**3. Results**

R L ˆ , *l ll J nk* @ - *B B* (41)

and use the algorithm previously designed in the study of Munoz‐Minjares and Shmaliy [22]

**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.

for the confidence masks based on the Laplace distribution.

130 Advanced Biosignal Processing and Diagnostic Methods

In this section, we test some CNA measurements and estimates by the algorithm developed in [22] based on the Laplace and Bessel approximations. In order to demonstrate the efficiency of the probabilistic masks and getting practically useful results, we exploit probes obtained by different technologies. First, we employ the results obtained with the HR‐CGH profile and test them by the probabilistic masks using Laplace distribution. We next demonstrate the efficiency of the Bessel‐based probabilistic masks versus the Laplace‐based masks for the probes obtained with the SNP profile.

### **3.1. HR‐CGH‐based probing**

The first test is conducted in the three‐sigma sense suggesting that the CNAs exist between the UB and LB masks with high probability of *P* =99.73*%*. The tested HR‐CGH array data are available from the representational oligonucleotide microarray analysis (ROMA) [23]. The breakpoint locations are also given in [23]. Voluntarily, we select data associated with potentially large jitter and large segmental errors. For clarity, we first compute some charac‐ teristics of the detected CNAs and notice that the segmental estimates found by averaging [18] are in a good correspondence with [23]. The database processed is a part of the seventh chromosome in archive "159A–vs–159D–cut" of ROMA a sample of B‐cell chronic lymphocytic leukemia (B‐CLL). It is shown to have 14 segments and 13 breakpoints (**Figure 4a** and **b**). Below, we shall show that, owing to large detection noise, there is a high probability that some breakpoints do not exist.

It follows from **Figure 4a** that the only breakpoint which location can be estimated with high accuracy is *i* 1. Jitter in *i* ^ <sup>6</sup> and *i* ^ <sup>7</sup> is moderate. All other breakpoints have large jitter. It is seen that the UB mask covering second to sixth segments is almost uniform. Thus, there is a probability that the second to fifth breakpoints do not exist. If to follow the LB mask, the locations of the second to fourth breakpoints can be predicted even with large errors. At least they can be supposed to exist. However, nothing definitive can be said about the fifth break‐ point and one may suppose that it does not exist. It is also hard to distinguish a true location of the eighth breakpoint. In **Figure 4b**, *i* 10, *i* <sup>12</sup>, and *i* 13 are well detectable owing to large segmental SNRs. The breakpoint *i* <sup>9</sup> has a moderate jitter. In turn, the location of *i* 11 is unclear. Moreover, there is a probability that *i* 11 does not exist.

### **3.2. SNP‐based probing**

Our purpose now is to apply the probabilistic mask with SNP profile that represents the CNA with low levels of SNR. Specifically, we employ the probes of the first chromosome available from "BLC\_B1\_T45.txt" a sample of primary breast carcinoma.

Inherently, the more accurate Bessel‐based approximation extends the jitter probabilistic boundaries with respect to the Laplace‐based ones, especially for low SNRs. We illustrate it in **Figure 5**, where the estimates of the first chromosome were tested by , , , and for *ϑ* =3 (confidence probability *P* =99.73*%*).

In **Figure 6**, the masks and are placed in the vicinity of segment *a* ^ <sup>18</sup> for several confidence probabilities: *ϑ* =0.6745(*P* =50*%*), *ϑ* =1(*P* =68.27*%*), *ϑ* =2(*P* =95.45*%*), and *ϑ* =3(*P* =99.73*%*). What the masks suggest here is that the CNA evidently exists with high probability, but the segmental levels and the breakpoint locations cannot be estimated with high accuracy, owing to low SNRs.

**Figure 5.** Jitter left boundaries **ℬ***<sup>l</sup>* <sup>ℬ</sup>L, *J<sup>l</sup>* L and right boundaries *J<sup>l</sup>* <sup>ℬ</sup>R, *J<sup>l</sup>* R for the breakpoint *i* ^ <sup>2</sup> of first chromosome from sample BLC\_B1\_T45.txt (primary breast carcinoma). The probabilistic masks detect a breakpoint with a confidence probability *ϑ* =3(*P* =99.73*%*).

**Figure 6.** The and masks placed around the segmental level *a*<sup>18</sup> for several confidence probabilities [20]. Here, the CNA exists with high probability, but the segmental levels and the breakpoint locations cannot estimate with high accuracy.

Enhancing Estimates of Breakpoints in Genome Copy Number Alteration using Confidence Masks http://dx.doi.org/10.5772/63913 133

In **Figure 6**, the masks and are placed in the vicinity of segment *a*

132 Advanced Biosignal Processing and Diagnostic Methods

to low SNRs.

**Figure 5.** Jitter left boundaries **ℬ***<sup>l</sup>*

probability *ϑ* =3(*P* =99.73*%*).

accuracy.

<sup>ℬ</sup>L, *J<sup>l</sup>*

L and right boundaries *J<sup>l</sup>*

<sup>ℬ</sup>R, *J<sup>l</sup>*

sample BLC\_B1\_T45.txt (primary breast carcinoma). The probabilistic masks detect a breakpoint with a confidence

**Figure 6.** The and masks placed around the segmental level *a*<sup>18</sup> for several confidence probabilities [20]. Here, the CNA exists with high probability, but the segmental levels and the breakpoint locations cannot estimate with high

R for the breakpoint *i*

^

<sup>2</sup> of first chromosome from

probabilities: *ϑ* =0.6745(*P* =50*%*), *ϑ* =1(*P* =68.27*%*), *ϑ* =2(*P* =95.45*%*), and *ϑ* =3(*P* =99.73*%*). What the masks suggest here is that the CNA evidently exists with high probability, but the segmental levels and the breakpoint locations cannot be estimated with high accuracy, owing

^

<sup>18</sup> for several confidence

**Figure 7.** The confidence masks placed around *a*10 for *ϑ* =0.6745(*P* =50*%*) and *ϑ* =3(*P* =99.73*%*). Masks and do not confirm an existence of segmental changes while and indicate a small change.

**Figure 8.** The confidence masks , ℒ*<sup>l</sup>* LB, **<sup>ℬ</sup>***<sup>l</sup>* UB and . placed around the breakpoint *i* ^ <sup>20</sup> for confidence probabilities *ϑ* =0.6745 and *ϑ* =3 of first chromosome from sample BLC\_B1\_T45.txt. The confidence masks based on Laplace distri‐ bution cannot detect the breakpoint *i* ^ 20.

A special case can also be noticed when the masks and are not able to confirm or deny an existence of segmental changes with high probability, owing to the inability of computing the Laplace‐based masks for extremely low SNRs. **Figures 7** and **8** illustrate such situations. Just on the contrary, masks and can be computed for any reasonable SNR.

A conclusion that can be made based on the results illustrated in **Figures 5**–**8** is that the Bessel‐ based probabilistic masks can be used to improve estimates of the chromosomal changes for the required probability.

We finally notice that the computation time required by the masks to process the first chro‐ mosome from sample "BLC B1 T45.txt" with a length of *n* = 905215 was 2.634599 s using MATLAB software on a personal computer with a processor Intel Core i5, 2.5 GHz.
