The key idea for improvement is based on the concept that each pattern and behavior in a fractal time series is repeated frequently in different scales. In a fractal time series in different scales shows similar patterns. That is a pattern in one small scale is expected to manifest in other larger scales with roughly the same fractality. Noise may affect the fractality of a pattern in one scale but its effect may not be considerable in other scales. Consequently, estimation of fractality of a time series in multiple scales can make it more robust to noise.[1]

Consider $x = [x_1,x_2,\ldots,x_N]$ as a time series with $N$ sample times, where $x_i$ indicates the $i^{th}$ sample time of $x$. The proposed improved PSVG algorithm consists of the following steps:

  1. Set $k=1$.
  2. Construct $k$ new sequence $x^k_m$ as follows:
    \begin{eqnarray}
    x^k_m = [x_m,x_{m+k},x_{m+2k},\ldots,x_{m+\lfloor\frac{N-m}{k}\rfloor k}](m = 1,2,\ldots,k) \nonumber
    \end{eqnarray}

    Where $\lfloor h \rfloor$ indicates the integer value of $h$, $k$ is called the scale which determines the delay between successive points in the sequence $x^k_m$.

    Calculate the PSVG for each sequence $x^k_m(m = 1,2,\ldots,k), PSVG_m(k)$. A least square fit is used to obtain the value of the slope known as Power of Scale-freeness.

  3. Compute the average PSVG for scale $k,PSVG(k)$, through averaging the $k$ calculated values $PSVG_m(k)$:
    \begin{eqnarray}
    PSVG(k) = \frac{1}{k} \sum_{m=1}^{k} PSVG_m(k) \nonumber
    \end{eqnarray}

    Where $PSVG(k)$ indicates fractality of the time series in scale $k$.

  4. Change $k = k + 1$ and if $k < k_{max}$ repeat steps 2 to 4. $k_{max}$ is considered to be the number when variation of $PSVG(k)$ in terms of $k$ remains roughly constant (less than a predefined small value).

References
  1. M. , H. and A. , "Improved visibility graph fractality with application for the diagnosis of Autism Spectrum Disorder", "Physica A: Statistical Mechanics and its Applications" "391"(0000), no. "20", 4720---4726.