**2. Methods**

#### **2.1. CNA model and problem statement**

Consider a chromosome section observed with some resolution Δ , bp at *M* discrete break‐ points, *n* ∈ 1, *M* . An example of the CNA probes with a single breakpoint and two segments is shown in **Figure 1**. Suppose that the copy numbers change at *K* breakpoints, 1<*i* <sup>1</sup> < … <*i <sup>K</sup>* <*M* , united in a vector *I* = *i* 1*i* <sup>2</sup>…*i K* T. The measurement can thus be represented with a vector *y* ∈ℛ*<sup>M</sup>* as

of alterations in the tumor genome, and absolute segment copy numbers. Thus, efficient estimators are required to extract information about the breakpoints with accuracy acceptable for medical needs. To produce CNA profile, several technologies have been developed such as comparative genomic hybridization (CGH) [8], high‐resolution CGH (HR‐CGH) [9], whole genome sequencing [10], and most recently single‐nucleotide polymorphism (SNP) [11]. The HR‐CGH technology is still used widely in spite of its low resolution [12]. It has been reported in [13] that the HR‐CGH arrays are accurate to detect structural variations (SVs) at the resolu‐ tion of 200 bp (*base pairs*). Most recently, the single‐nucleotide polymorphism technology was developedinthestudyofWangetal.[11]toprovidehigh‐resolutionmeasurementsoftheCNAs. In spite of their high resolution, the modern methods still demonstrate the inability in obtain‐ ing good estimates of the breakpoint locations because of the following factors: (1) the nature of biological material (tumor is contaminated by normal tissue, relative values, and unknown baseline for copy number estimation), (2) technological biases (quality of material and hybrid‐ ization/sequencing), and (3) intensive random noise. The HR‐CGH and SNP profiles have demonstrated deficiency in detecting the CNAs, but noise in the detected changes still re‐ mains at a high level [14] and accurate estimators are required to extract information about

In the HR‐CGH microarray technique, the CNAs are often normalized and plotted as log2*R* / *G* =log<sup>2</sup> ratio, where *R* and *G* are the fluorescent Red and Green intensities, respectively [12]. The CNA measurements using SNP technologies are represented by the Log‐*R* ratios (LRRs), which are the log‐transformed ratios of experimental and normal reference SNP intensities centered at zero for each sample [14]. From the standpoint of signal processing, the

**•** It is piecewise constant (PWC) and sparse with a small number of alterations on a long base‐

**•** The measurement noise in the log‐*R* ratio is highly intensive and can be modeled as additive

The CNA estimation problem is thus to predict the breakpoint locations and the segmental levels with a maximum possible accuracy and precision acceptable for medical applications. In this work, we developed our methods to two types of cancer: B‐cell chronic lymphocytic leukemia (B‐CLL) and BLC primary breast carcinoma. Nevertheless, the methods were

Consider a chromosome section observed with some resolution Δ , bp at *M* discrete break‐ points, *n* ∈ 1, *M* . An example of the CNA probes with a single breakpoint and two segments is shown in **Figure 1**. Suppose that the copy numbers change at *K* breakpoints,

**•** Constant values are integer, although this property is not survived in the log‐*R* ratio.

designed to any samples of cancer with the characteristics described above.

following properties of the CNA function are of importance [15]:

structural changes.

118 Advanced Biosignal Processing and Diagnostic Methods

pair length.

white Gaussian.

**2. Methods**

**2.1. CNA model and problem statement**

$$\mathbf{y} = \left[ \mathbf{y}\_1 \mathbf{y}\_2 \dots \mathbf{y}\_{i\_l} \mathbf{y}\_{i\_l} + 1 \dots \mathbf{y}\_{i\_1} \dots \mathbf{y}\_{i\_k} \dots \mathbf{y}\_M \right]^\nu. \tag{1}$$

**Figure 1.** Typical CNA measurements with white Gaussian noise with a single breakpoint, between two segments *l* and *l* + 1 having different segmental variances. The pdf for neighboring segments are depicted as *pl* (*x*) and *pl*+1(*x*).

Introduce a vector *a*∈ℛ*<sup>K</sup>* +1 of segmental levels,*<sup>a</sup>* <sup>=</sup> *<sup>a</sup>*1*a*2…*aK* +1 <sup>T</sup> , where *a*<sup>1</sup> corresponds to the interval 1, *i* <sup>1</sup> , *aK* +1 to *i <sup>K</sup>* , *M* , and *aK* , *k* ≥2, to *i <sup>k</sup>* <sup>−</sup>1, *i <sup>k</sup>* . In such a formulation, *y* obeys the linear regression model

$$\mathbf{y} = \mathcal{A}(I)\mathbf{a} + \mathbf{v} \tag{2}$$

where the regression matrix *A*∈ℛ*<sup>M</sup>* ×(*<sup>K</sup>* +1) is sparse,

$$\mathcal{A} = \left[ \begin{array}{c} \boldsymbol{A}\_1^{\boldsymbol{r}} \ \boldsymbol{A}\_2^{\boldsymbol{r}} \dots \boldsymbol{A}\_{K \times 1}^{\boldsymbol{r}} \end{array} \right]^{\boldsymbol{r}},\tag{3}$$

having a component

$$\mathcal{A}\_k = \begin{bmatrix} \mathbf{0} & \cdots & 1 & \cdots & \mathbf{0} \\ \mathbf{0} & \cdots & 1 & \cdots & \mathbf{0} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ \mathbf{0} & \cdots & 1 & \cdots & \mathbf{0} \end{bmatrix}, \tag{4}$$

in which the *k*th column is filled with unity and all others are zeros. The number of the columns in *A<sup>k</sup>* is exactly *K* + 1. However, the number or the row depends on the interval *i <sup>k</sup>* −*i <sup>k</sup>* <sup>−</sup>1. Thus, the row‐variant matrix (4) is *A<sup>k</sup>* ∈ℛ(*<sup>i</sup> <sup>k</sup>* −*i <sup>k</sup>* <sup>−</sup>1)×(*K* +1) . Additive noise *v* in Eq. (2) is zero mean, *E*{**v**}=0, and white Gaussian with the covariance *R* =*σ<sup>v</sup>* 2 *I*, where *I* ∈**ℛ***<sup>M</sup>* <sup>×</sup>*<sup>M</sup>* is an identity matrix and *σ<sup>v</sup>* 2 is a know variance.

The CNA estimation problem is thus to predict the breakpoint locations and evaluate the segmental changes *x* = *A*(*I*)*a* with a maximum possible accuracy and precision acceptable for medical applications. The problem is complicated by short number of the probes in each neighboring segment and indistinct edges. Therefore, an analysis of the estimation errors caused by the segmental noise and jitter in the breakpoints is required.

#### **2.2. Jitter probability in the breakpoints**

Consider a typical genomic measurement of two neighboring CNA segments in white Gaussian noise with different segmental variances as shown in **Figure 1**. A constant signal changes from level *al* to level *al*+1 around the *breakpoint i l* . In the presence of noise, the location of *i <sup>l</sup>* is not clear owing to commonly large segmental variances *σ<sup>l</sup>* 2 and *σl*+1 <sup>2</sup> . As an example, the Gaussian noise probability density functions (pdfs) *pl* (*x*) and *pl*+1(*x*) are shown in **Figure 1** for *σl* <sup>2</sup> <sup>&</sup>gt;*σl*+1 <sup>2</sup> . Let us notice that *pl* (*x*) and *pl*+1(*x*) cross each other in two points, *α<sup>l</sup>* and *β<sup>l</sup>* , provided that *σ<sup>l</sup>* <sup>2</sup> <sup>≠</sup>*σl*+1 2 .

Now considerer *N* probes in each segment neighboring to *i <sup>l</sup>* with an average resolution. We thus may assign an event *Alj* meaning that measurement at point *i <sup>l</sup>* − *N* ≤ *j* <*i <sup>l</sup>* belongs to the *l*th segment. Another event *Blj* means that measurement at *i <sup>l</sup>* ≤ *j* <*i <sup>l</sup>* + *N* −1 belongs to the (*l* + 1)th segment. In our approach, we think that a measured value belongs to one segment if the probability is larger than if it belongs to another segment. For example, any measurement point in the interval between *α<sup>l</sup>* and *β<sup>l</sup>* (**Figure 1**) is supposed to belong to the (*l* + 1)th segment. Following **Figure 1** and assuming different noise variances *σ<sup>l</sup>* 2 and *σl*+1 <sup>2</sup> , the events *Alj* and *Blj* can be specified as follows [16]:

$$A\_{\boldsymbol{\eta}} - N\_l \le j < i\_l \quad \begin{aligned} &A\_{\boldsymbol{\eta}} \\ &\text{ is} \\ &\sigma\_j > \sigma\_l, \\ &\sigma\_l < \mathbf{x}\_j < \beta\_l, \end{aligned} \qquad \begin{aligned} &(\sigma\_j < \boldsymbol{\beta}\_l), &\sigma\_l^2 > \sigma\_{l+1}^2, \\ &\mathbf{x}\_j > \mathbf{a}\_l, &\sigma\_l^2 = \sigma\_{l+1}^2, \\ &\sigma\_l < \mathbf{x}\_j < \beta\_l, &\sigma\_l^2 < \sigma\_{l+1}^2, \end{aligned} \tag{5}$$

$$\mathbf{x}\_{l} \le j < i\_l + N\_l - \mathbf{l} \begin{cases} \mathcal{B}\_l < \mathbf{x}\_j < \mathbf{a}\_l & \sigma\_l^2 < \sigma\_{l+1}^2, \\ & \mathbf{x}\_j < \mathbf{a}\_l, \\ (\mathbf{x}\_j < \mathbf{a}\_l) \vee \left(\mathbf{x}\_j > \mathcal{B}\_l\right), & \sigma\_l^2 > \sigma\_{l+1}^2. \end{cases} \tag{6}$$

Because each point can belong only to one segment, the inverse events are *A*¯ =1<sup>−</sup> *Alj* and *<sup>B</sup>*¯ =1<sup>−</sup> *Blj* .

Events *Alj* and *Blj* can be united into two blocks: *A<sup>l</sup>* ={*Al*(*<sup>i</sup> <sup>l</sup>*−*<sup>N</sup>* )*Al*(*<sup>i</sup> <sup>l</sup>*−*<sup>N</sup>* +1)… *Al*(*<sup>i</sup> <sup>l</sup>*<sup>−</sup>1)} and *B<sup>l</sup>* ={*Bl*(*<sup>i</sup> <sup>l</sup>*)*Bl*(*<sup>i</sup> <sup>l</sup>*+1)… *Bl*(*<sup>i</sup> <sup>l</sup>*+*<sup>N</sup>* <sup>−</sup>1)}.

If *A<sup>l</sup>* and *B<sup>l</sup>* occur simultaneously with unit probability each, then jitter at *i <sup>l</sup>* will never occur. However, some other events may be found, which do not obligatorily lead to jitter. We ignore such events and define approximately the probability *P*(*A<sup>l</sup>* , *B<sup>l</sup>* ) of the jitter‐free breakpoint as

$$P\left(\mathcal{A}\_l, \mathcal{B}\_l\right) = P\left(\mathbf{A}\_{\mathbf{i}\_l - N} \dots \mathbf{A}\_{\mathbf{i}\_l - 1} \mathbf{B}\_{\mathbf{i}\_l} \dots \mathbf{B}\_{\mathbf{i}\_l + N - 1}\right). \tag{7}$$

The inverse event *P*¯(*<sup>A</sup><sup>l</sup>* , *Bl*) =1−*P*(*A<sup>l</sup>* , *B<sup>l</sup>* ) can be called the *approximate jitter probability* [17].

#### **2.3. Jitter distribution in the breakpoints**

where the regression matrix *A*∈ℛ*<sup>M</sup>* ×(*<sup>K</sup>* +1)

120 Advanced Biosignal Processing and Diagnostic Methods

the row‐variant matrix (4) is *A<sup>k</sup>* ∈ℛ(*<sup>i</sup>*

**2.2. Jitter probability in the breakpoints**

<sup>2</sup> . Let us notice that *pl*

segment. Another event *Blj*

in the interval between *α<sup>l</sup>*

changes from level *al*

<sup>2</sup> <sup>≠</sup>*σl*+1 2 .

of *i*

*σl* <sup>2</sup> <sup>&</sup>gt;*σl*+1

that *σ<sup>l</sup>*

is a know variance.

*E*{**v**}=0, and white Gaussian with the covariance *R* =*σ<sup>v</sup>*

having a component

and *σ<sup>v</sup>* 2 is sparse,

12 1 , *<sup>T</sup> TT T <sup>K</sup>* <sup>+</sup> = ¼ é ù

> 010 <sup>010</sup> ,

é ù ê ú = ê ú ê ú ê ú ë û

L L L L MOMO M L L

010

in which the *k*th column is filled with unity and all others are zeros. The number of the columns

The CNA estimation problem is thus to predict the breakpoint locations and evaluate the segmental changes *x* = *A*(*I*)*a* with a maximum possible accuracy and precision acceptable for medical applications. The problem is complicated by short number of the probes in each neighboring segment and indistinct edges. Therefore, an analysis of the estimation errors

Consider a typical genomic measurement of two neighboring CNA segments in white Gaussian noise with different segmental variances as shown in **Figure 1**. A constant signal

*l*

(*x*) and *pl*+1(*x*) cross each other in two points, *α<sup>l</sup>*

2 and *σl*+1

*<sup>l</sup>* ≤ *j* <*i*

and *β<sup>l</sup>* (**Figure 1**) is supposed to belong to the (*l* + 1)th segment.

2

in *A<sup>k</sup>* is exactly *K* + 1. However, the number or the row depends on the interval *i*

*<sup>k</sup>* −*i <sup>k</sup>* <sup>−</sup>1)×(*K* +1)

caused by the segmental noise and jitter in the breakpoints is required.

to level *al*+1 around the *breakpoint i*

means that measurement at *i*

segment. In our approach, we think that a measured value belongs to one segment if the probability is larger than if it belongs to another segment. For example, any measurement point

*<sup>l</sup>* is not clear owing to commonly large segmental variances *σ<sup>l</sup>*

Gaussian noise probability density functions (pdfs) *pl*

Now considerer *N* probes in each segment neighboring to *i*

thus may assign an event *Alj* meaning that measurement at point *i*

*k*

ë û *A AA A* (3)

*A* (4)

. Additive noise *v* in Eq. (2) is zero mean,

*I*, where *I* ∈**ℛ***<sup>M</sup>* <sup>×</sup>*<sup>M</sup>* is an identity matrix

. In the presence of noise, the location

(*x*) and *pl*+1(*x*) are shown in **Figure 1** for

*<sup>l</sup>* − *N* ≤ *j* <*i*

<sup>2</sup> . As an example, the

*<sup>l</sup>* belongs to the *l*th

, provided

and *β<sup>l</sup>*

*<sup>l</sup>* with an average resolution. We

*<sup>l</sup>* + *N* −1 belongs to the (*l* + 1)th

*<sup>k</sup>* −*i*

*<sup>k</sup>* <sup>−</sup>1. Thus,

To determine the confidence limits for CNAs using high‐resolution genomic arrays, jitter in the breakpoints must be specified statistically for the segmental Gaussian distribution. This can be done approximately if to employ either the discrete skew Laplace distribution or, more accurately, the modified Bessel function of the second kind and zeroth order.

#### *2.3.1. Approximation with discrete skew Laplace distribution*

Following the definition of the *jitter probability* given in Section 2.1 and taking into considera‐ tion that all the events are independent in white Gaussian noise, Eq. (7) can be rewritten as: *P*(*A<sup>l</sup>* , *<sup>B</sup>l*) <sup>=</sup>*<sup>P</sup> <sup>N</sup>* (*Al*)*<sup>P</sup> <sup>N</sup>* (*Bl*), where, following Eqs. (5) and (6), the probabilities *P*(*Al* ) and *P*(*Bl* ) can be specified as, respectively,

$$P\left(A\_{l}\right) = \begin{cases} 1 - \int\_{\beta\_{l}}^{a\_{l}} p\_{l}\left(\mathbf{x}\right)d\mathbf{x}, & \sigma\_{l}^{2} > \sigma\_{l \approx 1'}^{2} \\ \int\_{a\_{l}}^{a} p\_{l}\left(\mathbf{x}\right)d\mathbf{x}, & \sigma\_{l}^{2} = \sigma\_{l \approx 1'}^{2} \\ \int\_{a\_{l}}^{b\_{l}} p\_{l}\left(\mathbf{x}\right)d\mathbf{x}, & \sigma\_{l}^{2} < \sigma\_{l \approx 1'}^{2} \end{cases} \tag{8}$$

$$P\left(B\_{l}\right) = \begin{cases} \int\_{\beta\_{l}}^{a\_{l}} p\_{l+1}\left(\mathbf{x}\right)d\mathbf{x}, & \sigma\_{l}^{2} > \sigma\_{l+1}^{2}, \\ \int\_{-a\_{l}}^{a\_{l}} p\_{l+1}\left(\mathbf{x}\right)d\mathbf{x}, & \sigma\_{l}^{2} = \sigma\_{l+1}^{2}, \\ \int\_{\beta\_{l}}^{\beta\_{l}} p\_{l+1}\left(\mathbf{x}\right)d\mathbf{x}, & \sigma\_{l}^{2} < \sigma\_{l+1}^{2}, \end{cases} \tag{9}$$

where *pl* (*x*)=1 / 2*πσ<sup>l</sup>* 2 *e* −((*x*−*al*)2)/*σ<sup>l</sup>* 2 is the Gaussian density.

Suppose that jitter occurs at some point *i <sup>l</sup>* ± *k*, 0≤*k* ≤ *N* , as shown, for example, in **Figure 1**, and assign two additional blocks of events *Alk* ={*Ai <sup>l</sup>*−*<sup>N</sup>* … *Ai <sup>l</sup>*−1−*<sup>k</sup>* } and *Blk* ={*Bi <sup>l</sup>*+*<sup>k</sup>* … *Bi <sup>l</sup>*+*<sup>N</sup>* <sup>−</sup>1}. The probability *Pk* −≜*Pk* − (*Alk <sup>A</sup>*¯ *l*(*i <sup>l</sup>*−*<sup>k</sup>* )… *<sup>A</sup>*¯ *i <sup>l</sup>*−1*Bl*) that jitter occurs at the *k*th point to the left from *<sup>i</sup> l* (left jitter) and the probability *Pk* <sup>+</sup>≜*Pk* + (*Al<sup>B</sup>*¯ *l*(*i l*+1)… *<sup>B</sup>*¯ *l*(*i <sup>l</sup>*+*<sup>k</sup>* <sup>−</sup>1)*B<sup>k</sup>* ) that jitter occurs at the *k*th point to the right from *i l* (right jitter) can thus be written as, respectively,

$$P\_k^- = P^{N-k} \left( A\_l \right) \left[ 1 - P \left( A\_l \right) \right]^k P^N \left( B\_l \right), \tag{10}$$

$$P\_k^\* = P^N\left(A\_l\right) \left[1 - P\left(B\_l\right)\right]^k P^{N-k}\left(B\_l\right),\tag{11}$$

By normalizing Eqs. (11) and (12) with Eq. (8), one can arrive at a function that turns out to be independent on *N* :

$$f\_i(k) = \begin{cases} [P^{-1}(A\_i) - 1]^{\mathbb{H}}, & k < 0, \, (\text{left})\\ \text{l} & k = 0\\ \left[P^{-1}(B\_i - 1)\right]^{\mathbb{k}}, & k < 0. \, (\text{right}) \end{cases} . \tag{12}$$

Further normalization of *f <sup>l</sup>* (*k*) to have a unit area leads to the pdf *pl* (*k*)= <sup>1</sup> *<sup>φ</sup><sup>l</sup> f <sup>l</sup>* (*k*), where *φ<sup>l</sup>* is the sum of *f <sup>l</sup>* (*k*) for all *k*,

Enhancing Estimates of Breakpoints in Genome Copy Number Alteration using Confidence Masks http://dx.doi.org/10.5772/63913 123

$$\phi\_l = \mathbf{l} + \sum\_{o}^{k-1} [\![\phi\_l^A \left(k\right) + \![\phi\_l^B \left(k\right)]\!],\tag{13}$$

where *φ<sup>l</sup> <sup>A</sup>*(*k*)= *P* <sup>−</sup><sup>1</sup> (*Al*) <sup>−</sup><sup>1</sup> *<sup>k</sup>* and *φ<sup>l</sup> <sup>B</sup>*(*k*)= *P* <sup>−</sup><sup>1</sup> (*Bl*) <sup>−</sup><sup>1</sup> *<sup>k</sup>* .

( )

( )

where *pl*

probability *Pk*

right from *i*

(*x*)=1 / 2*πσ<sup>l</sup>*

−≜*Pk* − (*Alk <sup>A</sup>*¯ *l*(*i <sup>l</sup>*−*<sup>k</sup>* )… *<sup>A</sup>*¯ *i*

jitter) and the probability *Pk*

*l*

independent on *N* :

Further normalization of *f <sup>l</sup>*

(*k*) for all *k*,

sum of *f <sup>l</sup>*

2 *e* −((*x*−*al*)2)/*σ<sup>l</sup>* 2

assign two additional blocks of events *Alk* ={*Ai*

<sup>+</sup>≜*Pk* + (*Al<sup>B</sup>*¯ *l*(*i l*+1)… *<sup>B</sup>*¯ *l*(*i*

( )

*l*

(right jitter) can thus be written as, respectively,

Suppose that jitter occurs at some point *i*

122 Advanced Biosignal Processing and Diagnostic Methods

( )

*p x dx*

<sup>ï</sup> <sup>=</sup> <sup>í</sup> ò =

*l l l l <sup>α</sup>*

ï ò <

*<sup>l</sup> l l <sup>α</sup>*

ï - ò >

1 ,,

*l ll <sup>β</sup>*

s s

+

2 2 1

2 2 1

2 2 1

2 2

, ,

s s

s s

, ,

2 2

2 2

*<sup>l</sup>*−*<sup>N</sup>* … *Ai*

*<sup>l</sup>*−1*Bl*) that jitter occurs at the *k*th point to the left from *<sup>i</sup>*


+ - = é- ù ë û (11)

*<sup>l</sup>* ± *k*, 0≤*k* ≤ *N* , as shown, for example, in **Figure 1**, and

*<sup>l</sup>*−1−*<sup>k</sup>* } and *Blk* ={*Bi*

*<sup>l</sup>*+*<sup>k</sup>* <sup>−</sup>1)*B<sup>k</sup>* ) that jitter occurs at the *k*th point to the

(*k*)= <sup>1</sup> *<sup>φ</sup><sup>l</sup> f <sup>l</sup>*

(*k*), where *φ<sup>l</sup>*

*<sup>l</sup>*+*<sup>k</sup>* … *Bi*

*<sup>l</sup>*+*<sup>N</sup>* <sup>−</sup>1}. The

*l* (left

(12)

is the

+

(8)

(9)

+

s s

, ,

, ,

s s

1 1

+ +

*l l l*

1 1

1 1

+ +

s s

*l l l*

( )

¥

*l l l*

*β*

*l l l*

a

b

a

*P B p x dx*

*l l*

b

a

*l l*

*α*

ì

ï

ï ï

*P A p x dx*

î

ì

ï

ï ï

ïî

( )

*p x dx*

( )

ï ò >

*p x dx*

( )

*l l l l*

ïï <sup>=</sup> <sup>í</sup> ò =

( )

ï - ò <

*p x dx*

is the Gaussian density.

( ) <sup>1</sup> ( ) ( ) , *<sup>k</sup> N k <sup>N</sup> P P A PA P B k l ll*

( ) <sup>1</sup> ( ) ( ) , *<sup>k</sup> <sup>N</sup> N k P P A PB P B kl l l*

( )

*l*

*f k k*

1


1


( )

<sup>ï</sup> - < <sup>î</sup>

*l*

By normalizing Eqs. (11) and (12) with Eq. (8), one can arrive at a function that turns out to be

[ 1] , 0, ( t)

*P A k lef*

*k*

<sup>ì</sup> - < <sup>ï</sup> = = <sup>í</sup>

*k*

[ 1 ] , 0. ( )

*P B k right*

(*k*) to have a unit area leads to the pdf *pl*

1 0 .

1 ,,

+ + -¥

It follows from the approximation admitted that *f <sup>l</sup>* (*k*) converges with *k* only if 0.5<*<sup>P</sup>*˜ ={*P*(*A*), *<sup>P</sup>*(*B*)}<1. Otherwise, if *P*˜ <0.5, the sum *φ<sup>l</sup>* is infinite and *f <sup>l</sup>* (*k*) cannot be trans‐ formed to *pl* (*k*). It has been shown in [18] that such a situation is practically rare. It can be observed with extremely small and different segmental SNRs when the probabilities are comparable that the measurement point belongs to one of another segment.

Accepting 0.5<*P*˜ ={*P*(*A*), *<sup>P</sup>*(*B*)}<1, one concludes that *P*˜ <0, ln(1−*<sup>P</sup>*˜)<0, and ln(1−*<sup>P</sup>*˜)<ln(*P*˜). Next, using a standard relation ∑ *k*=1 *∞ x <sup>k</sup>* =1 / (*x* <sup>−</sup><sup>1</sup> −1), where *x* <1, and after little transformations, Eq. (14) can be brought to

$$\phi\_l = \frac{P\left(A\_l\right) + P\left(B\_l\right) - 1}{\left[1 - 2P\left(A\_l\right)\right]\left[1 - 2P\left(B\_l\right)\right]}.\tag{14}$$

The *jitter pdf pl* (*k*) associated with the *lth* breakpoint can finally be found to be

$$p\_i\left(k\right) = \frac{1}{\phi\_l} \begin{cases} \left[P^{-1}\left(A\_l\right) - 1\right]^{\left[1\right]}, & k < 0\\ 1, & k = 0, \\ \left[P^{-1}\left(B\_l\right) - 1\right]^k & k > 0 \end{cases},\tag{15}$$

where *ϕ<sup>l</sup>* is specified by Eq. (15) and 0.5<*P*(*Al*), *P*(*Bl*) <1.

If now to substitute *ql* <sup>=</sup>*<sup>P</sup>* <sup>−</sup><sup>1</sup> (*Al*) <sup>−</sup>1 and *dl* <sup>=</sup>*<sup>P</sup>* <sup>−</sup><sup>1</sup> (*Bl*) −1, find *P*(*Al*) =1 / (1 + *ql* ) and *P*(*Bl*) =1 / (1 + *dl* ), and provide the transformations, then one may arrive at a conclusion that Eq. (16) is the discrete skew Laplace pdf [19].

$$p\left(k|d\_l q\_l\right) = \frac{(1-d\_l)(1-q\_l)}{1-d\_l q\_l} \begin{cases} p\_l^k, & k \ge 0\\ q\_l^{\|l\|} & k \le 0 \end{cases} \tag{16}$$

where *dl* =*e* −(*κl*/*νl*) ∈ 0, 1 and *ql* =*e* −(1/*κlνl*) ∈ 0, 1 and in which *κ<sup>l</sup>* and *ν<sup>l</sup>* >0 still need to be con‐ nected to Eq. (16). With this aim, consider Eqs. (16) and (17) at *k* = −1, *k* =0, and *k* =1. By equating Eqs. (16) and (17), first obtain ((1−*dl*)(1−*ql*)*dl* )/ (1−*dl ql* )=1 / *ϕ<sup>l</sup>* (1−*P*(*Bl* )) / *P*(*Bl* ) for *k* =1 and ((1−*dl*)(1−*ql*)*ql* ) / (1−*dl ql* )=1 / *ϕ<sup>l</sup>* (1−*P*(*Al* ))/ *P*(*Al* ) for *k* = −1 that yields

$$\nu\_l = \frac{1 - \kappa\_l^2}{\kappa\_l \ln \mu\_l},\tag{17}$$

where *μl* =(*P*(*Al*) 1−*P*(*Bl* ) ) / (*P*(*Bl* ) 1−*P*(*Al* ) ). For *k* =0 we have ((1−*dl*)(1−*ql*)) / (1−*dl ql* )=1 / *ϕ<sup>l</sup>* and transform it to the equation *xl* <sup>2</sup> <sup>−</sup>(*ϕ<sup>l</sup>* (1 + *μl* ))/ (1 + *ϕ<sup>l</sup>* )*x* −(1−*ϕ<sup>l</sup>* )/ (1 + *ϕl*)*μl* =0, where a proper solution is

$$\chi = \frac{\phi\_l (1 + \mu\_l)}{2(1 + \phi\_l)} \left( 1 - \sqrt{1 - \frac{4\mu\_l (1 - \phi\_l^2)}{\phi\_l^2 \left(1 + \mu\_l\right)^2}} \right) \tag{18}$$

and which *xl* =*μl* −(*κ<sup>l</sup>* 2)/(1−*κ<sup>l</sup>* 2 ) gives us

$$\kappa\_l = \sqrt{\ln \frac{\left(\chi\_l\right)}{\ln(\chi\_l / \mu\_l)}}\,\,\,\,\,\tag{19}$$

By combining Eq. (18) with Eq. (20), one may also get a simpler form for *ν<sup>l</sup>* , namely *ν<sup>l</sup>* = −*κ<sup>l</sup>* / ln*xl* .

Now, introduce the segmental signal‐to‐noise ratios (SNRs): *γ<sup>l</sup>* <sup>−</sup> <sup>=</sup> *<sup>Δ</sup><sup>l</sup>* 2 *σl* <sup>2</sup> , and *γ<sup>l</sup>* <sup>+</sup> <sup>=</sup> *<sup>Δ</sup><sup>l</sup>* 2 *σ<sup>l</sup>* +1 2 , where *Δ<sup>l</sup>* =*al*+1 −*al* , substitute the Gaussian pdf to Eqs. (9) and (10), provide the transformations, and rewrite Eqs. (9) and (10) as

$$P(A\_l) = \begin{cases} 1 + \frac{1}{2} \left[ \text{erf}\left(\mathbf{g}\_l^{\boldsymbol{\beta}}\right) - \text{erf}\left(\mathbf{g}\_l^{\boldsymbol{\alpha}}\right) \right], & \boldsymbol{\gamma}\_l^- < \boldsymbol{\gamma}\_l^+ \\\\ \frac{1}{2} \text{erfc}\left(\mathbf{g}\_l^{\boldsymbol{\alpha}}\right), & \boldsymbol{\gamma}\_l^- = \boldsymbol{\gamma}\_l^+, \\\\ \frac{1}{2} \left[ \text{erf}\left(\mathbf{g}\_l^{\boldsymbol{\beta}}\right) - \text{erf}\left(\mathbf{g}\_l^{\boldsymbol{\alpha}}\right) \right], & \boldsymbol{\gamma}\_l^- > \boldsymbol{\gamma}\_l^+ \end{cases} \tag{20}$$

$$P\left(B\_{l}\right) = \begin{cases} -\frac{1}{2} \left[ \text{erf}\left(h\_{l}^{a}\right) - \text{erf}\left(h\_{l}^{\rho}\right) \right], & \gamma\_{l}^{-} < \gamma\_{l}^{+} \\ & 1 - \frac{1}{2} \text{erfc}\left(h\_{l}^{a}\right), & \gamma\_{l}^{-} = \gamma\_{l}^{+} \\ & 1 + \frac{1}{2} \left[ \text{erf}\left(h\_{l}^{a}\right) - \text{erf}\left(h\_{l}^{\rho}\right) \right], & \gamma\_{l}^{-} > \gamma\_{l}^{+} \end{cases} \tag{21}$$

where *gl <sup>β</sup>* =(*β<sup>l</sup>* <sup>−</sup>*Δ<sup>l</sup>* ) / |*Δ<sup>l</sup>* | *γ<sup>l</sup>* <sup>−</sup> / 2, *gl <sup>α</sup>* =(*α<sup>l</sup>* <sup>−</sup>*Δ<sup>l</sup>* )/ |*Δ<sup>l</sup>* | *γ<sup>l</sup>* <sup>−</sup> / 2, *hl <sup>β</sup>* <sup>=</sup>*β<sup>l</sup>* / <sup>|</sup>*Δ<sup>l</sup>* <sup>|</sup> *<sup>γ</sup><sup>l</sup>* <sup>+</sup> / 2, *hl <sup>α</sup>* <sup>=</sup>*α<sup>l</sup>* / <sup>|</sup>*Δ<sup>l</sup>* <sup>|</sup> *<sup>γ</sup><sup>l</sup>* <sup>+</sup> / 2, erf(*x*) is the error function, erfc(*x*) is the complementary error function, and

$$\alpha\_{l}, \beta\_{l} = \frac{a\_{l}\boldsymbol{\gamma\_{l}^{-}} - a\_{l}\boldsymbol{\gamma\_{l}^{+}}}{\boldsymbol{\gamma\_{l}^{-}} - \boldsymbol{\gamma\_{l}^{+}}} \mp \frac{1}{\boldsymbol{\gamma\_{l}^{-}} - \boldsymbol{\gamma\_{l}^{+}}} \times \sqrt{(a\_{l} - a\_{l+1})\boldsymbol{\gamma\_{l}^{-}}\boldsymbol{\gamma\_{l}^{+}} + 2\Delta\_{l}^{2}\left(\boldsymbol{\gamma\_{l}^{-}} - \boldsymbol{\gamma\_{l}^{+}}\right) \ln\sqrt{\frac{\boldsymbol{\gamma\_{l}^{-}}}{\boldsymbol{\gamma\_{l}^{+}}}}} \,. \tag{22}$$

#### *2.3.2. Approximation of jitter distribution using the modified Bessel functions*

An analysis shows that the discrete skew Laplace pdf (17) gives good results only if SNR is >1. Otherwise, real measurements do not fit well, and a more accurate function is required. Below, we show that better approach to real jitter distribution can be provided using the modified Bessel functions.

#### *2.3.2.1. Modified Bessel function*

<sup>2</sup> <sup>1</sup> , ln *l*


<sup>=</sup> (19)

<sup>−</sup> <sup>=</sup> *<sup>Δ</sup><sup>l</sup>* 2 *σl*

<sup>2</sup> , and *γ<sup>l</sup>*

<sup>+</sup> <sup>=</sup> *<sup>Δ</sup><sup>l</sup>* 2 *σ<sup>l</sup>* +1

*ql* )=1 / *ϕ<sup>l</sup>*

)/ (1 + *ϕl*)*μl* =0, where a proper

and

(18)

, namely

2 , where

(20)

(21)

) ). For *k* =0 we have ((1−*dl*)(1−*ql*)) / (1−*dl*

)*x* −(1−*ϕ<sup>l</sup>*

2 2 2

( )

 m

*l l* k

(1 ) 4 (1 ) 1 1 2(1 ) 1 *l l l l l l l*

> ( ) ln . ln( / ) *l*

( ) ( )

*g g*

<sup>ì</sup> +- < é ù <sup>ï</sup> ë û <sup>ï</sup>

<sup>1</sup> 1 erf erf , <sup>2</sup>

b

( )

a

*l l l l*

<sup>ï</sup> = = <sup>í</sup>

( ) ( )

( ) ( )

<sup>ì</sup> é ù - < <sup>ï</sup> ë û <sup>ï</sup>

*h h*

*l l l l*

<sup>ï</sup> <sup>=</sup> <sup>í</sup> - =

<sup>1</sup> erf erf , <sup>2</sup>

a

<sup>1</sup> 1 erf erf , <sup>2</sup>

a

( )

a

<sup>1</sup> 1 erfc , , <sup>2</sup>

*l l ll*

 b

*l l ll*

 b

( ) ( )

*h h*

+- > é ù <sup>ï</sup> ë û <sup>î</sup>

<sup>ï</sup> é ù - > <sup>ï</sup> ë û <sup>î</sup>

*g g*

<sup>1</sup> erf erf , <sup>2</sup>

b

By combining Eq. (18) with Eq. (20), one may also get a simpler form for *ν<sup>l</sup>*

*l l x x*

m

, substitute the Gaussian pdf to Eqs. (9) and (10), provide the transformations, and

<sup>1</sup> erf , , <sup>2</sup>

*l l ll*

 a

*l l ll*

g g







g g

g g

g g

g g

g g

 a

æ ö + - = -- ç ÷ <sup>+</sup> <sup>+</sup> è ø

))/ (1 + *ϕ<sup>l</sup>*

 mf

f

k m

(1 + *μl*

*l*

n

) 1−*P*(*Al*

<sup>2</sup> <sup>−</sup>(*ϕ<sup>l</sup>*

fm

f

*l*

k

Now, introduce the segmental signal‐to‐noise ratios (SNRs): *γ<sup>l</sup>*

*P A c g*

*P B h*

ï ï

ï

( )

( )

) ) / (*P*(*Bl*

*x*

where *μl* =(*P*(*Al*) 1−*P*(*Bl*

solution is

and which *xl* =*μl*

*ν<sup>l</sup>* = −*κ<sup>l</sup>* / ln*xl*

*Δ<sup>l</sup>* =*al*+1 −*al*

.

rewrite Eqs. (9) and (10) as

transform it to the equation *xl*

124 Advanced Biosignal Processing and Diagnostic Methods

−(*κ<sup>l</sup>* 2)/(1−*κ<sup>l</sup>* 2 ) gives us

> **Figure 2** demonstrates the jitter pdf measured experimentally (dotted) for different SNRs. The breakpoint corresponds here to the peak density and the probability of the breakpoint location diminishes to the left and to the right of this point. Note that the discrete skew Laplace pdf (17) behaves linearly in such scales. Therefore, Eq. (17) cannot be applied when SNR is <1 and a more accurate function is required.

**Figure 2.** Experimentally defined one‐sided jitter probability densities (dotted) of the breakpoint location for equal seg‐ mental SNR *γ* in the range of *M* =400 points with a true breakpoint at *n* =200. The experimental density functions were found using the *Maximum Likelihood* (ML) estimator. The histogram was plotted over 50×10<sup>3</sup> runs repeated nine times and average. Approximations (continuous) are provided using the proposed Bessel‐based approximation depicted as MBA.

Among available functions demonstrating the pdf properties, the modified Bessel function of the second kind *K*<sup>0</sup> (*x*) and zeroth order is a most good candidate to fit the experimentally measured densities (**Figure 2**). The following form of *K*0(*x*) can be used:

$$\begin{aligned} \left[K\_0\right] \left[x\left(k\right)\right] &= \stackrel{\circ}{\int} \cos\left[x\left(k\right)\sinh t\right]dt\\ \stackrel{\circ}{\left[} = \stackrel{\circ}{\int} \frac{\cos\left[x\left(k\right)t\right]}{\sqrt{t^2+1}}\right]dt &> 0, x\left(k\right) > 0 \end{aligned} \tag{23}$$

in which a variable *x*(*k*) depends on index *k*, which represents a discrete departure from the assumed breakpoint location. Because *K*<sup>0</sup> *x*(*k*) is a positive‐valued for *x*(*k*)>0 smooth function decreasing with *x* to zero, it can be used to approximate the probability density.

#### *2.3.2.2. Approximation*

In order to use Eq. (24) as an approximating function

$$\mathbf{B}\left(k|\boldsymbol{\gamma}\right) = K\_0\left[\mathbf{x}\left(k\right)\right] \tag{24}$$

**Figure 3.** Simulated CNA with a single breakpoint at *n* = 200 and segmental standard deviations *σ<sup>l</sup>* and *σl*+1 corre‐ sponding to SNRs *γ<sup>l</sup>* <sup>−</sup> <sup>=</sup>*γ<sup>l</sup>* <sup>+</sup> =1.37: (a) measurement and (b) jitter distribution. Here, ML (circled) is the jitter pdf obtained experimentally using an ML estimator through a histogram over 50×10<sup>3</sup> runs, SkL (solid) is the Laplace distribution, and MBA (dashed) is the Bessel‐based approximation.

conditioned on *γ* for the one‐sided jitter probability densities shown in **Figure 2**, we represent a variable *x* via *k* as *x*(*k*, *γ*)=ln *Φ*(*k*, *γ*) in a way such that small *k* ≥0 correspond to large values *x* of and vice versa. Among several candidates, it has been found empirically that the following function *Φ*(*k*, *γ*) fits the histograms with highest accuracy:

$$\Phi\left(k,\mathcal{Y}\right) = \left(|k|+1\right)^{\rho+\varepsilon|k|} \left[\frac{1+\sqrt{\mathcal{Y}}}{\mathcal{Y}} - \epsilon\right],\tag{25}$$

if to set *γ* =*γ<sup>l</sup>* − for *<sup>k</sup>* <0, *<sup>γ</sup>* <sup>=</sup> *<sup>γ</sup><sup>l</sup>* <sup>−</sup> <sup>+</sup> *<sup>γ</sup><sup>l</sup>* + <sup>2</sup> for *k* =0, and *γ* =*γ<sup>l</sup>* + for *k* >0, and represent the coefficients and as *τ*(*γ*), *ρ*(*γ*), and (*γ*) as

$$
\pi\left(\boldsymbol{\gamma}\right) = a\_0 \boldsymbol{\gamma} + a\_1 \tag{26}
$$

$$
\rho\left(\boldsymbol{\gamma}\right) = \left.\gamma \left(\boldsymbol{b}\_{0}\boldsymbol{\gamma}^{\mathbb{N}} + \boldsymbol{a}\_{0}\right) + \boldsymbol{b}\_{2}\right|\tag{27}
$$

$$
\epsilon \left( \boldsymbol{\gamma} \right) = \mathbf{c}\_0 \boldsymbol{\gamma}^{\sphericalangle} + \mathbf{c}\_2 \tag{28}
$$

where *a*<sup>0</sup> <sup>=</sup>0.02737, *a*<sup>1</sup> <sup>=</sup> −4.5×10−3 , *b*<sup>0</sup> =0.3674, *b*<sup>1</sup> = −0.3137, *b*<sup>2</sup> =0.8066, *c*0=0.8865, *c*<sup>1</sup> = −1.033, and *c*<sup>2</sup> = −1.233 were found in the mean square error (MSE) sense. These values were found in several iterations until the MSE reached a minimum.

In summary, **Figure 3** gives a typical example of a simulated CNA, where the modified Bessel function‐based approximation (depicted as MBA) demonstrates better accuracy than the approximation obtained using the skew Laplace distribution (depicted as SkL).

#### **2.4. Probabilistic masks**

Among available functions demonstrating the pdf properties, the modified Bessel function of

measured densities (**Figure 2**). The following form of *K*0(*x*) can be used:

<sup>0</sup> <sup>0</sup>

*t*

0 2

cos

¥

In order to use Eq. (24) as an approximating function

( ) ( )

¥

*K x k x k t dt*

é ù=ò é ù ëû ë û

1

decreasing with *x* to zero, it can be used to approximate the probability density.

(*k K xk* |g

**Figure 3.** Simulated CNA with a single breakpoint at *n* = 200 and segmental standard deviations *σ<sup>l</sup>*

<sup>+</sup> =1.37: (a) measurement and (b) jitter distribution. Here, ML (circled) is the jitter pdf obtained

*xkt*

é ù ë û =ò > > +

( ) ( )

cos sinh

*dt x k*

in which a variable *x*(*k*) depends on index *k*, which represents a discrete departure from the assumed breakpoint location. Because *K*<sup>0</sup> *x*(*k*) is a positive‐valued for *x*(*k*)>0 smooth function

0, 0 ,

(*x*) and zeroth order is a most good candidate to fit the experimentally

) =é ù <sup>0</sup> ( ) ë û *B* (24)

(23)

and *σl*+1 corre‐

runs, SkL (solid) is the Laplace distribution,

the second kind *K*<sup>0</sup>

126 Advanced Biosignal Processing and Diagnostic Methods

*2.3.2.2. Approximation*

sponding to SNRs *γ<sup>l</sup>*

<sup>−</sup> <sup>=</sup>*γ<sup>l</sup>*

and MBA (dashed) is the Bessel‐based approximation.

experimentally using an ML estimator through a histogram over 50×10<sup>3</sup>

It follows from **Figure 3** that, in view of large noise, estimates of the CNAs may have low confidence, especially with small SNR *γ* ≤1. Thus, each estimate requires confidence bounda‐ ries within which it may exist with a given probability [20, 21].

Given an estimate *a* ^ *l* of the *l*th segmental level in white Gaussian noise, the probabilistic upper boundary (UB) and lower boundary (LB) can be specified for the given confidence probability *P*(*ϑ*) in the *ϑ*‐sigma sense as [20]

$$
\hat{a}\_l^{\text{UB}} \equiv \hat{a}\_l + \mathfrak{e} = \hat{a}\_l + \mathfrak{G} \sqrt{\frac{\sigma\_j^2}{N\_l}} = \hat{a}\_l + \mathfrak{G}\hat{\sigma}\_l \tag{29}
$$

$$
\hat{a}\_l^{LB} \equiv \hat{a}\_l - \varepsilon = \hat{a}\_l + \mathcal{G}\sqrt{\frac{\sigma\_l^2}{N\_l}} = \hat{a}\_l + \mathcal{G}\hat{\sigma}\_l \tag{30}
$$

where *ϑ* indicates the boundary wideness in terms of the segmental noise variance *σ* ^ *l* on an interval *Nl* points, from *n* ^ *<sup>l</sup>*−1 to *n* ^ *<sup>l</sup>* −1.

Likewise, detected the *l*th breakpoint location *n* ^ *l* , the jitter probabilistic left boundary *Jl <sup>L</sup>* and right boundary *Jl <sup>R</sup>* can be defined, following [20], as

$$J\_{\perp}^{L} \equiv \hat{n}\_{\perp} - k\_{\perp}^{R},\tag{31}$$

$$J\_l^{\mathcal{R}} \equiv \hat{n}\_l + k\_l^{L},\tag{32}$$

where *kl <sup>R</sup>*(*ϑ*) and *kl <sup>L</sup>* (*ϑ*) are specified by the jitter distribution in the *ϑ*‐sigma sense.

By combining Eqs. (30) and (31) with Eqs. (32) and (33), the probabilistic masks can be formed as shown in [20] to bound the CNA estimates in the *ϑ*‐sigma sense for the given confidence probability *P*(*ϑ*). An important property of these masks is that they can be used not only to bound the estimates and show their possible locations on a probabilistic field [20, 21] but also to remove supposedly wrong breakpoints. Such situations occur each time when the masks reveal double UB and LB uniformities in a gap of three neighboring detected breakpoints. If so, then the unlikely existing intermediate breakpoint ought to be removed.

Noticing that the segmental boundaries (30) and (31) remain the same irrespective of the jitter in the breakpoints, below we specify the masks for the jitter represented with the Laplace distribution (17) and Bessel‐based approximation (25).

#### *2.4.1. Masks for Laplace distribution*

For the Laplace distribution (17), the jitter left boundary *Jl <sup>L</sup>* (32) and right boundary *Jl <sup>R</sup>*(33) can be defined in the *ϑ*–‐sigma sense if to specify *kl <sup>R</sup>*(*ϑ*) and *kl <sup>L</sup>* (*ϑ*) as shown in [18],

$$k\_l^R = \left[\frac{\nu}{\kappa} \ln \frac{(1 - d\_l)(1 - q\_l)}{\xi'(1 - d\_l q\_l)}\right],\tag{33}$$

$$k\_l^L = \left[ \nu \kappa \ln \frac{(1 - d\_l)(1 - q\_l)}{\tilde{\varphi} \left(1 - d\_l q\_l \right)} \right],\tag{34}$$

where [*x*] means a maximum integer lower than or equal to *x*. Note that functions (34) and (35) were obtained in [18] by equating (17) to *ξ*(*Nl*) =erfc(*ϑ* / 2) and solving for *kl* .

The probabilistic UB mask and LB mask for the Laplace distribution were formed in [17,20,21] by the segmental upper boundary *a* ^ *l* UB and lower boundary *a* ^ *l* LB and by the jitter left boundary *Jl* L and jitter right boundary *Jl* R. The algorithm for computing and masks has been developed and applied to the CNA probes in [22].

#### *2.4.2. Masks for Bessel‐based approximation*

2

s

^ *l* *l*

 Js

(30)

, the jitter probabilistic left boundary *Jl*

*l ll J nk* @ - (31)

*l ll J nk* @ + (32)

*<sup>L</sup>* (32) and right boundary *Jl*

*<sup>L</sup>* (*ϑ*) as shown in [18],

^ *l* on an

*<sup>L</sup>* and

*<sup>R</sup>*(33)

(33)

(34)

ˆˆ ˆ ˆˆ *LB <sup>j</sup> ll l l l*

@ -= + = +

where *ϑ* indicates the boundary wideness in terms of the segmental noise variance *σ*

ˆ , *L R*

ˆ , *R L*

so, then the unlikely existing intermediate breakpoint ought to be removed.

distribution (17) and Bessel‐based approximation (25).

For the Laplace distribution (17), the jitter left boundary *Jl*

*l*

*l*

can be defined in the *ϑ*–‐sigma sense if to specify *kl*

*2.4.1. Masks for Laplace distribution*

*<sup>L</sup>* (*ϑ*) are specified by the jitter distribution in the *ϑ*‐sigma sense.

By combining Eqs. (30) and (31) with Eqs. (32) and (33), the probabilistic masks can be formed as shown in [20] to bound the CNA estimates in the *ϑ*‐sigma sense for the given confidence probability *P*(*ϑ*). An important property of these masks is that they can be used not only to bound the estimates and show their possible locations on a probabilistic field [20, 21] but also to remove supposedly wrong breakpoints. Such situations occur each time when the masks reveal double UB and LB uniformities in a gap of three neighboring detected breakpoints. If

Noticing that the segmental boundaries (30) and (31) remain the same irrespective of the jitter in the breakpoints, below we specify the masks for the jitter represented with the Laplace

> ( )( ) ( ) 1 1 ln , <sup>1</sup>

( )( ) ( ) 1 1 ln , <sup>1</sup>

*l l*

*l l*

*d q*

*d q*

*R l l*

é ù - - <sup>=</sup> ê ú - ë û

 x

*L l l*

x

é ù - - <sup>=</sup> ê ú - ë û

*d q <sup>k</sup>*

*d q <sup>k</sup>*

n

k

nk

*<sup>R</sup>*(*ϑ*) and *kl*

 J

*aa a a <sup>N</sup>*

e

*<sup>R</sup>* can be defined, following [20], as

interval *Nl*

where *kl*

*<sup>R</sup>*(*ϑ*) and *kl*

right boundary *Jl*

points, from *n*

128 Advanced Biosignal Processing and Diagnostic Methods

^ *<sup>l</sup>*−1 to *n* ^ *<sup>l</sup>* −1.

Likewise, detected the *l*th breakpoint location *n*

The UB mask and LB mask for the Bessel‐based approximation can be formed using the same equations as for the Laplace distribution. Suppose that the Laplace pdf (17) is equal to the approximating function **Β***<sup>l</sup>* (*k*) at *k* =0,

$$p\left(k=0|d\_{\perp},q\_{\perp}\right)=\mathbf{B}\_{\perp}\left(k=0\right),\tag{35}$$

that yields **Β***<sup>l</sup>* (*<sup>k</sup>* =0)= <sup>1</sup> *ϕl* . Then, define the probabilities *<sup>P</sup> <sup>B</sup>*(*Al* ) at *<sup>k</sup>* <sup>=</sup> <sup>−</sup>1 and *<sup>P</sup> <sup>Β</sup>*(*Bl* ) at *k* =1 as

$$P^{\mathbf{B}}\left(A\_{l}\right) = \frac{\mathbf{B}\_{l}\left(k=0\right)}{\mathbf{B}\_{l}\left(k=-1\right) + \mathbf{B}\_{l}\left(k=0\right)},\tag{36}$$

$$P^{\mathbf{B}}\left(B\_{l}\right) = \frac{\mathbf{B}\_{l}\left(k=0\right)}{\mathbf{B}\_{l}\left(k=1\right) + \mathbf{B}\_{l}\left(k=0\right)}.\tag{37}$$

Next, substitute Eqs. (37) and (38) into Eqs. (19) and (20) to obtain *κ<sup>l</sup> <sup>B</sup>* and *ν<sup>l</sup> <sup>B</sup>*. The right‐hand jitter *kl <sup>B</sup>*<sup>R</sup> and left‐hand jitter *kl <sup>B</sup>*L can now be specified by, respectively,

$$k\_{l}^{\mathbf{z}\mathbf{z}} = \left[\frac{\nu\_{l}^{\mathbf{z}}}{\kappa\_{l}^{\mathbf{z}}} \ln \frac{1}{\xi^{\mathbf{z}} \mathbf{B}\_{l}(k=0)}\right],\tag{38}$$

$$k\_{l}^{\text{BL}} = \left[\nu\_{l}^{\text{B}}\kappa\_{l}^{\text{B}}\ln\frac{1}{\mathcal{E}\_{l}\mathcal{B}\_{l}(k=0)}\right].\tag{39}$$

Finally, define the jitter left boundary *Jl <sup>B</sup>*L and right boundary *Jl <sup>B</sup>*R as, respectively,

$$J\_l^{\mathbf{B}1} \equiv \hat{n}\_l - k\_l^{\mathbf{B}\mathbf{R}},\tag{40}$$

$$J\_l^{\mathbf{z}\mathbf{R}} \equiv \hat{n}\_l - k\_l^{\mathbf{z}\mathbf{1}},\tag{41}$$

and use the algorithm previously designed in the study of Munoz‐Minjares and Shmaliy [22] for the confidence masks based on the Laplace distribution.

**Figure 4.** and masks for the seventh chromosome taken from "159A–vs–159D–cut" of ROMA: (a) genomic location from 130 to 146 Mb and (b) genomic location from 146 to 156 Mb. Breakpoints *î*1, *î*6, *î*7, *î*9, *î*10, *î*12, and *î*13 are well detectable because jitter is moderate. Owing to large jitter the breakpoints *î*2, *î*3, *î*4, *î*5, *î*8, *î*9, and *î*11 cannot be estimated correctly. There is a probability that the breakpoints *î*2, *î*3, *î*4, *î*5, and *î*11 do not exist. There is a high proba‐ bility that breakpoint *î*5 does not exist.
