least square approximations

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December 7th 2011, 08:55 PM
alphabeta89

least square approximations

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

$P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$
is the least squares approximation to the data points $(t_j,x_j)$ for $j=0,...,99$ .
December 10th 2011, 01:07 AM
CaptainBlack

Re: least square approximations

Quote:

Originally Posted by alphabeta89

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

$P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$
is the least squares approximation to the data points $(t_j,x_j)$ for $j=0,...,99$ .

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
December 10th 2011, 06:02 AM
alphabeta89

Re: least square approximations

Quote:

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

Sorry! I realised I left out some details!

Let $t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $j=0,...,99$ . Define
$f(t)= \sum_{k=0}^{99}(a_{k}\textup{cos}(2\pi{kt})+b_{k}{ sin}(2\pi{kt}))$
.
Define $f(t_{j})$ by $x_{j}$ for $j=0,...,99.$ Determine the values of $c_{l},d_{m}$ for $l=0,...,5,m=1,...,4$ , so that

$P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$
is the least squares approximation to the data points $(t_j,x_j)$ for $j=0,...,99$ .
December 10th 2011, 08:43 AM
CaptainBlack

Re: least square approximations

Quote:

Originally Posted by alphabeta89

Sorry! I realised I left out some details!

Let $t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $j=0,...,99$ . Define
$f(t)= \sum_{k=0}^{99}(a_{k}\textup{cos}(2\pi{kt})+b_{k}{ sin}(2\pi{kt}))$
.
Define $f(t_{j})$ by $x_{j}$ for $j=0,...,99.$ Determine the values of $c_{l},d_{m}$ for $l=0,...,5,m=1,...,4$ , so that

$P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$
is the least squares approximation to the data points $(t_j,x_j)$ for $j=0,...,99$ .

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

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

CB