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Methodological framework for estimating the correlation dimension in HRV signals.

Bolea J, Laguna P, Remartínez JM, Rovira E, Navarro A, Bailón R - Comput Math Methods Med (2014)

Bottom Line: Each approach for slope estimation leads to a correlation dimension estimate, called D₂, D(2(⊥)), and D(2(max)).D₂ and D(2(max)) estimate the theoretical value of correlation dimension for the Lorenz attractor with relative error of 4%, and D(2(⊥)) with 1%.D₂ keeps the 81% of accuracy previously described in the literature while D(2(⊥)) and D(2(max)) approaches reach 91% of accuracy in the same database.

View Article: PubMed Central - PubMed

Affiliation: Communications Technology Group (GTC), Aragón Institute for Engineering Research (I3A), IIS Aragón, University of Zaragoza, 50018 Zaragoza, Spain ; CIBER de Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), 50018 Zaragoza, Spain.

ABSTRACT
This paper presents a methodological framework for robust estimation of the correlation dimension in HRV signals. It includes (i) a fast algorithm for on-line computation of correlation sums; (ii) log-log curves fitting to a sigmoidal function for robust maximum slope estimation discarding the estimation according to fitting requirements; (iii) three different approaches for linear region slope estimation based on latter point; and (iv) exponential fitting for robust estimation of saturation level of slope series with increasing embedded dimension to finally obtain the correlation dimension estimate. Each approach for slope estimation leads to a correlation dimension estimate, called D₂, D(2(⊥)), and D(2(max)). D₂ and D(2(max)) estimate the theoretical value of correlation dimension for the Lorenz attractor with relative error of 4%, and D(2(⊥)) with 1%. The three approaches are applied to HRV signals of pregnant women before spinal anesthesia for cesarean delivery in order to identify patients at risk for hypotension. D₂ keeps the 81% of accuracy previously described in the literature while D(2(⊥)) and D(2(max)) approaches reach 91% of accuracy in the same database.

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In the left panel, vector differences of any two reconstructed vectors are shown (i.e., for m = 2, Δyi,j2), where solid circles and dashed lines represent the points whose ℓ2-norm and ℓ1-norm are equal to 1, respectively. The dots are the differences below ℓ2-norm unity and the dots with circles are below ℓ1-norm unity. In the right panel log-log curves of one HRV signal (300 samples) used in the study are shown for a m = 10 and ℓ1-, ℓ2- and ℓ∞-norms.
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fig8: In the left panel, vector differences of any two reconstructed vectors are shown (i.e., for m = 2, Δyi,j2), where solid circles and dashed lines represent the points whose ℓ2-norm and ℓ1-norm are equal to 1, respectively. The dots are the differences below ℓ2-norm unity and the dots with circles are below ℓ1-norm unity. In the right panel log-log curves of one HRV signal (300 samples) used in the study are shown for a m = 10 and ℓ1-, ℓ2- and ℓ∞-norms.

Mentions: The correlation dimension is considered norm invariant [14]. However, the effect of selecting the norm in correlation dimension estimates deserves further attention when applied to a finite data set. The norm of the difference vector Δyi,jm defines the distance di,jm in (2). Norms can be defined from ℓ1 (//·//1) to ℓ∞ (//·//∞). Left panel in Figure 8 shows norm unity for ℓ1 and for ℓ2. Moreover, it is illustrated how a distance di,j2 can be lower than the norm unity or not depending on which norm is used. The norm unity is chosen as an example of any threshold r used in the correlation dimension algorithm. Therefore, by fixing the set of thresholds, the appearance of the linear region of the log-log curve can be compromised.


Methodological framework for estimating the correlation dimension in HRV signals.

Bolea J, Laguna P, Remartínez JM, Rovira E, Navarro A, Bailón R - Comput Math Methods Med (2014)

In the left panel, vector differences of any two reconstructed vectors are shown (i.e., for m = 2, Δyi,j2), where solid circles and dashed lines represent the points whose ℓ2-norm and ℓ1-norm are equal to 1, respectively. The dots are the differences below ℓ2-norm unity and the dots with circles are below ℓ1-norm unity. In the right panel log-log curves of one HRV signal (300 samples) used in the study are shown for a m = 10 and ℓ1-, ℓ2- and ℓ∞-norms.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3926396&req=5

fig8: In the left panel, vector differences of any two reconstructed vectors are shown (i.e., for m = 2, Δyi,j2), where solid circles and dashed lines represent the points whose ℓ2-norm and ℓ1-norm are equal to 1, respectively. The dots are the differences below ℓ2-norm unity and the dots with circles are below ℓ1-norm unity. In the right panel log-log curves of one HRV signal (300 samples) used in the study are shown for a m = 10 and ℓ1-, ℓ2- and ℓ∞-norms.
Mentions: The correlation dimension is considered norm invariant [14]. However, the effect of selecting the norm in correlation dimension estimates deserves further attention when applied to a finite data set. The norm of the difference vector Δyi,jm defines the distance di,jm in (2). Norms can be defined from ℓ1 (//·//1) to ℓ∞ (//·//∞). Left panel in Figure 8 shows norm unity for ℓ1 and for ℓ2. Moreover, it is illustrated how a distance di,j2 can be lower than the norm unity or not depending on which norm is used. The norm unity is chosen as an example of any threshold r used in the correlation dimension algorithm. Therefore, by fixing the set of thresholds, the appearance of the linear region of the log-log curve can be compromised.

Bottom Line: Each approach for slope estimation leads to a correlation dimension estimate, called D₂, D(2(⊥)), and D(2(max)).D₂ and D(2(max)) estimate the theoretical value of correlation dimension for the Lorenz attractor with relative error of 4%, and D(2(⊥)) with 1%.D₂ keeps the 81% of accuracy previously described in the literature while D(2(⊥)) and D(2(max)) approaches reach 91% of accuracy in the same database.

View Article: PubMed Central - PubMed

Affiliation: Communications Technology Group (GTC), Aragón Institute for Engineering Research (I3A), IIS Aragón, University of Zaragoza, 50018 Zaragoza, Spain ; CIBER de Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), 50018 Zaragoza, Spain.

ABSTRACT
This paper presents a methodological framework for robust estimation of the correlation dimension in HRV signals. It includes (i) a fast algorithm for on-line computation of correlation sums; (ii) log-log curves fitting to a sigmoidal function for robust maximum slope estimation discarding the estimation according to fitting requirements; (iii) three different approaches for linear region slope estimation based on latter point; and (iv) exponential fitting for robust estimation of saturation level of slope series with increasing embedded dimension to finally obtain the correlation dimension estimate. Each approach for slope estimation leads to a correlation dimension estimate, called D₂, D(2(⊥)), and D(2(max)). D₂ and D(2(max)) estimate the theoretical value of correlation dimension for the Lorenz attractor with relative error of 4%, and D(2(⊥)) with 1%. The three approaches are applied to HRV signals of pregnant women before spinal anesthesia for cesarean delivery in order to identify patients at risk for hypotension. D₂ keeps the 81% of accuracy previously described in the literature while D(2(⊥)) and D(2(max)) approaches reach 91% of accuracy in the same database.

Show MeSH
Related in: MedlinePlus