Hello forum members,

I am happy to release the first pass at a simple, local approximation of the Hurst exponent. This is a work in progress and this post is already taking up too much of my lunch hour.

### What is the Hurst Exponent?

It's a measure of a time-series' feedback on itself, or its memory. If the exponent is found to be 1/2, the time series is moving "truly" randomly, and towards 1 or 0 it is either displaying positive feedback or negative feedback, respectively. It was developed by the hydrologist Harol Edwin Hurst in the 1950's to figure out how large to make a dam to deal with all of the possible flooding outcomes. It is directly related to the fractal dimension

*D*. Please read more from

Wikipedia or

*(Mis)Behavior of Markets* by Mandelbrot.

The equation from Wikipedia is as follows:

Hurst used the following equation for the Nile. He found it to be ~0.73 for that river.

*R( n)* is the range of the first *n* cumulative deviations from the mean, and *S*( n) their standard deviation, in the case of Hurst's research, the regular standard distribution is used. Mandelbrot cautions again and again in his book about being wary of such a measure because he claims markets are not governed by Gaussians, but by a mix of Cauchy and Gaussian distributions.
**E** s the expected value
*n *is the time span of the observation (number of data points in a time series)
*C *is a constant.

Mandelbrot suggests the following for

*R/S *from p. 288 of the referenced book

*:*
(LaTeX because it was a pain to write: \frac{R( n)}{S( n)} = \frac {\max\limits_{1 \leq k \leq n} \sum\limits_{j=1}^k (r_j - \bar{r_n}) - \min\limits_{1 \leq k \leq n} \sum\limits_{j=1}^k (r_j - \bar{r_n})} {\bigg( \sum\limits_{j} (r_j - \bar{r_n})^2\bigg)^\frac{1}{2}} )

*r *is the return
*r *bar is the average return
*n *is the max time range
*k* is the index for the sums

### ... What?

Okay what does it mean that it's related to the fractal dimension? It basically means that for a time series, the value of

*H* corresponds to how rough the time series is. Let me break it down:

- When
*H *is near 1 is implies **persistency:**
- positive feedback is at work
- all investors are bullish or bearish
- what to watch for: volume dropping off since everyone has made their bets, the market losing steam and going sideways

- When
*H *is exactly 1/2 is implies **random:**
- The stochastic time-series events are independent of each other. No memory or feedback
- What to watch for: conventional finance theories rejoicing temporarily

- When
*H *is near 0 it implies **reversion:**
- negative feedback
- tug-of-war, choppy, rough
- what to watch for: chartists mentioning the market coiling up, possibly a change in volatility

This measurement is taken over multiple timescales, ideally disparate-scaled ones extending deep, deep into the past.

### That seems like that math takes ages to do in thinkscript, is there a faster way?

As mentioned in another post, here is a local approximation of the Hurst exponent which can be used:

- Where
*h* is the *highest high* in the time range,
*l* is the *lowest low* in the time range,
*t *is the time period over which measurements are taken
- average_*true*_range is the average true
*price range*
*C* is a correction factor... ExtremeHurst has some other part that considers log-periodic oscillations... not going into that

Local to what? That time range. The Hurst exponent for one time range is hardly insightful. It is more useful to see what it is at multiple timescales. Which, finally brings us to:

### Plotting Hurst in ThinkorSwim

Here is the thinkscript study:

Rich (BB code):

```
input length = 4;
input averageType = AverageType.EXPONENTIAL;
def ll = Lowest(low[1], length);
def hh = Highest(high[1], length);
def atr = MovingAverage(averageType, TrueRange(high, close, low), length);
plot H;
def tmp_H = (Log(hh - ll) - Log(atr)) / (Log(length));
if H > 1 then {
H = 1;
} else if H < 0 {
H = 0;
} else {
H = tmp_H;
}
```

I suggest you run it in thermo mode. I've been using a range of 3-32. I tried making defaults in the above script but it breaks ToS...

**TL;DR: Shared link.** This is on top of

@YungTraderFromMontana's wonderful charting setup, FYI:

https://tos.mx/mFd0SQZ
### What about that correction factor? Where's the _______ ?

This script ****s. I have the following improvements to make:

- Make it work over multiple time frames. Right now it just descends some
*length *ticks into the past at the current timescale. This limits the utility
- Make defaults. Especially the mid color and range for the thermo plot
- Make it display semilog data; we want to peek deep deep into the time series and see what's frothing in the short term.
- Make alerts for when persistency or antipersistency is appearing across multiple timescales.
- Model log-periodic oscillations, esp. near binary events
- Account for the multifractal nature of trading time