Linear Regression with Ordinary Least Squares

I'm a bit confused here, so help me understand. When you do a linear regression line of best fit, you're minimizing the sum of the squares of the perpendicular distances from the data points to the line of best fit. Therefore, the points are?ÿalways "perpendicular" to the line. (A point can't be perpendicular to anything, but one of the many lines that can be drawn from a point to the line of best fit is perpendicular to the line of best fit.)

Also, I would like to see a line of best fit that has all of the data points below it. A mathematical constraint of a linear regression line is that it passes through the mean of the data points, so there should always be at least one point on the opposite side of the line from the others. Could you share the link to the video that you saw?

Are you finding the best fit of redundant measurements of the corners, or are you finding the best fit straight line from one corner to another?

Draw us a simple diagram, nothing fancy, just enough for this non-surveyor to visualize the problem.

I thought the standard linear regression minimized sum of squares of the distances from points to line in the Y axis direction. If you reverse the roles of X and Y you get a different line. More greatly different for data less correlated.

Best fit on perpendicular distances would be of interest in surveying but I don't recall seeing it treated in statistics books. Intuitively, I would expect that line to be between the y-based fit and the x-based fit lines.

Posted by: @mathteacher

(A point can't be perpendicular to anything, but one of the many lines that can be drawn from a point to the line of best fit is perpendicular to the line of best fit.)

Thanks for the real explanation, it reminds me of the worst lame (but they do close) metes and bounds legal descriptions which describe a line terminus as a "point". There is no such thing as a "point" (mathwise) in surveying descriptions. Linear regression is mostly used in R/W boundaries but can lead to big problems if not incorporating all found monuments. ?????ÿ

@bill93?ÿ

You're correct about the y-difference rather than the perpendicular distance. I knew that, but I've been off working with some analytic geometry problems where it was the other way round. It's supposed to work like this:

I've never been sure about how such a line would be calculated with survey data and I'm not sure what its purpose is. Are we trying to see if a group of measured points are collinear, or the accuracy of individual points, or something else?

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Field Dog,?ÿ good to hear you are trying to learn. For your problem you would want to read from CRC Concise Encyclopedia of Mathematics, 2nd edition by Eric W. Weisstein. Read, starting on page 1717; Least Squares Fitting. At the bottom of page 1717 and on the next page you will start the Math that you will need to know to understand "HOW" to solve for the LS fitting of the perpendicular offsets.?ÿ(The article ends on page 1721).

Hope this helps

JOHN NOLTON

Posted by: @mathteacher

Could you share the link to the video that you saw?

3.2: Linear Regression with Ordinary Least Squares Part 1 - Intelligence and Learning

Posted by: @mathteacher

Are you finding the best fit of redundant measurements of the corners, or are you finding the best fit straight line from one corner to another?

The later. Carlson Survey 2021 calculated the following, holding Pts. 248 and 254 as end points:

Pt.: 248 Wt.: 1 Offset: 0.24655 RT

Pt.: 249 Wt.: 1 Offset: 0.24033 RT

Pt.: 251 Wt.: 1 Offset: 0.78342 LT

Pt.: 252 Wt.: 1 Offset: 0.06476 RT

Pt.: 254 Wt.: 1 Offset: 0.23177 RT

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I haven't studied these pages, butthey are on point for this thread.

https://mathworld.wolfram.com/LeastSquaresFittingPerpendicularOffsets.html

https://www.toddreed.name/articles/line-fitting/

@bill93?ÿ

After reading this I will assume that best fit line line examples are using vertical instead of perpendicular offsets.

@john-nolton?ÿ

There is a copy at a local college library. Where did you get your copy from?

I presume you have located property monuments and want to use a best fit line calculation? Why? Why have software determine the r/w line? To call all points ƒ??offline, which is a best fit, is reprehensible. Make and use your own calculation. Some points will be off by a bit and they shouldnƒ??t be considered.

Posted by: @robertusa

Why have software determine the r/w line?

It may be just an exercize, and not an actual deliverable product.?ÿ Sounds interesting

@field-dog?ÿ ?ÿI bought the book (2nd Edition) when it first came out (2003?). I see the 3rd edition, 3 vol. is over $750

I might have to think about that. Dave Lindell also has the book and knows all about what you are doing. I was hoping either he or Math Teacher would post on what I gave reference to.?ÿ

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@field-dog?ÿ

Notice that one of the offsets is left and the other 4 are right, so they are not all on one side of the line. Note also that, if you add them up making the left negative and right positive, they sum to nearly zero. Thus the mean of the offset distances is zero and the line passes through the mean coordinate value.

As to the perpendicular, horizontal or vertical offset, it could be any of these. Consider this from Microsurvey: Line - Distance Offset (microsurvey.com) ?ÿ

I didn't find similar stuff for Carlson, but it seems likely that Carlson has a setting as Microsurvey does.

If you give us the measured coordinates, we may be able to determine which it is.

Also, a heckuva nice and smart PhD professor at UNC taught me how to use Excel to do a wide variety of least squares calculations as we were discussing and calculating Covid spread models. Perhaps I could use that with your data and help us all learn something new.

@robertusa?ÿ

I have located the property monuments and I want to use a best fit line calculation to determine if the calculated R/W widths match the plat R/W widths. I'm interested to know how you would approach the problem.