*Andrew J. Connolly, Jacob T. VanderPlas, Alexander Gray, Andrew J. Connolly, Jacob T. VanderPlas, and Alexander Gray*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691151687
- eISBN:
- 9781400848911
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691151687.003.0008
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Regression is a special case of the general model fitting and selection procedures discussed in chapters 4 and 5. It can be defined as the relation between a dependent variable, y, and a set of ...
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Regression is a special case of the general model fitting and selection procedures discussed in chapters 4 and 5. It can be defined as the relation between a dependent variable, y, and a set of independent variables, x, that describes the expectation value of y given x: E [y¦x]. The purpose of obtaining a “best-fit” model ranges from scientific interest in the values of model parameters (e.g., the properties of dark energy, or of a newly discovered planet) to the predictive power of the resulting model (e.g., predicting solar activity). This chapter starts with a general formulation for regression, list various simplified cases, and then discusses methods that can be used to address them, such as regression for linear models, kernel regression, robust regression and nonlinear regression.Less

Regression is a special case of the general model fitting and selection procedures discussed in chapters 4 and 5. It can be defined as the relation between a dependent variable, *y*, and a set of independent variables, *x*, that describes the expectation value of *y* given *x*: *E* [*y*¦*x*]. The purpose of obtaining a “best-fit” model ranges from scientific interest in the values of model parameters (e.g., the properties of dark energy, or of a newly discovered planet) to the predictive power of the resulting model (e.g., predicting solar activity). This chapter starts with a general formulation for regression, list various simplified cases, and then discusses methods that can be used to address them, such as regression for linear models, kernel regression, robust regression and nonlinear regression.

*Theodore R. Holford*

- Published in print:
- 2002
- Published Online:
- September 2009
- ISBN:
- 9780195124408
- eISBN:
- 9780199864270
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195124408.003.0006
- Subject:
- Public Health and Epidemiology, Public Health, Epidemiology

This chapter considers the problem of fitting binary response models to data in which there are multiple regressor variables that may be either discrete or continuous in nature. The linear logistic ...
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This chapter considers the problem of fitting binary response models to data in which there are multiple regressor variables that may be either discrete or continuous in nature. The linear logistic model, the most commonly used model for this type of response, provides estimates of parameters that are assumed to have linear effects on the log odds ratio, thus yielding values that can be interpreted as log odds ratios. The more flexible generalized linear models family that can readily be adapted for fitting many alternative forms for the relationship between exposure and disease outcome are considered: the log-linear hazard, the probit model, the linear odds model, and the linear power of the odds model. Exercises are provided at the end of the chapter.Less

This chapter considers the problem of fitting binary response models to data in which there are multiple regressor variables that may be either discrete or continuous in nature. The linear logistic model, the most commonly used model for this type of response, provides estimates of parameters that are assumed to have linear effects on the log odds ratio, thus yielding values that can be interpreted as log odds ratios. The more flexible generalized linear models family that can readily be adapted for fitting many alternative forms for the relationship between exposure and disease outcome are considered: the log-linear hazard, the probit model, the linear odds model, and the linear power of the odds model. Exercises are provided at the end of the chapter.

*Ray Huffaker, Marco Bittelli, and Rodolfo Rosa*

- Published in print:
- 2017
- Published Online:
- February 2018
- ISBN:
- 9780198782933
- eISBN:
- 9780191826153
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198782933.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Phenomenological models mathematically describe relationships among empirically observed phenomena without attempting to explain underlying mechanisms. Within the context of NLTS, phenomenological ...
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Phenomenological models mathematically describe relationships among empirically observed phenomena without attempting to explain underlying mechanisms. Within the context of NLTS, phenomenological modeling goes beyond phase space reconstruction to extract equations governing real-world system dynamics from a single or multiple observed time series. Phenomenological models provide several benefits. They can be used to characterize the dynamics of variable interactions; for example, whether an incremental increase in one variable drives a marginal increase/decrease in the growth rate of another, and whether these dynamic interactions follow systematic patterns over time. They provide an analytical framework for data driven science still searching for credible theoretical explanation. They set a descriptive standard for how the real world operates so that theory is not misdirected in explaining fanciful behavior. The success of phenomenological modeling depends critically on selection of governing parameters. Model dimensionality, and the time delays used to synthesize dynamic variables, are guided by statistical tests run for phase space reconstruction. Other regression and numerical integration parameters can be set on a trial and error basis within ranges providing numerical stability and successful reproduction of empirically-detected dynamics. We illustrate phenomenological modeling with solutions of the Lorenz model so that we can recognize the dynamics that need to be reproduced.Less

Phenomenological models mathematically describe relationships among empirically observed phenomena without attempting to explain underlying mechanisms. Within the context of NLTS, phenomenological modeling goes beyond phase space reconstruction to extract equations governing real-world system dynamics from a single or multiple observed time series. Phenomenological models provide several benefits. They can be used to characterize the dynamics of variable interactions; for example, whether an incremental increase in one variable drives a marginal increase/decrease in the growth rate of another, and whether these dynamic interactions follow systematic patterns over time. They provide an analytical framework for data driven science still searching for credible theoretical explanation. They set a descriptive standard for how the real world operates so that theory is not misdirected in explaining fanciful behavior. The success of phenomenological modeling depends critically on selection of governing parameters. Model dimensionality, and the time delays used to synthesize dynamic variables, are guided by statistical tests run for phase space reconstruction. Other regression and numerical integration parameters can be set on a trial and error basis within ranges providing numerical stability and successful reproduction of empirically-detected dynamics. We illustrate phenomenological modeling with solutions of the Lorenz model so that we can recognize the dynamics that need to be reproduced.