How can I make an existing time series more volatile? 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.

This is the original.

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

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.

