Lars / Regressor Layer

Least Angle Regression (LARS): Forward stagewise additive modeling.

Mathematical formulation:

Start with all coefficients = 0
Find feature most correlated with current residual
Take step in equiangular direction: $\overset{μ}{^} = X \hat{β}$ $u_{A} = X_{A} (X_{A}^{T} X_{A})^{- 1} 1_{A}$ where A is active set

Key characteristics:

Forward selection variant
Equiangular direction
Solution path computation
Controlled feature entry
Efficient regularization path

Advantages:

Computationally efficient
Solution path visualization
Feature selection order
Statistical properties
Connection to Lasso

Common applications:

High-dimensional data
Feature selection
Model interpretation
Signal processing
Genomics analysis

Outputs:

Predicted Table: Results with predictions
Validation Results: Cross-validation metrics
Test Metric: Hold-out performance
Feature Importances: Model coefficients

Table

Predicted Table

Validation Results

Test Metric

Feature Importances

SelectFeatures

[column, ...]

Feature column selection for LARS Regression:

Data requirements:

Preprocessing needs:
- Standardization crucial
- Correlation structure
- Missing value handling
- Outlier treatment
Feature properties:
- Numeric values
- Linear relationships
- Independence preferred
- Collinearity handling
Quality checks:
- Correlation analysis
- Signal strength
- Noise levels
- Feature relevance
Special considerations:
- Active set dynamics
- Entry order importance
- Path stability
- Computational feasibility

Note: If empty, uses all numeric columns except target

SelectTarget

column

Target column specification for LARS Regression:

Requirements:

Data characteristics:
- Numeric continuous
- No missing values
- Finite values
- Scale consideration
Statistical properties:
- Linear relationships
- Error structure
- Distribution shape
- Noise characteristics
Preprocessing needs:
- Standardization check
- Outlier treatment
- Scale adjustments
- Transform consideration
Quality controls:
- Range verification
- Distribution analysis
- Relationship linearity
- Signal-to-noise ratio

Note: Must be a single numeric column

Params

oneof

DefaultParams

Default configuration for LARS Regression:

Core settings:

Model structure:
- Intercept included
- Max 500 non-zero coefficients
- Complete solution path
Numerical stability:
- eps = machine precision
- No jittering
- Standard computation
Computation control:
- Full path computation
- Deterministic behavior
- Optimal precision

Best suited for:

Medium-sized problems
Feature selection studies
Path analysis needs
Well-conditioned data

CustomParams

Customizable parameters for LARS Regression:

Parameter categories:

Model complexity:
- Non-zero coefficient limit
- Intercept inclusion
- Solution path control
Numerical stability:
- Precision control
- Jittering options
- Computation robustness
Computation efficiency:
- Path calculation
- Memory usage
- Processing speed

FitIntercept

bool

true

Intercept calculation control:

Model forms: With intercept: $y = Xβ + b + ϵ$ Without intercept: $y = Xβ + ϵ$

Effects when True:

Centers predictions
Accounts for bias
Better general fit
Standard approach

Effects when False:

Forces origin fitting
No bias term
Domain-specific needs
Centered data assumed

NNonzeroCoefs

u32

500

Target number of non-zero coefficients.

Control aspects:

Model sparsity
Solution complexity
Feature selection
Path length

Guidelines:

0: No limit
n < p: Sparse solution
Small: Feature selection
Large: Full model

Impact:

Computation time
Memory usage
Model interpretability

Eps

f64

Cholesky computation precision:

Purpose:

Numerical stability
Matrix conditioning
Computation reliability

Values:

1.0: Standard precision
>1.0: More regularization
Based on machine epsilon
Larger for ill-conditioned data

Note: Not an optimization tolerance

FitPath

bool

true

Solution path computation control:

When True:

Computes full path
All lambda values
Complete solution
More memory usage

When False:

Final solution only
Memory efficient
Faster computation
Large problem suitable

Jitter

f64

Noise addition for stability:

Purpose:

Breaks ties
Improves stability
Handles degeneracy
Satisfies assumptions

Usage:

None: No noise
>0: Uniform noise bound
Small values preferred
Problem-dependent

RandomState

u64

Determines random number generation for jittering.

Affects:

Jitter generation
Reproducibility
Stability analysis

Usage:

Fixed: Reproducible results
Different: Stability check
None: System random
Ignored if no jitter

GridSearch

Hyperparameter optimization for LARS Regression:

Search space organization:

Model structure:
- Intercept options
- Coefficient limits
- Solution paths
Numerical stability:
- Precision control
- Jitter parameters
- Computation robustness
Performance settings:
- Path computation
- Memory usage
- Speed optimization

Computational impact:

Time: O(n_params * n_features³)
Memory: O(n_features²)
Storage: O(n_features * steps)

FitIntercept

[bool, ...]

true

Intercept inclusion search:

Options:

Single mode:
- [true]: Standard modeling
Complete search:
- [true, false]: Compare both

Selection impact:

Model flexibility
Path computation
Solution stability
Feature scaling needs

NNonzeroCoefs

[u32, ...]

500

Active feature limit search space:

Search patterns:

Feature-based:
- [p/4, p/2, 3p/4] (p=features)
- Fraction of features
Fixed sizes:
- [10, 50, 100, 500]
- Specific limits
Problem-specific:
- Based on domain knowledge
- Resource constraints

Impact:

Solution sparsity
Computation time
Memory usage

Eps

[f64, ...]

Numerical precision search space:

Search ranges:

Standard scale:
- [1.0, 2.0, 5.0]
- Machine epsilon multiples
Ill-conditioned data:
- [5.0, 10.0, 20.0]
- Higher regularization
Well-conditioned data:
- [1.0, 1.5, 2.0]
- Lower regularization

FitPath

[bool, ...]

true

Path computation evaluation:

Options:

Single mode:
- [true]: Full path
- [false]: Final only
Comparison:
- [true, false]: Both methods

Selection criteria:

Problem size
Memory availability
Path analysis needs
Computation time

Jitter

[f64, ...]

Stability noise parameter search:

Search ranges:

No/low noise:
- [0.0, 1e-6, 1e-5]
- Minimal perturbation
Higher noise:
- [1e-4, 1e-3, 1e-2]
- More stability

Trade-offs:

Stability vs accuracy
Noise vs precision
Computation effects

RandomState

u64

Random seed configuration:

Usage:

Development:
- Fixed seed
- Reproducible results
- Debugging capability
Validation:
- Multiple seeds
- Stability testing
- Robustness checks

RefitScore

enum

R2score

Regression model evaluation metrics:

Purpose:

Model performance evaluation
Error measurement
Quality assessment
Model comparison

Selection criteria:

Error distribution
Scale sensitivity
Domain requirements
Business objectives

Default ~

Model's native scoring method:

Typically R² score
Model-specific implementation
Standard evaluation
Quick assessment

R2score ~

Coefficient of determination (R²):

Formula: $R^{2} = 1 - \frac{\sum ( y _{i} - y ^ _{i} ) ^{2}}{\sum ( y _{i} - y ˉ ) ^{2}}$

Properties:

Range: (-∞, 1]
1: Perfect prediction
0: Constant model
Negative: Worse than mean

Best for:

General performance
Variance explanation
Model comparison
Standard reporting

ExplainedVariance ~

Explained variance score:

Formula: $1 - \frac{Va r ( y - y ^ )}{Va r ( y )}$

Properties:

Range: (-∞, 1]
Accounts for bias
Variance focus
Similar to R²

Best for:

Variance analysis
Bias assessment
Model stability

MaxError ~

Maximum absolute error:

Formula: $max (∣ y_{i} - \overset{y}{^}_{i} ∣)$

Properties:

Worst case error
Original scale
Sensitive to outliers
Upper error bound

Best for:

Critical applications
Error bounds
Safety margins
Risk assessment

NegMeanAbsoluteError ~

Negative mean absolute error:

Formula: $- \frac{1}{n} \sum ∣ y_{i} - \overset{y}{^}_{i} ∣$

Properties:

Linear error scale
Robust to outliers
Original units
Negated for optimization

Best for:

Robust evaluation
Interpretable errors
Outlier presence

NegMeanSquaredError ~

Negative mean squared error:

Formula: $- \frac{1}{n} \sum (y_{i} - \overset{y}{^}_{i})^{2}$

Properties:

Squared error scale
Outlier sensitive
Squared units
Negated for optimization

Best for:

Standard optimization
Large error penalty
Statistical analysis

NegRootMeanSquaredError ~

Negative root mean squared error:

Formula: $- \frac{1}{n} \sum (y_{i} - \overset{y}{^}_{i})^{2}$

Properties:

Original scale
Outlier sensitive
Interpretable units
Negated for optimization

Best for:

Standard reporting
Interpretable errors
Model comparison

NegMeanSquaredLogError ~

Negative mean squared logarithmic error:

Formula: $- \frac{1}{n} \sum (lo g (1 + y_{i}) - lo g (1 + \overset{y}{^}_{i}))^{2}$

Properties:

Relative error scale
For positive values
Sensitive to ratios
Negated for optimization

Best for:

Exponential growth
Relative differences
Positive predictions

NegMedianAbsoluteError ~

Negative median absolute error:

Formula: $- m e d ian (∣ y_{i} - \overset{y}{^}_{i} ∣)$

Properties:

Highly robust
Original scale
Outlier resistant
Negated for optimization

Best for:

Robust evaluation
Heavy-tailed errors
Outlier presence

NegMeanPoissonDeviance ~

Negative Poisson deviance:

Formula: $- 2 \sum (y_{i} lo g (\frac{y _{i}}{y ^ _{i}}) - (y_{i} - \overset{y}{^}_{i}))$

Properties:

For count data
Non-negative values
Poisson assumption
Negated for optimization

Best for:

Count prediction
Event frequency
Rate modeling

NegMeanGammaDeviance ~

Negative Gamma deviance:

Formula: $- 2 \sum (lo g (\frac{y _{i}}{y ^ _{i}}) + \frac{y _{i} - y ^ _{i}}{y ^ _{i}})$

Properties:

For positive continuous data
Constant CV assumption
Relative errors
Negated for optimization

Best for:

Positive continuous data
Multiplicative errors
Financial modeling

NegMeanAbsolutePercentageError ~

Negative mean absolute percentage error:

Formula: $- \frac{100}{n} \sum ∣ \frac{y _{i} - y ^ _{i}}{y _{i}} ∣$

Properties:

Percentage scale
Scale independent
For non-zero targets
Negated for optimization

Best for:

Relative performance
Scale-free comparison
Business metrics

D2AbsoluteErrorScore ~

D² score with absolute error:

Formula: $1 - \frac{\sum ∣ y _{i} - y ^ _{i} ∣}{\sum ∣ y _{i} - y ˉ ∣}$

Properties:

Range: (-∞, 1]
Robust version of R²
Linear error scale
Outlier resistant

Best for:

Robust evaluation
Non-normal errors
Alternative to R²

D2PinballScore ~

D² score with pinball loss:

Properties:

Quantile focus
Asymmetric errors
Risk assessment
Distribution modeling

Best for:

Quantile regression
Risk analysis
Asymmetric costs
Distribution tails

D2TweedieScore ~

D² score with Tweedie deviance:

Properties:

Compound Poisson-Gamma
Flexible dispersion
Mixed distributions
Insurance modeling

Best for:

Insurance claims
Mixed continuous-discrete
Compound distributions
Specialized modeling

Split

oneof

DefaultSplit

Standard train-test split configuration optimized for general classification tasks.

Configuration:

Test size: 20% (0.2)
Random seed: 98
Shuffling: Enabled
Stratification: Based on target distribution

Advantages:

Preserves class distribution
Provides reliable validation
Suitable for most datasets

Best for:

Medium to large datasets
Independent observations
Initial model evaluation

Splitting uses the ShuffleSplit strategy or StratifiedShuffleSplit strategy depending on the field stratified. Note: If shuffle is false then stratified must be false.

CustomSplit

Configurable train-test split parameters for specialized requirements. Allows fine-tuning of data division strategy for specific use cases or constraints.

Use cases:

Time series data
Grouped observations
Specific train/test ratios
Custom validation schemes

RandomState

u64

Random seed for reproducible splits. Ensures:

Consistent train/test sets
Reproducible experiments
Comparable model evaluations

Same seed guarantees identical splits across runs.

Shuffle

bool

true

Data shuffling before splitting. Effects:

true: Randomizes order, better for i.i.d. data
false: Maintains order, important for time series

When to disable:

Time dependent data
Sequential patterns
Grouped observations

TrainSize

f64

0.8

Proportion of data for training. Considerations:

Larger (e.g., 0.8-0.9): Better model learning
Smaller (e.g., 0.5-0.7): Better validation

Common splits:

0.8: Standard (80/20 split)
0.7: More validation emphasis
0.9: More training emphasis

Stratified

bool

false

Maintain class distribution in splits. Important when:

Classes are imbalanced
Small classes present
Representative splits needed

Requirements:

Classification tasks only
Cannot use with shuffle=false
Sufficient samples per class

Cv

oneof

DefaultCv

Standard cross-validation configuration using stratified 3-fold splitting.

Configuration:

Folds: 3
Method: StratifiedKFold
Stratification: Preserves class proportions

Advantages:

Balanced evaluation
Reasonable computation time
Good for medium-sized datasets

Limitations:

May be insufficient for small datasets
Higher variance than larger fold counts
May miss some data patterns

CustomCv

Configurable stratified k-fold cross-validation for specific validation requirements.

Features:

Adjustable fold count with NFolds determining the number of splits.
Stratified sampling
Preserved class distributions

Use cases:

Small datasets (more folds)
Large datasets (fewer folds)
Detailed model evaluation
Robust performance estimation

NFolds

u32

Number of cross-validation folds. Guidelines:

3-5: Large datasets, faster training
5-10: Standard choice, good balance
10+: Small datasets, thorough evaluation

Trade-offs:

More folds: Better evaluation, slower training
Fewer folds: Faster training, higher variance

Must be at least 2.

KfoldCv

K-fold cross-validation without stratification. Divides data into k consecutive folds for iterative validation.

Process:

Splits data into k equal parts
Each fold serves as validation once
Remaining k-1 folds form training set

Use cases:

Regression problems
Large, balanced datasets
When stratification unnecessary
Continuous target variables

Limitations:

May not preserve class distributions
Less suitable for imbalanced data
Can create biased splits with ordered data

NSplits

u32

Number of folds for cross-validation. Selection guide: Recommended values:

5: Standard choice (default)
3: Large datasets/quick evaluation
10: Thorough evaluation/smaller datasets

Trade-offs:

Higher values: More thorough, computationally expensive
Lower values: Faster, potentially higher variance

Must be at least 2 for valid cross-validation.

RandomState

u64

Random seed for fold generation when shuffling. Important for:

Reproducible results
Consistent fold assignments
Benchmark comparisons
Debugging and validation

Set specific value for reproducibility across runs.

Shuffle

bool

true

Whether to shuffle data before splitting into folds. Effects:

true: Randomized fold composition (recommended)
false: Sequential splitting

Enable when:

Data may have ordering
Better fold independence needed

Disable for:

Time series data
Ordered observations

StratifiedKfoldCv

Stratified K-fold cross-validation maintaining class proportions across folds.

Key features:

Preserves class distribution in each fold
Handles imbalanced datasets
Ensures representative splits

Best for:

Classification problems
Imbalanced class distributions
When class proportions matter

Requirements:

Classification tasks only
Sufficient samples per class
Categorical target variable

NSplits

u32

Number of stratified folds. Guidelines: Typical values:

5: Standard for most cases
3: Quick evaluation/large datasets
10: Detailed evaluation/smaller datasets

Considerations:

Must allow sufficient samples per class per fold
Balance between stability and computation time
Consider smallest class size when choosing

RandomState

u64

Seed for reproducible stratified splits. Ensures:

Consistent fold assignments
Reproducible results
Comparable experiments
Systematic validation

Fixed seed guarantees identical stratified splits.

Shuffle

bool

false

Data shuffling before stratified splitting. Impact:

true: Randomizes while maintaining stratification
false: Maintains data order within strata

Use cases:

true: Independent observations
false: Grouped or sequential data

Class proportions maintained regardless of setting.

ShuffleSplitCv

Random permutation cross-validator with independent sampling.

Characteristics:

Random sampling for each split
Independent train/test sets
More flexible than K-fold
Can have overlapping test sets

Advantages:

Control over test size
Fresh splits each iteration
Good for large datasets

Limitations:

Some samples might never be tested
Others might be tested multiple times
No guarantee of complete coverage

NSplits

u32

Number of random splits to perform. Consider: Common values:

5: Standard evaluation
10: More thorough assessment
3: Quick estimates

Trade-offs:

More splits: Better estimation, longer runtime
Fewer splits: Faster, less stable estimates

Balance between computation and stability.

RandomState

u64

Random seed for reproducible shuffling. Controls:

Split randomization
Sample selection
Result reproducibility

Important for:

Debugging
Comparative studies
Result verification

TestSize

f64

0.2

Proportion of samples for test set. Guidelines: Common ratios:

0.2: Standard (80/20 split)
0.25: More validation emphasis
0.1: More training data

Considerations:

Dataset size
Model complexity
Validation requirements

It must be between 0.0 and 1.0.

StratifiedShuffleSplitCv

Stratified random permutation cross-validator combining shuffle-split with stratification.

Features:

Maintains class proportions
Random sampling within strata
Independent splits
Flexible test size

Ideal for:

Imbalanced datasets
Large-scale problems
When class distributions matter
Flexible validation schemes

NSplits

u32

Number of stratified random splits. Guidelines: Recommended values:

5: Standard evaluation
10: Detailed analysis
3: Quick assessment

Consider:

Sample size per class
Computational resources
Stability requirements

RandomState

u64

Seed for reproducible stratified sampling. Ensures:

Consistent class proportions
Reproducible splits
Comparable experiments

Critical for:

Benchmarking
Research studies
Quality assurance

TestSize

f64

0.2

Fraction of samples for stratified test set. Best practices: Common splits:

0.2: Balanced evaluation
0.3: More thorough testing
0.15: Preserve training size

Consider:

Minority class size
Overall dataset size
Validation objectives

It must be between 0.0 and 1.0.

Time Series cross-validator. Provides train/test indices to split time series data samples that are observed at fixed time intervals, in train/test sets. It is a variation of k-fold which returns first k folds as train set and the k + 1th fold as test set. Note that unlike standard cross-validation methods, successive training sets are supersets of those that come before them. Also, it adds all surplus data to the first training partition, which is always used to train the model. Key features:

Maintains temporal dependence
Expanding window approach
Forward-chaining splits
No future data leakage

Use cases:

Sequential data
Financial forecasting
Temporal predictions
Time-dependent patterns

Note: Training sets are supersets of previous iterations.

NSplits

u32

Number of temporal splits. Considerations: Typical values:

5: Standard forward chaining
3: Limited historical data
10: Long time series

Impact:

Affects training window growth
Determines validation points
Influences computational load

MaxTrainSize

u64

Maximum size of training set. Should be strictly less than the number of samples. Applications:

0: Use all available past data
>0: Rolling window of fixed size

Use cases:

Limit historical relevance
Control computational cost
Handle concept drift
Memory constraints

TestSize

u64

Number of samples in each test set. When 0:

Auto-calculated as n_samples/(n_splits+1)
Ensures equal-sized test sets

Considerations:

Forecast horizon
Validation requirements
Available future data

Gap

u64

Number of samples to exclude from the end of each train set before the test set.Gap between train and test sets. Uses:

Avoid data leakage
Model forecast lag
Buffer periods

Common scenarios:

0: Continuous prediction
>0: Forward gap for realistic evaluation
Match business forecasting needs