Least Squares: Overdetermined Systems

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Questions and Answers

What is the assumption made about the values of m and n in the context provided?

  • *m* is equal to *n*
  • *n* is less than *m*
  • *m* is greater than or equal to *n* (correct)
  • *m* is less than *n*

What happens to the matrix A when m is equal to n?

  • The matrix *A* becomes nonsingular. (correct)
  • The matrix *A* becomes undefined.
  • The matrix *A* becomes singular.
  • The matrix *A* becomes the identity matrix.

What is the least squares solution in the case when m = n?

  • The solution to the quadratic equation, where *x* is the unknown.
  • The solution to the homogeneous system *Ax* = 0
  • The solution to the equation *A*x = *b*
  • The solution to the linear system, *A*−1*b* (correct)

In the given example, what is the value of m?

<p>3 (A)</p> Signup and view all the answers

What is the value of n in the given example?

<p>2 (A)</p> Signup and view all the answers

What does the least squares problem aim to achieve?

<p>Find the vector that minimizes the sum of squares of the errors corresponding to the equations. (C)</p> Signup and view all the answers

What is the specific mathematical expression representing the least squares problem in the given example?

<p>min (x1 + 2x2 - 1)² + (2x1 + x2 - 1)² + (3x1 + 2x2 - 1)² (C)</p> Signup and view all the answers

Under what condition is the least squares solution guaranteed to be the solution of the linear system?

<p>When <em>m</em> is equal to <em>n</em> (D)</p> Signup and view all the answers

What is the primary purpose of the regularization function R(·) in the penalized problem?

<p>To improve the generalization performance of the model by preventing overfitting (D)</p> Signup and view all the answers

What is the effect of increasing the regularization parameter λ in the RLS problem?

<p>It increases the weight given to the regularization function. (B)</p> Signup and view all the answers

What is the common choice for the regularization function R(x) in many applications of RLS?

<p>R(x) = ∥Dx∥2 (C)</p> Signup and view all the answers

What is the primary goal of the quadratic regularization function given by R(x) = ∥Dx∥2?

<p>To control the norm of the vector Dx and prevent overfitting (A)</p> Signup and view all the answers

How can the RLS problem with quadratic regularization be equivalently written?

<p>min ∥Ax − b∥2 + λ∥Dx∥2 (D)</p> Signup and view all the answers

What is the role of the matrix D in the regularization function R(x) = ∥Dx∥2 ?

<p>D is a matrix that transforms the solution vector x into a new vector. (A)</p> Signup and view all the answers

What happens to the solution of the RLS problem as the regularization parameter λ approaches infinity?

<p>The solution becomes increasingly simple, possibly approaching the zero vector. (D)</p> Signup and view all the answers

What is a potential downside of using a very large regularization parameter λ in the RLS problem?

<p>It can lead to underfitting, where the model fails to capture the underlying patterns in the data. (D)</p> Signup and view all the answers

What is the basic structure of the linear system as described?

<p>A matrix with coefficients of variables and constants on different sides (C)</p> Signup and view all the answers

Which operation is performed on the matrix to obtain the least squares solution?

<p>Matrix inversion of XT X (D)</p> Signup and view all the answers

In the least squares solution, what does XT represent?

<p>The transpose of the original variable matrix (D)</p> Signup and view all the answers

What is the outcome when applying (XT X)−1 to XT y?

<p>The best fit solution for the variables (D)</p> Signup and view all the answers

Which of the following is NOT a characteristic of the least squares solution?

<p>Provides an exact solution for all data points (B)</p> Signup and view all the answers

What does the vector 'y' represent in the context of the linear system?

<p>The observed dependent variable outcomes (C)</p> Signup and view all the answers

What does the expression (XT X)−1 indicate about its components?

<p>The matrix XT X must be square and invertible (A)</p> Signup and view all the answers

Why might a least squares solution be preferred over other methods?

<p>It efficiently handles situations with more equations than unknowns (C)</p> Signup and view all the answers

What is the main objective of the least squares approach as described?

<p>To minimize the residual sum of squares between the observed and predicted values. (B)</p> Signup and view all the answers

Which mathematical expression represents the goal of the least squares method?

<p>$ ext{min } (Sx - t)^2$ (D)</p> Signup and view all the answers

What does the variable $t_i$ represent in the linear relation?

<p>The observed value. (A)</p> Signup and view all the answers

In the context of the least squares approach, what does the matrix S represent?

<p>The matrix formed from the vectors $s_i$. (B)</p> Signup and view all the answers

How is the least squares problem reformulated in vector notation?

<p>$ ext{min } (Sx - t)$ (B)</p> Signup and view all the answers

What does the notation $ ext{min}_{x ext{ in } R^n}$ indicate in this context?

<p>Minimization for parameter vectors of any dimension. (A)</p> Signup and view all the answers

Which assumption is made about the linear relationship in the least squares formulation?

<p>It holds approximately. (D)</p> Signup and view all the answers

What does the expression $(s_i^T x - t_i)^2$ represent geometrically?

<p>The square of the distance between the predicted and observed value. (D)</p> Signup and view all the answers

In the least squares optimization problem, what does the expression $Sx$ signify?

<p>The linear transformations of the vector x using matrix S. (C)</p> Signup and view all the answers

What condition must be satisfied for the RLS solution to be valid?

<p>AT A + λDT D must be positive definite. (B)</p> Signup and view all the answers

How is the Hessian of the objective function represented in this context?

<p>∇2 fRLS (x) = 2(AT A + λDT D) (A)</p> Signup and view all the answers

What does the stationary point of the function satisfy?

<p>(AT A + λDT D)x = AT b. (C)</p> Signup and view all the answers

What does λ represent in this context?

<p>A weight used to regularize the solution. (B)</p> Signup and view all the answers

What mathematical operation is performed to derive xRLS?

<p>xRLS is computed using matrix inversion. (A)</p> Signup and view all the answers

What is implied if the Hessian is positive semidefinite?

<p>There is a unique global minimum. (A)</p> Signup and view all the answers

What is the main purpose of using RLS in the given context?

<p>To enhance the accuracy of predictions. (D)</p> Signup and view all the answers

Given the matrix A, what can be inferred about the term 10^{-3} in this context?

<p>It is a small regularization term to reduce overfitting. (A)</p> Signup and view all the answers

What is the primary purpose of using the squared version of the norm function in the circle fitting problem?

<p>To ensure the function is differentiable (D)</p> Signup and view all the answers

In the circle fitting problem, what does the variable $r$ represent?

<p>The radius of the circle (C)</p> Signup and view all the answers

What does the notation $∥x - a_i∥^2 ≈ r^2$ imply in the context of the circle fitting problem?

<p>The distance from point $x$ to $a_i$ is approximately equal to $r$ (D)</p> Signup and view all the answers

What type of problem does the circle fitting problem represent in mathematical optimization?

<p>Nonlinear least squares (NLS) problem (A)</p> Signup and view all the answers

Which of the following fields does NOT typically apply the circle fitting problem?

<p>Nuclear science (D)</p> Signup and view all the answers

What mathematical function is used to minimize the squared difference in the nonlinear least squares problem for circle fitting?

<p>$∑_{i=1}^{m} (∥x - a_i∥^2 - r^2)^2$ (D)</p> Signup and view all the answers

What kind of equations does the circle fitting problem associate with the points $a_i$ and the radius $r$?

<p>Approximate nonlinear equations (C)</p> Signup and view all the answers

In the equation $∥x - a_i∥^2 ≈ r^2$, what does the symbol $a_i$ represent?

<p>Data points on the circle (C)</p> Signup and view all the answers

Flashcards

Nonsingular Matrix

A matrix A is nonsingular if it has an inverse, or when m = n.

Least Squares Solution

The least squares solution minimizes the sum of squared errors in a system of equations.

Inconsistent Linear System

A system of equations that has no solution due to conflicting equations.

Coefficients Matrix

Matrix A containing the coefficients of the variable terms in a linear system.

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Right-Hand-Side Vector

Vector b in a system of equations representing the constants on the right side.

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Sum of Squares

The total of squared differences between observed and predicted values in least squares.

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Error in Equations

The difference between the actual values and predicted values in a model.

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Minimal Sum of Squares

The least value of the sums of squared errors in the least squares problem.

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Regularized Least Squares (RLS)

An optimization problem form using a regularization function with least squares.

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Objective Function

The function that is minimized in an optimization problem, often denoting error.

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Regularization Parameter (λ)

A positive constant that controls the weight of the regularization function in RLS.

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Quadratic Regularization Function

A common form of regularization where the function is based on the norm of a transformed vector, R(x) = ∥Dx∥².

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Norm of a Vector

A measure of the length or magnitude of a vector, often denoted as ∥x∥.

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Matrix D

A matrix used in regularization to control the norm of transformed vector x.

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Minimization Problem

A situation where the goal is to find the lowest value of a function, often used in optimization.

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Formulation of RLS

The specific representation of the RLS problem as min ∥Ax − b∥² + λ∥Dx∥².

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Least Squares Method

A statistical technique to minimize the sum of squares of the differences between observed and predicted values.

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Parameters Vector

A vector that contains the parameters to be estimated in a model.

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Minimization

The process of reducing the value of a function to its lowest point.

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Cost Function

A function that measures the cost associated with a specific choice of parameters.

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Residuals

The difference between observed values and the values predicted by the model.

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Norm

A function that assigns a strictly positive length or size to vectors in a vector space.

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Linear Relation

A relationship that can be graphically represented as a straight line.

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Matrix Notation

The compact representation of systems of equations and transformations using matrices.

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sTi x

The inner product of vectors sTi and x, representing a transformation of the input.

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Linear system

A collection of linear equations involving the same variables.

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XT

The transpose of matrix X, flipping rows and columns.

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Inverse of a matrix

A matrix that, when multiplied by the original, yields the identity matrix.

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(XT X)−1

The inverse of the product of XT and X used in least squares.

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Best-fit line

A straight line that minimizes the distance from all data points.

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Circle Fitting Problem

A mathematical challenge to best fit a circle to a set of points.

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Applications

Various fields where circle fitting is useful, including archeology and petroleum engineering.

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Nonlinear Equations

Equations where the variable appears in a non-linear manner, complicating solutions.

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Norm Function

A function that assigns a size or length to a vector, but is not differentiable.

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Squared Version

Refers to squaring terms like distances in the circle fitting problem for differentiability.

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NLS Problem

Nonlinear least squares problem, minimizing the difference between predicted and observed values.

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Minimization Process

The method of finding the smallest value of a function in optimization problems.

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m Points

The number of observations or data points used in circle fitting.

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RLS Objective Function

min{ fRLS (x) = xT (AT A + λDT D)x - 2bT Ax + ∥b∥2 }

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Hessian Matrix

∇2 fRLS (x) = 2(AT A + λDT D) ⪰ 0 indicates the function's curvature.

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Stationary Points

Points satisfying ∇ fRLS (x) = 0 are potential minima.

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Global Minimum

A stationary point that is the lowest in the entire domain.

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RLS Solution

xRLS = (AT A + λDT D)−1 AT b is the computed solution.

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Positive Definite

AT A + λDT D ≻ 0 indicates a matrix is positive definite.

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Solving Linear Equations

Involves finding a vector that satisfies the equation AT A + λDT D = AT b.

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Study Notes

Least Squares

  • Overdetermined Systems: A linear system is overdetermined if there are more equations than unknowns. This often occurs in situations where data points are fitted to a model.
  • Solution: The exact solution to an overdetermined system is typically not available. Instead, the least squares approach finds the solution that minimizes the sum of squared errors.
  • Normal Equations: The approach to finding the least squares solution involves solving the normal equations, which are derived from minimizing the squared Euclidean norm of the residual.
  • Optimal Solution: The optimal least squares solution is expressible as XLS = (ATA)⁻¹Ab. Alternatively, this solution is equivalent to solving the system (ATA) XLS = Ab.
  • Full Column Rank: The matrix A in the system Ax = b must have a full column rank for the least squares solution to be unique.
  • Data Fitting: The process of finding the best-fitting curve or model to a set of data points, using the least squares method to approximate the data.

Data Fitting

  • Problem Definition: Given a set of data points (sᵢ, tᵢ), i = 1, 2, ..., m, where sᵢ ∈ Rⁿ and tᵢ ∈ R, find the parameters x ∈ Rⁿ that minimizes the sum of squared errors, such as ∑ᵢ(sᵢx - tᵢ)² .
  • Linear Relation: A common case is a linear relation between s and t: tᵢ = sᵢx.
  • Objective: Find the optimal parameters vector x that minimizes the sum of squared differences between the calculated values and the measured values for all the data points.
  • Matrix Formulation: Using vectors and matrices to formalize the optimization problem.

Regularized Least Squares

  • Problem: Used to find a good solution even when the system is underdetermined. It introduces a regularization term to the least squares problem. In particular, it forces the least squares solution to be as small as possible.
  • Objective: Minimizing a penalized problem with a regularization term, which is typically a quadratic norm of a matrix multiplied on x.
  • Regularization Parameter: A positive constant, λ, controlling the influence of the regularization term. Larger values of λ result in smoother solutions.

Denoising

  • Problem: Used to reduce noise in data, often applied to signal processing/time series data. A noisy measurement, b, is equal to the original signal, x, plus a noise term, w.
  • Objective: To find the best estimate of the original signal, x.
  • Noise Reduction: Adds a penalty term to the problem that encourages small differences between consecutive signal elements which reflects the signal's smoothness.
  • Minimization: The solution involves minimizing the sum of the squared differences between the measured values and the estimated values for the data points along with a penalty proportional to the norm of either the difference between consecutive elements, or a derivative operator, depending on the approach.

Nonlinear Least Squares

  • Problem: Fitting a model to data where the relationship between the variables is not linear.
  • Approach: The problem is reformulated as a minimization of the sum of squared errors with the corresponding nonlinear equations. The optimization problem is solved through iterative methods, like the Gauss-Newton method.

Circle Fitting

  • Problem: Finding the best-fitting circle to a set of points in an n-dimensional space.
  • Approach: Reformulated as a nonlinear least squares problem and reformulated as a linear least squares problem.
  • Solution: The linear least squares solution is found followed by finding the variables.

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