The formulas (5.63) apply only when the sample times are...

The formulas (5.63) apply only when the sample times are symmetric around 0 . When the sample points $t_1, \ldots, t_n$ are equally spaced, so $t_{i+1}-t_i=h$ for all $i=1, \ldots, n-1$, then there is a simple trick to convert the least squares problem into a symmetric form. (a) Show that the translated sample points $s_i=t_i-\bar{t}$, where $\bar{t}=\frac{1}{n} \sum_{i=1}^n t_i$ is the average, are symmetric around 0 . (b) Suppose $q(s)$ is the least squares polynomial for the data points $\left(s_i, y_i\right)$. Prove that $p(t)=q(t-\bar{t})$ is the least squares polynomial for the original data $\left(t_i, y_i\right)$. (c) Apply this method to find the least squares polynomials of degrees 1 and 2 for the following data: \begin{tabular}{c|cccccc} $t_i$ & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline$y_i$ & -8 & -6 & -4 & -1 & 1 & 3 \end{tabular}

The formulas (5.63) apply only when the sample times are symmetric around 0 . When the sample points $t_1, \ldots, t_n$ are equally spaced, so $t_{i+1}-t_i=h$ for all $i=1, \ldots, n-1$, then there is a simple trick to convert the least squares problem into a symmetric form.
(a) Show that the translated sample points $s_i=t_i-\bar{t}$, where $\bar{t}=\frac{1}{n} \sum_{i=1}^n t_i$ is the average, are symmetric around 0 . (b) Suppose $q(s)$ is the least squares polynomial for the data points $\left(s_i, y_i\right)$. Prove that $p(t)=q(t-\bar{t})$ is the least squares polynomial for the original data $\left(t_i, y_i\right)$. (c) Apply this method to find the least squares polynomials of degrees 1 and 2 for the following data:
\begin{tabular}{c|cccccc}
$t_i$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline$y_i$ & -8 & -6 & -4 & -1 & 1 & 3
\end{tabular}

Key Concepts

Least Squares Approximation

This concept is about finding a function, often a polynomial, that minimizes the sum of the squared differences (errors) between the given data points and the values predicted by the function. It is a fundamental approach in regression analysis, allowing one to obtain the best fitting curve in the presence of noise or measurement errors.

Data Translation and Shifting

Data translation involves shifting the independent variable values by subtracting a constant—typically the mean—to center the data. This operation often simplifies the structure of the least squares problem by producing a symmetric distribution of sample points around zero, which can lead to simpler computations and better numerical properties.

Equivalence Under Translation

This principle states that a least squares solution obtained on transformed (translated) data can be mapped back to the original variable space without loss of optimality. Specifically, if a best-fit polynomial is found for the centered data, translating that polynomial back by the same shift produces the best-fit polynomial for the original data. This invariance under translation leverages the linearity of the least squares method.

Polynomial Regression

Polynomial regression is a form of regression analysis where the relationship between the independent variable and the dependent variable is modeled as an nth degree polynomial. It is applied when the data suggests a curvilinear relationship, and the fitted polynomial is found using methods like least squares to minimize the error between predicted and actual values.

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