pyclsp 1.1.0

Modular Two-Step Convex Optimization Estimator for Ill-Posed Problems

CLSP — Convex Least Squares Programming

The Convex Least Squares Programming (CLSP) estimator is a two-step method for solving underdetermined, ill-posed, or structurally constrained least-squares problems. It combines pseudoinverse-based estimation with convex-programming correction (e.g., Lasso, Ridge, Elastic Net) to ensure numerical stability, structural coherence, and enhanced interpretability.

Installation

pip install clsp

Quick Example

import numpy as np
import matplotlib.pyplot as plt
from clsp import CLSP

# Example allocation problem: row sums (first 5) + col sums (last 5)
b = np.array([22, 23, 26, 27, 21, 28, 24, 22, 24, 21], dtype=float)

# Initialize estimator
model = CLSP()

# Solve the system (allocation problem)
result = model.solve(problem='ap', b=b, m=5, p=5, final=True)
print("Estimated matrix:")
print(result.x)

# Access diagnostics
print("NRMSE:", model.nrmse)

# Correlogram: RMSA sensitivity by constraint
corr = model.corr()
plt.figure(figsize=(8, 4))
plt.grid(True, linestyle="--", alpha=0.6)
plt.bar(range(len(corr["rmsa_i"])), corr["rmsa_i"])
plt.xlabel("Constraint index")
plt.ylabel("RMSA (row deletion effect)")
plt.title("CLSP Correlogram")
plt.tight_layout()
plt.show()

# Hypothesis test
print("t-test on NRMSE:", model.ttest())

User Reference

For comprehensive information on the estimator’s capabilities, advanced configuration options, and implementation details, please refer to the docstrings provided in each of the individual .py source files. These docstrings contain complete descriptions of available methods, their parameters, expected input formats, and output structures.

The CLSP Class

self.__init__()

Stores the solution, goodness-of-fit statistics, and ancillary parameters.

The class has three core methods: solve(), corr(), and ttest().

Selected attributes:
self.A : np.ndarray
design matrix A = [C | S; M | Q], where Q is either a zero matrix or S_residual.

self.b : np.ndarray
vector of the right-hand side.

self.zhat : np.ndarray
vector of the first-step estimate.

self.r : int
number of refinement iterations performed in the first step.

self.z : np.ndarray
vector of the final solution. If the second step is disabled, it equals self.zhat.

self.x : np.ndarray
m x p matrix or vector containing the variable component of z.

self.y : np.ndarray
vector containing the slack component of z.

self.kappaC : float
spectral κ() for C_canon.

self.kappaB : float
spectral κ() for B = C_canon^+A.

self.kappaA : float
spectral κ() for A.

self.rmsa : float
total root mean square alignment (RMSA).

self.r2_partial : float
R^2 for the M block in A.

self.nrmse : float
mean square error calculated from A and normalized by standard deviation (NRMSE).

self.nrmse_partial : float
mean square error calculated from the M block in A and normalized by standard deviation (NRMSE).

self.z_lower : np.ndarray
lower bound of the diagnostic interval (confidence band) based on κ(A).

self.z_upper : np.ndarray
upper bound of the diagnostic interval (confidence band) based on κ(A).

self.x_lower : np.ndarray
lower bound of the diagnostic interval (confidence band) based on κ(A).

self.x_upper : np.ndarray
upper bound of the diagnostic interval (confidence band) based on κ(A).

self.y_lower : np.ndarray
lower bound of the diagnostic interval (confidence band) based on κ(A).

self.y_upper : np.ndarray
upper bound of the diagnostic interval (confidence band) based on κ(A).

Solver Method: solve()

self.solve(problem, C, S, M, b, m, p, i, j, zero_diagonal, r, Z, tolerance, iteration_limit, final, alpha)

Solves the Convex Least Squares Programming (CLSP) problem.

This method performs a two-step estimation: (1) a pseudoinverse-based solution using either the Moore–Penrose or Bott–Duffin inverse, optionally iterated for refinement; (2) a convex-programming correction using Lasso, Ridge, or Elastic Net regularization (if enabled).

Parameters:
problem : str, optional
Structural template for matrix construction. One of:
- 'ap' or 'tm' : allocation (transaction) matrix problem (AP).
- 'cmls' or 'rp' : constrained-model least squares (regression) problem.
- anything else: general CLSP problem (user-defined C and/or M).

C, S, M : np.ndarray or None
Blocks of the design matrix A = [C | S; M | Q]. If C and/or M are provided, the matrix A is constructed accordingly (please note that for AP, C is constructed automatically and known values are specified in M).

b : np.ndarray or None
Right-hand side vector. Must have as many rows as A (please note that for AP, it should start with row sums). Required.

m, p : int or None
Dimensions of X ∈ ℝ^{m×p}, relevant for AP.

i, j : int, default = 1
Grouping sizes for row and column sum constraints in AP.

zero_diagonal : bool, default = False
If True, enforces structural zero diagonals.

r : int, default = 1
Number of refinement iterations for the pseudoinverse-based estimator.

Z : np.ndarray or None
A symmetric idempotent matrix (projector) defining the subspace for Bott–Duffin pseudoinversion. If None, the identity matrix is used, reducing the Bott–Duffin inverse to the Moore–Penrose case.

tolerance : float, default = square root of machine epsilon
Convergence tolerance for NRMSE change between refinement iterations.

iteration_limit : int, default = 50
Maximum number of iterations allowed in the refinement loop.

final : bool, default = True
If True, a convex programming problem is solved to refine zhat. The resulting solution z minimizes a weighted L1/L2 norm around zhat subject to Az = b.

alpha : float, default = 1.0
Regularization parameter (weight) in the final convex program:
- α = 0: Lasso (L1 norm)
- α = 1: Tikhonov Regularization/Ridge (L2 norm)
- 0 < α < 1: Elastic Net

*args, **kwargs : optional
CVXPY arguments passed to the CVXPY solver.

Returns:
self

Correlogram Method: corr()

self.corr(reset, threshold)

Computes the structural correlogram of the CLSP constraint part.

This method performs a row-deletion sensitivity analysis on the canonical constraint matrix [C | S], denoted as C_canon, and evaluates the marginal effect of each constraint row on numerical stability, angular alignment, and estimator sensitivity.

For each row i in C_canon, it computes:
- The Root Mean Square Alignment (RMSA_i) with all other rows j ≠ i.
- The change in condition numbers κ(C), κ(B), and κ(A) when row i is deleted.
- The effect on estimation quality: changes in nrmse, zhat, z, and x when row i is deleted.

Additionally, it computes the total rmsa statistic across all rows, summarizing the overall angular alignment of C_canon.

Parameters:
reset : bool, default = False
If True, forces recomputation of all diagnostic values (the results are preserved for eventual reproduction after the method is called).

threshold : float, default = 0
If positive, limits the output to constraints with RMSA_i ≥ threshold.

Returns:
dict of list
A dictionary containing per-row diagnostic values:
{
"constraint" : [1, 2, ..., k], # 1-based indices
"rmsa_i" : list of RMSA_i values,
"rmsa_dkappaC" : list of Δκ(C) after deleting row i,
"rmsa_dkappaB" : list of Δκ(B) after deleting row i,
"rmsa_dkappaA" : list of Δκ(A) after deleting row i,
"rmsa_dnrmse" : list of Δnrmse after deleting row i,
"rmsa_dzhat" : list of Δzhat after deleting row i,
"rmsa_dz" : list of Δz after deleting row i,
"rmsa_dx" : list of Δx after deleting row i,
}

T-Test Method: ttest

self.ttest(reset, sample_size, seed, distribution)

Performs a Monte Carlo-based one- or two-sided t-test on the NRMSE statistic.

This function simulates right-hand side vectors b using a user-defined or default distribution and recomputes the estimator for every new b. It tests whether the observed NRMSE significantly deviates from the null distribution (under H₀) of simulated NRMSE values. The quality of the test depends on the size of the simulated sample.

Parameters:
reset : bool, default = False
If True, forces recomputation of the NRMSE null distribution (under H₀) (the results are preserved for eventual reproduction after the method is called).

sample_size : int, default = 50
Size of the Monte Carlo simulated sample under H₀.

seed : int or None, optional
Optional random seed to override the default.

distribution : str or None, default = ’normal’
Distribution for generating simulated b vectors. One of (standard): 'normal', 'uniform', or 'laplace'.

Returns:
dict
Dictionary with test results and null distribution statistics:
{
'p_one_left' : P(nrmse ≤ null mean),
'p_one_right' : P(nrmse ≥ null mean),
'p_two_sided' : 2-sided t-test p-value,
'nrmse' : observed value,
'mean_null' : mean of the null distribution (under H₀),
'std_null' : standard deviation of the null distribution (under H₀)
}

Bibliography

To be added.

License

MIT License — see the LICENSE file.