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Whole set of α ^ estimated. On the x -axis, α is the true value of the exponents, the values computed in the generator. On the y -axis, the estimated values of α ^ computed by the corresponding analysis method are located. Red curves represent true alpha values α , and blue curves represent the estimated alpha values α ^ . The first column (A,C,E) corresponds to the signals generated via the Cholesky method, and the second column (B,D,F) corresponds to the signals generated via ARFIMA filtering. The first row (A,B) presents the α ^ values computed using fGn-based Whittle’s maximum likelihood estimator, the second row (C,D) presents those computed using ARFIMA-based Whittle’s maximum likelihood estimator, and the third row (E,F) presents the α ^ values computed using DFA.

Journal: Frontiers in Network Physiology

Article Title: A guide to Whittle maximum likelihood estimator in MATLAB

doi: 10.3389/fnetp.2023.1204757

Figure Lengend Snippet: Whole set of α ^ estimated. On the x -axis, α is the true value of the exponents, the values computed in the generator. On the y -axis, the estimated values of α ^ computed by the corresponding analysis method are located. Red curves represent true alpha values α , and blue curves represent the estimated alpha values α ^ . The first column (A,C,E) corresponds to the signals generated via the Cholesky method, and the second column (B,D,F) corresponds to the signals generated via ARFIMA filtering. The first row (A,B) presents the α ^ values computed using fGn-based Whittle’s maximum likelihood estimator, the second row (C,D) presents those computed using ARFIMA-based Whittle’s maximum likelihood estimator, and the third row (E,F) presents the α ^ values computed using DFA.

Article Snippet: .................................................................. MATLAB code 6 : Whittle’s log-likelihood function with fGn theoretical PSD function lwHfgn = WLLFfgn (H,w,P,N) Tp = sin (pi*H)*gamma ((2*H)+1)*(abs(w).^(1-(2*H))); c = sum(P)/sum (Tp); T = c *Tp; lwHfgn=(2/N)*sum (log(T)+(P./T)); .................................................................. MATLAB code 7 : Whittle’s log-likelihood function with ARFIMA (0,d,0) theoretical PSD function lwHarf = WLLFarf (H,w,P,N) d = H-0.5; Tp=(1/(2*pi))*(2*sin (w/2)).^-(2*d); c = sum(P)/sum (Tp); T = c *Tp; lwHarf=(2/N)*sum (log(T)+(P./T)); .................................................................. In the first line, the function is used to declare the functions WLLFfgn and WLLFarf .

Techniques: Generated

Periodogram of choleskyfgn (A) , arfima0d0 (B) , whitenoise (C) , and empirical (D) signals with the theoretical power spectral density of fGn (orange curve) and ARFIMA (0, d ,0) (yellow curve). The theoretical power spectral densities were computed with the estimated values of H and d obtained via whittle.m. Those values, entered in MATLAB code 2 and 3, are presented in .

Journal: Frontiers in Network Physiology

Article Title: A guide to Whittle maximum likelihood estimator in MATLAB

doi: 10.3389/fnetp.2023.1204757

Figure Lengend Snippet: Periodogram of choleskyfgn (A) , arfima0d0 (B) , whitenoise (C) , and empirical (D) signals with the theoretical power spectral density of fGn (orange curve) and ARFIMA (0, d ,0) (yellow curve). The theoretical power spectral densities were computed with the estimated values of H and d obtained via whittle.m. Those values, entered in MATLAB code 2 and 3, are presented in .

Article Snippet: .................................................................. MATLAB code 6 : Whittle’s log-likelihood function with fGn theoretical PSD function lwHfgn = WLLFfgn (H,w,P,N) Tp = sin (pi*H)*gamma ((2*H)+1)*(abs(w).^(1-(2*H))); c = sum(P)/sum (Tp); T = c *Tp; lwHfgn=(2/N)*sum (log(T)+(P./T)); .................................................................. MATLAB code 7 : Whittle’s log-likelihood function with ARFIMA (0,d,0) theoretical PSD function lwHarf = WLLFarf (H,w,P,N) d = H-0.5; Tp=(1/(2*pi))*(2*sin (w/2)).^-(2*d); c = sum(P)/sum (Tp); T = c *Tp; lwHarf=(2/N)*sum (log(T)+(P./T)); .................................................................. In the first line, the function is used to declare the functions WLLFfgn and WLLFarf .

Techniques:

Periodogram of choleskyfgn (A) , arfima0d0 (B) , whitenoise (C) , and empirical (D) signals (blue curve) with the adjusted theoretical power spectral density of fGn (orange curve) and ARFIMA (0, d ,0) (yellow curve). The H and d values are the same as those used in the previous figure.

Journal: Frontiers in Network Physiology

Article Title: A guide to Whittle maximum likelihood estimator in MATLAB

doi: 10.3389/fnetp.2023.1204757

Figure Lengend Snippet: Periodogram of choleskyfgn (A) , arfima0d0 (B) , whitenoise (C) , and empirical (D) signals (blue curve) with the adjusted theoretical power spectral density of fGn (orange curve) and ARFIMA (0, d ,0) (yellow curve). The H and d values are the same as those used in the previous figure.

Article Snippet: .................................................................. MATLAB code 6 : Whittle’s log-likelihood function with fGn theoretical PSD function lwHfgn = WLLFfgn (H,w,P,N) Tp = sin (pi*H)*gamma ((2*H)+1)*(abs(w).^(1-(2*H))); c = sum(P)/sum (Tp); T = c *Tp; lwHfgn=(2/N)*sum (log(T)+(P./T)); .................................................................. MATLAB code 7 : Whittle’s log-likelihood function with ARFIMA (0,d,0) theoretical PSD function lwHarf = WLLFarf (H,w,P,N) d = H-0.5; Tp=(1/(2*pi))*(2*sin (w/2)).^-(2*d); c = sum(P)/sum (Tp); T = c *Tp; lwHarf=(2/N)*sum (log(T)+(P./T)); .................................................................. In the first line, the function is used to declare the functions WLLFfgn and WLLFarf .

Techniques:

Whittle’s log-likelihood functions of choleskyfgn (A) , arfima0d0 (B) , whitenoise (C) , and empirical (D) signals. The blue curves correspond to Whittle’s likelihood function calculated using the fGn theoretical spectrum, while the orange curves correspond to the same function calculated using the ARFIMA (0, d ,0) theoretical spectrum.

Journal: Frontiers in Network Physiology

Article Title: A guide to Whittle maximum likelihood estimator in MATLAB

doi: 10.3389/fnetp.2023.1204757

Figure Lengend Snippet: Whittle’s log-likelihood functions of choleskyfgn (A) , arfima0d0 (B) , whitenoise (C) , and empirical (D) signals. The blue curves correspond to Whittle’s likelihood function calculated using the fGn theoretical spectrum, while the orange curves correspond to the same function calculated using the ARFIMA (0, d ,0) theoretical spectrum.

Article Snippet: .................................................................. MATLAB code 6 : Whittle’s log-likelihood function with fGn theoretical PSD function lwHfgn = WLLFfgn (H,w,P,N) Tp = sin (pi*H)*gamma ((2*H)+1)*(abs(w).^(1-(2*H))); c = sum(P)/sum (Tp); T = c *Tp; lwHfgn=(2/N)*sum (log(T)+(P./T)); .................................................................. MATLAB code 7 : Whittle’s log-likelihood function with ARFIMA (0,d,0) theoretical PSD function lwHarf = WLLFarf (H,w,P,N) d = H-0.5; Tp=(1/(2*pi))*(2*sin (w/2)).^-(2*d); c = sum(P)/sum (Tp); T = c *Tp; lwHarf=(2/N)*sum (log(T)+(P./T)); .................................................................. In the first line, the function is used to declare the functions WLLFfgn and WLLFarf .

Techniques:

Box plot of α ^ squared error values obtained via fGn-based Whittle’s likelihood (A) , ARFIMA-based Whittle’s likelihood (B) , and DFA (C) . The lower and upper edges of the boxes represent the 25 and 75 percentiles, respectively. The horizontal black line represents the median. The whiskers extend to the most extreme points not considered as outliers. The outliers are plotted as individual points. The orange diamond represents the MSE value.

Journal: Frontiers in Network Physiology

Article Title: A guide to Whittle maximum likelihood estimator in MATLAB

doi: 10.3389/fnetp.2023.1204757

Figure Lengend Snippet: Box plot of α ^ squared error values obtained via fGn-based Whittle’s likelihood (A) , ARFIMA-based Whittle’s likelihood (B) , and DFA (C) . The lower and upper edges of the boxes represent the 25 and 75 percentiles, respectively. The horizontal black line represents the median. The whiskers extend to the most extreme points not considered as outliers. The outliers are plotted as individual points. The orange diamond represents the MSE value.

Article Snippet: .................................................................. MATLAB code 6 : Whittle’s log-likelihood function with fGn theoretical PSD function lwHfgn = WLLFfgn (H,w,P,N) Tp = sin (pi*H)*gamma ((2*H)+1)*(abs(w).^(1-(2*H))); c = sum(P)/sum (Tp); T = c *Tp; lwHfgn=(2/N)*sum (log(T)+(P./T)); .................................................................. MATLAB code 7 : Whittle’s log-likelihood function with ARFIMA (0,d,0) theoretical PSD function lwHarf = WLLFarf (H,w,P,N) d = H-0.5; Tp=(1/(2*pi))*(2*sin (w/2)).^-(2*d); c = sum(P)/sum (Tp); T = c *Tp; lwHarf=(2/N)*sum (log(T)+(P./T)); .................................................................. In the first line, the function is used to declare the functions WLLFfgn and WLLFarf .

Techniques:

Box plot of α ^ squared error values obtained using ARFIMA-based Whittle’s likelihood for four sets of length: 32 (A) , 64 (B) , 128 (C) , and 256 (D) . The lower and upper edges of the boxes represent the 25 and 75 percentiles, respectively. The horizontal black line represents the median. The whiskers extend to the most extreme points not considered as outliers. The outliers are plotted as individual points. The orange diamond represents the MSE value. The vertical scale of the top left graph is 20 times larger than the other three panels.

Journal: Frontiers in Network Physiology

Article Title: A guide to Whittle maximum likelihood estimator in MATLAB

doi: 10.3389/fnetp.2023.1204757

Figure Lengend Snippet: Box plot of α ^ squared error values obtained using ARFIMA-based Whittle’s likelihood for four sets of length: 32 (A) , 64 (B) , 128 (C) , and 256 (D) . The lower and upper edges of the boxes represent the 25 and 75 percentiles, respectively. The horizontal black line represents the median. The whiskers extend to the most extreme points not considered as outliers. The outliers are plotted as individual points. The orange diamond represents the MSE value. The vertical scale of the top left graph is 20 times larger than the other three panels.

Article Snippet: .................................................................. MATLAB code 6 : Whittle’s log-likelihood function with fGn theoretical PSD function lwHfgn = WLLFfgn (H,w,P,N) Tp = sin (pi*H)*gamma ((2*H)+1)*(abs(w).^(1-(2*H))); c = sum(P)/sum (Tp); T = c *Tp; lwHfgn=(2/N)*sum (log(T)+(P./T)); .................................................................. MATLAB code 7 : Whittle’s log-likelihood function with ARFIMA (0,d,0) theoretical PSD function lwHarf = WLLFarf (H,w,P,N) d = H-0.5; Tp=(1/(2*pi))*(2*sin (w/2)).^-(2*d); c = sum(P)/sum (Tp); T = c *Tp; lwHarf=(2/N)*sum (log(T)+(P./T)); .................................................................. In the first line, the function is used to declare the functions WLLFfgn and WLLFarf .

Techniques:

Comparison of α ^ estimated via ARFIMA-based Whittle’s likelihood and DFA on gait data made available by J. Hausdorff on the PhysioNet platform ( <xref ref-type= Goldberger et al., 2000 ; Hausdorff, 2001 ). The results highlighted in red correspond to anti-persistent series, i.e., with α lower than 0.5." width="100%" height="100%">

Journal: Frontiers in Network Physiology

Article Title: A guide to Whittle maximum likelihood estimator in MATLAB

doi: 10.3389/fnetp.2023.1204757

Figure Lengend Snippet: Comparison of α ^ estimated via ARFIMA-based Whittle’s likelihood and DFA on gait data made available by J. Hausdorff on the PhysioNet platform ( Goldberger et al., 2000 ; Hausdorff, 2001 ). The results highlighted in red correspond to anti-persistent series, i.e., with α lower than 0.5.

Article Snippet: .................................................................. MATLAB code 6 : Whittle’s log-likelihood function with fGn theoretical PSD function lwHfgn = WLLFfgn (H,w,P,N) Tp = sin (pi*H)*gamma ((2*H)+1)*(abs(w).^(1-(2*H))); c = sum(P)/sum (Tp); T = c *Tp; lwHfgn=(2/N)*sum (log(T)+(P./T)); .................................................................. MATLAB code 7 : Whittle’s log-likelihood function with ARFIMA (0,d,0) theoretical PSD function lwHarf = WLLFarf (H,w,P,N) d = H-0.5; Tp=(1/(2*pi))*(2*sin (w/2)).^-(2*d); c = sum(P)/sum (Tp); T = c *Tp; lwHarf=(2/N)*sum (log(T)+(P./T)); .................................................................. In the first line, the function is used to declare the functions WLLFfgn and WLLFarf .

Techniques: Comparison

log-power spectral density of Hausdorff data rearranged into three groups: persistent behavior (A) , anti-persistent behavior (B) , and mixed behavior (C) . log-power spectral density of an artificial ARFIMA(p,d,q) signal with parameters (2, −0.35.1) generated using ARFIMApdq.m (D) .

Journal: Frontiers in Network Physiology

Article Title: A guide to Whittle maximum likelihood estimator in MATLAB

doi: 10.3389/fnetp.2023.1204757

Figure Lengend Snippet: log-power spectral density of Hausdorff data rearranged into three groups: persistent behavior (A) , anti-persistent behavior (B) , and mixed behavior (C) . log-power spectral density of an artificial ARFIMA(p,d,q) signal with parameters (2, −0.35.1) generated using ARFIMApdq.m (D) .

Article Snippet: .................................................................. MATLAB code 6 : Whittle’s log-likelihood function with fGn theoretical PSD function lwHfgn = WLLFfgn (H,w,P,N) Tp = sin (pi*H)*gamma ((2*H)+1)*(abs(w).^(1-(2*H))); c = sum(P)/sum (Tp); T = c *Tp; lwHfgn=(2/N)*sum (log(T)+(P./T)); .................................................................. MATLAB code 7 : Whittle’s log-likelihood function with ARFIMA (0,d,0) theoretical PSD function lwHarf = WLLFarf (H,w,P,N) d = H-0.5; Tp=(1/(2*pi))*(2*sin (w/2)).^-(2*d); c = sum(P)/sum (Tp); T = c *Tp; lwHarf=(2/N)*sum (log(T)+(P./T)); .................................................................. In the first line, the function is used to declare the functions WLLFfgn and WLLFarf .

Techniques: Generated