Geometry of regression, Other Management

Geometry of Regression

Since graphical portrayal brings an added dimension to the understanding of the regression line we will go through the regression line for various values of 'a' and 'b'. As you know the regression line is Y = a + bx. Here 'a' is the Y-intercept and 'b' is the slope.

Figure 1

1797_geometry of regression.png

Since the regression line intersects the 'Y' axis at the point 'a', 'a' is called the 'Y' intercept. That is, the value of 'a' determines at what point the line intersects the 'Y' axis.

Let us understand the concept better, by taking the value of 'b' as zero, and calculating the regression line for various values of 'a'.

Case 1

What happens when a = 0, b = 0?

The regression line is Y = 0.

Figure 2

330_geometry of regression1.png

Case 2

Now let us take the value of a as 1, and b as 0.

The regression line will be Y = 1.

Figure 3

1532_geometry of regression2.png

So, one can see that when b is 0, the regression line is a straight line parallel to the X-axis, and the line intercepts the Y-axis at point 1 which is the value of a.

Case 3

What happens when a = 2, b = 0?

The regression line Y = 2.

The regression line is a straight line parallel to the X-axis, and intercepts the Y-axis at point 2, which is equal to 'a'.

Figure 4

1918_geometry of regression3.png

Case 4

What happens when a is negative. Let us see when a = -1 and b = 0.

The regression line will be equal to Y = -1.

Thus, we see that the regression line intercepts the y-axis at -1, which is equal to 'a'.

Figure 5

1303_geometry of regression4.png

Case 5

Let us see what happens when a = 0 and b = 1. The regression equation becomes Y = X.

Figure 6

2437_geometry of regression5.png


The line can be found in the I and III quadrants. This line goes through the origin. This is because a = 0. Had 'a' been equal to 1, the line would have intercepted the Y-axis at 1 as shown in the figure.

Figure 7

554_geometry of regression6.png

When 'a' changes from 0 to 1, the entire line has shifted upward by 1 unit.

Case 6

Let us see what happens when a = 0 and b = 2

Figure 8

2251_geometry of regression7.png

The regression line now becomes Y = 2X. Let us now plot the regression line, when the values of X are 1, 2, 3, 4, then the corresponding values of Y = 2, 4, 6, 8.

Now compare this graph with the previous one. What can you notice. Yes, the line has become steeper. Why? Because, previously the value of b was 1, now the value of b = 2. Thus, the value of 'b' determines how steep the line is. In other words 'b' determines the slope of the line.

Can the slope be only positive? No, it can be negative too.

Case 7

What happens when a = 0 and b = -1? The regression line becomes Y = -X.

Figure 9

2162_geometry of regression8.png

What do we observe? Now, the line slopes downward from the left to the right. What does this imply? It shows that for a unit change in 'X', the value of 'Y' declines by a unit 'b'.


We know that any straight line, with the exception of the vertical line, can be characterized by its slope. So, what do we mean by 'slope'. By 'slope' we mean basically the inclination of a line - whether it rises or falls as you move from left to right along the X-axis and the rate at which the line rises or falls (in other words, how steep the line is).

We have seen that the slope of a line may be positive, negative, zero or undefined. A line with a positive slope rises from left to right. For such a line the value of 'Y' increases as 'X' increases (or Y decreases as X decreases). A line having a negative slope falls from left to right. For such a line the value of Y decreases as X increases (or Y increases as X decreases). This means that X and Y behave in an inverse manner. A line having a zero slope is horizontal. As X increases or decreases, Y stays constant. Vertical lines have a slope which is undefined.

The slope of a line is quantified by a number. The sign of the slope indicates whether the line is rising or falling. The magnitude (absolute value) of the slope indicates the relative steepness of the line. The slope tells us the rate at which the value of Y changes relatively to changes in the value of X. The larger the absolute value of the slope, the steeper the angle at which the line rises or falls.

Posted Date: 9/17/2012 1:11:37 AM | Location : United States

Related Discussions:- Geometry of regression, Assignment Help, Ask Question on Geometry of regression, Get Answer, Expert's Help, Geometry of regression Discussions

Write discussion on Geometry of regression
Your posts are moderated
Related Questions
Translation Service:   One of the constraints in the flow of information is the language barrier. About half of world's literature is published in languages other than English

Non Conventional Indexing and Filing Equipment -libraray equipment-libraray management Use of mechanical devices for sorting information recorded on special kinds of cards lik

Case Study:  In a case study the survey  is based on one or more samples. Study on the  respondent(s) can be  investigated in  depth, which  is not possible with  large number of

Question: (a) Managers and other change agents use various means for mobilising power, exerting influence over other people, and getting them to support planned change effort

Question 1 In the recent times, operations are considered from end to end of value chain which means the operations that start from sourcing of materials and other inputs to succe

Question 1) What is fibre? Explain classification of natural fibers Question 2) Narrate the characteristics of Polyester fabrics Question 3) Write short notes on c

Proposal or Renewal Sub System: Type                                         Web-based Application Role                                          Software Engineer Team s

3. "Patient self-determination and autonomy are under the law ancillary and derivative from the well-articulated right of privacy. Under this well established right, procreative fr

Factors in designing  a project structure The project's success is defined by a properly designed organisation chart. An organisation chart shows where each person is positione

Problem 1 Strong employee relations equals solid organizations. Companies that interact effectively with workers financially outperform those that don't. Identify and descri