Least squares adjustment/regression - two points known distance apart

In summary, the conversation discusses a problem with finding a solution to an adjustment for a kinematic time series of X,Y positions for two points with a known constraint of a certain distance between them. The conversation also explores different methods for solving this problem, such as changing coordinates or using least squares. The issue of having too many constraints is also raised, making it difficult to have a viable problem to solve.
  • #1
gge
12
0
Hi All,

I'm struggling with finding a solution to an adjustment I'm working on. Thought someone else may have some thoughts?

I have a kinematic time series of X,Y positions for two points (X1,Y1,X2,Y2). I know that the two points were a distance D (e.g., 100 m) apart from each other (the constraint).

I'm attempting to find a solution that finds a best fit to X1,Y1 and to X2,Y2 based on the constraint that the distance between them is D. The path formed by the points is not linear -- I'm starting with a quadratic model.

Application: Picture two GPS receivers rigidly mounted on top of a car. They both have positions and both have noise in their position. I'd like to best fit to both of their positions, but with the constraint on that known distance between them.

In coming up with observation equations, my initial thought was something like:

X1(t) = At^2+Bt+C
Y1(t) = Dt^2+Et+F
X2(t) = Gt^2+Ht+I
Y2(t) = Jt^2+Kt+L

Then:

sqrt((X1-X2)^2+(Y1-Y2)^2) = 100

So this leaves me with 12 parameters. I have ample observations to solve this (observations are X1, Y1, X2, Y2).

Does it appear that I'm on the right track?

I'm not sure what the best method is to proceed from here (i.e. how to set up the LS problem). Any suggestions?

Thanks in advance for any help!

GGE
 
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  • #2
You could change your coordinates to include the constraint there. One example would be to track the central position and the angle of the line between both points.

If that is not useful, I think many fitting programs have some way to consider external constraints in some way.

I would expect that your uncertainties for the positions will be heavily correlated, as the atmosphere is the same for measurements close together in time and space. Differential GPS can get cm-accurary.
 
  • #3
You have two variables that could be in error - the x and y coordinates.

To set up the least squares line for this you calculate the perpendicular distance from each point to the assumed line and minimise the sum of the squares of these distances.
 
  • #4
The first part of the below link describes how to solve least squares problems with constraints.

http://kom.aau.dk/~borre/kalman/lecture2/p403.pdf

However, if the constraint applies to each time stamp I don't think you have a viable problem. Let's say you want to use 20 observations to solve for 12 variables in a least squares sense. That would work, but you also have 20 constraints. For a constrained least squares problem involving n variables, you can have at most n-1 constraints. Otherwise it is impossible to satisfy all of the constraints and have a meaningful least squares problem to solve.
 
Last edited:
  • #5
hotvette said:
However, if the constraint applies to each time stamp I don't think you have a viable problem. Let's say you want to use 20 observations to solve for 12 variables in a least squares sense. That would work, but you also have 20 constraints. For a constrained least squares problem involving n variables, you can have at most n-1 constraints. Otherwise it is impossible to satisfy all of the constraints and have a meaningful least squares problem to solve.
Good point. The quadratic model for Xi, Yi will not satisfy the constraints for each step in time apart from some special conditions (like no rotation).
Alternative parameters solve this problem.
 

What is a least squares adjustment/regression?

A least squares adjustment/regression is a statistical method used to find the best-fit line or curve through a set of data points. It minimizes the sum of squared residuals to determine the most accurate relationship between the variables.

Why is least squares adjustment/regression important in scientific research?

Least squares adjustment/regression is important in scientific research because it allows us to analyze and interpret the relationships between variables in a quantitative manner. It also helps us to make predictions and identify trends in data.

How does least squares adjustment/regression work?

Least squares adjustment/regression works by minimizing the sum of squared residuals, which are the differences between the actual data points and the predicted values on the best-fit line or curve. This is achieved through mathematical calculations and iterative processes.

What is the difference between least squares adjustment and least squares regression?

The terms "least squares adjustment" and "least squares regression" are often used interchangeably, but there is a subtle difference between the two. Least squares adjustment is used to determine the best-fit line or curve through a set of data points, while least squares regression is used to predict the value of a dependent variable based on the value of an independent variable.

What are the assumptions of least squares adjustment/regression?

The main assumptions of least squares adjustment/regression are that the relationship between the variables is linear, the errors are normally distributed, and the errors have equal variances. These assumptions should be checked before performing a least squares adjustment/regression to ensure the accuracy of the results.

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