Error-in-variables estimation using Compact Genetic Algorithms in Julia.

The package initially implements the CGA method proposed in the given references. Since Orthogonal regression and the Deming estimator are related to the subject, these methods are also implemented.

The plan is to implement some other methods in this area.

Suppose the linear regression model is

$$ y = \beta_0 + \beta_1 x^* + \varepsilon $$

where $y$ is n-vector of the response variable, $\beta_0$ and $\beta_1$ are unknown regression parameteres, $\varepsilon$ is the iid. error term, $x^*$ is the unknown n-vector of the independent variable, and $n$ is the number of observations.

We call $x^*$ unknown because in some situations the true values of the variable cannot be visible or directly observable, or observable with some measurement error. Now suppose that $x$ is the observable version of the true values and it is defined as

$$ x = x^* + \delta $$

where $\delta$ is the measurement error and $x$ is the erroneous version of the true $x^*$. If the estimated model is

$$ \hat{y} = \hat{\beta_0} + \hat{\beta_1}x $$

then the ordinary least squares (OLS) estimates are no longer unbiased and even consistent.

Eive-cga is an estimator devised for this problem. The aim is to reduce the errors-in-variable bias with some cost of increasing the variance. At the end, the estimator obtains lower Mean Square Error (MSE) values defined as

$$ MSE(\hat{\beta_1}) = Var(\hat{\beta_1}) + Bias^2(\hat{\beta_1}) $$

for the Eive-cga estimator. For more detailed comparisons, see the original paper given in the Citation part.

For the single variable case

julia> eive(dirtyx = dirtyx, y = y, otherx = nothing) 

and for the multiple regression

julia> eive(dirtyx = dirtyx, y = y, otherx = matrixofotherx) 

Note that the method assumes there is only one erroneous variable in the set of independent variables.

Lets generate data from the model $y = 20 + 10x^* + \varepsilon$

import Random
using ErrorsInVariables

rng = Random.MersenneTwister(1234)
n = 30
deltax = randn(rng, n) * sqrt(3.0)
cleanx = randn(rng, n) * sqrt(7.0)
e = randn(rng, n) * sqrt(5.0)
y = 20.0 .+ 10.0 .* cleanx .+ e
dirtyx = cleanx + deltax

and assume that

$$ x^* $$

is unobservable and it is observed as

$$ x = x^* + \delta$$

. We can calculate an unbiased estimate of the slope parameter using

eive(dirtyx = dirtyx, y = y, otherx = nothing) 

The result is

EiveResult([20.28458307772922, 9.456757289676714])

whereas OLS estimates are

julia> X = hcat(ones(n), dirtyx);

julia> X \ y
2-element Vector{Float64}:
 17.94867860059858
  5.8099584879737876

and clearly biased towards to zero.

@article{satman2015reducing,
  title={Reducing errors-in-variables bias in linear regression using compact genetic algorithms},
  author={Satman, M Hakan and Diyarbakirlioglu, Erkin},
  journal={Journal of Statistical Computation and Simulation},
  volume={85},
  number={16},
  pages={3216--3235},
  year={2015},
  publisher={Taylor \& Francis}
}

@article{Satman_Diyarbakirlioglu_2024, 
title={A Solution to Errors-in-variables Bias in Multivariate Linear Regression using Compact Genetic Algorithms},
volume={4}, 
url={https://journals.gen.tr/index.php/jame/article/view/2293}, 
DOI={10.53753/jame.2293}, 
number={1}, 
journal={JOURNAL OF APPLIED MICROECONOMETRICS}, 
author={Satman, Mehmet Hakan and Diyarbakırlıoğlu, Erkin}, 
year={2024}, month={Jun.}, 
pages={31–64} 
}