Figure 1.
The integrated scheme of HWKS method for fast CP detection, which includes three parts: construction of two BSTs, namely TcA and TcD; CP detection of HWKS in terms of two search criteria; and evaluation of HWKS method.
Figure 2.
The diagram of multi-level HW for time-series signal Z, it is composed of k-level cA and cD vectors, i.e., the average and difference coefficients vectors.
Figure 3.
The diagrams of TcA and TcD, derived from a diagnosed time-series Z by means of k-level HW.
Figure 4.
The scheme of two search criteria for CP detection of HWKS.
(a) Criterion1 based on TcA, and (b) Criterion 2 based on TcD ensure that an optimal path of abrupt CP can be detected from root to leaf nodes in TcA.
Figure 5.
The scheme of search strategy in Criterion 1 in terms of distribution distance between X and Z.
The dotted red line refers to a supposed CP, namely Zc, the solid black line, and green one stands for the points of zk-1,2j, and zk-1,2j-1, respectively.
Figure 6.
The results of single simulation on single CP test position, with constant variance v = 2, different sample size N, and CP test position k, by HWKS, KS, HW, and T, respectively.
(A1) The estimated CP from diagnosed sample Z1, with N = 32, k = 5. (A2) The estimated CP from diagnosed sample Z2, with N = 64, k = 9. (A3) The estimated CP from diagnosed sample Z3, with N = 128, k = 113. (A4) The estimated CP from diagnosed sample Z4, with N = 256, k = 225.
Table 1.
The summarized results of single simulation with single CP test position.
Figure 7.
The results of multiple 600 simulations on single CP test position, with v = 2, different N, and k, by HWKS, KS, HW, and T, respectively.
For samples with N = 32, k = 5; N = 64, k = 9; N = 128, k = 113; and N = 256, k = 225, (A1)–(A4) the distribution of e-CP, (B1)–(B4) the PDF of e-CP, and (C1)–(C4) the averaged e-CP, by HWKS, KS, HW, and T, respectively.
Table 2.
The summarized results of multiple 600 simulation on single CP test position.
Figure 8.
The analysis of e-CP, hit rate, error and accuracy for multiple 100 simulations on different CP test positions, with different N, and k, by HWKS, KS, HW, and T, respectively.
(A) The results of multiple samples Z1, with N = 16; (B) the results of multiple samples Z2, with N = 32; (C) the results of multiple samples Z3, with N = 64; (D) the results of multiple samples Z4, with N = 128.
Figure 9.
The analysis of computation time, hit rate, error and accuracy on different sample size, for HWKS, KS, HW, and T, respectively.
(A) The trend analysis for different sample size from N = 23 to 210, and (B) the histogram analysis for the averaged computation time, hit rate, error and accuracy. In (B), ‘1’ stands for HWKS, ‘2’ stands for KS, ‘3’ stands for HW, and ‘4’ stands for T.
Table 3.
The summary of multiple simulations on different CP test positions.
Figure 10.
The results of CP detection from assembled ECG time series of size N = 2k, k = 9, 10, …, 14, by HWKS, KS, HW, and T, respectively.
(A1)–(A6) the assembled ECG sample Z1–Z6; (B1)–(B6),(C1)–(C6) ,(D1)–(D6) ,(E1)–(E6) the e-CP detected from Z1–Z6, by HWKS, KS, HW, and T, respectively; (F1)–(F6) the diagram analysis for the computation time, (G1)–(G6) the error of e-CP, and (H1)–(H6) the accuracy for Z1–Z6, respectively. In (F)-(H), ‘1’ stands for HWKS, ‘2’ stands for KS, ‘3’ stands for HW, and ‘4’ stands for T.
Table 4.
The summary of CP detection from the assembled ECG samples.
Figure 11.
The results of CP detection from abnormal ECG time series of size N = 2k, k = 10, 11, …, 15, by HWKS, KS, HW, and T, respectively.
(A1)–(A6) the abnormal ECG sample Z1–Z6; (B1)–(B6), (C1)–(C6), (D1)–(D6), (E1)–(E6) the e.c.d.f derived from two segments of Z1–Z6, by HWKS, KS, HW, and T, respectively; (F1)–(F6) the diagram analysis of the distance of e.c.d.f, and (G1)–(G6) the computation time of HWKS, KS, HW, and T in Z1–Z6, respectively. In (F)–(G), ‘1’ stands for HWKS, ‘2’ stands for KS, ‘3’ stands for HW, and ‘4’ stands for T.
Table 5.
The summary of CP detection from abnormal ECG samples.