Episode 14 The Simple Explanation of Least Squares Part 1 of 3

welcome to another episode of the 10 minute land surveyor i'm dave wooley today we're talking about these square adjustments most people associate a least square adjustment with post-processing gps and yes that's true that's how we do process the post-process gps but least-squared adjustments have many practical purposes even in resolving boundaries i'm going to talk today about what a least square adjustment is put it in real simple terms and then the next episode i'm going to talk about using it for practical purposes so what are we talking about in least square adjustments well when you take a series of measurements let's say a hundred feet nominally and you measure that several times and you have a variation in that measurement what is the best fit of that adjustment well let's go ahead and look at the screen here here i have measurements of a hundred feet and each one of those measurements are my measurements of a hundred feet and you'll see i have variations in there the first one's 100.05 and if you go up you'll see that we we get it all the way down to a 99.69 here so on and so forth we took 10 measurements the average of those measurements is 99.95 shown right here so that is the best fit of all those measurements and in its simplest terms each measurement is equal we didn't use a better chain or a worse chain and so in the simplest terms each measurement holds the equal amount of weight so we take the average shown here and the average is taken from the measurement given and that difference shown here is a residual and some are plus and some are minus and that's the residuals of that measurement now this residuals are squared you end up with they lose their negative positive and negative sign when you square them and so you have a sum of residuals squared which is 0.21 you divide that by the number of measurements taken 10 in this instance and of course you end up with 0.021 equals and you take that 0.021 and you take the square root of that and you end up with your standard deviation here standard deviation is 68 of your measurements will fall within 15 hundredths well let's see if that's true i have 1 2 3 4 5 six seven what do you know seven out of ten measurements fall within fifteen hundreds as they should 68 percent that is the math now to check ourselves uh and just here in uh excel you'll see the formula up there for standard deviation and the standard deviation of those measurements is 15 hundredths this is the long hand don't write this down you don't need to know it it's just to give you a better sense of what a least square adjustment does let's take a look at a linear regression or a least square adjustment let's say just for the sake of discussion here that we have these points here and these these points shown as blue circles and those blue circles are meant to be on the same line they could be front lock corners they could be top of curve so on and so forth there's no scale so this could be a hundredth for every shot it could be a tenth who knows doesn't matter so what you see here is you see the residual from the line shown here the residual is actually how far left or how far right the the actual measured point is from the best fit line or the linear regression and the fit depending on whether it's a plus or minus really is as simple as what's the direction of the line if the directional line is going left to right as you see here then the residuals are positive to the right negative to the left if i went the other direction and i went right to left it would just change the sign on these that's how unimportant it is it's just the direction of a line so if you go through and you look at this here and let's say that this line running through here now i have a sum of residuals squared and let's say that this one is uh negative two hundredths and this one's positive three hundredths and this one's uh positive uh six hundredths positive three hundredths negative uh seven hundredths so on and so forth and what it does is it it tests the line through these points this one just let's say this one was zero zero say it fell on there so when it comes through whatever line fits the best going through here let's say that this this is the line let's see here that was actually the best fit line what it means is that i the the residual squared are the smallest hence least square that's all it is is a least square adjustment is just saying that this this solution creates the smallest number of residuals squared interesting right so the program if you're running a program and and the program runs through and you see it comes up with a line it's the least square it's just saying that it makes the residuals the smallest now you always have an option to remove a point like this point here because it just doesn't work so you could remove the point that would reduce the number of residuals squared you do not want to reduce a point unless you can identify all points being equal unless you can identify a reason to remove that point you can remove enough points you get down to two and you have no residual squared so you want to be very careful when you do an at least square adjustment that not to remove measurements unless you can identify a particular blunder so the program will do one line as you see here it'll test it and then the the second line it will test it and let's say that the first test comes through and it's right here it comes through it does the sum of residuals squared and it says that's a baseline it comes through again and it tests this line and it does a sum of a residual squared and now it compares it says okay that's my least square and then it'll do another one now if you're doing a least square adjustment just so you know each time you see that adjustment say iteration each one of these potential solutions shown here are iterations so when you see an adjustment going through iterations it's testing solutions now this is a line solution you can do a point solution and you can do a network solution but it all is the same it's testing to try to hold those measurements in a severe sense to the least amount of adjustment required so that you have the smallest residual squared least square and that's your solution now when you see your iterations going up it's having data it's having a hard time fitting the data into the parameters that you have given it and it walks those solutions in kind of like a mortar like it test test test and then it says okay each test i go through i'm not yielding any residuals or results and you'll usually have to set a parameter in there and say you know when you get down to 100 don't try to do any better cap my cap my iterations and so when you do an adjustment you say this is what i expect don't do any more iterations after you reach a hundredth on a network solution and that is my friend's least square adjustment i'm going to give you the longhand formula of what what was happening on the excel spreadsheet and this is what the longhand formula looks like this symbol here is the sigma symbol lowercase this symbol here is the uppercase sigma symbol which means sum this is the symbol that you use for standard deviation so when you say it's one sigma it's it's the lowercase sigma uppercase sigma x is residuals squared we all recognize that divided by the number of measurements and then you take the square root of it let me show you i took my residuals my best fit was an average in this case i came up with my residuals which is just the difference from my average to my measurement i squared them i totaled them big sigma here uh summed them divided by 10 came up with .021 i did a square root and i end up with the standard deviation now to check myself i just went in and you can see the formula shown right up here the the highlighted box just shown here at 15 hundredths and there there's the formula in excel that shows you what the numbers i took i took b 2 through 11 said give me the standard deviation 1500s checks my longhand work and that my friends is least square adjustment easy peasy everybody can do it it's time to put away our clown shoes pack away our mini bikes and take up some least square adjustments thank you to everybody who has taken the time to send me emails and to comment on the videos i really appreciate it the orange county chapter appreciates it have a nice day

Channel: Orange County Chapter, CLSA

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Id: qR9P0E-QqN4

Length: 10min 52sec (652 seconds)

Published: Wed Nov 03 2021