An Overview of Signal Quality Indices on Dynamic ECG Signal Quality Assessment

Abstract

With the rapid development of wearable ECG medical devices, it is an imperious demand to evaluate the quality of dynamic ECG signals. Thus, a lot of signal quality indices (SQIs) have been proposed in the past few years. In this chapter, we review the analysis performances of SQIs from time-domain, frequency-domain, joint time-frequency, self-correlation, cross-correlation, entropy methods. We then illustrate the SQI performances using real clinical data, allowing a comparison of the SQIs. The performance of 26 SQIs was analyzed and discussed systematically.

Appendix

1.1 A.1 ApEn_SQI

To solve the problems of short and noisy recordings in physiological signals, Pincus [45] presented approximate entropy (ApEn) as a measure of complexity that is applicable to noisy, medium-sized datasets. For an N sample time series {u(i) : 1 ≤ i ≤ N}, given m, form vector sequences ${\mathbf{X}}_{\mathbf{1}}^{\boldsymbol{m}}$ through ${\mathbf{X}}_{\boldsymbol{N}-\boldsymbol{m}+\mathbf{1}}^{\boldsymbol{m}}$ as

$$ {X}_i^m=\left\{u(i),u\left(i+1\right),\dots, u\left(i+m-1\right)\right\},\kern1.6em i=1,\dots, N-m+1 $$

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where m is the length of compared window. For each i ≤ N − m + 1, let ${\boldsymbol{C}}_{\boldsymbol{i}}^{\boldsymbol{m}}\left(\boldsymbol{r}\right)$ be (N – m + 1)⁻¹ times the number of vectors ${\mathbf{X}}_{\boldsymbol{j}}^{\boldsymbol{m}}$ within r of ${\mathbf{X}}_{\boldsymbol{i}}^{\boldsymbol{m}}$. By defining

$$ {\phi}^m(r)={\left(N-m+1\right)}^{-1}\sum \limits_{i=1}^{N-m+1}\ln {C}_i^m(r) $$

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where ln is the natural logarithm, Pincus defined the parameter:

$$ \mathrm{ApEn}\left(m,r\right)=\underset{x\to \infty }{\lim}\left[{\phi}^m(r)-{\phi}^{m+1}(r)\right] $$

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In this chapter, ApEn_SQI was defined as:

$$ \mathrm{ApEn}\_\mathrm{SQI}=\mathrm{ApEn}\left(m,r\right) $$

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where m = 2, r is equal to the 0.15 times of standard deviation of the signal.

1.2 SampEn_SQI

Sample entropy (SampEn) is a modification of approximate entropy (ApEn), used for assessing the complexity of physiological time-series signals, diagnosing diseased states [46]. Now assume we have a time-series data set of length N = {x₁, x₂, x₃, …, x_N} with a constant time interval τ. We define a template vector of length m, such that X_m(i) = {x_i, x_i + 1, x_i + 2, …, x_{i + m − 1}} and the distance function d[X_m(i), X_m(j)](i ≠ j) is to be the Chebyshev distance. We define the sample entropy to be

$$ \mathrm{SampEn}\left(m,r,N\right)=-\ln \left[\frac{A^m(r)}{B^m(r)}\right] $$

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where ${B}_i^m$ is (N − m − 1)⁻¹ times the number of vectors ${X}_j^m$ within r of ${X}_i^m$, where j ranges from 1 to N − m, and j ≠ i to exclude self-matches, and then define

$$ {B}^m(r)={\left(N-m\right)}^{-1}\sum \limits_{i=1}^{N-m}{B}_i^m(r) $$

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where ${A}_i^m$ is (N − m − 1)⁻¹ times the number of vectors ${X}_j^{m+1}$ within r of ${X}_i^{m+1}$, where j ranges from 1 to N − m, and j ≠ i to exclude self-matches, and then define

$$ {A}^m(r)={\left(N-m\right)}^{-1}\sum \limits_{i=1}^{N-m}{A}_i^m(r) $$

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In this chapter, ApEn_SQI was defined as:

$$ \mathrm{SampEn}\_\mathrm{SQI}=\mathrm{SampEn}\left(m,r,N\right) $$

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where N is equal to the length of the signal, m = 2, r is equal to the 0.15 times of standard deviation of the signal.

1.3 FuzzyEn_SQI

FuzzyEn [46] excludes self-matches and considers only the first N − m vectors of length m to ensure that ${X}_i^m$ and ${X}_i^{m+1}$are defined for all 1 ≤ i ≤ N − m. For times series {u(i) : 1 ≤ i ≤ N}, form vectors:

$$ \left\{{X}_i^m=\left\{u(i),u\left(i+1\right),\dots, u\left(i+m-1\right)\right\}-u0(i),i=1,\dots, N-m+1\right\} $$

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where $u0(i)={m}^{-1}{\sum}_{j=0}^{m-1}u\left(i+j\right)$_.

For finite datasets, FuzzyEn can be estimated by the statistic:

$$ \mathrm{FuzzyEn}\left(m,r,N\right)=\ln {\varphi}^m(r)-\ln {\varphi}^{m+1}(r) $$

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where ${\varphi}^m(r)={\left(N-m\right)}^{-1}{\sum}_{i=1}^{N-m}{\phi}_i^m(r)$ and ${\varphi}^{m+1}(r)={\left(N-m\right)}^{-1}{\sum}_{i=1}^{N-m}{\phi}_i^{m+1}(r)$. And ${\phi}_i^m(r)={\left(N-m-1\right)}^{-1}{\sum}_{j=1,j\ne i}^{N-m}{D}_{ij}^m$.

Given vector ${X}_i^m$, calculate the similarity degree ${D}_{ij}^m$ of its neighboring vector ${X}_j^m$ to it through the similarity degree defined by a fuzzy function:

$$ {D}_{ij}^m=\mu \left({d}_{ij}^m,r\right) $$

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where ${d}_{ij}^m$ is the maximum absolute difference of the corresponding scalar components of ${X}_i^m$ and ${X}_j^m$. For each vector ${X}_i^m\left(i-1,\dots, N-m+1\right)$, averaging all the similarity degree of its neighboring vectors ${X}_j^m\left(j-1,\dots, N-m+1,\mathrm{and}\ j\ne i\right)$, we get ${\phi}_i^m(r)$.

For a ECG signal,

$$ \mathrm{FuzzyEn}\_\mathrm{SQI}=\mathrm{FuzzyEn}\left(m,r,N\right) $$

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where N is equal to the length of the signal, m = 2, r is equal to the 0.15 times of standard deviation of the signal.

1.4 DistEn_SQI

Distribution entropy was established by Li et al. [39]. For times series {u(i) : 1 ≤ i ≤ N}, form vectors:

$$ \left\{X(i)=\left\{u(i),u\left(i+1\right),\dots, u\left(i+m-1\right)\right\}i=1,\dots, N-m\right\} $$

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Here, m indicates the embedding dimension. Define the distance matrix D = {d_{i, j}} among vectors X(i) and X(j) for all 1 ≤ i, j ≤ N − m, wherein d_{i, j} =  max {|u(i + k) − u(j + k)|, 0 ≤ k ≤ m − 1} is the Chebyshev distance between X(i) and X(j). The distribution characteristics of all d_{i, j} for 1 ≤ i, j ≤ N − m should be complete quantification of the information underlying the distance matrix D. We here apply the histogram approach to estimate the empirical probability density function of D. If the histogram has M bins, we use p_t, t = 1, 2, …, M to denote the probability of each bin. To reduce bias, elements with i = j are excluded when estimating the empirical probability density function.

Define the DistEn of u(i) by the classical formula of Shannon entropy, that is

$$ \mathrm{DistEn}(m)=-\frac{1}{\log_2(M)}\sum \limits_{t=1}^M{p}_t{\log}_2\left({p}_t\right) $$

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1.5 LZ Complexity

The calculation process of LZ complexity is summarized as follows [17, 40]. For CLZ complexity, the coarse-graining process is performed by comparing signal X with a threshold to transform X into a binary sequence R. That is, whenever the signal is larger than the threshold, one maps the signal to 1, otherwise, to 0. The mean or median of the signal is usually selected as the threshold. The MLZ converts signal X = x₁,x₂, …, x_n to a 0, 1, 2, …, γ-sequence S, where γ is an integer number higher than 3. After the coarse-graining process, the LZ complexity counter c(n) of the new symbol sequence can be calculated according to the rules. Let S and Q, respectively, denote two strings, and SQ is the concatenation of S and Q, whereas string SQπ is derived from SQ after its last character is deleted (π means the operation to delete the last character in the string). Let v(SQπ) denote the vocabulary of all different substrings of SQπ. Initially, c(n) = 1, S = s₁, and Q = s₂, and thus SQπ = s₁. In summary, S = s_1, s₂, …, s_r, and Q = s_r+1, and thus SQπ = s₁s₂, …, s_r. If Q belongs to v(SQπ), then s_r+1, that is, Q is a substring of SQπ, and so S does not change, and renew Q to be s_r+1s_r+2, and then judge if Q belongs to v(SQπ) or not. This process is repeated until Q does not belong to v(SQπ). Next, Q = s_r+1s_r+2, …, s_r+i, which is not a substring of SQπ = s_1,s₂, …, s_rs_r+1,…, s_r+i−1; therefore, c(n) is increased by 1. Subsequently, S is renewed to be S = s₁s₂, …, s_r+i, and Q = s_r+i+1. The procedures are repeated until Q is the last character. Concurrently, c(n) is the number of different substrings (new pattern) contained in the new sequence. Finally, c(n) can be normalized as:

$$ C(n)=c(n)\frac{\log_a(n)}{n} $$

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where n is the length of signal X, α is the number of possible symbols contained in the new sequence, and C(n) is the normalized LZ complexity and denotes the arising rate of new patterns within the sequence. In practice, the normalized complexity C(n), instead of c(n), is considered.

1.6 ELZ Complexity

ELZ transforms each x_i contained within the original signal [17] X = x₁, x₂, …, x_n into a three-bit binary symbol b₁(i)b₂(i)b₃(i), and the process is described as.

The first binary digit b₁(i) is determined by comparing x_i with a threshold T_mean which is the mean of signal X, and it is defined as:

$$ {b}_1=\left\{\begin{array}{cc}0& \mathrm{if}\kern0.5em {x}_i<{T}_{\mathrm{mean}}\\ {}1& \mathrm{if}\kern0.5em {x}_i\ge {T}_{\mathrm{mean}}\end{array}\right.,\kern1em i=1,2,\dots n $$

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The second binary digit b₂(i) is determined by the difference between x_i and x_i−1, and it is defined as:

$$ {b}_2=\left\{\begin{array}{ll}0& \mathrm{if}\;{x}_i-{x}_{i-1}<0\\ {}1& \mathrm{if}\;{x}_i-{x}_{i-1}\ge 0\end{array}\right.,\kern1em i=1,2,\dots n $$

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where b₂(1) is set to 0.

For the third binary digit b₃(i), a variable Flag is first denoted as:

$$ \mathrm{Flag}(i)=\left\{\begin{array}{ll}0& \mathrm{if}\;\left|{x}_i-{x}_{i-1}\right|<\mathrm{dm}\\ {}1& \mathrm{if}\;\left|{x}_i-{x}_{i-1}\right|\ge \mathrm{dm}\end{array}\right.,\kern1em i=2,3,\dots n $$

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where dm is the mean distance between adjacent points within signal X. If Flag(i) is 0, point x_i is relatively close to point x_i−1; otherwise, the two points are relatively far away. Subsequently, b₃(i) is calculated as:

$$ {b}_3(i)=\mathrm{NOT}\left({b}_2(i)\mathrm{XOR}\ \mathrm{Flag}(i)\right),\kern1em i=2,3,\dots, n $$

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where b₃(1) is 0. Moreover, b₂(i) = 1 and Flag(i) = 1 mean that x_i is not only higher than x_i−1 but also relatively farther from x_i−1 compared with b₂(i) = 1 and Flag(i) = 0.