A non-iterated general implementation of the LPLS estimator for cOLS, TM, and custom cases

The program implements the LPLS (linear programming through least squares) estimator with the help of the Moore-Penrose inverse (pseudoinverse), calculated using singular value decomposition (SVD), with emphasis on the estimation of OLS constrained in values (cOLS), Transaction Matrix (TM), and custom (user-defined) cases. The pseudoinverse offers a unique minimum-norm least-squares solution, which is the best linear unbiased estimator (BLUE); see Albert (1972, Chapter VI). (Over)determined problems are accompanied by regression analysis, which is feasible in their case. For such and especially all remaining cases, a Monte Carlo-based ttest of mean NRMSE (normalized by the standard deviation of the RHS) is performed, the sample being drawn from a uniform or user-provided distribution (via a Python function).

OLS constrained in values (cOLS) is an estimation based on constraints in the model and/or data but not in parameters. Typically, such models are of size ≤ kN, where N is the number of observations, since the number of their constraints may vary in the LHS (e.g., level, derivatives, etc.).

Example of a cOLS problem:
Estimate the trend and the cyclical component of a country's GDP given the textbook or any other definition of its peaks, troughs, and saddles. For a pre-LPLS approach to this problem, see (Bolotov, 2014).

Transaction Matrix (TM) of size (M x N) is a formal model of interaction (allocation, assignment, etc.) between M and N elements in any imaginable system, such as intercompany transactions (netting tables), industries within/between economies (input-output tables), cross-border trade/investment (trade/investment matrices), etc., where row and column sums are known, but individual elements of the TM may not be:

• a netting table is a type of TM where M = N and the elements are subsidiaries of a MNC;
• an input-output table (IOT) is a type of TM where M = N and the elements are industries;
• a matrix of trade/investment is a type of TM where M = N and the elements are countries or (macro)regions, where diagonal elements may be equal to zero;
• a country-product matrix is a type of TM where M ≠ N and the elements are of different types;
  ...

Example of a TM problem:
Estimate the matrix of trade/investment with/without zero diagonal elements, the country shares in which are unknown. For a pre-LPLS approach to this problem, see (Bolotov, 2015).

Methods and formulas:

The LP problem in the LPLS estimator is a matrix equation a @ x = b, loosely based on the structure of the Simplex tableau, where a consists of coefficients for CONSTRAINTS, LP-type CHARACTERISTIC and/or SPECIFIC, and for SLACK/SURPLUS VARIABLES (the upper part) as well as for the MODEL (the lower part), as illustrated in Figure 1. Each part of a can be omitted to accommodate a particular case:

• cOLS problems require SPECIFIC CONSTRAINTS, no LP-type CHARACTERISTIC CONSTRAINTS, and a MODEL;
• TM requires LP-type CHARACTERISTIC CONSTRAINTS, no SPECIFIC CONSTRAINTS, and an optional MODEL;
• SLACK/SURPLUS VARIABLES are included only for inequality constraints and should be set to 1 or -1;
  ...

Figure 1: Matrix equation a @ x = b

Source: self-prepared

The solution to the equation, x = pinv(a) @ b, is estimated with the help of SVD and is a minimum-norm least-squares generalized solution if the rank of a is not full. To check if a is within the computational limits, its (maximum) dimensions can be calculated using the formulas:

• (k * N) x (K + K*) cOLS - without slack/surplus variables;
• (k * N) x (K + K* + l) cOLS - with slack/surplus variables;
• (M + N) x (M * N) TM - without slack/surplus variables;
• (M + N) x (M * N + l) TM - with slack/surplus variables;
• M x N custom - without slack/surplus variables;
• M x (N + l) custom - with slack/surplus variables;

where, in cOLS problems, K is the number of independent variables in the model (including the constant), K* is the number of eventual extra variables in CONSTRAINTS, and N is the number of observations; in TM, M and N are the dimensions of the matrix; and in custom cases, M and N or M x (N + l) are the dimensions of a.

Parameters:

Type of the LP problem

• cols=False : bool, OLS constrained in values (cOLS)
• tm=False : bool, Transaction Matrix (TM), options cols and tm are mutually exclusive (not specifying any of them equals custom)
• rhs=np.empty((0,0)) : array_like, right-hand side of the problem (rowsums first for TM problems)

Constructing the LHS

• model=np.empty((0,0)) : array_like, the MODEL part of a, including an eventual user-provided constant as a matrix column in cOLS (see Methods and formulas)
• constraints=np.empty((0,0)) : array_like, the CONSTRAINTS part of a
• slackvars=np.empty((0,0)) : array_like, the SLACK/SURPLUS VARIABLES part of a
• zero_diagonal=False : bool, set all the diagonal elements of a to 0

SVD-based estimation

• tolerance=None : float, scipy.linalg.lstsq's cutoff for `small' singular values; used to determine effective rank of a. Singular values smaller than cond * largest_singular_value are considered zero
• level=95 : int, confidence level (by default: 95)

Monte-Carlo-based t-test

• seed=123456789 : int, random-number seed, # is any number between 0 and 2^31-1 (or 2,147,483,647) (by default: 123456789)
• iterate=300 : int, number of iterations, # must be divisible by 50 (by default: 300)
• distribution=runiform: fucntion, random-variable generating function, name of an earlier declared Python object returning an array_like (r x c) with two arguments, real scalars r and c (by default: lppinv.runiform, see Examples on how to pass np.random.uniform to lppinv)
• mc=True : bool, skip the Monte Carlo-based t-test
• trace=True: bool, hide any output with the exception of dots

Results:

class lppinv.LPpinvResult(

• OLSResults : statsmodels.regression.linear_model.OLSResults or None, statsmodels' results class for for an OLS model
• TtestResult : (scipy.stats._result_classes.TtestResult, scipy.stats._result_classes.TtestResult, scipy.stats._result_classes.TtestResult) or (None, None, None), scipy's result of a t-test
• solution : array_like, the solution obtained from the LPLS estimator
• a : array_like, the a matrix
• nrmse : float, root mean square error normalized by the standard deviation of `b'
• r2_c : float or np.nan, R-squared for CONSTRAINTS in TM

)

Errors:

• LPpinvError : lppinv.LPpinvError, specific Python-exception-derived object raised by lppinv functions
• LinAlgError : scipy.linalg.LinAlgError, generic Python-exception-derived object raised by linalg functions

Examples:

import numpy  as np
import lppinv as lp

# cOLS problem
print('\n', lp.solve(
    cols=True,
    rhs=np.array([[0, 0, 0, 0, -1, 0.2, 0.9, 2.1]]).T,
    constraints=np.array([np.ones(4), [0, 1, 2, 3], [0, 5, 2, 8]]).T,
    model=np.array([np.ones(4), [0, 1, 2, 3], [0, 5, 2, 8]]).T,
    slackvars=np.array([[-1, 1, -1, 0]]).T,
    mc=False
))

# TM problem
def rnormal(r=None, c=None):
    return (np.random.normal(loc=0.0, scale=1.0, size=(r, c)))

print('\n', lp.solve(
    tm=True,
    rhs=np.array([[4, 5, 3, 4, 6], [1, 2, 0, np.nan, np.nan]]).T,
    zero_diagonal=True,
    distribution=rnormal
))

# custom problem
print('\n', lp.solve(
    rhs=np.array([[2, 3, 9], [5, 7, 9]]).T,
    model=np.vstack([[0, 1, 1], [1, 0, 1], [1, 1, 0]]),
    tolerance=10**-10,
    trace=False
))

References:

1. Albert, A., 1972. Regression And The Moore-Penrose Pseudoinverse. New York: Academic Press.

2. Bolotov, I. 2014. Modeling of Time Series Cyclical Component on a Defined Set of Stationary Points and its Application on the US Business Cycle. [Paper presentation]. The 8th International Days of Statistics and Economics: Prague. https://msed.vse.cz/msed_2014/article/348-Bolotov-Ilya-paper.pdf

3. Bolotov, I. 2015. Modeling Bilateral Flows in Economics by Means of Exact Mathematical Methods. [Paper presentation]. The 9th International Days of Statistics and Economics: Prague. https://msed.vse.cz/msed_2015/article/111-Bolotov-Ilya-paper.pdf

PS Please also check the Web of Science (WoS) for new research on LPLS.