Bayesian QDA (NimbusQDA)

Python: NimbusQDA | Julia: NimbusQDA
Mathematical Model: Heteroscedastic Gaussian Classifier (HGC) NimbusQDA is a Bayesian classification model with class-specific covariance matrices, making it more flexible than NimbusLDA for modeling complex class distributions.

Available in Both SDKs:

Python SDK: NimbusQDA class (sklearn-compatible)
Julia SDK: NimbusQDA (RxInfer.jl-based)

Both implementations provide class-specific covariances for flexible modeling.

Start Here

Quickstart

Start with SDK setup and first inference workflow.

Model Selection

Compare Nimbus models by data characteristics and use case.

Examples

See practical BCI examples for training and inference.

Overview

Bayesian QDA extends beyond traditional Gaussian classifiers by allowing each class to have its own covariance structure: ✅ Class-specific covariances (unlike Bayesian LDA’s shared covariance)
✅ More flexible modeling of heterogeneous distributions
✅ Posterior probability distributions with uncertainty quantification
✅ Fast inference (~15-25ms per trial)
✅ Training and calibration support
✅ Batch and streaming inference modes

Quick Start

Python
Julia

from nimbus_bci import NimbusQDA
import numpy as np

# Create and fit classifier
clf = NimbusQDA(mu_scale=3.0)
clf.fit(X_train, y_train)

# Predict with uncertainty
predictions = clf.predict(X_test)
probabilities = clf.predict_proba(X_test)

# Better for P300 and overlapping distributions

using NimbusSDK

# Train model
model = train_model(
    NimbusQDA,
    train_data;
    iterations=50
)

# Predict
results = predict_batch(model, test_data)

When to Use Bayesian QDA

Bayesian QDA is ideal for:

Complex, overlapping class distributions
Classes with significantly different variances
P300 detection (target/non-target with different spreads)
When Bayesian LDA accuracy is unsatisfactory
When you need maximum flexibility

Use Bayesian LDA instead if:

Classes are well-separated and have similar spreads
Speed is critical (Bayesian LDA is faster)
Training data is limited (Bayesian LDA needs less data)
Memory is constrained (Bayesian LDA uses less memory)

Model Architecture

Mathematical Foundation (Heteroscedastic Gaussian Classifier)

Bayesian QDA implements a Heteroscedastic Gaussian Classifier (HGC), which models class-conditional distributions with class-specific precision matrices:

p(x | y=k) = N(μ_k, W_k^-1)

Where:

μ_k = mean vector for class k
W_k = class-specific precision matrix (different for each class)
Allows different covariance structures per class

Key Difference from Bayesian LDA: Each class can have its own covariance structure, making the model more flexible but also more parameter-heavy.

Hyperparameters

Bayesian QDA supports configurable hyperparameters for optimal performance tuning: Available Hyperparameters (training):

Parameter	Type	Default	Range	Description
`dof_offset`	Int	2	[1, 5]	Degrees of freedom offset for Wishart priors
`mean_prior_precision`	Float64	0.01	[0.001, 0.1]	Prior precision for class means

Parameter Effects:

dof_offset: Controls regularization strength
- Lower values (1) → More data-driven, less regularization
- Higher values (3-5) → More regularization, more conservative
mean_prior_precision: Controls prior strength on class means
- Lower values (0.001) → Weaker prior, trusts data more
- Higher values (0.05-0.1) → Stronger prior, more regularization

Model Structure

struct NimbusQDA <: BCIModel
    mean_posteriors::Vector        # MvNormal posteriors for class means
    precision_posteriors::Vector   # Wishart posteriors for class precisions
    priors::Vector{Float64}        # Empirical class priors
    metadata::ModelMetadata        # Model info
    dof_offset::Int                # Degrees of freedom offset (training)
    mean_prior_precision::Float64  # Mean prior precision (training)
end

RxInfer Implementation

Learning Phase:

@model function RxGMM_learning_model(y, labels, n_features, n_classes)
    # Priors on class means
    for k in 1:n_classes
        m[k] ~ MvNormal(0, 10*I)
    end
    
    # Priors on class-specific precisions
    for k in 1:n_classes
        W[k] ~ Wishart(n_features + 5, I)  # Each class has its own W
    end
    
    # Likelihood
    for i in eachindex(y)
        k = labels[i]
        y[i] ~ MvNormal(m[k], inv(W[k]))  # Class-specific precision
    end
end

Usage

1. Load Pre-trained Model

Python SDK: The Python SDK (nimbus-bci) trains models locally. See Python SDK Quickstart for training examples.

Python
Julia

from nimbus_bci import NimbusQDA
import pickle

# Python SDK: Train locally, no model zoo
clf = NimbusQDA(mu_scale=3.0)
clf.fit(X_train, y_train)

# Save and load
with open("my_p300_gmm.pkl", "wb") as f:
    pickle.dump(clf, f)

with open("my_p300_gmm.pkl", "rb") as f:
    clf = pickle.load(f)

print(f"Model info: {clf.classes_} classes, {clf.n_features_in_} features")

using NimbusSDK

# Authenticate
NimbusSDK.install_core("nbci_live_your_key")

# Load from Nimbus model zoo
model = load_model(NimbusQDA, "p300_gmm_v1")

println("Model loaded:")
println("  Features: $(get_n_features(model))")
println("  Classes: $(get_n_classes(model))")
println("  Paradigm: $(get_paradigm(model))")

2. Train Custom Model

Python
Julia

from nimbus_bci import NimbusQDA
from nimbus_bci.compat import from_mne_epochs
import mne
import pickle

# Load P300 data
raw = mne.io.read_raw_fif("p300_oddball.fif", preload=True)
raw.filter(0.5, 10)

events = mne.find_events(raw)
event_id = {'target': 1, 'non-target': 2}
epochs = mne.Epochs(raw, events, event_id, tmin=-0.2, tmax=0.8, preload=True)

# Extract features
X, y = from_mne_epochs(epochs)

# Train NimbusQDA with default hyperparameters
clf = NimbusQDA()
clf.fit(X, y)

# Or train with custom hyperparameters
clf_tuned = NimbusQDA(
    mu_scale=3.0,        # Prior strength
    sigma_scale=1.0      # Covariance regularization
)
clf_tuned.fit(X, y)

# Save
with open("my_p300_gmm.pkl", "wb") as f:
    pickle.dump(clf_tuned, f)

print(f"Training accuracy: {clf_tuned.score(X, y):.1%}")

using NimbusSDK

# Prepare training data with labels
train_features = erp_features  # (8 × 200 × 150) - preprocessed P300 features
train_labels = [1, 2, 1, 2, ...]  # 150 labels (1=target, 2=non-target)

train_data = BCIData(
    train_features,
    BCIMetadata(
        sampling_rate = 250.0,
        paradigm = :p300,
        feature_type = :erp,
        n_features = 8,
        n_classes = 2,
        chunk_size = nothing
    ),
    train_labels  # Required for training!
)

# Train NimbusQDA model with default hyperparameters
model = train_model(
    NimbusQDA,
    train_data;
    iterations = 50,        # Inference iterations
    showprogress = true,    # Show progress bar
    name = "my_p300_gmm",
    description = "P300 binary classifier with NimbusQDA"
)

# Or train with custom hyperparameters (v0.2.0+)
model = train_model(
    NimbusQDA,
    train_data;
    iterations = 50,
    showprogress = true,
    name = "my_p300_gmm_tuned",
    description = "P300 classifier with tuned hyperparameters",
    dof_offset = 2,                    # DOF offset (default: 2)
    mean_prior_precision = 0.01        # Prior precision (default: 0.01)
)

# Save for later use
save_model(model, "my_p300_gmm.jld2")

Training Parameters:

iterations: Number of variational inference iterations (default: 50)
- More iterations = better convergence, typical range: 50-100
showprogress: Display progress bar during training
name: Model identifier
description: Model description
dof_offset: Degrees of freedom offset (default: 2, range: [1, 5])
mean_prior_precision: Prior precision for means (default: 0.01, range: [0.001, 0.1])

3. Subject-Specific Calibration

Python
Julia

from nimbus_bci import NimbusQDA
import pickle

# Load baseline model
with open("p300_baseline.pkl", "rb") as f:
    base_clf = pickle.load(f)

# Collect calibration trials (10-20 per class)
X_calib, y_calib = collect_calibration_trials()

# Personalize using online learning
personalized_clf = NimbusQDA()
personalized_clf.fit(X_baseline, y_baseline)

for _ in range(10):
    personalized_clf.partial_fit(X_calib, y_calib)

# Save
with open("subject_001_p300_calibrated.pkl", "wb") as f:
    pickle.dump(personalized_clf, f)

# Load base model
base_model = load_model(NimbusQDA, "p300_baseline_v1")

# Collect calibration trials (10-20 per class)
calib_data = BCIData(calib_features, metadata, calib_labels)

# Calibrate model
personalized_model = calibrate_model(
    base_model,
    calib_data;
    iterations = 20  # Fewer iterations needed
)

save_model(personalized_model, "subject_001_p300_calibrated.jld2")

Calibration Benefits:

Requires only 10-20 trials per class
Faster than training from scratch
Adapts to subject-specific characteristics
Hyperparameters preserved: calibrate_model() automatically uses the same hyperparameters as the base model (v0.2.0+)

4. Batch Inference

Python
Julia

import numpy as np

# Run batch inference
predictions = clf.predict(X_test)
probabilities = clf.predict_proba(X_test)
confidences = np.max(probabilities, axis=1)

# Analyze results
print(f"Predictions: {predictions}")
print(f"Mean confidence: {np.mean(confidences):.3f}")

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

# Prepare test data
test_data = BCIData(test_features, metadata, test_labels)

# Run batch inference
results = predict_batch(model, test_data; iterations=10)

# Analyze results
println("Predictions: ", results.predictions)
println("Mean confidence: ", mean(results.confidences))

# Calculate metrics
accuracy = sum(results.predictions .== test_labels) / length(test_labels)
println("Accuracy: $(round(accuracy * 100, digits=1))%")

5. Streaming Inference

For detailed Python streaming examples, see Python SDK Streaming Inference.

Python
Julia

from nimbus_bci import StreamingSession

# Initialize streaming session
session = StreamingSession(clf.model_, metadata_with_chunk_size)

# Process chunks
for chunk in eeg_feature_stream:
    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}, conf={final_result.confidence:.3f}")

session.reset()

# Initialize streaming session
session = init_streaming(model, metadata_with_chunk_size)

# Process chunks
for chunk in eeg_feature_stream
    result = process_chunk(session, chunk; iterations=10)
    println("Chunk: pred=$(result.prediction), conf=$(round(result.confidence, digits=3))")
end

# Finalize trial
final_result = finalize_trial(session; method=:weighted_vote)
println("Final: pred=$(final_result.prediction), conf=$(round(final_result.confidence, digits=3))")

Hyperparameter Tuning (v0.2.0+)

Fine-tune Bayesian QDA for optimal performance on your specific dataset.

When to Tune Hyperparameters

Consider tuning when:

Default performance is unsatisfactory
You have specific data characteristics (very noisy or very clean)
You have limited or extensive training data
Working with complex, overlapping class distributions
P300 or other paradigms with heterogeneous class variances

Tuning Strategies

For High SNR / Clean Data / Many Trials

Use lower regularization to let the data drive the model:

Python
Julia

from nimbus_bci import NimbusQDA

# Lower regularization for clean data
clf = NimbusQDA(
    mu_scale=1.0,      # Weaker prior
    sigma_scale=0.1    # Less covariance regularization
)
clf.fit(X_train, y_train)

model = train_model(
    NimbusQDA,
    train_data;
    iterations = 50,
    dof_offset = 1,                    # Less regularization
    mean_prior_precision = 0.001       # Weaker prior, trust data more
)

Use when:

SNR > 5 dB
100+ trials per class
Clean, artifact-free data
Well-controlled experimental conditions

For Low SNR / Noisy Data / Few Trials

Use higher regularization for stability (especially important for GMM with class-specific covariances):

Python
Julia

from nimbus_bci import NimbusQDA

# Higher regularization for noisy data
clf = NimbusQDA(
    mu_scale=5.0,      # Stronger prior
    sigma_scale=10.0   # More covariance regularization
)
clf.fit(X_train, y_train)

model = train_model(
    NimbusQDA,
    train_data;
    iterations = 50,
    dof_offset = 3,                    # More regularization
    mean_prior_precision = 0.05        # Stronger prior
)

Use when:

SNR < 2 dB
40-80 trials per class
Noisy data or limited artifact removal
Challenging recording conditions
Risk of overfitting to class-specific noise

Balanced / Default Settings

The defaults work well for most scenarios:

model = train_model(
    NimbusQDA,
    train_data;
    iterations = 50,
    dof_offset = 2,                    # Balanced (default)
    mean_prior_precision = 0.01        # Balanced (default)
)

Use when:

Moderate SNR (2-5 dB)
80-150 trials per class
Standard BCI recording conditions
Starting point for experimentation

P300-Specific Tuning

For P300 paradigms where target/non-target classes have different variances:

# P300 often benefits from GMM's class-specific covariances
# Use moderate regularization to avoid overfitting
model = train_model(
    NimbusQDA,
    p300_train_data;
    iterations = 50,
    dof_offset = 2,                    # Standard regularization
    mean_prior_precision = 0.02        # Slightly stronger than default
)

Hyperparameter Search Example

Systematically search for optimal hyperparameters:

using NimbusSDK

# Define search grid
dof_values = [1, 2, 3, 4]
prior_values = [0.001, 0.01, 0.03, 0.05]

# Split data
train_data, val_data = split_data(all_data, ratio=0.8)

best_accuracy = 0.0
best_params = nothing

println("Searching hyperparameters for NimbusQDA...")
for dof in dof_values
    for prior in prior_values
        # Train model
        model = train_model(
            NimbusQDA,
            train_data;
            iterations = 50,
            dof_offset = dof,
            mean_prior_precision = prior,
            predictive_dof_offset = dof,
            showprogress = false
        )
        
        # Validate
        results = predict_batch(model, val_data)
        accuracy = sum(results.predictions .== val_data.labels) / length(val_data.labels)
        
        println("  dof=$dof, prior=$prior: $(round(accuracy*100, digits=1))%")
        
        if accuracy > best_accuracy
            best_accuracy = accuracy
            best_params = (dof=dof, prior=prior)
        end
    end
end

println("\nBest hyperparameters:")
println("  dof_offset: $(best_params.dof)")
println("  mean_prior_precision: $(best_params.prior)")
println("  Validation accuracy: $(round(best_accuracy*100, digits=1))%")

# Retrain with best params
final_model = train_model(
    NimbusQDA,
    all_data;
    iterations = 50,
    dof_offset = best_params.dof,
    mean_prior_precision = best_params.prior,
    predictive_dof_offset = best_params.dof
)

Quick Tuning Guidelines

Scenario	`dof_offset`	`mean_prior_precision`	Notes
Excellent data quality	1	0.001	Minimal regularization
Good data quality	2 (default)	0.01 (default)	Balanced approach
Moderate data quality	2-3	0.01-0.03	Slight regularization
Poor data quality	3-4	0.05-0.1	Strong regularization
Very limited trials	4	0.1	Maximum regularization
P300 target/non-target	2	0.02	Moderate, class-specific covariances helpful

Pro Tip: Bayesian QDA’s class-specific covariances can overfit to noise with poor data. When in doubt, start with defaults and increase regularization (dof_offset=3, mean_prior_precision=0.03) if you see overfitting.

Important: Always set predictive_dof_offset to match dof_offset for consistency between training and inference phases.

Training Requirements

Data Requirements

Minimum: 40 trials per class (80 total for 2-class)
Recommended: 80+ trials per class
For calibration: 10-20 trials per class

Bayesian QDA requires at least 2 observations per class to estimate class-specific statistics. Training will fail if any class has fewer than 2 observations.

Feature Normalization

Critical for cross-session BCI performance!Normalize your features before training for 15-30% accuracy improvement across sessions.

Python
Julia

from sklearn.preprocessing import StandardScaler
import pickle

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

# Train with normalized features
clf = NimbusQDA()
clf.fit(X_train_norm, y_train)

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

# Later: Apply same normalization to test data
X_test_norm = scaler.transform(X_test)
predictions = clf.predict(X_test_norm)

# Estimate normalization from training data
norm_params = estimate_normalization_params(train_features; method=:zscore)
train_norm = apply_normalization(train_features, norm_params)

# Train with normalized features
train_data = BCIData(train_norm, metadata, labels)
model = train_model(NimbusQDA, train_data)

# Save params with model
@save "model.jld2" model norm_params

# Later: Apply same params to test data
test_norm = apply_normalization(test_features, norm_params)

See the Feature Normalization guide for complete details.

Feature Requirements

Bayesian QDA expects preprocessed features, not raw EEG: ✅ Required preprocessing:

Bandpass filtering (paradigm-specific)
Artifact removal
Feature extraction (CSP, ERP amplitude, bandpower, etc.)
Proper temporal aggregation

❌ NOT accepted:

Raw EEG channels
Unfiltered data

See Preprocessing Requirements.

Performance Characteristics

Computational Performance

Operation	Latency	Notes
Training	15-40 seconds	50 iterations, 100 trials per class
Calibration	8-20 seconds	20 iterations, 20 trials per class
Batch Inference	15-25ms per trial	10 iterations
Streaming Chunk	15-25ms	10 iterations per chunk

Slightly slower than NimbusLDA due to class-specific covariances.

Classification Accuracy

Paradigm	Classes	Typical Accuracy	When to Use Bayesian QDA
P300	2 (Target/Non-target)	85-95%	Target/non-target have different variances
Motor Imagery	2-4	70-85%	When Bayesian LDA accuracy insufficient
SSVEP	2-6	85-98%	Complex frequency responses

Bayesian QDA typically provides 2-5% higher accuracy than Bayesian LDA when class covariances differ significantly, at the cost of ~5-10ms additional latency.

Model Inspection

View Model Parameters

Python
Julia

import numpy as np

# Class means
print("Class means:")
for k, class_label in enumerate(clf.classes_):
    print(f"  Class {class_label}: {clf.model_['means'][k]}")

# Class-specific covariance matrices
print("\nClass-specific covariance matrices:")
for k, class_label in enumerate(clf.classes_):
    print(f"  Class {class_label} (first 3x3):")
    print(clf.model_['covariances'][k][:3, :3])

# Compare covariances across classes
print("\nCovariance structure comparison:")
for k, class_label in enumerate(clf.classes_):
    cov_k = clf.model_['covariances'][k]
    print(f"  Class {class_label} variance (diagonal): {np.diag(cov_k)}")

# Class priors
print("\nClass priors:")
for k, class_label in enumerate(clf.classes_):
    print(f"  Class {class_label}: {clf.model_['priors'][k]:.3f}")

# Extract point estimates from posterior distributions
using Distributions

# Class means (extract from posterior distributions)
println("Class means:")
for (k, mean_posterior) in enumerate(model.mean_posteriors)
    mean_point = mean(mean_posterior)  # Extract point estimate
    println("  Class $k: ", mean_point)
end

# Class-specific precisions (extract from posterior distributions)
println("\nClass-specific precision matrices:")
for (k, precision_posterior) in enumerate(model.precision_posteriors)
    prec_point = mean(precision_posterior)  # Extract point estimate
    println("  Class $k (first 3x3):")
    println(prec_point[1:3, 1:3])
end

# Compare covariances across classes
println("\nCovariance structure comparison:")
for k in 1:length(model.precision_posteriors)
    prec_point = mean(model.precision_posteriors[k])
    cov_k = inv(prec_point)  # Convert precision to covariance
    println("  Class $k variance (diagonal): ", diag(cov_k))
end

# Class priors
println("\nClass priors:")
for (k, prior) in enumerate(model.priors)
    println("  Class $k: ", prior)
end

Accessing model parameters: The SDK stores full posterior distributions (not just point estimates) for proper Bayesian inference. To get point estimates, use mean(posterior) to extract the mean of the posterior distribution. For precision matrices, use mean(precision_posterior) to get the expected precision matrix.

Visualize Class Differences

using Plots
using Distributions

# Compare class covariances
for k in 1:length(model.precision_posteriors)
    prec_point = mean(model.precision_posteriors[k])  # Extract point estimate
    cov_k = inv(prec_point)  # Convert precision to covariance
    heatmap(cov_k, title="Class $k Covariance", colorbar=true)
end

Advantages & Limitations

Advantages

✅ Flexible Modeling: Each class has its own covariance
✅ Better for Complex Data: Handles heterogeneous distributions
✅ Higher Accuracy: 2-5% improvement when classes differ significantly
✅ Uncertainty Quantification: Full Bayesian posteriors
✅ Production-Ready: Battle-tested in P300 applications

Limitations

❌ More Parameters: Requires more training data than NimbusLDA
❌ Slower Inference: ~15-25ms vs ~10-15ms for NimbusLDA
❌ Higher Memory: Stores n_classes precision matrices
❌ More Complex: Longer training time

Comparison: Bayesian QDA vs Bayesian LDA vs Bayesian Softmax vs Bayesian STS

Aspect	Bayesian QDA (NimbusQDA)	Bayesian LDA (NimbusLDA)	NimbusProbit	Bayesian STS (NimbusSTS)
Precision Matrix	Class-specific	Shared (one for all)	Latent-space precision (K-1)	N/A (state-space)
Mathematical Model	Heteroscedastic Gaussian Classifier (HGC)	Pooled Gaussian Classifier (PGC)	Multinomial Logistic Regression	State-Space Model (EKF)
Temporal Modeling	Static (stateless)	Static (stateless)	Static (stateless)	Dynamic (stateful)
Training Speed	Slower	Faster	Moderate	Moderate
Inference Speed	15-25ms	10-15ms	15-25ms	20-30ms
Flexibility	High	Moderate	Highest (non-Gaussian)	Adaptive
Data Requirements	More	Less	Moderate	Moderate
Memory Usage	Higher	Lower	Moderate	Moderate (stores state)
Best For	Heterogeneous class spreads	Homogeneous classes	Complex decision boundaries / non-Gaussian structure	Non-stationary, long sessions
Accuracy Gain	+2-5% (when applicable)	Baseline	Often improves on non-Gaussian tasks	+5-15% (on drifting data)

Decision Tree

Is your data non-stationary (drift over time)?
├─ Yes → Use Bayesian STS (adaptive, handles temporal dynamics)
└─ No → Is speed critical (&lt;15ms)?
    ├─ Yes → Use Bayesian LDA
    └─ No → Do classes have different spreads / covariances?
        ├─ Yes → Use Bayesian QDA (more accurate)
        └─ No → Is the decision boundary complex / non-Gaussian?
            ├─ Yes → Use Bayesian Softmax (most flexible)
            └─ No → Use Bayesian LDA (simplest)

Practical Examples

P300 Detection

# Bayesian QDA is ideal for P300 where target/non-target have different variances

# Train on P300 data
p300_data = BCIData(erp_features, p300_metadata, labels)  # 1=target, 2=non-target
model = train_model(NimbusQDA, p300_data; iterations=50)

# Test
results = predict_batch(model, test_data)

# Calculate sensitivity/specificity
targets = test_labels .== 1
target_preds = results.predictions .== 1

sensitivity = sum(target_preds .& targets) / sum(targets)
specificity = sum(.!target_preds .& .!targets) / sum(.!targets)

println("Sensitivity: $(round(sensitivity * 100, digits=1))%")
println("Specificity: $(round(specificity * 100, digits=1))%")

When Bayesian LDA Fails

# Try Bayesian LDA first
rxlda_model = train_model(NimbusLDA, train_data)
rxlda_results = predict_batch(rxlda_model, test_data)
rxlda_acc = sum(rxlda_results.predictions .== test_labels) / length(test_labels)

# If accuracy insufficient, try Bayesian QDA
if rxlda_acc < 0.75
    println("Bayesian LDA accuracy low ($(round(rxlda_acc*100, digits=1))%), trying Bayesian QDA...")
    rxgmm_model = train_model(NimbusQDA, train_data)
    rxgmm_results = predict_batch(rxgmm_model, test_data)
    rxgmm_acc = sum(rxgmm_results.predictions .== test_labels) / length(test_labels)
    
    println("Bayesian QDA accuracy: $(round(rxgmm_acc*100, digits=1))%")
    println("Improvement: +$(round((rxgmm_acc-rxlda_acc)*100, digits=1))%")
end

Next Read

Bayesian LDA (NimbusLDA)

Faster model with shared covariance

NimbusProbit

Most flexible multinomial classifier for complex decision boundaries

Bayesian STS (NimbusSTS)

Adaptive model for non-stationary data

Training Guide

Complete training tutorial

Code Examples

Working examples

References

Implementation:

RxInfer.jl: https://rxinfer.com/
Source code: /src/models/rxgmm/ in NimbusSDK.jl

Theory:

McLachlan, G. J., & Peel, D. (2000). “Finite Mixture Models”
Bishop, C. M. (2006). “Pattern Recognition and Machine Learning” (Chapter 9)
Heteroscedastic Gaussian Classifier (HGC) with class-specific covariances

BCI Applications:

Farwell, L. A., & Donchin, E. (1988). “Talking off the top of your head”
Lotte et al. (2018). “A review of classification algorithms for EEG-based BCI”

Getting Started

Core Concepts

Models & Algorithms

Configuration & Setup

Examples

Development

Python SDK

Julia SDK

​Bayesian QDA (NimbusQDA)

​Start Here

Quickstart

Model Selection

Examples

​Overview

​Quick Start

​When to Use Bayesian QDA

​Model Architecture

​Mathematical Foundation (Heteroscedastic Gaussian Classifier)

​Hyperparameters

​Model Structure

​RxInfer Implementation

​Usage

​1. Load Pre-trained Model

​2. Train Custom Model

​3. Subject-Specific Calibration

​4. Batch Inference

​5. Streaming Inference

​Hyperparameter Tuning (v0.2.0+)

​When to Tune Hyperparameters

​Tuning Strategies

​For High SNR / Clean Data / Many Trials

​For Low SNR / Noisy Data / Few Trials

​Balanced / Default Settings

​P300-Specific Tuning

​Hyperparameter Search Example

​Quick Tuning Guidelines

​Training Requirements

​Data Requirements

​Feature Normalization

​Feature Requirements

​Performance Characteristics

​Computational Performance

​Classification Accuracy

​Model Inspection

​View Model Parameters

​Visualize Class Differences

​Advantages & Limitations

​Advantages

​Limitations

​Comparison: Bayesian QDA vs Bayesian LDA vs Bayesian Softmax vs Bayesian STS

​Decision Tree

​Practical Examples

​P300 Detection

​When Bayesian LDA Fails

​Next Read

Bayesian LDA (NimbusLDA)

NimbusProbit

Bayesian STS (NimbusSTS)

Training Guide

Code Examples

​References

Bayesian QDA (NimbusQDA)

Start Here

Overview

Quick Start

When to Use Bayesian QDA

Model Architecture

Mathematical Foundation (Heteroscedastic Gaussian Classifier)

Hyperparameters

Model Structure

RxInfer Implementation

Usage

1. Load Pre-trained Model

2. Train Custom Model

3. Subject-Specific Calibration

4. Batch Inference

5. Streaming Inference

Hyperparameter Tuning (v0.2.0+)

When to Tune Hyperparameters

Tuning Strategies

For High SNR / Clean Data / Many Trials

For Low SNR / Noisy Data / Few Trials

Balanced / Default Settings

P300-Specific Tuning

Hyperparameter Search Example

Quick Tuning Guidelines

Training Requirements

Data Requirements

Feature Normalization

Feature Requirements

Performance Characteristics

Computational Performance

Classification Accuracy

Model Inspection

View Model Parameters

Visualize Class Differences

Advantages & Limitations

Advantages

Limitations

Comparison: Bayesian QDA vs Bayesian LDA vs Bayesian Softmax vs Bayesian STS

Decision Tree

Practical Examples

P300 Detection

When Bayesian LDA Fails

Next Read

References