## Extending a sine regression to forecast

on 6 May 2012

### Wayne King (view profile)

I would like to know the best way to extend my newly constructed sine wave fit to my data into the future another 20 days.

I have constructed a regression of 151 different sine waves to a data set across 501 days at a sample of 1 day.

```y = Score(:);
n = 501;
t = (1:501)';
games = 1:501;
```
```data(1:151) = struct('X',NaN(501,3),'bhat',NaN(3,1),'yhat',NaN);
```
```for ii = 1:151
tmp = 2*pi*(sincos(ii))*t;
data(ii).X = rand(501,3);
data(ii).X(:,2) = cos(tmp)';
data(ii).X(:,3) = sin(tmp)';
data(ii).bhat = data(ii).X\y;
data(ii).yhat = data(ii).bhat(1)+data(ii).bhat(2)*cos(tmp)+data(ii).bhat(3)*sin(tmp);
end
```

After that, I have combined or superposed all of the 151 different sin waves into one sine wave

```sum(horzcat(data.yhat),2) ./ 151
```

So ideally, I would like the combined sine wave to extend to 521 days (for a 20 day forecast) while my data set remains to be only 501 days long and plot them both. Help much appreciated!

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### Wayne King (view profile)

on 6 May 2012
```   n = (0:500)';
% create some fake data
x = 1.5*cos(2*pi*(1/4)*n)+randn(size(n));
X = ones(501,3);
X(:,2) = cos(2*pi*(1/4)*n);
X(:,3) = sin(2*pi*(1/4)*n);
beta = X\x;
nn = 0:520;
xhat = beta(1)+beta(2)*cos(2*pi*(1/4)*n)+beta(3)*sin(2*pi*(1/4)*n);
xhatpred = beta(1)+beta(2)*cos(2*pi*(1/4)*(501:520)')+beta(3)*sin(2*pi*(1/4)*(501:520)');
xhat = [xhat; xhatpred];
plot(n,x,'k');
hold on;
plot(1:521,xhat,'r');
set(gca,'xlim',[490 512]);
legend('original data','prediction for next 21 days');
grid on;```

Clifford Shelton

### Clifford Shelton (view profile)

on 6 May 2012

thanks! crunching it all now to work with the code I have already written...

Clifford Shelton

### Clifford Shelton (view profile)

on 6 May 2012

Does it seem strange to you that the 'original' data and 'prediction' data aren't in line with each other at all before the forecast? I thought they should be more in line so as to have a more accurate forecast. U?

Wayne King

### Wayne King (view profile)

on 6 May 2012

I would not expect them to agree exactly. You have to keep in mind that the regression model will not fit the data exactly. The regression model parameters minimize the overall sum of squares of the residuals.

### Wayne King (view profile)

on 6 May 2012

Once you have the parameters, which are the amplitudes of the cosines and sines, then you can easily extend your model to make predictions by simply increasing the length of the time vector. Of course how far into the future you can make reasonable predictions is a tricky question.

```   n = (0:500)';
% create some fake data
x = 1.5*cos(2*pi*(1/4)*n-pi/4)+randn(size(n));
X = ones(501,3);
X(:,2) = cos(2*pi*(1/4)*n);
X(:,3) = sin(2*pi*(1/4)*n);
beta = X\x;
nn = 0:520;
xhat = beta(1)+beta(2)*cos(2*pi*(1/4)*nn)+beta(3)*sin(2*pi*(1/4)*nn);```

Clifford Shelton

### Clifford Shelton (view profile)

on 6 May 2012

ok..but I'm still not sure how to plot the extension.
When I try to plot it...the plot ends where my last data point is in the original data. :-(

Clifford Shelton

### Clifford Shelton (view profile)

on 6 May 2012

umm..furthermore when I add the nn variable and add it to my calculations of the xhat varable..it affects the fit drastically.

I like the fit I am able to generate originally. Is there a simpler way to keep the current fit and then just project the xhat line forward a few more steps?

Thanks a bunch!

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