Infractal geometry, theHiguchi dimension (orHiguchi fractal dimension (HFD)) is an approximate value for thebox-counting dimension of the graph of a real-valued function or time series. This value is obtained via an algorithmic approximation so one also talks about theHiguchi method. It has many applications in science and engineering and has been applied to subjects like characterizing primary waves inseismograms,^[1] clinical neurophysiology^[2] and analyzing changes in the electroencephalogram in Alzheimer's disease.^[3]

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${\displaystyle k_{\mathrm{m}\mathrm{a}\mathrm{x}}\ge 2}$ the Higuchi Fractal dimension (HFD) of${\displaystyle X}$ is calculated in the following way: For each${\displaystyle k\in \{1,\dots ,k_{\mathrm{m}\mathrm{a}\mathrm{x}}}\}$ 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 theWeierstrass 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]).

• Fractals
• Algorithms

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