**4. Discussion**

We evaluate the breakpoints obtained by the projects representational oligonucleotide microarray analysis [23] and GAP [14] with the confidence masks. As has been shown before, not all of the detected chromosomal changes have the same confidence to mean that there is a probability that some breakpoints do not exist. In order to improve the CNA estimates for the required confidence, the following process can be used:


Application of this methodology to the CNA structure detected in frames of the Project GAP is shown in **Figure 9**. Its special feature is a number of hardly recognized small chromosomal changes (**Figure 9a**). We test them by the proposed masks **ℬ***<sup>l</sup>* UB and **ℬ***<sup>l</sup>* LB. To this end, we first start with equal confidence probabilities of *P* =50*%* for each estimate to exist or not exist and find out that three breakpoints demonstrate no detectability. We remove these breakpoints and depict their locations with "×". Reasoning similarly, we remove four breakpoints to retain only probable changes, by *P* =75*%*, nine breakpoints to show a picture combined with almost certain changes, by *P* =93*%*, and 10 breakpoints in the three‐sigma sense, *P* =99.73*%*. Observing the results, we infer that the masks are able to correct only the estimates obtained under the low SNRs. The relevant chromosomal sections S1–S7 are circled in **Figure 9**. It is not surprising because changes existing with high SNRs are seen visually. An estimator thus can easily detect them with high confidence.

Enhancing Estimates of Breakpoints in Genome Copy Number Alteration using Confidence Masks http://dx.doi.org/10.5772/63913 135

**Figure 9.** Improving estimates of the CNAs obtained in Project GAP [25] by removing some unlikely existing break‐ points: (a) original estimates, (b) even changes, *P* =50*%*, (c) probable changes, *P* =75*%*, (d) almost certain changes, *P* =93*%*, and (e) three‐sigma sense, *P* =99.73*%*.

#### **5. Conclusions**

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.

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

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

We evaluate the breakpoints obtained by the projects representational oligonucleotide microarray analysis [23] and GAP [14] with the confidence masks. As has been shown before, not all of the detected chromosomal changes have the same confidence to mean that there is a probability that some breakpoints do not exist. In order to improve the CNA estimates for the

**1.** Obtain estimates of the CNA using the standard CBS algorithm [24, 25] or any other

**2.** Compute masks and for the given confidence probability *P*, *%* and bound the

**3.** If the masks reveal double uniformities, in UB and LB, in a GAP of any three neighboring breakpoints, then remove the intermediate breakpoint and estimate the segmental level between the survived breakpoints by simple averaging. The CNAs estimated in such a

Application of this methodology to the CNA structure detected in frames of the Project GAP is shown in **Figure 9**. Its special feature is a number of hardly recognized small chromosomal

start with equal confidence probabilities of *P* =50*%* for each estimate to exist or not exist and find out that three breakpoints demonstrate no detectability. We remove these breakpoints and depict their locations with "×". Reasoning similarly, we remove four breakpoints to retain only probable changes, by *P* =75*%*, nine breakpoints to show a picture combined with almost certain changes, by *P* =93*%*, and 10 breakpoints in the three‐sigma sense, *P* =99.73*%*. Observing the results, we infer that the masks are able to correct only the estimates obtained under the low SNRs. The relevant chromosomal sections S1–S7 are circled in **Figure 9**. It is not surprising because changes existing with high SNRs are seen visually. An estimator thus can easily detect

UB and **ℬ***<sup>l</sup>*

LB. To this end, we first

Just on the contrary, masks and can be computed for any reasonable SNR.

MATLAB software on a personal computer with a processor Intel Core i5, 2.5 GHz.

required confidence, the following process can be used:

way will be valid for the given confidence *P*, *%*.

changes (**Figure 9a**). We test them by the proposed masks **ℬ***<sup>l</sup>*

the required probability.

134 Advanced Biosignal Processing and Diagnostic Methods

**4. Discussion**

algorithm.

estimates.

them with high confidence.

Modern technologies developed to produce the CNA profiles with high resolution still admit intensive white Gaussian noise. Accordingly, not one estimator even ideal is able to provide jitter‐free estimation of segmental changes. Thus, in order to avoid wrong decisions, the estimates must be bounded for the confidence probability. Jitter exists in the CNA's break‐ points fundamentally. When SNR is >1, it can statistically be described using the discrete skew Laplace distribution. Otherwise, if SNR is <1, the Bessel‐based approximation produces more accuracy. By the jitter distribution, it is easy to find a region within which the breakpoint exists for the required probability. Of practical importance are the confidence UB and LB masks, which can be created based on the segmental and jitter distributions for the given confidence probability. The masks can serve as an auxiliary tool for medical experts to make decisions about the CNA structures. Applications to probes obtained using the HR‐CGH and SNP technologies confirm efficiency of the confidence masks.
