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Relying on multifractal behavior of pulsar timing residuals ({\it PTR}s), we examine the capability of Multifractal Detrended Fluctuation Analysis (MF-DFA) and Multifractal Detrending Moving Average Analysis (MF-DMA) modified by Singular Value Decomposition (SVD) and Adaptive Detrending (AD), to detect footprint of gravitational waves (GWs) superimposed on {\it PTR}s. Mentioned methods enable us to clarify the type of GWs which is related to the value of Hurst exponent.
%We compare scaling exponents determined by multifractal analyses of synthetic {\it PTR}s induced by a robust model for GWs with that of computed for pure pulsar timing residuals.
%Combining the mentioned methods enables us to clarify the type of GWs which is related to the value of Hurst exponent.
We introduce three strategies based on generalized Hurst exponent and width of singularity spectrum, to determine the dimensionless amplitude of GWs. For a stochastic gravitational wave background with characteristic strain spectrum as $\mathcal{H}_c(f)\sim \mathcal{A}f^{\zeta}$, the dimensionless amplitude greater than $\mathcal{A}\gtrsim 10^{-17}$ can be recognized irrespective to value of $\zeta$.
%}
%\txg{we applied our simulation in predicted interval for $\mathcal{A}$ . Maybe our strategies detect smaller values of $\mathcal{A}$ but we dont need them because they are not predicted for GWs. So} \txb{our relations are reliable for $\mathcal{A}\gtrsim 10^{-17}$}
We also utilize MF-DFA and MF-DMA to explore 20 millisecond pulsars observed by Parkes Pulsar Timing Array (PPTA). Our analysis demonstrates that there exists a cross-over in fluctuation function versus time scale for observed timing residuals representing a universal property and equates to $s_{\times}\sim60$ days. To asses multifractal nature of observed timing residuals, we apply AD and SVD algorithms on time series as pre-processes to remove superimposed trends as much as possible. The scaling exponents determined by MF-DFA and MF-DMA confirm that, all data are classified in non-stationary class elucidating second universality feature. The value of corresponding Hurst exponent is in interval $H \in [0.35,0.85]$. The $q$-dependency of generalized Hurst exponent demonstrates observed {\it PTR}s have multifractal behavior and the source of this multifractality is mainly devoted to correlation of data which is another universality of observed data sets.
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