Can NNLS algorithms solve overdetermined systems with positive constraints?

In summary, there are specific algorithms such as least squares with non-negative constraints and NNLS (non-negative least squares) that have been developed to solve overdetermined systems of linear equations with known coefficients and the requirement that all unknowns are positive. These algorithms can be useful for fitting experimental data and avoiding issues such as oscillations or negative coefficients. Resources for these algorithms can be found through online sources such as MathWorks and NASA.
  • #1
Sergei_G
5
0
Hello everyone,

I'd like to solve overdetermined system of linear equations (in fact to fit experimental data)
(like y1=C1*X11+C2*X12+...+Cm1*X1m)
y2=C1*X21+C2*X22+..+Cm*X2m
...
yn=C1*Xn1+C2*Xn2+...Cm*Xnm)
sometimes n>>m sometimes n>~m , yi and xij are known coefficients
and I know ab initio that all unknowns C1...Cm are positive. Are there specific algorithms developed for such problem? I tried to solve it with simplest least square, but I always get something like oscillations with increase of m - Positive C are compensated by negative C and fit becomes perfect but it does not have physical sence.

Thanks,

Sergei.
 
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  • #3
Thanks!
 
  • #4

1. What is the concept of least squares with constraints?

Least squares with constraints is a statistical method used to find the best fit line or curve that minimizes the sum of squared residuals while also satisfying certain constraints or conditions. It is commonly used in regression analysis when there are limitations on the values of the independent variables.

2. How does least squares with constraints differ from regular least squares?

In regular least squares, the goal is to minimize the sum of squared residuals without any restrictions or constraints. In contrast, least squares with constraints adds additional conditions that must be met, such as limiting the range of the independent variables or requiring the coefficients to have a specific relationship.

3. What are some common types of constraints used in least squares?

Some common types of constraints used in least squares include upper and lower bounds on the independent variables, equality or inequality constraints on the coefficients, and linear or nonlinear relationships between the coefficients.

4. How is least squares with constraints typically solved?

There are several approaches to solving least squares with constraints, including the use of optimization algorithms such as linear programming, quadratic programming, or interior point methods. These methods involve iteratively adjusting the values of the coefficients until the constraints are satisfied and the sum of squared residuals is minimized.

5. What are the benefits of using least squares with constraints?

Using least squares with constraints allows for more accurate and reliable results in situations where there are limitations on the values of the independent variables. It also allows for the incorporation of additional prior knowledge or assumptions about the relationship between the variables, leading to more meaningful and interpretable results.

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