ISYE 6414 - Midterm 1 Prep (2023/2024) Already Graded A

ISYE 6414 - Midterm 1 Prep (2023/2024) Already Graded A If λ=1 we do not transform non-deterministic Regression analysis is one of the simplest ways we have in statistics to investigate the relationship between two or more variables in a ___ way random The response variable is a ___ variable, because it varies with changes in the predicting variable, or with other changes in the environment fixed The predicting variable is a ___ variable. It is set fixed, before the response is measured. simple linear regression regression analysis involving one independent variable and one dependent variable in which the relationship between the variables is approximated by a straight line Multiple Linear Regression A statistical method used to model the relationship between one dependent (or response) variable and two or more independent (or explanatory) variables by fitting a linear equation to observed data polynomial regression a regression model which does not assume a linear relationship; a curvilinear correlation coefficient is computed (we can think of X and X-squared as two different predicting variables) three objectives in regression 1) Prediction 2) Modeling 3) Testing hypothesis Prediction We want to see how the response variable behaves in different settings. For example, for a different location, if we think about a geographic prediction, or in time, if we think about temporal prediction Modeling modeling the relationship between the response variable and the explanatory variables, or predicting variables Testing hypotheses of association relationships useful representation of reality We do not believe that the linear model represents a true representation of reality. Rather, we think that, perhaps, it provides a ___ β0 intercept parameter (the value at which the line intersects the y-axis) β1 slope parameter (slope of the line we are trying to fit) epsilon (ε) is the deviance of the data from the linear model to find β0 and β1 to find the line that describes a linear relationship, such that we fit this model. simple linear regression data structure pairs of data consisting of a value for the response variable,and a value for the predicting variable. And we have n such pairs modeling framework for the simple linear regression: 1) identifying data structure 2) clearly stating the model assumptions linear regression assumptions 1) linearity 2) constant variance assumption 3) independence assumption linearity assumption mean zero assumption, means that the expected value of the errors is zero. A violation of this assumption will lead to difficulties in estimating β0, and means that your model does not include a necessary systematic component. constant variance assumption which means that the variance (σ^2) of the error terms or deviances is constant for the given population. A violation of this assumption means that the estimates are not as efficient as they could be in estimating the true parameters Independence Assumption which means that the deviances are independent random variables. Violation of this assumption can lead to misleading assessments of the strength of the regression. normality assumption errors (ε) are normally distributed. This is needed for statistical inference, for example, confidence or prediction intervals, and hypothesis testing. If this assumption is violated, hypothesis tests and confidence and prediction intervals can be misleading.v third parameter the variance of the error terms (σ^2) One approach is to minimize the sum of squared residuals or errors with respect to β0 and β1. This translated into finding the line such that the total squared deviances from the line is minimum. How can we get estimates of the regression coefficients or parameters in linear regression analysis? fitted values to be the regression line where the parameters are replaced by the estimated values of the parameters. Residuals are simply the difference between observed response and fitted values, and they are proxies of the error terms in the regression model MSE