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I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32.5$$ using the least squares method.

So we are looking for a cubic polynomial $p(x)$ such that $$\sum_{i=0}^4\left (p(x_i)-y_i\right )^2$$ is minimal, right?

Let $p(x)=a_3x^3+a_2x^2+a_1x+a_0$. Then we get the following sum:

$$\left (-8a_3+4a_2-2a_1+a_0+33\right )^2+\left (-a_3+a_2-a_1+a_0+20\right )^2+\left (a_0+20.1\right )^2+\left (27a_3+9a_2+3a_1+a_0+4.3\right )^2+\left (42.875a_3+12.25a_2+3.5a_1+a_0-32.5\right )^2$$

Now we want to calculate the values of $a_0, a_1, a_2, a_3$ such that this sum is minimal, right?

How could we do that? Could you give me a hint?

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    $\begingroup$ Hint: take the partial derivatives with respect to each $a_i$, set to $0$, and solve. $\endgroup$ – Robert Israel Feb 3 at 2:17
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So you want to minimize $S = \sum_{i=0}^4\left (p(x_i)-y_i\right )^2 $ where $p(x) =\sum_{k=0}^3 a_kx^k $.

The parameters you want to find are the $a_k$. You need to differentiate $S$ with respect to each $a_k$ and set that expression equal to zero.

This will give you $4$ equations in the $4$ $a_k$s.

Here is a typical one:

$\begin{array}\\ \dfrac{\partial S}{\partial a_k} &=\dfrac{\partial }{\partial a_k}\sum_{i=0}^4\left( p(x_i)-y_i\right)^2\\ &=\sum_{i=0}^4 \dfrac{\partial }{\partial a_k}\left(p(x_i)-y_i\right)^2\\ &=\sum_{i=0}^4 2\dfrac{\partial }{\partial a_k}(p(x_i)-y_i)(p(x_i)-y_i)\\ &=\sum_{i=0}^4 2\dfrac{\partial }{\partial a_k}(\sum_{j=0}^3 a_jx_i^j)(p(x_i)-y_i)\\ &=\sum_{i=0}^4 2( x_i^k)(\sum_{j=0}^3 a_jx_i^j-y_i)\\ &=2(\sum_{j=0}^3 a_j\sum_{i=0}^4 x_i^{j+k}-\sum_{i=0}^4 x_i^ky_i)\\ \end{array} $

Setting $\dfrac{\partial S}{\partial a_k} = 0$, this gives $\sum_{j=0}^3 a_j\sum_{i=0}^4 x_i^{j+k} =\sum_{i=0}^4 x_i^ky_i $ for $k = 0$ to $3$.

These are the equations that determine the $a_j$.

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To find the polynomial of order $k$ given $N$ observations ($x_i$, $y_i$) it reduces to solving the following set of linear equations:

\begin{eqnarray} \begin{bmatrix} N & \sum_{i=1}^{N} x_i & \cdots & \sum_{i=1}^{N} x_i^k \\ \sum_{i=1}^{N} x_i & \sum_{i=1}^{N} x_i^2 & \cdots & \sum_{i=1}^{N} x_i^{k+1} \\ \vdots & \vdots & \vdots & \vdots \\ \sum_{i=1}^{N} x_i^k & \sum_{i=1}^{N} x_i^{k+1} & \cdots & \sum_{i=1}^{N} x_i^{2k} \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_k \\ \end{bmatrix}= \begin{bmatrix} \sum_{i=1}^{N} y_i \\ \sum_{i=1}^{N} x_i y_i \\ \vdots \\ \sum_{i=1}^{N} x_i^k y_i \\ \end{bmatrix} \end{eqnarray}

One method to solve this is Cramer's Rule which allows us to solve the above equation of the form $Ma = b$ for $a_i$ as:

$$a_i = \frac{\mathrm{det}(M_i)}{\mathrm{det}(M)}$$

where where $M_{i}$ the matrix formed by replacing the $i$-th column of $M$ by the column vector $b$.

For your above observations we would be solving: \begin{equation} \begin{bmatrix} 5 & 3.5 & 26.25 & 60.875 \\ 3.5 & 26.25 & 60.875 & 248.0625 \\ 26.25 & 60.875 & 248.0625 & 735.21875 \\ 60.875 & 248.0625 & 735.21875 & 2632.265625 \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \\ \end{bmatrix}= \begin{bmatrix} -44.9 \\ 186 \\ 207.425 \\ 1561.3375 \\ \end{bmatrix} \end{equation}

Using Cramer's rule for $a_0$ we have

\begin{equation} a_o = \frac{\mathrm{det} \begin{bmatrix} -44.9 & 3.5 & 26.25 & 60.875 \\ 186 & 26.25 & 60.875 & 248.0625 \\ 207.425 & 60.875 & 248.0625 & 735.21875 \\ 15613375 & 248.0625 & 735.21875 & 2632.265625 \end{bmatrix}}{\mathrm{det} \begin{bmatrix} 5 & 3.5 & 26.25 & 60.875 \\ 3.5 & 26.25 & 60.875 & 248.0625 \\ 26.25 & 60.875 & 248.0625 & 735.21875 \\ 60.875 & 248.0625 & 735.21875 & 2632.265625 \end{bmatrix}}\approx -23.0087 \end{equation}

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  • $\begingroup$ Ahh ok!! But then we would get $p(0)\approx -23.0087$, or not? But it is given that $y_2=-20.1$. Shouldn't it be the same? $\endgroup$ – Mary Star Feb 9 at 16:07
  • $\begingroup$ @MaryStar, not necessarily. Thought that is the actual data point, the $a_k$ are such that sum of the the squared residual is minimized across all points. So though $a_0 = -20.1$ would minimize the residual for that point, it would lead to an overall higher sum of the square residual across all points, which is why they are not necessarily the same. $\endgroup$ – ALollz Feb 9 at 18:19

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