RBA Regression Model Pseudocode
Main Experiment Flow
ALGORITHM: RBA Regression Experiment
INPUT: data_path, random_state
OUTPUT: comprehensive_results, visualizations
PROCEDURE main_experiment():
1. INITIALIZE experiment environment
SET random seeds for reproducibility
CONFIGURE matplotlib for publication quality (Times New Roman, 14pt)
2. LOAD and PREPROCESS data
LOAD California Housing dataset from data_path
HANDLE missing values and outliers
SPLIT data into train/test sets (80/20)
APPLY StandardScaler to features and targets
3. CREATE model architectures
INITIALIZE RBA(input_dim, hidden_dim=128, heads=8, layers=3)
INITIALIZE Transformer(input_dim, hidden_dim=128, heads=8, layers=3)
4. TRAIN models
FOR each model:
TRAIN using Adam optimizer with early stopping
APPLY learning rate scheduling
MONITOR validation loss for convergence
5. EVALUATE performance
COMPUTE regression metrics (RMSE, MAE, R², CV, MAPE)
ANALYZE uncertainty quantification (RBA only)
PERFORM cross-validation analysis
6. CONDUCT ablation study
TRAIN ablation variants (NoGP, NoResidual, NoUncertainty, NoLayerNorm)
COMPARE component importance
7. GENERATE results
PRINT comprehensive statistical analysis
CREATE geographic visualizations
SAVE publication-quality figures
Core Model Architecture
CLASS ResidualBayesianAttention:
INPUTS: input_dim, hidden_dim, num_heads, num_layers, dropout, gp_kernel_type
COMPONENTS:
input_embedding: Linear(input_dim → hidden_dim)
attention_layers: List[BayesianMultiHeadAttention]
layer_norms: List[LayerNorm]
feedforward_layers: List[BayesianFeedForward]
residual_weights: List[Parameter([0.5, 0.5])]
output_projection: Linear(hidden_dim → 1)
uncertainty_head: Linear(hidden_dim → 1) + Softplus
FORWARD(x):
h = input_embedding(x)
attention_uncertainties = []
FOR layer_i in range(num_layers):
residual_input = h
h_norm = layer_norms[i](h)
attention_output, uncertainty = attention_layers[i](h_norm)
attention_uncertainties.append(uncertainty)
alpha, beta = softmax(residual_weights[i])
h = alpha * residual_input + beta * attention_output
residual_input = h
h_norm = ff_layer_norms[i](h)
ff_output = feedforward_layers[i](h_norm)
h = alpha * residual_input + beta * ff_output
h_pooled = mean(h, dim=sequence)
prediction = output_projection(h_pooled)
uncertainty = uncertainty_head(h_pooled)
total_uncertainty = uncertainty + mean(attention_uncertainties)
RETURN prediction, total_uncertainty
Bayesian Multi-Head Attention
CLASS BayesianMultiHeadAttention:
INPUTS: hidden_dim, num_heads, dropout, gp_kernel_type
COMPONENTS:
q_proj, k_proj, v_proj, o_proj: Linear projections
length_scale: Parameter(ones(num_heads))
signal_variance: Parameter(ones(num_heads))
dropout: Dropout(dropout)
COMPUTE_GP_KERNEL(x):
batch_size, seq_len, hidden_dim = x.shape
x_expanded = unsqueeze(x, dim=2)
x_tiled = unsqueeze(x, dim=1)
distances = norm(x_expanded - x_tiled, dim=-1)
kernel_matrices = []
FOR head_h in range(num_heads):
kernel = signal_variance[h]² * exp(-distances² / (2 * length_scale[h]²))
kernel_matrices.append(kernel)
RETURN stack(kernel_matrices, dim=1)
FORWARD(x):
batch_size, seq_len, _ = x.shape
Q = reshape_multihead(q_proj(x))
K = reshape_multihead(k_proj(x))
V = reshape_multihead(v_proj(x))
attention_scores = matmul(Q, transpose(K, -2, -1)) * scale
gp_kernel = compute_gp_kernel(x)
enhanced_scores = attention_scores + gp_kernel
attention_weights = softmax(enhanced_scores, dim=-1)
attention_weights = dropout(attention_weights)
attention_output = matmul(attention_weights, V)
output = o_proj(reshape_back(attention_output))
attention_entropy = -sum(attention_weights * log(attention_weights + ε), dim=-1)
uncertainty = mean(attention_entropy, dim=(1, 2))
RETURN output, uncertainty
Training Procedure
PROCEDURE train_model(model, train_loader, val_loader, epochs, lr):
optimizer = Adam(model.parameters(), lr=lr, weight_decay=1e-5)
scheduler = ReduceLROnPlateau(optimizer, patience=10, factor=0.5)
best_val_loss = infinity
patience_counter = 0
patience = 20
FOR epoch in range(epochs):
model.train()
train_loss = 0
FOR batch_x, batch_y in train_loader:
optimizer.zero_grad()
IF isinstance(model, ResidualBayesianAttention):
prediction, uncertainty = model(batch_x)
loss = MSE(squeeze(prediction), batch_y)
uncertainty_loss = mean(uncertainty)
loss = loss + 0.01 * uncertainty_loss
ELSE:
prediction = model(batch_x)
loss = MSE(squeeze(prediction), batch_y)
loss.backward()
clip_grad_norm(model.parameters(), max_norm=1.0)
optimizer.step()
train_loss += loss.item()
model.eval()
val_loss = 0
WITH no_grad():
FOR batch_x, batch_y in val_loader:
IF isinstance(model, ResidualBayesianAttention):
prediction, _ = model(batch_x)
ELSE:
prediction = model(batch_x)
loss = MSE(squeeze(prediction), batch_y)
val_loss += loss.item()
train_loss /= len(train_loader)
val_loss /= len(val_loader)
scheduler.step(val_loss)
IF val_loss < best_val_loss:
best_val_loss = val_loss
patience_counter = 0
save_model(model, 'best_model.pth')
ELSE:
patience_counter += 1
IF patience_counter >= patience:
BREAK
load_model(model, 'best_model.pth')
RETURN train_losses, val_losses
Evaluation and Analysis
PROCEDURE evaluate_comprehensive(model, test_loader, X_test, y_test_original):
model.eval()
predictions = []
uncertainties = []
WITH no_grad():
FOR batch_x, _ in test_loader:
IF isinstance(model, ResidualBayesianAttention):
pred, uncertainty = model(batch_x)
IF len(uncertainty.shape) > 1:
uncertainty = squeeze(uncertainty)
uncertainties.extend(uncertainty.cpu().numpy())
ELSE:
pred = model(batch_x)
predictions.extend(squeeze(pred).cpu().numpy())
predictions = array(predictions)
predictions_original = inverse_transform(predictions.reshape(-1, 1)).flatten()
metrics = {
'MSE': mean_squared_error(y_test_original, predictions_original),
'RMSE': sqrt(MSE),
'MAE': mean_absolute_error(y_test_original, predictions_original),
'R²': r2_score(y_test_original, predictions_original),
'MAPE': mean(abs((y_test_original - predictions_original) / y_test_original)) * 100,
'CV': (RMSE / mean(y_test_original)) * 100,
'Explained_Variance': 1 - (var(y_test_original - predictions_original) / var(y_test_original))
}
IF len(uncertainties) > 0:
uncertainties = array(uncertainties)
uncertainties_scaled = uncertainties * target_scaler.scale_[0]
prediction_intervals = {
'lower_95': predictions_original - 1.96 * uncertainties_scaled,
'upper_95': predictions_original + 1.96 * uncertainties_scaled,
'mean_interval_width': mean(3.92 * uncertainties_scaled)
}
metrics['prediction_intervals'] = prediction_intervals
RETURN metrics
Cross-Validation Analysis
PROCEDURE cross_validation_analysis(X, y):
kf = KFold(n_splits=5, shuffle=True, random_state=random_state)
rba_cv_scores = []
transformer_cv_scores = []
rba_cv_values = []
transformer_cv_values = []
fold = 1
FOR train_idx, val_idx in kf.split(X):
X_train_cv, X_val_cv = X[train_idx], X[val_idx]
y_train_cv, y_val_cv = y[train_idx], y[val_idx]
scaler_X = StandardScaler()
scaler_y = StandardScaler()
X_train_cv_scaled = scaler_X.fit_transform(X_train_cv)
X_val_cv_scaled = scaler_X.transform(X_val_cv)
y_train_cv_scaled = scaler_y.fit_transform(y_train_cv.reshape(-1, 1)).flatten()
train_loader_cv, val_loader_cv = create_torch_datasets(
X_train_cv_scaled, X_val_cv_scaled, y_train_cv_scaled, y_train_cv_scaled[:len(X_val_cv_scaled)])
rba_model = ResidualBayesianAttention(input_dim=X.shape[1], hidden_dim=128, num_heads=8, num_layers=3)
train_model(rba_model, train_loader_cv, val_loader_cv, epochs=50)
transformer_model = StandardTransformer(input_dim=X.shape[1], hidden_dim=128, num_heads=8, num_layers=3)
train_model(transformer_model, train_loader_cv, val_loader_cv, epochs=50)
rba_metrics = evaluate_comprehensive(rba_model, val_loader_cv, X_val_cv, y_val_cv)
transformer_metrics = evaluate_comprehensive(transformer_model, val_loader_cv, X_val_cv, y_val_cv)
rba_cv_scores.append(rba_metrics['R²'])
transformer_cv_scores.append(transformer_metrics['R²'])
rba_cv_values.append(rba_metrics['CV'])
transformer_cv_values.append(transformer_metrics['CV'])
fold += 1
RETURN {
'rba_r2_scores': rba_cv_scores,
'transformer_r2_scores': transformer_cv_scores,
'rba_cv_values': rba_cv_values,
'transformer_cv_values': transformer_cv_values
}
Ablation Study Analysis
PROCEDURE ablation_study_analysis(X, y):
ablation_models = {
'Full RBA': ResidualBayesianAttention,
'No GP Kernel': RBA_NoGPKernel,
'No Residual': RBA_NoResidual,
'No Uncertainty': RBA_NoUncertainty,
'No LayerNorm': RBA_NoLayerNorm,
'Transformer': StandardTransformer
}
results = {}
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=random_state)
scaler_X = StandardScaler()
scaler_y = StandardScaler()
X_train_scaled = scaler_X.fit_transform(X_train)
X_test_scaled = scaler_X.transform(X_test)
y_train_scaled = scaler_y.fit_transform(y_train.reshape(-1, 1)).flatten()
train_loader, test_loader = create_torch_datasets(X_train_scaled, X_test_scaled, y_train_scaled, y_train_scaled[:len(X_test_scaled)])
FOR model_name, model_class in ablation_models.items():
model = model_class(input_dim=X.shape[1], hidden_dim=128, num_heads=8, num_layers=3)
train_losses, val_losses = train_model(model, train_loader, test_loader, epochs=50)
IF model_name == 'No Uncertainty':
predictions = []
WITH no_grad():
FOR batch_x, _ in test_loader:
pred = model(batch_x)
predictions.extend(squeeze(pred).cpu().numpy())
predictions = array(predictions)
predictions_original = scaler_y.inverse_transform(predictions.reshape(-1, 1)).flatten()
metrics = {
'MSE': mean_squared_error(y_test, predictions_original),
'RMSE': sqrt(MSE),
'MAE': mean_absolute_error(y_test, predictions_original),
'R²': r2_score(y_test, predictions_original),
'MAPE': mean(abs((y_test - predictions_original) / y_test)) * 100,
'CV': (RMSE / mean(y_test)) * 100,
'predictions': predictions_original
}
ELSE:
metrics = evaluate_comprehensive(model, test_loader, X_test_scaled, y_test)
results[model_name] = metrics
RETURN results
Statistical Analysis
PROCEDURE statistical_analysis(rba_metrics, transformer_metrics):
rba_errors = abs(rba_metrics['residuals'])
trans_errors = abs(transformer_metrics['residuals'])
t_statistic, p_value = paired_t_test(trans_errors, rba_errors)
effect_size = t_statistic / sqrt(len(rba_errors))
significance_level = IF p_value < 0.001 THEN "***"
ELSE IF p_value < 0.01 THEN "**"
ELSE IF p_value < 0.05 THEN "*"
ELSE "ns"
IF rba_metrics['prediction_intervals']:
actual = y_test_original
intervals = rba_metrics['prediction_intervals']
coverage_95 = mean((actual >= intervals['lower_95']) & (actual <= intervals['upper_95'])) * 100
RETURN {
't_statistic': t_statistic,
'p_value': p_value,
'effect_size': effect_size,
'significance': significance_level,
'coverage': coverage_95 if available else None
}
Visualization Generation
PROCEDURE plot_focused_analysis():
figure = create_figure(size=(20, 12))
# CV Comparison Box Plot
subplot1 = subplot(2, 3, 1)
cv_data = [cv_results['rba_cv_values'], cv_results['transformer_cv_values']]
boxplot(cv_data, labels=['RBA', 'Transformer'])
title('CV Comparison Across 5-Fold Cross-Validation')
# Geographic Distribution Plots
subplot2 = subplot(2, 3, 2)
scatter(coordinates[:, 0], coordinates[:, 1], c=y_true, cmap='viridis')
title('True House Values (Geographic Distribution)')
subplot3 = subplot(2, 3, 3)
scatter(coordinates[:, 0], coordinates[:, 1], c=rba_predictions, cmap='viridis')
title('RBA Predictions (Geographic Distribution)')
subplot4 = subplot(2, 3, 4)
scatter(coordinates[:, 0], coordinates[:, 1], c=transformer_predictions, cmap='viridis')
title('Transformer Predictions (Geographic Distribution)')
# Error Analysis
subplot5 = subplot(2, 3, 5)
error_difference = transformer_errors - rba_errors
scatter(coordinates[:, 0], coordinates[:, 1], c=error_difference, cmap='RdBu_r')
title('Error Improvement (Transformer Error - RBA Error)')
# Performance Summary Table
subplot6 = subplot(2, 3, 6)
create_performance_table(rba_metrics, transformer_metrics)
save_figure('Focused_RBA_vs_Transformer_Analysis', formats=['png', 'pdf'])
show()
Component Importance Analysis
PROCEDURE analyze_component_importance(ablation_results):
full_rba_metrics = ablation_results['Full RBA']
component_analysis = []
FOR model_name, metrics in ablation_results.items():
IF model_name != 'Full RBA':
rmse_change = ((metrics['RMSE'] - full_rba_metrics['RMSE']) / full_rba_metrics['RMSE']) * 100
r2_change = ((metrics['R²'] - full_rba_metrics['R²']) / full_rba_metrics['R²']) * 100
cv_change = ((metrics['CV'] - full_rba_metrics['CV']) / full_rba_metrics['CV']) * 100
impact = IF abs(rmse_change) > 10 THEN "极高"
ELSE IF abs(rmse_change) > 5 THEN "高"
ELSE IF abs(rmse_change) > 2 THEN "中等"
ELSE "低"
component_analysis.append({
'component': model_name,
'rmse_change': rmse_change,
'r2_change': r2_change,
'cv_change': cv_change,
'impact': impact,
'abs_impact': abs(rmse_change)
})
sort(component_analysis, key=lambda x: x['abs_impact'], reverse=True)
RETURN component_analysis
