Higuchi dimension

The original formulation of the method is due to T. Higuchi.[4] Given a time series ${\displaystyle X:\{1,\dots ,N\}\to \mathbb {R} }$ consisting of ${\displaystyle N}$ data points and a parameter ${\displaystyle k_{\mathrm {max} }\geq 2}$ the Higuchi Fractal dimension (HFD) of ${\displaystyle X}$ is calculated in the following way: For each ${\displaystyle k\in \{1,\dots ,k_{\mathrm {max} }}\}$ and ${\displaystyle m\in \{1,\dots ,k}\}$ define the length ${\displaystyle L_{m}(k)}$ by

${\displaystyle L_{m}(k)={\frac {N-1}{\lfloor {\frac {N-m}{k}}\rfloor k^{2}}}\sum _{i=1}^{\lfloor {\frac {N-m}{k}}\rfloor }|X_{N}(m+ik)-X_{N}(m+(i-1)k)|.}$

The length ${\displaystyle L(k)}$ is defined by the average value of the ${\displaystyle k}$ lengths ${\displaystyle L_{1}(k),\dots ,L_{k}(k)}$,

${\displaystyle L(k)={\frac {1}{k}}\sum _{m=1}^{k}L_{m}(k).}$

The slope of the best-fitting linear function through the data points ${\displaystyle \left\{\left(\log {\frac {1}{k}},\log L(k)\right)\right\}}$ is defined to be the Higuchi fractal dimension of the time-series ${\displaystyle X}$.

For a real-valued function ${\displaystyle f:[0,1]\to \mathbb {R} }$ one can partition the unit interval ${\displaystyle [0,1]}$ into ${\displaystyle N}$ equidistantly intervals ${\displaystyle [t_{j},t_{j+1})}$ and apply the Higuchi algorithm to the times series ${\displaystyle X(j)=f(t_{j})}$. This results into the Higuchi fractal dimension of the function ${\displaystyle f}$. It was shown that in this case the Higuchi method yields an approximation for the box-counting dimension of the graph of ${\displaystyle f}$ as it follows a geometrical approach (see Liehr & Massopust 2020[5]).

Applications to fractional Brownian functions and the Weierstrass function reveal that the Higuchi fractal dimension can be close to the box-dimension.[4][5] On the other hand, the method can be unstable in the case where the data ${\displaystyle X(1),\dots ,X(N)}$ are periodic or if subsets of it lie on a horizontal line (see Liehr & Massopust 2020[5]).