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I have an existing time series that I would like to make more volatile, or more variance.
I would like the highs to be higher and the lows to be lower.
The time series is somewhat stationary and I would like the amplification of the numbers in the series to keep the same slope. In other words, I would like the mean of the series to remain the same and the standard deviation of the series to increase.

Below is my attempt.
I fit a line to it with linear regression. This part works okay. The problem comes when I try to make every number above the regression line higher and every number below it low at a scale. See the picture below for how this code works. It's not giving me a spikey, volatile series I want. It's just dividing the highs and lows further apart.
Sorry, I don't know math notation, and I'm trying to do this in Python, so I'd really appreciate it if you could make your answer legible to a math illiterate like me.

def slope(series):
    x = [x for x in range(int(len(series)))]
    fit = np.polyfit(x, series, 1)
    fit_fn = np.poly1d(fit)
    return fit_fn(x)

ss = slope(time_series)

tu = []
w = 0
for n, z in enumerate(time_series):
    b = z * 1.3
    if ss[n] < z:
        r = (z - b)
    else:
        r = (z + b) 
    w += r
    tu.append(r)

This was my attempt.

enter image description here

This is the original.

enter image description here

What I want is just a time series with more extreme swings. I want everything to scale from the mean trend line.

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    $\begingroup$ Isn't wanting the standard deviation to stay the same while volatility increases contradictory? If you drop the former requirement, what about fitting a linear trend, obtaining the residuals, multiplying them by $c>1$ and adding them back to the fitted trend? That would increase the amplitude of the swings $c$ times. $\endgroup$ Commented Sep 6, 2021 at 14:53
  • $\begingroup$ Thank you. I fixed the standard deviation part. As for your solution, I'm not sure where to begin with that. $\endgroup$
    – Renoldus
    Commented Sep 6, 2021 at 16:36
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    $\begingroup$ You can begin by fitting a linear trend to your data. (I do not speak Python, so I cannot suggest a code bit.) $\endgroup$ Commented Sep 6, 2021 at 16:50
  • $\begingroup$ I appreciate you paying attention to my lonely post. Yes, I fit a line with linear regression in the code. I tried a few combinations of multiplying and subtracting or adding to the time series if the the number was above or below the fit line. It's how I got the first picture in my post. It was the point I got lost at. $\endgroup$
    – Renoldus
    Commented Sep 6, 2021 at 17:43
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    $\begingroup$ Are you able to extract residuals after fitting a linear trend? If yes, multiply them by $c$ and add to the fitted line. It should do the trick. You do not need to condition on whether the residual is positive or negative; maybe the conditioning is messing things up. $\endgroup$ Commented Sep 6, 2021 at 18:33

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Probably the quickest/easiest way to do this is to:

  • Stationarise the time series by taking differences or returns: e.g. $x_{diff}(t) = x(t)-x(t-1)$.
  • Subtract the mean and divide by the standard deviation: $x_{norm}(t) = \frac{x_{diff}(t)-\mu(x_{diff})}{\sigma(x_{diff})}$
  • You now have something resembling data drawn from a Gaussian distribution of mean 0 and standard deviation 1.
  • Now just multiply the data by whatever you want the new standard deviation to be, add back in the mean, and do a cumulative sum/product to convert from difference/returns back to original values.
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