Estimation of Linear Regression with the Dimensional Analysis Method (original) (raw)

Dimensional Analysis and Its Applications in Statistics

Dimensional Analysis (DA) is a well-developed widely-employed methodology in the physical and engineering sciences. The application of dimensional analysis in statistics leads to three ad-vantages: (1) the reduction of the number of potential causal factors that we need to consider, (2) the analytical insights into the relations among variables that it generates, and (3) the scala-bility of results. The formalization of the dimensional analysis method in statistical design and analysis gives a clear view of its generality and overlooked significance. In this paper, we first provide general procedures for dimensional analysis prior to statistical design and analysis. We illustrate the use of dimensional analysis with three practical examples. In the first example, we demonstrate the basic dimensional analysis process in connection with a study of factors that affect vehicle stopping distance. The second example integrates dimensional analysis into the regression analysis of the pine ...

Dimensional analysis of models and data sets

American Journal of Physics, 2003

The angle of the line, Џnond, the dependent variable. ͑2͒ The time, tЏm 0 l 0 t 1 , the independent variable. ͑3͒ The mass of the bob, M Џm 1 l 0 t 0 , a parameter. ͑4͒ The length of the line, LЏm 0 l 1 t 0 , a parameter. ͑5͒ The acceleration of gravity, gЏm 0 l 1 t Ϫ2 , a parameter. ͑6͒ The initial angle, 0 Џnond, a parameter.

Use of dimensional analysis in social science research

Socio-economic Planning Sciences, 1981

Dimensional analysis is a branch of applied mathematics which is only now beginning to emerge as an important methodology in social and behavioral science research. Unlike regression analysis, dimensional analysis "discovers" the functional relationship in the data from the defined dimensions; hence its ability to extend regression analysis in developing predictive models makes it particularly useful in social science research. In addition, many of the limitations with the use of regression analysis in social science which limit its usefulness in applications, such as multicollinearity, normality assumptions, homoscedasticity assumptions, etc. are not limitations in dimensional analysis. In this study, the application of dimensional analysis to the study of population migrations, specifically white flight in a large urban school district, is presented. Finally, a review of relevant literature and the use of dimensional analysis in other social science areas are presented.

Bias and variance reduction in estimation of model dimension

Proceedings of the American Mathematical Society, 1994

The problem of estimating the number of regressors to include in a linear regression model is considered. Estimators based on the final prediction error and Akaike's criterion frequently have large positive bias. Shrinkage correction factors and bootstrapping are used to produce new estimators with reduced bias. The asymptotic bias and mean-squared errors of these estimators are derived analytically. Finite-sample estimates are obtained by simulation.

The ABC of linear regression analysis: What every author and editor should know

Regression analysis is a widely used statistical technique to build a model from a set of data on two or more variables. Linear regression is based on linear correlation, and assumes that change in one variable is accompanied by a proportional change in another variable. Simple linear regression, or bivariate regression, is used for predicting the value of one variable from another variable (predictor); however, multiple linear regression, which enables us to analyse more than one predictor or variable, is more commonly used. This paper explains both simple and multiple linear regressions illustrated with an example of analysis and also discusses some common errors in presenting the results of regression, including inappropriate titles, causal language, inappropriate conclusions, and misinterpretation.

An Insight of Linear Regression Analysis

Scientific Research Journal

Regression models are developed in various field of applications to help researchers to predict certain variables based on other predictor variables. The dependent variables in the regression model are estimated by a number of independent variables. Model utility test is a hypothesis testing procedure in regression to verify if there is a useful relationship between the dependent variable and the independent variable. The hypothesis testing procedure that involves p-value is commonly used in model utility test. A new technique that involves coefficient of determination R2 in model utility test is developed in this paper. The effectiveness of the model utility test in testing the significance of regression model is evaluated using simple linear regression model with the significance level α = 0.01, 0.025 and 0.05. The study in this paper shows that a regression model that is declared to be a significant model by using model utility test, however it fails to guarantee a strong linear ...

Data visualization and dimension reduction for metric-valued response regression

arXiv (Cornell University), 2023

Data visualization and dimension reduction for regression between a general metric space-valued response and Euclidean predictors is proposed. Current Fréchét dimension reduction methods require that the response metric space be continuously embeddable into a Hilbert space, which imposes restriction on the type of metric and kernel choice. We relax this assumption by proposing a Euclidean embedding technique which avoids the use of kernels. Under this framework, classical dimension reduction methods such as ordinary least squares and sliced inverse regression are extended. An extensive simulation experiment demonstrates the superior performance of the proposed method on synthetic data compared to existing methods where applicable. The real data analysis of factors influencing the distribution of COVID-19 transmission in the U.S. and the association between BMI and structural brain connectivity of healthy individuals are also investigated.

Using linear smoothers to assess the structural dimension of regressions

2001

Sliced Inverse Regression (Li (1991)) is a simple nonparametric estima- tion method for the structural dimension of a regression, that is, for the dimension of the linear subspace spanned by projections of the multidimensional regressor vec- tor X that contains part or all of the modelling information about the regression of a random variable Y on X. In this paper, the nonparametric estimation method is extended to include the family of linear smoothers. No restrictions are placed on the distribution of the regressors except for the linearity condition and exis- tence of second moments. An asymptotic chi-square test for dimension is obtained. Theoretical results are illustrated with a small comparative simulation study.