# Why should Gaussian noise have fractal dimension of 1.5?

In a paper I'm trying to understand, the following time series is generated as "simulated data":

$$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,...,N)$$

where $Z(j)$ is a Gaussian noise with mean $0$ and standard deviation $1$.

The paper is about estimating the fractal dimension of $Y$ (not to mention some other series). The author says that the fractal dimension of $Y$ is $1.5$, but doesn't explain why.

My first question is why should we expect the fractal dimension of $Y$ to be $1.5$?

My second question is: can anyone think of a reason why I keep getting a fractal dimension of about $2$ for $Y$, when using the technique described in the paper?

That technique, summarized, is:

1) Create a new set of time series based on the original $Y$ as follows:

$$X^m_k=Y(m),Y(m+k),Y(m+2k),...,Y\left(m+\left \lfloor \frac{N-m}{k}\right \rfloor k\right) \:\:\:\: ; \:\:\:\: (m=1,2,...,k)$$

where the notation $\left \lfloor \right \rfloor$ denotes the floor function, and $k=1,2,3,4,...$. But for $k>4$, $k=\lfloor 2^{(j-1)/4} \rfloor$ where $j=11,12,13,...$

2) Define and calculate the "length" of each "curve" $X^m_k$ as follows:

$$L_m(k)=\frac{1}{k} \frac{N-1}{Qk} \left( \sum_{i=1}^{Q} \left | X(m+ik)-X(m+(i-1)k) \right | \right)$$

where $Q=\left \lfloor\frac{N-m}{k} \right \rfloor$

3) For each $k$, define the average $L_m(k)$ (averaged over $m$) as $y(k)=\langle L_m(k) \rangle$, and then scatter plot $\ln(y(k))$ against $\ln(k)$ and fit a line via least squares. The line should be straight. And the slope of the line should be about $-1.5$, the negative of which is then interpreted as an estimation of the fractal dimension.

When I follow these steps, I get a straight line, but slope of $-2$.

The paper is Higuchi. 1988. "Approach to an irregular time series on basis of fractal dimension."

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Is there suddenly a problem with my answer, explaining that you (silently) unaccept it? –  Did Mar 5 at 22:07
@Did Yes. Thanks again for that answer, which is instructive in and of itself. However, recently I have had occasion to revisit this matter in light of new readings and must reconsider its acceptance. It is fractal Brownian noise (a.k.a. red noise) which should have a FD of 1.5. The term "Gaussian noise," which I used to think referred exclusively to white noise, can also refer to fBn. Even so, the noise formula $Y(i)$ Higuchi gives yields an FD of 2, not 1.5. This is confirmed by plotting the periodogram and seeing that it is flat (i.e. power law index a=0). –  ben Mar 6 at 4:01
Which part of the argument in this answer do you have trouble with? –  Did Mar 6 at 6:27
@Did Your answer asserts that the series $Y$ has FD=1.5 when in fact it has FD=2.0. –  ben Mar 6 at 12:14
No my answer does not "assert" that, it proves it. You, on the contrary, are invoking results which might or might not correspoond to the setting you described in your question. Please refer to what is written in my answer (did you actually read it?) to mention its specific faulty steps, if any. –  Did Mar 6 at 16:02

T. Higuchi, Approach to an irregular time series on basis of fractal dimension, Physica D: Nonlinear Phenomena, Volume 31, Issue 2, June 1988, Pages 277–283.

My second question is: can anyone think of a reason why I keep getting a fractal dimension of about 2 for Y, when using the technique described in the paper?

This question is difficult to answer since you do not show your computations but $D=\frac32$.

To show this, note that for every process with stationary increments such as $(Y(i))_i$, $$\mathbb E(L(k))=\frac1k\frac{N-1}{Qk}Q\,\mathbb E(|Y(k)-Y(0)|),$$ and that in the present case $\mathbb E(|Y(k)-Y(0)|)=\sqrt{k}\,\mathbb E(|Z|)$ since $Y(k)-Y(0)$ is the sum of $k$ i.i.d. standard normal random variables hence equals $\sqrt{k}|Z|$ in distribution with $Z$ standard normal. Thus, $D=\frac32$ since $$\mathbb E(L(k))=\frac{N-1}{k\sqrt{k}}\,\mathbb E(|Z|).$$ The same computation yields $D=2$ when $(Y(i))_i$ is i.i.d. and $D=1$ when $(Y(i))_i$ is regular, say $Y(i)=Y(0)+iT$ for some integrable random variable $T$.

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My first question is why should we expect the fractal dimension of $Y$ to be $1.5$ ?

There are several algorithm used to calculate dimensions from data. See google.

For your second question you certainly have an error, either in the steps or in numerical calculations. I notice that the topic of chaos theory and fractals is very complicated and needs a deep learning to understood notions and algorithms. So, be carreful about what you reading.

See this:

http://www.ll.mit.edu/publications/journal/pdf/vol03_no1/3.1.4.estimatingfractals.pdf

http://www.hindawi.com/journals/mpe/2010/157264/

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