least square approximations

Let $\displaystyle t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $\displaystyle j=0,...,99$. Determine the values of $\displaystyle c_{l},d_{m}$ for $\displaystyle l=0,...,5,m=1,...,4$, so that

$\displaystyle P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$

is the least squares approximation to the data points $\displaystyle (t_j,x_j)$ for $\displaystyle j=0,...,99$.

Quote:

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Originally Posted by alphabeta89

Let $\displaystyle t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $\displaystyle j=0,...,99$. Determine the values of $\displaystyle c_{l},d_{m}$ for $\displaystyle l=0,...,5,m=1,...,4$, so that

$\displaystyle P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$

is the least squares approximation to the data points $\displaystyle (t_j,x_j)$ for $\displaystyle j=0,...,99$.

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More context would help with this, like is the data known and you want the actual coefficients, or do you want something like the regression equations?

CB

Quote:

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Originally Posted by CaptainBlack

More context would help with this, like is the data known and you want the actual coefficients, or do you want something like the regression equations?

CB

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Sorry! I realised I left out some details!

Let $\displaystyle t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $\displaystyle j=0,...,99$. Define

$\displaystyle f(t)= \sum_{k=0}^{99}(a_{k}\textup{cos}(2\pi{kt})+b_{k}{ sin}(2\pi{kt}))$

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Define $\displaystyle f(t_{j})$ by $\displaystyle x_{j}$ for $\displaystyle j=0,...,99.$Determine the values of $\displaystyle c_{l},d_{m}$ for $\displaystyle l=0,...,5,m=1,...,4$, so that

$\displaystyle P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$

is the least squares approximation to the data points $\displaystyle (t_j,x_j)$ for $\displaystyle j=0,...,99$.

Quote:

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Originally Posted by alphabeta89

Sorry! I realised I left out some details!

Let $\displaystyle t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $\displaystyle j=0,...,99$. Define

$\displaystyle f(t)= \sum_{k=0}^{99}(a_{k}\textup{cos}(2\pi{kt})+b_{k}{ sin}(2\pi{kt}))$

.
Define $\displaystyle f(t_{j})$ by $\displaystyle x_{j}$ for $\displaystyle j=0,...,99.$Determine the values of $\displaystyle c_{l},d_{m}$ for $\displaystyle l=0,...,5,m=1,...,4$, so that

$\displaystyle P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$

is the least squares approximation to the data points $\displaystyle (t_j,x_j)$ for $\displaystyle j=0,...,99$.

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This is now a numerical problem, consider using the Excel solver to find the best fit.

(Fourier theory suggests that $\displaystyle c_0\approx 0$ and $\displaystyle c_i\approx 100$ for $\displaystyle i=1..5$ and $\displaystyle d_i\approx 100$ for $\displaystyle i=1 .. 4$ )

CB