SIMPLE LINEAR REGRESSION AND CORRELATION.pptx
- 1. SIMPLE LINEAR REGRESSION AND CORRELATION Group 1
- 3. EMPIRICAL MODEL • Models can take various forms. Some incorporate physical and chemical principles with adjustable parameters. These are fundamental models. • Often, it is expedient to use models that are simple mathematically, such as linear or polynomial equations. These are empirical models. • Occasionally, models are a mix of the two, semi-empirical models • When choosing the form of a purely empirical model, it is important to make a selection that has a good potential of “fitting” the behavior of the data.
- 4. EMPIRICAL MODEL • A first step in choosing an empirical model is to look at the behavior of the process via collected data sets and depicted graphically. • The simplest of models would be a straight line. And the expansion of this model into a polynomial model to capture evident curvature in the data is most common.
- 5. EMPIRICAL MODEL • There are many other empirical models available. Here are common examples: Exponential Power Logarithmic
- 6. EMPIRICAL MODEL • The appropriate way to know whether one of these models is a candidate to explain your data is to observe its behavior for different parameter values. • The assumptions we make in specifying a model is that the mean (or the expected value) of the output variable is a function of the input variable, for example, Since , then x is sometimes called the regressor variable. Also, which describes the variance of Y at any given value of x.
- 7. REGRESSION; MODELING LINEAR RELATIONSHIP • REGRESSION is a statistical method used to describe the relationship between two variables. • There are two types of relationships: simple and multiple. In a simple relationship, there are two variables under study. In multiple relationships, many variables are under study. Simple relationships can be further classified as positive or negative. A positive relationships exists when both variables increase or decrease at the same time. A negative relationship exists when one variable increases while the other decreases, or vice versa. • A scatter plot is a graph of the ordered pair (x,y) of numbers consisting of the independent variable, x, and the dependent variable, y.
- 8. LINEAR REGRESSION • After a scatter plot is constructed and the value of the correlation coefficient is deemed to be significant, then an equation of the regression line is determined. The regression line is the data's line of best fit. FORMULA FOR THE REGRESSION LINE where and
- 9. CORRELATION: ESTIMATING THE STRENGTH OF LINEAR RELATION • A correlation is a statistical method used to determine if there is a relationship between variables and the strength of relationship. • Correlation coefficient measures how closely the points in a scattered diagram are spread around a line. It is detonated by r. FORMULA FOR CORRELATION COEFFICIENT where n is the number of data pairs and r should be round off to three decimal places.
- 10. RANGE CORRELATION COEFFICIENT The range of the coefficient correlation is from -1 to +1. If the linear correlation coefficient r is positive, the relationship between the variables has a positive correlation. In this case, if one variable increases, the other variable also tends to increase. If r is negative, the linear relationship between the variables has a negative correlation. In this case, if one variable increases, the other variable tends to decrease. Value of r Relationship Close to +1 Strong positive linear relationship Close to 0 Weak or no linear relationship Close to -1 Strong negative linear relationship
- 11. The figures show some scatter diagrams along with the type of linear correlation that exists between the x and y variables. The closer is to 1, the stronger the linear relationship between the variables.
- 12. EXAMPLE A Statistics professor at a state university wants to see how strong the relationship is between a student’s score on a test and his or her grade point average . The data obtained from the sample follow. Also, find the equation of the regression line. Solution: It will be easier to solve for r if a table of values will be constructed. Find the values of xy, x, and y and place these values in the corresponding columns of the table. Test Score, x 98 105 100 100 106 95 116 112 GPA, y 2.1 2.4 3.2 2.7 2.2 2.3 3.8 3.4
- 13. LINEAR CORRELATION Subject Test score, x GPA, y xy 1 98 2.1 205.8 9604 4.41 2 105 2.4 252 11025 5.76 3 100 3.2 320 10000 10.24 4 100 2.7 270 10000 7.29 5 106 2.2 233.2 11236 4.84 6 95 2.3 218.5 9025 5.29 7 116 3.8 440.8 13456 14.44 8 112 3.4 380.8 12544 11.56 Total
- 16. SHORT QUIZ A teacher gives a test to the students to see how strong the relationship is between how long his/her study for the test and his or her grade to the test. The data obtained from the sample follow. Also, find the equation of the regression line. No. of hours, x 1 2 3 3.5 4 5 6 Grades, y 80 83 88 90 94 96 99
- 17. THANK YOU !