Chapter 06.03: Lesson: Linear Regression with Zero Intercept: Derivation

Learn how to regress data to a linear polynomial with zero constant term (no intercept). This segment shows you the derivation and also explains why using the formula for a general straight line is not valid for this case. For more videos and resources on this topic, please visit nm.mathforcolle...

[@victorwidiputra4819]

hello, can i know why does the residual need to be squared?

[@numericalmethodsguy]

One cannot minimize each residual as it would make other residuals large. So one would think sum of residuals could be minimized. This does not give a unique line. So one may choose sum of absolute value of residuals. That also does not give unique line. Minimizing sum of square of residuals gives unique line. See here for complete information! mathforcollege.com/nm/mws/gen/06reg/mws_gen_reg_txt_straightline.pdf See all videos mathforcollege.com/nm/videos/youtube/06reg/linear/linear_06reg_background1.html

[@SuperDangerousMouse]

Hi Victor, think about the following example: x = 0 , y = 0 x = 1 , y = 1 x = 2 , y = 2 x = 3, y = 3.5 x =4, y = 3.5 from this trend, you can see that y *should* be 0, 1, 2, 3, 4 (y = x), but instead the last 2 points are off. The first point is 0.50 ABOVE 3, and the last point is 0.50 BELOW 4. x = 0 , y = 0 , diff = 0 x = 1 , y = 1 , diff = 0 x = 2 , y = 2 , diff = 0 x = 3, y = 3.5 , diff = +0.50 x =4, y = 3.5 , diff = -0.50 So if you simply add up the differences, the last 2 cancel out (and you would think that the values fit). Just by plotting them, you will see, that they are not a straight line. This is why we square the differences. The idea behind linear regression of scatted points is to find a straight line that "fits" the given data. You want to find a line that MINIMIZES the error (or the difference). If simply minimize the sum of the differences (in the example) above, we can see that we are already are zero (the minimum possible value). Zero means the data is a perfect line (we know from the observation that it's not a perfect line). Therefore, we have to sum the square of the difference (and try to minimize that).

[@psibarpsi]

Can't believe this was posted 13 years ago!

[@nataliaecheverry9474]

THANK YOU!!!!! my semester appreciates it :D

[@KidsPoems274]

you are life saver

[@numericalmethodsguy]

Thank you. Please subscribe and ask your friends to subscribe - our goal is to get to 100,000 subscribers by the end of 2021. To get even more help, subscribe to the numericalmethodsguy channel kzbin.info, and go to MathForCollege.com/nm and MathForCollege.com/ma for more resources. Follow the numerical methods blog at AutarKaw.org. You can also take a free massive open online course (MOOC) at canvas.instructure.com/enroll/KYGTJR Please share these links with your friends and fellow students through social media and email. Support the channel if you able to do so at kzbin.info/store

[@ismailabdelrazig981]

explained it very well, thanks a lot for your help

[@welliamrony9790]

thanks for your help, GOD Bless you

[@lindsayanderson9662]

how would you derive the formula for the uncertainty in a1?

[@jassmitha137]

Thank you....U save my life

[@darkside1928]

What is the standard error of a1?

[@Pumbear]

How do you calculate the correlation though?

[@isaaclumley]

Hello, what happens if the x value becomes the independent variable and y is the dependent one? ( with new coefficients x=c0*y + c1)

[@numericalmethodsguy]

If x-c0+c1*y is used, then x is the predicted variable (dependent) and y is the predictor variable (independent). So you have switched what is what. Now if we have y and x as two variables which are measured at several points, how do we choose which one should be independent and which one dependent. Generally, the variable which is claimed to be known more accurately would be the independent variable. statistics.laerd.com/spss-tutorials/linear-regression-using-spss-statistics.php

[@isaaclumley]

numericalmethodsguy will this influence our calculations, giving us a different linear function in the end?

[@numericalmethodsguy]

yes

[@MrRazimcool]

hello, i want to ask that how we know that line regression function is zero intercept ?

[@numericalmethodsguy]

Mohd Fazim Ahmad The intercept being zero is dependent on the physics of the problem. Imagine one is using a force vs deflection data to find the spring constant of a linear spring with no residual stresses. In that case F=kx would be the model.

[@atxlax]

@@numericalmethodsguy For this exact example you give, the instruction given in this lab manual says explicitly to NOT model the data forcing the line to zero intercept (see pages 10 and 11: www.baylor.edu/physics/doc.php/110769.pdf). Can you provide comment?

[@numericalmethodsguy]

@@atxlax I do not know what to say? Yes, if the spring has residual stress, we may use the general straight line, but is the spring measuring that residual stress. If we say that at x=0, F-0 as per F=kx, then it does not matter if there is residual stress in it. In a mousetrap spring, torque is required to open it to an angle of just 0+

[@exkillerd]

I am the 4th people like this. It seems calculus kills the most.

[@SuperDangerousMouse]

jump to 5:40 to follow the derivation

[@numericalmethodsguy]

Don't jump. You will lose on learning a very important concept on why you cannot just use the general straight line formula.

[@SuperDangerousMouse]

@@numericalmethodsguy Correct... don't jump if you're new to the subject. But if you want to see the derivation ONLY (and you ALREADY know how to derive the standard regression formula), go to 5:40.

[@numericalmethodsguy]

@@SuperDangerousMouse Point taken. You must be smart. Several students do not understand why we are deriving a formula for y=b*x model when we already have the formula for y=a+b*x. They simply say a=0. And that is wrong!

[@SuperDangerousMouse]

@@numericalmethodsguy Thanks. A friend asked me to derive the formula for him. But I found that it's easier to use your video (instead of scan the paper and send it). So thank you for positing.