
    
    z_history.append(z_mean.copy())
    uncertainty_history.append(np.trace(z_cov))
    
    pred = clf.predict(x_t.reshape(1, -1))

# Plot state evolution
z_history = np.array(z_history)
plt.figure(figsize=(12, 4))

plt.subplot(1, 2, 1)
plt.plot(z_history)
plt.xlabel('Time (trials)')
plt.ylabel('Latent State')
plt.title('State Evolution Over Time')
plt.grid(True)

plt.subplot(1, 2, 2)
plt.plot(uncertainty_history)
plt.xlabel('Time (trials)')
plt.ylabel('Uncertainty (trace of cov)')
plt.title('State Uncertainty Over Time')
plt.grid(True)

plt.tight_layout()
plt.show()

View Model Parameters

# Feature weights (W)
W = clf.model_.params['W']
print(f"Feature weights shape: {W.shape}")  # (n_classes, n_features)

# State projection (H)
H = clf.model_.params['H']
print(f"State projection shape: {H.shape}")  # (n_classes, state_dim)

# Transition matrix (A)
A = clf.model_.params['A']
print(f"Transition matrix:\n{A}")

# Process noise (Q)
Q = clf.model_.params['Q']
print(f"Process noise covariance:\n{Q}")

Advantages & Limitations

Advantages

✅ Handles Non-Stationarity: Explicitly models temporal drift
✅ Adaptive: Continuously learns from feedback
✅ Cross-Session Transfer: State persistence across days
✅ Uncertainty Quantification: For both predictions and states
✅ Delayed Feedback Support: Natural for BCI paradigms
✅ Production-Ready: Real-time capable with <30ms latency
✅ sklearn-Compatible: Works with pipelines and CV

Limitations

❌ More Complex API: State management requires careful usage
❌ Slightly Slower: 5-10ms overhead vs static models
❌ Requires More Tuning: transition_cov is critical
❌ Not Ideal for Stationary Data: Use NimbusLDA if data is stable
❌ Memory: Maintains state history (minimal overhead)

Comparison: NimbusSTS vs Static Models

Aspect	NimbusSTS (Stateful)	NimbusLDA/GMM/Softmax (Stateless)
Temporal Modeling	✅ Explicit state dynamics	❌ No temporal modeling
Non-Stationarity	✅ Adapts to drift	❌ Degrades over time
State Management	✅ Explicit API	❌ Not applicable
Inference Speed	~20-30ms	~10-25ms (faster)
API Complexity	Higher	Lower
Best For	Long sessions, cross-day	Short sessions, stable data
Online Learning	✅ Natural with delayed feedback	✅ Supported but stateless
Cross-Session Transfer	✅ State persistence	❌ Retrain from scratch

Decision Tree

Is your data non-stationary (drift over time)?
├─ Yes → How long are your sessions?
│   ├─ >30 min → NimbusSTS (essential)
│   └─ &lt;30 min → Try NimbusSTS, compare with static models
└─ No → Do classes have different covariances?
    ├─ Yes → NimbusGMM
    └─ No → NimbusLDA (fastest)

Rule of thumb: If accuracy degrades >10% from start to end of session, use NimbusSTS.

Practical Examples

Example 1: Detecting and Adapting to Drift

from nimbus_bci import NimbusSTS, NimbusLDA
import numpy as np

# Compare static vs adaptive on drifting data
def generate_drifting_data(n_samples, n_features, drift_rate=0.05):
    X, y = [], []
    means = np.random.randn(4, n_features) * 2.0
    
    for t in range(n_samples):
        means += np.random.randn(4, n_features) * drift_rate  # Drift!
        label = t % 4
        x = np.random.randn(n_features) + means[label]
        X.append(x)
        y.append(label)
    
    return np.array(X), np.array(y)

# Generate non-stationary data
X_train, y_train = generate_drifting_data(200, 16, drift_rate=0.03)
X_test, y_test = generate_drifting_data(100, 16, drift_rate=0.03)

# Static model
clf_static = NimbusLDA()
clf_static.fit(X_train, y_train)
acc_static = clf_static.score(X_test, y_test)

# Adaptive model
clf_sts = NimbusSTS(transition_cov=0.05, num_steps=100)
clf_sts.fit(X_train, y_train)

# Evaluate with state propagation
preds = []
for x in X_test:
    clf_sts.propagate_state()
    pred = clf_sts.predict(x.reshape(1, -1))[0]
    preds.append(pred)

acc_sts = np.mean(preds == y_test)

print(f"Static model accuracy: {acc_static:.1%}")
print(f"Adaptive model accuracy: {acc_sts:.1%}")
print(f"Improvement: {(acc_sts - acc_static)*100:+.1f}%")

Example 2: Cross-Day Transfer Learning

from nimbus_bci import NimbusSTS
import pickle

# Day 1: Initial session
clf_day1 = NimbusSTS(transition_cov=0.01)
clf_day1.fit(X_day1_full, y_day1_full)  # Full training

# Save state
z_day1, P_day1 = clf_day1.get_latent_state()

# Day 2: Start with minimal calibration
clf_day2 = NimbusSTS(transition_cov=0.01)
clf_day2.fit(X_day2_calib, y_day2_calib)  # Just 20 trials

# Option A: No transfer (baseline)
acc_no_transfer = clf_day2.score(X_day2_test, y_day2_test)

# Option B: Transfer state from day 1
clf_day2.set_latent_state(
    z_mean=z_day1 * 0.7,  # Partial transfer
    z_cov=P_day1 * 1.5    # Increase uncertainty
)
acc_with_transfer = clf_day2.score(X_day2_test, y_day2_test)

print(f"Day 2 accuracy (no transfer): {acc_no_transfer:.1%}")
print(f"Day 2 accuracy (with transfer): {acc_with_transfer:.1%}")
print(f"Improvement: {(acc_with_transfer - acc_no_transfer)*100:+.1f}%")

Example 3: Real-Time Adaptive BCI Loop

from nimbus_bci import NimbusSTS
import time

# Setup
clf = NimbusSTS(transition_cov=0.05, learning_rate=0.1)
clf.fit(X_calibration, y_calibration)

# Real-time loop
accuracy_history = []
window_size = 10

for i, (features, true_label) in enumerate(online_trials):
    # 1. Propagate state forward in time
    clf.propagate_state(n_steps=1)
    
    # 2. Make prediction
    t0 = time.time()
    prediction = clf.predict(features.reshape(1, -1))[0]
    latency_ms = (time.time() - t0) * 1000
    
    # 3. Execute action (not shown)
    execute_bci_command(prediction)
    
    # 4. Get feedback after action completes
    feedback_label = wait_for_user_feedback()
    
    # 5. Update model
    clf.partial_fit(features.reshape(1, -1), [feedback_label])
    
    # Track performance
    correct = (prediction == feedback_label)
    accuracy_history.append(correct)
    
    if len(accuracy_history) > window_size:
        accuracy_history.pop(0)
    
    recent_acc = np.mean(accuracy_history)
    
    print(f"Trial {i+1}: pred={prediction}, true={feedback_label}, "
          f"correct={correct}, recent_acc={recent_acc:.1%}, "
          f"latency={latency_ms:.1f}ms")

Next Steps

Bayesian LDA (RxLDA)

Faster static model for stationary data

Bayesian GMM (RxGMM)

Static model with class-specific covariances

Python SDK Streaming

Real-time streaming with NimbusSTS

Advanced Applications

Complete tutorials and use cases

References

Implementation:

Extended Kalman Filter: https://en.wikipedia.org/wiki/Extended_Kalman_filter
Source code: /nimbus_bci/models/nimbus_sts/ in nimbus-bci

Theory:

Durbin, J., & Koopman, S. J. (2012). “Time Series Analysis by State Space Methods”
Harvey, A. C. (1990). “Forecasting, Structural Time Series Models and the Kalman Filter”
Särkkä, S. (2013). “Bayesian Filtering and Smoothing”

BCI Applications:

Vidaurre, C., et al. (2011). “Co-adaptive calibration to improve BCI efficiency”
Shenoy, P., et al. (2006). “Towards adaptive classification for BCI”
Kirchner, E. A., et al. (2013). “On the applicability of brain reading for predictive human-machine interfaces in robotics”

Getting Started

Core Concepts

Models & Algorithms

Configuration & Setup

Examples

Development

Python SDK

Julia SDK & Cloud API

​Bayesian STS - Bayesian Structural Time Series

​Overview

​Quick Start

​When to Use Bayesian STS

​Model Architecture

​Mathematical Foundation (State-Space Model)

​Inference with Extended Kalman Filter

​Hyperparameters

​Model Structure

​Usage

​1. Basic Training and Prediction

​2. Stateful Prediction with Time Propagation

​3. Online Learning with Delayed Feedback

​4. State Inspection and Transfer

​5. State Reset and Management

​6. Batch Inference

​7. Streaming Inference with StreamingSessionSTS

​Hyperparameter Tuning

​When to Tune Hyperparameters

​Tuning transition_cov (Critical Parameter)

​For Stable, Short Sessions

​For Standard Sessions with Gradual Drift

​For Highly Non-Stationary Environments

​Auto-Estimation

​Hyperparameter Search Example

​Quick Tuning Guidelines

​Training Requirements

​Data Requirements

​Feature Normalization

​Feature Requirements

​Performance Characteristics

​Computational Performance

​Classification Accuracy

​Latency Trade-offs

​Model Inspection

​View Current State

​Monitor State Evolution

​View Model Parameters

​Advantages & Limitations

​Advantages

​Limitations

​Comparison: NimbusSTS vs Static Models

​Decision Tree

​Practical Examples

​Example 1: Detecting and Adapting to Drift

​Example 2: Cross-Day Transfer Learning

​Example 3: Real-Time Adaptive BCI Loop

​Next Steps

Bayesian LDA (RxLDA)

Bayesian GMM (RxGMM)

Python SDK Streaming

Advanced Applications

​References

Bayesian STS - Bayesian Structural Time Series

Overview

Quick Start

When to Use Bayesian STS

Model Architecture

Mathematical Foundation (State-Space Model)

Inference with Extended Kalman Filter

Hyperparameters

Model Structure

Usage

1. Basic Training and Prediction

2. Stateful Prediction with Time Propagation

3. Online Learning with Delayed Feedback

4. State Inspection and Transfer

5. State Reset and Management

6. Batch Inference

7. Streaming Inference with StreamingSessionSTS

Hyperparameter Tuning

When to Tune Hyperparameters

Tuning `transition_cov` (Critical Parameter)

For Stable, Short Sessions

For Standard Sessions with Gradual Drift

For Highly Non-Stationary Environments

Auto-Estimation

Hyperparameter Search Example

Quick Tuning Guidelines

Training Requirements

Data Requirements

Feature Normalization

Feature Requirements

Performance Characteristics

Computational Performance

Classification Accuracy

Latency Trade-offs

Model Inspection

View Current State

Monitor State Evolution

View Model Parameters

Advantages & Limitations

Advantages

Limitations

Comparison: NimbusSTS vs Static Models

Decision Tree

Practical Examples

Example 1: Detecting and Adapting to Drift

Example 2: Cross-Day Transfer Learning

Example 3: Real-Time Adaptive BCI Loop

Next Steps

References