Algorithms & Methods Deep Dive
This document provides detailed mathematical and algorithmic foundations underlying Lavoisier’s high-performance mass spectrometry analysis capabilities.
Table of Contents
- Algorithms & Methods Deep Dive
Spectral Processing Algorithms
Peak Detection using Continuous Wavelet Transform
Lavoisier employs an advanced peak detection algorithm based on continuous wavelet transform (CWT) that significantly outperforms traditional methods in noisy spectral environments.
Mathematical Foundation
The CWT of a signal $f(t)$ is defined as:
\[CWT_f(a,b) = \frac{1}{\sqrt{a}} \int_{-\infty}^{\infty} f(t) \psi^*\left(\frac{t-b}{a}\right) dt\]Where:
- $a$ is the scale parameter (inversely related to frequency)
- $b$ is the translation parameter (time localization)
- $\psi(t)$ is the mother wavelet
- $\psi^*(t)$ is the complex conjugate of the mother wavelet
Implementation Details
def continuous_wavelet_peak_detection(spectrum, scales, wavelet='mexh'):
"""
Advanced peak detection using continuous wavelet transform
Args:
spectrum: Input spectral data
scales: Array of scales for wavelet analysis
wavelet: Mother wavelet type ('mexh', 'morl', 'cgau8')
Returns:
detected_peaks: Peak positions with confidence scores
"""
# Compute CWT coefficients
coefficients = signal.cwt(spectrum, signal.wavelets.mexh, scales)
# Ridge line extraction using dynamic programming
ridges = _extract_ridge_lines(coefficients, scales)
# Peak validation using multi-scale consistency
validated_peaks = _validate_peaks_multiscale(ridges, coefficients)
# Confidence scoring based on ridge strength and consistency
confidence_scores = _calculate_peak_confidence(validated_peaks, coefficients)
return PeakDetectionResult(
peak_positions=validated_peaks,
confidence_scores=confidence_scores,
wavelet_coefficients=coefficients
)
Ridge Line Extraction Algorithm
The ridge line extraction uses dynamic programming to find optimal paths through the wavelet coefficient matrix:
\[R(i,j) = \max_{k \in [j-w, j+w]} \{R(i-1,k) + |CWT(i,j)|\}\]Where:
- $R(i,j)$ is the ridge score at scale $i$ and position $j$
- $w$ is the allowed deviation window
-
$ CWT(i,j) $ is the absolute value of the wavelet coefficient
Adaptive Baseline Correction
Asymmetric Least Squares (AsLS) Method
Lavoisier implements an enhanced version of the AsLS algorithm for robust baseline correction:
\[\min_z \sum_{i=1}^n w_i(y_i - z_i)^2 + \lambda \sum_{i=2}^{n-1}(\Delta^2 z_i)^2\]Subject to: $w_i = p$ if $y_i > z_i$, $w_i = 1-p$ if $y_i \leq z_i$
Where:
- $y_i$ are the observed intensities
- $z_i$ are the baseline estimates
- $\lambda$ is the smoothness parameter
- $p$ is the asymmetry parameter
- $\Delta^2$ is the second-order difference operator
Implementation
class AdaptiveBaselineCorrector:
def __init__(self, lambda_=1e6, p=0.01, max_iterations=50):
self.lambda_ = lambda_
self.p = p
self.max_iterations = max_iterations
def correct_baseline(self, spectrum):
"""
Asymmetric least squares baseline correction with adaptive parameters
"""
n = len(spectrum)
# Initialize weights
w = np.ones(n)
# Create difference matrix
D = sparse.diags([1, -2, 1], [0, 1, 2], shape=(n-2, n))
W = sparse.diags(w, format='csc')
for iteration in range(self.max_iterations):
# Solve weighted least squares problem
A = W + self.lambda_ * D.T @ D
baseline = sparse.linalg.spsolve(A, w * spectrum)
# Update weights based on residuals
residuals = spectrum - baseline
w_new = np.where(residuals > 0, self.p, 1 - self.p)
# Check convergence
if np.allclose(w, w_new, rtol=1e-6):
break
w = w_new
W = sparse.diags(w, format='csc')
return BaselineCorrectionResult(
corrected_spectrum=spectrum - baseline,
baseline=baseline,
iterations=iteration + 1
)
Mass Spectral Matching Algorithms
Spectral Similarity Scoring
Dot Product Similarity
The fundamental similarity metric used in spectral matching:
\[\text{Similarity}(A, B) = \frac{\sum_{i} I_A(m_i) \cdot I_B(m_i)}{\sqrt{\sum_{i} I_A(m_i)^2} \cdot \sqrt{\sum_{i} I_B(m_i)^2}}\]Where $I_A(m_i)$ and $I_B(m_i)$ are the intensities at mass $m_i$ for spectra A and B.
Enhanced Weighted Similarity
Lavoisier implements an enhanced similarity metric that accounts for peak intensity, mass accuracy, and biological relevance:
\[\text{Enhanced Similarity} = \alpha \cdot S_{dot} + \beta \cdot S_{mass} + \gamma \cdot S_{bio}\]Where:
- $S_{dot}$ is the normalized dot product similarity
- $S_{mass}$ is the mass accuracy component
- $S_{bio}$ is the biological relevance weight
- $\alpha + \beta + \gamma = 1$
Implementation
class EnhancedSpectralMatcher:
def __init__(self, mass_tolerance=0.01, intensity_threshold=0.05):
self.mass_tolerance = mass_tolerance
self.intensity_threshold = intensity_threshold
self.biological_weights = self._load_biological_weights()
def calculate_similarity(self, query_spectrum, library_spectrum):
"""
Calculate enhanced spectral similarity with multiple scoring components
"""
# Normalize spectra
query_norm = self._normalize_spectrum(query_spectrum)
library_norm = self._normalize_spectrum(library_spectrum)
# Align peaks within mass tolerance
aligned_peaks = self._align_peaks(query_norm, library_norm)
# Calculate dot product similarity
dot_similarity = self._calculate_dot_product(aligned_peaks)
# Calculate mass accuracy component
mass_accuracy = self._calculate_mass_accuracy(aligned_peaks)
# Calculate biological relevance
biological_relevance = self._calculate_biological_relevance(
aligned_peaks, self.biological_weights
)
# Combine scores with adaptive weights
weights = self._adaptive_weight_selection(query_spectrum, library_spectrum)
enhanced_similarity = (
weights['dot'] * dot_similarity +
weights['mass'] * mass_accuracy +
weights['bio'] * biological_relevance
)
return SimilarityResult(
total_score=enhanced_similarity,
component_scores={
'dot_product': dot_similarity,
'mass_accuracy': mass_accuracy,
'biological_relevance': biological_relevance
},
weights=weights,
aligned_peaks=aligned_peaks
)
Multi-Database Fusion Algorithm
Bayesian Evidence Combination
Lavoisier combines evidence from multiple databases using Bayesian inference:
\[P(compound|evidence) = \frac{P(evidence|compound) \cdot P(compound)}{P(evidence)}\]For multiple evidence sources:
\[P(compound|E_1, E_2, ..., E_n) \propto P(compound) \prod_{i=1}^n P(E_i|compound)\]Implementation
class BayesianDatabaseFusion:
def __init__(self):
self.database_weights = self._initialize_database_weights()
self.compound_priors = self._load_compound_priors()
def fuse_database_results(self, database_results):
"""
Combine results from multiple databases using Bayesian inference
"""
# Extract unique compounds across all databases
all_compounds = self._extract_unique_compounds(database_results)
posterior_scores = {}
for compound in all_compounds:
# Get prior probability
prior = self.compound_priors.get(compound.id, 1e-6)
# Calculate likelihood for each database
likelihood_product = 1.0
for db_name, results in database_results.items():
if compound.id in results:
# Database-specific likelihood
likelihood = self._calculate_likelihood(
results[compound.id], db_name
)
likelihood_product *= likelihood
else:
# Penalty for absence in database
likelihood_product *= 0.1
# Calculate posterior probability (unnormalized)
posterior_scores[compound.id] = prior * likelihood_product
# Normalize probabilities
total_score = sum(posterior_scores.values())
normalized_scores = {
comp_id: score / total_score
for comp_id, score in posterior_scores.items()
}
return FusedDatabaseResult(
compound_probabilities=normalized_scores,
evidence_summary=self._summarize_evidence(database_results),
confidence_intervals=self._calculate_confidence_intervals(normalized_scores)
)
Machine Learning Algorithms
Deep Learning for MS/MS Prediction
Transformer Architecture for Spectral Prediction
Lavoisier employs a specialized transformer architecture for predicting MS/MS spectra from molecular structures:
\[\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V\]Where molecular graphs are embedded using graph neural networks before transformer processing.
Implementation
class SpectralTransformer(nn.Module):
def __init__(self, d_model=512, nhead=8, num_layers=6):
super().__init__()
self.molecular_encoder = MolecularGraphEncoder(d_model)
self.positional_encoding = PositionalEncoding(d_model)
encoder_layer = nn.TransformerEncoderLayer(
d_model=d_model,
nhead=nhead,
dim_feedforward=2048,
dropout=0.1
)
self.transformer = nn.TransformerEncoder(encoder_layer, num_layers)
self.spectrum_decoder = SpectrumDecoder(d_model, max_mz=2000)
def forward(self, molecular_graph):
# Encode molecular structure
mol_embedding = self.molecular_encoder(molecular_graph)
# Add positional encoding
mol_embedding = self.positional_encoding(mol_embedding)
# Transform through attention layers
transformed = self.transformer(mol_embedding)
# Decode to spectrum
predicted_spectrum = self.spectrum_decoder(transformed)
return predicted_spectrum
class MolecularGraphEncoder(nn.Module):
def __init__(self, d_model):
super().__init__()
self.atom_embedding = nn.Embedding(120, d_model) # 120 atom types
self.bond_embedding = nn.Embedding(12, d_model) # 12 bond types
self.graph_conv_layers = nn.ModuleList([
GraphConvLayer(d_model) for _ in range(4)
])
def forward(self, molecular_graph):
# Embed atoms and bonds
atom_features = self.atom_embedding(molecular_graph.atoms)
bond_features = self.bond_embedding(molecular_graph.bonds)
# Graph convolution layers
for layer in self.graph_conv_layers:
atom_features = layer(atom_features, bond_features, molecular_graph.edges)
return atom_features
Continuous Learning with Knowledge Distillation
Knowledge Distillation Loss Function
\[\mathcal{L}_{KD} = \alpha \mathcal{L}_{CE}(y, \sigma(z_s)) + (1-\alpha) \mathcal{L}_{KL}(\sigma(z_t/T), \sigma(z_s/T))\]Where:
- $\mathcal{L}_{CE}$ is the cross-entropy loss with true labels
- $\mathcal{L}_{KL}$ is the KL divergence between teacher and student
- $T$ is the temperature parameter
- $\alpha$ balances the losses
Implementation
class ContinuousLearningEngine:
def __init__(self, teacher_model, student_model, temperature=4.0, alpha=0.3):
self.teacher_model = teacher_model
self.student_model = student_model
self.temperature = temperature
self.alpha = alpha
self.experience_buffer = ExperienceBuffer(capacity=100000)
def distillation_loss(self, student_logits, teacher_logits, true_labels):
"""
Calculate knowledge distillation loss combining hard and soft targets
"""
# Soft targets from teacher
teacher_probs = F.softmax(teacher_logits / self.temperature, dim=1)
student_log_probs = F.log_softmax(student_logits / self.temperature, dim=1)
# KL divergence loss
kl_loss = F.kl_div(student_log_probs, teacher_probs, reduction='batchmean')
kl_loss *= (self.temperature ** 2)
# Hard target loss
ce_loss = F.cross_entropy(student_logits, true_labels)
# Combined loss
total_loss = self.alpha * ce_loss + (1 - self.alpha) * kl_loss
return total_loss, {'ce_loss': ce_loss.item(), 'kl_loss': kl_loss.item()}
def incremental_update(self, new_data):
"""
Perform incremental model update with new experience
"""
# Add new experience to buffer
self.experience_buffer.add(new_data)
# Sample balanced batch for training
batch = self.experience_buffer.sample_balanced_batch(batch_size=128)
# Generate teacher predictions
with torch.no_grad():
teacher_logits = self.teacher_model(batch['inputs'])
# Train student model
self.student_model.train()
student_logits = self.student_model(batch['inputs'])
loss, loss_components = self.distillation_loss(
student_logits, teacher_logits, batch['labels']
)
# Backward pass and optimization
self.optimizer.zero_grad()
loss.backward()
self.optimizer.step()
return loss_components
Optimization Algorithms
Adaptive Parameter Optimization
Bayesian Optimization for Hyperparameter Tuning
Lavoisier uses Bayesian optimization to automatically tune analysis parameters:
\[\alpha(x) = \mathbb{E}[I(x)] = \mathbb{E}[\max(f(x) - f(x^+), 0)]\]Where $I(x)$ is the improvement function and $f(x^+)$ is the current best value.
Implementation
class BayesianParameterOptimizer:
def __init__(self, parameter_space, acquisition_function='ei'):
self.parameter_space = parameter_space
self.acquisition_function = acquisition_function
self.gp_model = GaussianProcessRegressor(
kernel=Matern(nu=2.5),
alpha=1e-6,
normalize_y=True
)
self.evaluated_points = []
self.evaluated_scores = []
def optimize_parameters(self, objective_function, n_iterations=50):
"""
Optimize analysis parameters using Bayesian optimization
"""
# Initialize with random samples
for _ in range(5):
params = self._sample_random_params()
score = objective_function(params)
self.evaluated_points.append(params)
self.evaluated_scores.append(score)
for iteration in range(n_iterations):
# Fit Gaussian process to observed data
X = np.array(self.evaluated_points)
y = np.array(self.evaluated_scores)
self.gp_model.fit(X, y)
# Find next point using acquisition function
next_params = self._optimize_acquisition()
# Evaluate objective function
score = objective_function(next_params)
# Update observations
self.evaluated_points.append(next_params)
self.evaluated_scores.append(score)
# Early stopping if converged
if self._check_convergence():
break
# Return best parameters
best_idx = np.argmax(self.evaluated_scores)
return OptimizationResult(
best_parameters=self.evaluated_points[best_idx],
best_score=self.evaluated_scores[best_idx],
convergence_iteration=iteration,
optimization_history=list(zip(self.evaluated_points, self.evaluated_scores))
)
def _expected_improvement(self, X):
"""
Calculate expected improvement acquisition function
"""
mu, sigma = self.gp_model.predict(X, return_std=True)
mu_best = np.max(self.evaluated_scores)
with np.errstate(divide='warn'):
improvement = mu - mu_best
Z = improvement / sigma
ei = improvement * norm.cdf(Z) + sigma * norm.pdf(Z)
ei[sigma == 0.0] = 0.0
return ei
Memory-Efficient Processing
Streaming Algorithms for Large Datasets
For datasets that exceed available memory, Lavoisier implements streaming algorithms:
Online Statistics Calculation
\[\mu_n = \mu_{n-1} + \frac{x_n - \mu_{n-1}}{n}\] \[\sigma_n^2 = \frac{(n-1)\sigma_{n-1}^2 + (x_n - \mu_{n-1})(x_n - \mu_n)}{n}\]Implementation
class StreamingStatistics:
def __init__(self):
self.n = 0
self.mean = 0.0
self.variance = 0.0
self.min_val = float('inf')
self.max_val = float('-inf')
def update(self, value):
"""
Update statistics with new value using Welford's online algorithm
"""
self.n += 1
delta = value - self.mean
self.mean += delta / self.n
delta2 = value - self.mean
self.variance += delta * delta2
self.min_val = min(self.min_val, value)
self.max_val = max(self.max_val, value)
def get_statistics(self):
"""
Return current statistics
"""
if self.n < 2:
variance = 0.0
else:
variance = self.variance / (self.n - 1)
return {
'count': self.n,
'mean': self.mean,
'variance': variance,
'std': np.sqrt(variance),
'min': self.min_val,
'max': self.max_val
}
class StreamingSpectrumProcessor:
def __init__(self, chunk_size=1000):
self.chunk_size = chunk_size
self.stats = StreamingStatistics()
self.peak_detector = StreamingPeakDetector()
def process_stream(self, spectrum_stream):
"""
Process spectral data stream with constant memory usage
"""
results = []
for chunk in self._chunk_stream(spectrum_stream, self.chunk_size):
# Process chunk
chunk_results = self._process_chunk(chunk)
# Update global statistics
for spectrum in chunk:
self.stats.update(spectrum.total_intensity)
# Append results
results.extend(chunk_results)
# Optional: yield intermediate results for real-time processing
yield chunk_results
return StreamingResult(
peak_detections=results,
global_statistics=self.stats.get_statistics(),
processing_metadata=self._get_processing_metadata()
)
Quality Control Algorithms
Statistical Process Control
Control Chart Implementation
Lavoisier implements statistical process control for monitoring analysis quality:
\(UCL = \mu + 3\sigma\) \(LCL = \mu - 3\sigma\)
Where $\mu$ and $\sigma$ are estimated from historical data.
Implementation
class QualityControlMonitor:
def __init__(self, control_limits_factor=3.0, min_history=30):
self.control_limits_factor = control_limits_factor
self.min_history = min_history
self.historical_data = deque(maxlen=1000)
self.control_statistics = {}
def monitor_analysis_quality(self, analysis_result):
"""
Monitor analysis quality using statistical process control
"""
# Extract quality metrics
quality_metrics = self._extract_quality_metrics(analysis_result)
# Update historical data
self.historical_data.append(quality_metrics)
# Calculate control limits if sufficient history
if len(self.historical_data) >= self.min_history:
self._update_control_limits()
# Check for out-of-control conditions
control_status = self._check_control_status(quality_metrics)
return QualityControlResult(
metrics=quality_metrics,
control_status=control_status,
control_limits=self.control_statistics,
recommendations=self._generate_recommendations(control_status)
)
else:
return QualityControlResult(
metrics=quality_metrics,
control_status='insufficient_history',
control_limits=None,
recommendations=['Collecting baseline data for control limits']
)
def _update_control_limits(self):
"""
Update control limits based on historical data
"""
historical_array = np.array([
[metrics[key] for key in metrics.keys()]
for metrics in self.historical_data
])
means = np.mean(historical_array, axis=0)
stds = np.std(historical_array, axis=0, ddof=1)
metric_names = list(self.historical_data[0].keys())
for i, metric_name in enumerate(metric_names):
self.control_statistics[metric_name] = {
'mean': means[i],
'std': stds[i],
'ucl': means[i] + self.control_limits_factor * stds[i],
'lcl': means[i] - self.control_limits_factor * stds[i]
}
This comprehensive algorithms and methods documentation provides the mathematical and computational foundations that enable Lavoisier’s exceptional performance in mass spectrometry analysis. The combination of advanced signal processing, machine learning, and optimization techniques creates a robust and efficient analytical platform.