## 数学代写|数值分析代写numerical analysis代考|MATH2722

2023年1月4日

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## 数学代写|数值分析代写numerical analysis代考|Least squares problems

In this chapter, we consider problems where there are more linear equations than unknowns, or equivalently, more constraints than variables. Such problems arise, for example, when we try to find parameters that “best fit” a model to a data set.

To help with the discussion, consider a representative example of finding a line of best fit for given data points $\left(x_1, y_1\right),\left(x_2, y_2\right), \ldots,\left(x_n, y_n\right)$. To determine such a line, we want to determine $a_0$ and $a_1$ such that $l(x)=a_0+a_1 x$ “best fits” the data. For this problem, we have 2 unknowns, $a_0$ and $a_1$, but we have $n$ equations we wish to satisfy:
$$y_i=l\left(x_i\right), \quad i=1,2,3, \ldots, n$$
Combining these equations into a matrix equation, we obtain
$$\left(\begin{array}{cc} 1 & x_1 \ 1 & x_2 \ \vdots & \vdots \ 1 & x_n \end{array}\right)\left(\begin{array}{l} a_0 \ a_1 \end{array}\right)=\left(\begin{array}{c} y_1 \ y_2 \ \vdots \ y_n \end{array}\right) \Longleftrightarrow \mathbf{A x}=\mathbf{b}$$
It is highly unlikely that there can be a solution to these $n$ equations, unless the $n$ points happened to all lie on a single line. So instead of looking for an exact solution, we will look for a “least squares” solution in the sense that we want to find $a_0$ and $a_1$ so that the least squares error
$$e\left(a_0, a_1\right)=\sum_{i=1}^n\left(y_i-\left(a_0+a_1 x_i\right)\right)^2$$
is minimized. Note there are different ways to minimize, but typically for data fitting problems, least squares error minimization is both intuitive and has good mathematical properties.

## 数学代写|数值分析代写numerical analysis代考|Solving LSQ problems with the normal equations

We consider now a linear system of $n$ equations and $m$ unknowns that can be written as
$$\mathbf{A}{n \times m} \mathbf{x}{m \times 1} \cong \mathbf{b}_{n \times 1} .$$
We assume that $n>m$, and $\mathbf{A}$ has full column $\operatorname{rank}$ (i. e., $\operatorname{rank}(\mathbf{A})=m$ ). This assumption corresponds to the parameters being independent of each other, which is typically a safe assumption. For example, in a line of best fit, one wants to find $a_0$ and $a_1$ that best fit a line
$$l(x)=a_0+a_1 x$$
to a data set. Here, $m=2$ and we would have full column rank (see the matrix $\mathbf{A}$ above). But if we changed the problem to instead look for coefficients of $1, x$, and also $(x+1)$, then we would have $m=3$ but a column rank of only 2 . This is because if we tried to use $a_0, a_1$, and $a_2$ to best fit a line
$$l(x)=a_0+a_1 x+a_2(x+1)$$
to a set of data, column 3 of the matrix would be a linear combination of the first two columns. Hence, for LSQ problems, the assumption of full column rank is reasonable since if the columns are not linearly independent, then typically this can be fixed by eliminating redundant columns and unknowns.

We now derive a solution method for the LSQ problem $\mathbf{A x} \cong \mathbf{b}$. Since this problem is defined to be finding $\mathbf{x}$ that minimizes $|\mathbf{A} \mathbf{x}-\mathbf{b}|_2$ and with that also $|\mathbf{A} \mathbf{x}-\mathbf{b}|_2^2$, we define the function
$$h(\mathbf{x})=|\mathbf{A} \mathbf{x}-\mathbf{b}|_2^2 .$$

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Least squares problems

$$y_i=l\left(x_i\right), \quad i=1,2,3, \ldots, n$$

$$e\left(a_0, a_1\right)=\sum_{i=1}^n\left(y_i-\left(a_0+a_1 x_i\right)\right)^2$$

## 数学代写|数值分析代写numerical analysis代考|Solving LSQ problems with the normal equations

$$\mathbf{A} n \times m \mathbf{x} m \times 1 \cong \mathbf{b}_{n \times 1} .$$

$$l(x)=a_0+a_1 x$$

$$l(x)=a_0+a_1 x+a_2(x+1)$$

$$h(\mathbf{x})=|\mathbf{A} \mathbf{x}-\mathbf{b}|_2^2 .$$

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## MATLAB代写

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