# Variance Ratio test for 3-D random walks

The variance ratio test proposed by Lo and MacKinlay (1988) is used to detect 1-D random-walk-like-behaviour. 1-D works great for time-series data, but I'd like to adapt this test for imaging data to detect 3-D random walks. Is there a way to extend this test to detect 3-D random walks?

For reference, the variance ratio of a $$k$$ time-step interval is:

$$V(k)=\frac{\operatorname{Var}\left(x_{t}+x_{t-1}+\ldots+x_{t-k+1}\right) / k}{\operatorname{Var}\left(x_{t}\right)}$$

The proposed variance ratio is given by:

$$V R(k)=\frac{\hat{\sigma}^{2}(k)}{\hat{\sigma}^{2}(1)}$$

where $$\hat{\sigma}^{2}(1)$$ is a one time-step interval variance.

The statistic follows a standard normal distribution:

$$Z(k)=\frac{V R(k)-1}{\sqrt{\phi(k)}}$$