Question

Fit by the method of least squares the plane \(z=a+b x+c y\) to the five points \((x, y, z):(-1,-2,5),(0,-2,4),(0,0,4),(1,0,2),(2,1,0)\). Let the \(\mathrm{R}\) vectors \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) contain the values for \(x, y\), and \(z .\) Then the \(\mathrm{LS}\) fit is computed by \(\operatorname{lm}(z-x+y)\)

Short answer

The short answer would be the coefficients a, b, and c of the plane's equation, obtained by solving the linear system which should be the optimal solution when employing the least squares method. They present an equation of a plane that best fits the given points.

Step-by-step solution

Setting Up the Vectors

The x, y, and z vectors each holds the corresponding x, y, and z values from the given points. This means, the x vector would be \([-1, 0, 0, 1, 2]\), the y vector would be \([-2, -2, 0, 0, 1]\), and the z vector would contain \([5, 4, 4, 2, 0]\). These vectors will be used to facilitate the computation of the least squares fit.

Building the Coefficient Matrix

To apply the least squares method, it's needed to create a coefficient matrix, A. Starting with a column of ones (to help calculate the intercept term), append the x and y vectors. This results in a 5x3 matrix A:\n\[\begin{bmatrix} 1 & -1 & -2 \ 1 & 0 & -2 \ 1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 2 & 1 \end{bmatrix}\]

Apply the Least Squares Method

The least squares method requires the computation of \(A^T A\) and \(A^T z\). First compute \(A^T A\), a 3x3 matrix, then compute \(A^T z\), a 3x1 matrix. Now, solve the linear system \(A^T A x = A^T z\) for vector x, where x = [a, b, c] represents the coefficients of the equation of the plane.

Solve for Coefficients

With the linear system set up, the final step is to solve for the coefficients a, b, and c. The three equations derived from the linear system could be solved using various methods; substitution, elimination or using technology like scientific calculators or software like MATLAB, Python etc.

Key concepts

Linear Algebra

Linear algebra is a field of mathematics that deals with the study of vectors, vector spaces (also known as linear spaces), linear transformations, and systems of linear equations. It provides a way of concisely representing and operating on sets of linear equations. For example, linear algebra is the backbone for solving a system of equations like the one found in our least squares exercise, where we aim to find the best-fitting plane through a set of points.

In the context of the exercise, linear algebraic concepts such as vectors, matrices, and matrix multiplication come into play when building the coefficient matrix and vectors that represent the points in space. The method of least squares also involves solving a linear system, which is a fundamental operation in linear algebra. Tools from this mathematical field make it relatively straightforward to handle large datasets and complex relationships in a computationally efficient manner.

Method of Least Squares

The method of least squares is a statistical technique to determine the best fit line or plane in the context of regression problems. It addresses the question of how best to represent data with a model when there is variability or noise in the measurements. The least squares approach minimizes the sum of the squares of the residuals, which are the differences between observed values and the values predicted by the model.

In our exercise, the plane's equation is represented by \(z=a+bx+cy\), where \(a, b,\) and \(c\) are the coefficients we aim to determine. By minimizing the squared differences between the given z values and those predicted by our plane for the corresponding x and y values, we calculate these coefficients. The resulting least squares fit provides the coefficients such that the plane is the best approximation of the data points in the least squares sense.

Statistical Fitting

Statistical fitting is the process of constructing a curve, or a mathematical function, which best approximates the relationship between two or more variables in a given set of data. The goal is to depict the overall trend of the data, despite the possible presence of randomness or outliers in the sample.

This concept is essential for understanding our exercise because fitting the plane \(z=a+bx+cy\) is fundamentally a problem of statistical fitting. We are seeking the parameter values for \(a, b,\) and \(c\) that produce the curve – in this case, a plane – that has the closest possible agreement with the observed data points. Each statistical fitting scenario may involve different types of models, such as linear, polynomial, or more complex ones, but the core purpose is to derive a model that suffices to summarize the relationship within the observed data.

Regression Analysis

Regression analysis is a predictive modeling technique used to investigate the relationship between a dependent (target) variable and one or more independent (predictor) variables. The aim is to understand whether and how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed.

In the given exercise, regression analysis involves identifying the values of \(a, b,\) and \(c\) in the plane equation \(z=a+bx+cy\) based on the observed 3D points. This type of regression is known as 'multiple linear regression,' since we're working with more than one predictor variable (x and y) to estimate the dependent variable (z). The method of least squares, as applied here, is a fundamental strategy employed in regression analysis to derive the model parameters. It ensures that our plane is the best linear representation of the relationship between the x, y, and z values observed.