In statistics, linear regression is a linear approach to modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
If the goal is prediction, forecasting, or error reduction, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables. After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response.
If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the "lack of fit" in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares cost function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.
Consider this situation. There is an exam designed in such a way that it appears that the pass/failure rate of the exam has a linear relationship to the age of the exam taker. The older the test taker, the higher the pass rate. I'm not interested in the exact scores of the exam, only pass or...
Simple linear regression:
Y = β0 + β1 *X + ε , where ε is random error
Fitted (predicted) value of Y for each X is:
^
Y = b0 + b1 *X (e.g. Y hat = 7.2 + 2.6 X)
Consider
^
X = b0' + b1' *Y
[the b0,b1,b0', and b1' are least-square estimates of the β's]
Prove whether or not...
1) "Simple linear regression model:
Y = β0 + β1X + ε
E(Y) = β0 + β1X
A linear model means that it is linear in β's, and not necessarily a linear function of X.
The independent variable X could be W2 or ln(W), and so on, for some other independent variable W."
I have some trouble...
1) "Simple linear regression model: Yi = β0 + β1Xi + εi , i=1,...,n where n is the number of data points, εi is random error
We want to estimate β0 and β1 based on our observed data. The estimates of β0 and β1 are denoted by b0 and b1, respectively."
I don't understand the difference...
1) "In regression models, there are two types of variables:
X = independent variable
Y = dependent variable
Y is modeled as random.
X is sometimes modeled as random and sometimes it has fixed value for each observation."
I don't understand the meaning of the last line. When is X random...
Homework Statement
I've been given a set of data
x 0 0.5 0.7 1.5 1.75
y 0.5 0.72 0.51 1.5 1.63
Given y=ax+b
for this data points of linear model, I have to
1. minimize the sum of the absolute values of deviations between experimental value of Y and value predicted by the...
Homework Statement
Consider model of linear regression:
Y_i = \beta_0 + x_i \beta_1 + \epsilon_i
i = 1, ..., 5, where \epsilon_i \sim \mathcal{N}(0, \sigma^2) are independent. Find expected value and variance of predicted values \widehat{Y}_i considering that observations are...
I'm interested in fitting a line to some data. There is a built-in function in R lm() that gives me both the best-fit slope and intercept, however, I would like to determine the best fit intercept GIVEN a specified value of the slope. Is there an easy way to do this?
I apologize if this is in...
AFAIK, there are two basic type of linear regression:
y=ax+b and y=a2 + bx + c
But I have to do the same with the function y = asin(x)+bcos(x).
Here is what I have done:
We have:
\begin{array}{l}
\frac{{\partial L}}{{\partial a}} = 0
\frac{{\partial L}}{{\partial b}} = 0Continue...
[SOLVED] Multivariate Linear Regression With Coefficient Constraint
I'm attempting a multivariate linear regression (mvlr) by method of least squares. Basically, I'm solving a matrix of the following form for \beta_p,
$ \begin{bmatrix} \sum y \\ \sum x_1 y \\ \sum x_2 y \\ \sum x_3 y...
For my chemistry lab, in order to computer change in temperature for a calorimetry experiment we're suggested to take the line of best fit from the peak temperature onwards (excluding the initial data) and extrapolate to y = 0. For example, here's some of the data I gathered:
A, Trial 2...
1) To test the quality of a tennis ball you drop it onto the floor from a hieght of 4 m. it rebounds to a hieght of 2 m. if the ball is in contact with the floor for 12 ms, what is the magnitude of its average acceleration during contact and is the average acceleration up or down.
What i did...
My problem in short:
I have a set of data, and I want to calculate the linear regression, and the uncertainty of the slope of the linear regression line, based on the uncertainties of the variables
My problem in detail:
My data is from an experiment and the uncertainties (errors) are...
Hi,
I've got what should be a very easy simple linear regression problem, but I can't seem to be able to get my head around it. Here it is:
So far I've been trying to sub these values into a regression equation like this one:
Y = 5B + (-0.003)B^2
Where "B" is my Beta1 value. I...
linear regression where am i going wrong??
Linear regression using least square fit method for the
determination of cocaine sample
Cocaine (mg/ml) Peak height
X= 2.75 Y=27377 X squared=0.9625 X x Y=3241.272
M = 10 x 3241.272 – 2.75 x 27377 / 10 x 0.9625 – 7.5625
=...
Im having some trouble with this, and I was hoping someone could help me.
I have a data set from which I've determined the \widehat{a} and \widehat{b} values and determined where the line of best fit should go using linear regression. The next thing I have to do is work out the varience using...
It seems to me that Linear Regression and Linear Least Squares are often used interchangeably, but I believe there to be subtle differences between the two. From what I can tell (for simplicity let's assume the uncertainity is in y only), Linear Regression refers to the general case of fitting...
Hi,
Complicated stats question, but maybe someone out there knows how to proceed. I am trying to perform regression on two variables, the samples of which have significant, but known error components. Ordinary least squares regression cannot be used as it is assumed that measurements are made...