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Overdetermined System

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The Perplexing Puzzle of Overdetermined Systems: When Too Much Information Leads to Trouble



Imagine you're a detective investigating a crime. You have several eyewitness accounts, forensic evidence, and security footage – all pointing towards the culprit. This is a well-determined system; the evidence converges on a single, consistent solution. But what if some of your witnesses contradict each other, or the forensic evidence clashes with the security footage? You now have an overdetermined system – more information than is strictly necessary, and some of it is conflicting. This scenario highlights the core challenge posed by overdetermined systems: too much data can lead to inconsistencies and computational headaches. This article delves into the nature of overdetermined systems, exploring their causes, consequences, and how to handle them effectively.

Understanding Overdetermined Systems: A Mathematical Perspective



In the realm of mathematics, an overdetermined system is a system of equations where there are more equations than unknowns. Consider a simple example:

x + y = 5
2x - y = 1
x + 2y = 6

We have three equations (more than the two unknowns, x and y). Solving the first two equations gives us x = 2 and y = 3. However, if we plug these values into the third equation, we find 2 + 2(3) = 8 ≠ 6. The system is inconsistent; there's no solution that satisfies all three equations simultaneously. This inconsistency arises because the equations are not linearly independent; one equation can be derived from the others (or there's contradictory information).

Causes of Overdetermination: Why Do These Systems Arise?



Overdetermined systems frequently emerge from:

Redundant measurements: In scientific experiments or engineering projects, multiple sensors or instruments might measure the same quantity. While redundancy improves robustness against single-point failures, it can also lead to inconsistent readings due to noise or calibration errors. Think of multiple GPS receivers providing slightly different location coordinates.
Inconsistent models: A system might be modelled using several independent equations or approaches. If these models are not perfectly compatible or if they rely on differing assumptions, the resulting system can become overdetermined and inconsistent. For instance, economic models based on different macroeconomic assumptions can produce conflicting predictions.
Inaccurate data: Errors in data collection or transcription can create inconsistencies within an otherwise well-defined system. In a survey, inconsistencies in respondent answers can lead to an overdetermined system when analyzing the data.
Implicit constraints: Sometimes, implicit relationships between variables are overlooked, leading to more equations than necessary. For example, in a geometric problem, if we inadvertently include an equation that’s already implied by other geometric relationships, we create an overdetermined system.


Handling Overdetermined Systems: Strategies and Techniques



Several strategies are employed to deal with the challenges posed by overdetermined systems:

Least Squares Method: This is a common approach for finding the best-fit solution when dealing with inconsistent equations. It minimizes the sum of the squares of the differences between the observed and predicted values. This technique is widely used in data fitting and regression analysis. For instance, in geodetic surveying, least squares are used to adjust measurements from multiple stations to obtain the best estimate of the position of survey markers.
Regularization Techniques: These techniques add penalty terms to the objective function to discourage overly complex solutions and to improve the robustness of the solution to noise. Techniques like Ridge Regression and Lasso Regression are frequently used in machine learning and statistics to handle overdetermined systems.
Data Filtering and Preprocessing: Identifying and removing outliers or noisy data points can improve the consistency of the system. Careful data validation and cleaning are crucial steps before attempting to solve an overdetermined system.
Model Refinement: Reviewing the underlying assumptions and equations of the model can reveal inconsistencies or redundant information. Modifying the model to incorporate only the essential and consistent equations can eliminate the overdetermination.
Bayesian Approach: This probabilistic approach incorporates prior knowledge and uncertainties into the solution process. It allows for a more nuanced understanding of the inconsistencies and helps to quantify the uncertainty in the resulting estimates.


Real-World Examples: Overdetermination in Action



Overdetermined systems are prevalent in diverse fields:

Global Navigation Satellite Systems (GNSS): GNSS receivers use signals from multiple satellites to determine location. Because more satellites than necessary are often available, least-squares techniques are used to process the redundant information and obtain the best position estimate.
Image Processing: In image reconstruction, multiple images of the same scene, potentially from different angles or wavelengths, provide redundant information. Overdetermined systems emerge, and techniques like compressed sensing are employed to reconstruct a high-quality image.
Financial Modelling: Constructing portfolio optimization models might involve many constraints and objectives, leading to an overdetermined system. Optimization techniques are then used to find the best compromise solution.


Conclusion



Overdetermined systems present a common challenge across numerous disciplines. Recognizing the sources of overdetermination – redundant measurements, inconsistent models, and inaccurate data – is crucial for effectively addressing the problem. By employing appropriate techniques like least squares, regularization, data filtering, and model refinement, we can manage the inconsistencies and extract meaningful insights from these complex systems. Understanding these strategies empowers scientists, engineers, and analysts to navigate the challenges of excessive information and derive robust, reliable results.


FAQs



1. How can I tell if my system is overdetermined? Count the number of equations and the number of unknowns. If the number of equations exceeds the number of unknowns, the system is potentially overdetermined.

2. What if the least squares method doesn't provide a satisfactory solution? Consider other techniques like regularization or exploring data preprocessing to improve the data quality. You might also need to revise your underlying model.

3. Is overdetermination always a problem? Not necessarily. Redundancy can improve robustness and accuracy. The problem arises when the redundant information is inconsistent or noisy.

4. Can I use software to solve overdetermined systems? Yes, many mathematical and statistical software packages (e.g., MATLAB, R, Python libraries like NumPy and SciPy) offer tools and functions for solving overdetermined systems, including least squares and regularization techniques.

5. How do I choose the best method for handling an overdetermined system? The optimal method depends on the specific context, including the nature of the data, the underlying model, and the desired accuracy. Experimentation and comparative analysis of different techniques are often necessary.

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How to solve an overdetermined system of equations? $\begingroup$ If a system is overdetermined, you just need to solve it regularly, and then check that the solutions found satisfy the remaining equations. $\endgroup$ – YoTengoUnLCD Commented Aug 5, 2016 at 10:01

Solving an overdetermined system of nonlinear equations If your system were linear (for instance, because of the change of variables suggested by @anon), then the right way to go would be to solve the system in the least-squares sense. In general, an over-determined system has no solution, so you want to get "as close as possible", i.e., minimize the squared $\ell_2$-norm of the residual.

Check whether an overdetermined linear equation system is … In the unlikely scenario that the overdetermined system has a solution, then we in fact have $||Ax-b||_2=0$. The x solving the least squares problem can be found via the Moore-Penrose pseudoinverse :

Solving overdetermined, well posed, linear system of PDEs 13 Jul 2021 · Now, by the Leibniz rule the solution in my previous comment seems to satisfy the system, but I'm trying to check if this is the same that you kindly sketched. $\endgroup$ – DanielKatzner Commented Jul 20, 2021 at 19:19

Does an overdetermined system always have no solutions? 5 Nov 2014 · An overdetermined system (more equations than unknowns) is not necessarily a system with no solution. If one or more of the equations in the system (or one or more rows of its corresponding coefficient matrix) is/are (a) linear combination of the other equations, so the such a system might or might not be inconsistent.

Solving overdetermined system by QR decomposition 12 Apr 2019 · This is an overdetermined system. That is, it has more equations than needed for a unique solution. I need to find $\min ||Ax-b||$. How should I solve it using QR? I know that QR can be used to reduce the problem to $$\Vert Ax - b \Vert = \Vert QRx - b \Vert = \Vert Rx - Q^{-1}b \Vert.$$ but what do I do after this?

Solve an overdetermined system of linear equations $\begingroup$ The system is indeed overdetermined and will only have a solution if the constants in the right-hand sides of the equation satisfy certain conditions. $\endgroup$ – StackTD Commented Jul 15, 2016 at 14:10

statistics - Over-determined and Under-determined systems 10 Apr 2015 · I believe that, as pointed out in Overdetermined and underdetermined systems of equation put simply, thinking of the equations in a system making up a set of requests (equations) to a certain number of people (unknowns) is helpful to understand why systems can be overdetermined or underdetermined in the first place.

Can overdetermined systems have infinitely many solutions? 17 Jun 2016 · A system with more equations than variables is called overdetermined. It can be either consistent or inconsistent. If one or more equations are a linear combination of the other, you can always obtain a consistent system.

Overdetermined linear system? - Mathematics Stack Exchange Understanding Overdetermined System. 0. Solving system of overdetermined linear equations with $2$-norm. 0.