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IEEE/ACM Trans Comput Biol Bioinform. Author manuscript; available in PMC 2017 May 1.

Published in final edited form as:

Published online 2015 June 22. doi: 10.1109/TCBB.2015.2448077

PMCID: PMC4905595

NIHMSID: NIHMS722963

Catherine Stamoulis, Department of Radiology, Harvard Medical School and Boston Children’s Hospital, Boston, MA 02115;

Catherine Stamoulis: ude.tim@aniretac; Rebecca A. Betensky: ude.dravrah.hpsh@yksneteb

The publisher's final edited version of this article is available at IEEE/ACM Trans Comput Biol Bioinform

We aim to improve the performance of the previously proposed signal decomposition matched filtering (SDMF) method [26] for the detection of copy-number variations (CNV) in the human genome. Through simulations, we show that the modified SDMF is robust even at high noise levels and outperforms the original SDMF method, which indirectly depends on CNV frequency. Simulations are also used to develop a systematic approach for selecting relevant parameter thresholds in order to optimize sensitivity, specificity and computational efficiency. We apply the modified method to array CGH data from normal samples in the cancer genome atlas (TCGA) and compare detected CNVs to those estimated using circular binary segmentation (CBS) [19], a hidden Markov model (HMM)-based approach [11] and a subset of CNVs in the Database of Genomic Variants. We show that a substantial number of previously identified CNVs are detected by the optimized SDMF, which also outperforms the other two methods.

Copy-number variations (CNV) are relatively large (≥1000 base pairs) structural changes in the genome that reflect either normal genetic heterogeneity [15], [18], [22], [23], or various pathologies [8], [16], [17], [24], [29]. CNVs vary considerably in length. Although they may be located in genomic regions that encode proteins, they are often found in regions that either contain functional non-coding DNA elements or have currently unknown function.

Genomic data are typically contaminated by various types of noise making it difficult to identify CNVs. Over the last decade various methods have been developed for CNV detection [4], [5], [7], [10], [11], [13], [19], [21], [25], [28]. A few signal processing methods have also been proposed [3], [6], [14], [20], [26]. Circular binary segmentation (CBS), a change-point detection method [19], is popular for CNV detection in array comparative genomic hybridization (aCGH) data. Other methods, such as the much faster HaarSeg segmentation [6], hold promise for robust CNV detection at low computational cost. We recently proposed a signal processing-inspired methodology (signal decomposition matched filtering (SDMF)) for improving the signal-to-noise ratio (SNR) in genomic data and consequently facilitating CNV detection [26]. As a signal processing approach, SDMF is appropriate for any signals that are either continuous or can be treated as such, e.g., aCGH and high-resolution next-generation sequencing (NGS) data.

It is anticipated that in the near future CNVs will be detected using multiple high-resolution platforms. NGS is becoming less expensive for investigating the entire genome. Consequently, the role of the lower resolution aCGH may eventually change from a detection to a validation tool, i.e., aCGH-based CNVs may be used to validate CNVs detected in NGS data. However, some genomic studies may still use aCGH since it directly estimates a relative measure between test and reference sequences (the ratio of fluorescence intensities), whereas NGS requires the identification of a reference genome for comparison to the genome of interest. The selection objectivity, estimation and robustness of this signal are important technical issues that will eventually be resolved as NGS evolves. Finally, large volumes of publicly available aCGH data, e.g., in the cancer genome atlas [1] remain valuable in genomic studies. Given the value of aCGH and promise of NGS, it is desirable to develop CNV detection methods that are applicable, robust and computationally efficient across platforms. A common methodology also eliminates method-related uncertainties that arise when distinct approaches are applied to distinct data types. Advantages and shortcomings of various methods developed for aCGH and applied to NGS data were recently shown in [9]. Overall, it is expected that signal processing methods, which are inherently based on principles of analysis of high-dimensional and correlated signals, may be more appropriate for CNV detection in aCGH and NGS data, particularly as progressively larger volumes of continuous NGS data from longer segments of the genome become available.

SDMF aims to improve the data SNR first by eliminating noisy signal components, followed by matched-filtering [27] resulting in local SNR changes. By design, SDMF compares pairs of signals to assess their waveform similarity/dissimilarity. The original study [26] focused on the development of the SDMF method and optimization of the denoising process (the first step of the method). It was tested with normal aCGH samples from the TCGA. Given that CNVs in the healthy genome are relatively sparse (often with frequencies ≤10%), typically a large number of pairwise sequence comparisons will result in SNR changes that reflect dissimilarity (mismatch). Thus, in the original SDMF, SNR changes following matched-filtering between each sequence and all others in the dataset were averaged and compared to a mismatch threshold. For relatively rare CNVs, this averaged SNR reflects the overall mismatch between each sequence in the small subset of the dataset that contains a CNV with the majority of sequences that have no CNV in a particular segment. Although this approach is reasonable in this setting, it makes SDMF indirectly dependent on CNV frequency. As the latter increases, the average of pairwise SNR changes may no longer reflect overall mismatch and may not meet the mismatch threshold. Thus, as CNV frequency increases, the sensitivity of SDMF is likely to decrease. It is more appropriate to eliminate this averaging, use the traditional application of matched filtering where SNR gain due to signal match is the relevant measure, and estimate relevant thresholds for SNR gain. Finally, for SDMF to become a computationally efficient and useful tool for analysis of genomic data, its parameters also need to be optimized, primarily the processing window length and the SNR gain threshold.

To address the limitations of the original method and expand the capability of SDMF to detect CNVs independently of their frequency of occurrence in a dataset, we have developed a modified SDMF method that uses pairwise SNR gains to identify genomic regions potentially containing CNVs in sequence pairs. Therefore, the proposed method can detect both rare CNVs that may occur in as few as one pair of sequences as well as CNVs that may occur in as many as the entire sequence dataset. In addition, we have developed a systematic approach for parameter and relevant SNR threshold selection. Using extensive simulations we have investigated the effects of noise level, processing window length, CNV number, frequency, length and distance between CNVs on SNR changes. These simulations provide important information into the effects of various parameters on the performance of SDMF. Results from these simulations may be used to select an optimal combination of a processing window length and SNR threshold for improved CNV detection. We have applied the improved SDMF method to 429 normal sequences from the TCGA, and have compared detected CNVs to i) those obtained using the original SDMF, ii) CNVs identified using CBS and the Hidden Markov modeling (HMM) approach proposed by [11] and iii) CNVs in the database for genomic variants (DGV) [2]. We have selected these two approaches based on their wide application to array CGH data for CNV detection. Finally, using simulations, we have systematically compared the computational efficiency of SDMF and CBS.

SDMF is a two-step algorithm. The first step involves filtering through sequence decomposition. The empirical mode decomposition (EMD) approach [12] is used to decompose each aCGH sequence *y*(*k*) into its dominant components (modes) in an unsupervised fashion. When linearly superimposed, these modes yield a signal that is an accurate representation of the original signal, i.e., the mean square error (MAE) or reconstruction error between *y*(*k*) and the estimated signal *ŷ*(*k*) is very small. The EMD method fits spline functions through local extrema of the original signal *y*(*k*) to obtain upper and lower envelopes of the signal. At the first iteration, the mean of these envelopes *m*_{1,1}(*k*) is subtracted from *y*(*k*) to obtain the residual *h*_{1,1}(*k*). If this residual’s mean is zero, then *h*_{1,1} represents the first mode. Typically this is rarely the case, and the process of envelope estimation and subtraction from *y*(*k*) is repeated several times until convergence, i.e., until the mean of *h*(·) reaches a pre-defined threshold *ε* 1. The mode *d*_{1}(*k*)= *h*_{1,}* _{q}*(

The second step of SDMF involves comparison of pairs of sequences through matched filtering. The match-filter (MF) maximizes SNR, i.e., the ratio
$20{log}_{10}{\scriptstyle \frac{<{y}_{(.)}{>}_{\mathit{rms}}}{<{\nu}_{(.)}{>}_{\mathit{rms}}}}$, where *ν*(*k*) is Gaussian noise and <·>* _{rms}* denotes root-mean square, by linear convolution of a known signal (the template) with an unknown signal (the measurement). SNR increases in regions of template-measurement waveform match. The resulting increase in SNR following this process is referred to as SNR gain. By definition, the filter

The primary difference of the two versions of SDMF is in how they identify regions potentially containing CNVs based on SNR changes following matched filtering. For a particular genomic segment of interest, once every pair from a set of *N* sequences has been convolved with each other, the ratio of pre- over post-filtering SNR
$\mathrm{\Delta}\mathit{SNR}=20{log}_{10}{\scriptstyle \frac{<{y}_{MF}{>}_{\mathit{rms}}}{<{y}_{\mathit{obs}}{>}_{\mathit{rms}}}}$ is computed. In the original SDMF, for each sequence, the *N* − 1 Δ*SNR*s resulting from its convolution with *N* − 1 sequences are averaged to obtain
$\overline{\mathrm{\Delta}\mathit{SNR}}$. A segment is further examined for potential CNVs if
$\overline{\mathrm{\Delta}\mathit{SNR}}$ is less than a pre-established threshold. The original SDMF further assumes that CNVs are relatively rare and thus the majority of sequences will contain noise and only a few will contain a CNV. Consequently, for a sequence that contains a CNV, its comparison with sequences that do not contain that CNV will result in a
$\overline{\mathrm{\Delta}\mathit{SNR}}$ that will reflect an overall mismatch. The threshold set for SNR associated with mismatch was set based on a simple simulation and is somewhat arbitrary. The limitation of the original SDMF is that the above assumptions may cause SDMF to miss CNVs with high frequencies of occurrence, since in these cases the
$\overline{\mathrm{\Delta}\mathit{SNR}}$ may reflect an overall match of a particular sequence with several others that also contain a common CNV and will not meet the threshold for mismatch. In addition to pathological CNVs, which may have high frequencies of occurrence in particular patient cohorts, even the healthy genome includes a few common CNVs with relatively high frequencies.

To address the limitations of the original SDMF and eliminate the indirect dependence of the detection on CNV frequency, the optimized SDMF does not use SNR averaging following matched filtering, but instead examines SNR changes at the sequence pair level. The proposed modification allows the detection of CNVs of any frequency, from a single pair of sequences to the entire dataset. However, it still requires setting a threshold for SNR gain. Using extensive simulations the present study investigated various potential factors affecting SNR, including the processing window length, number, length and space between CNVs. A practical approach to the selection of the SNR threshold is proposed. Fig. 1 summarizes the optimized SDMF.

SNR gain is proportional to the amplitudes of the signals that are being compared and the length of the window over which the convolution is computed. Once thresholds have been established for detection, the raw (globally denoised) data in pairs of segments that exceed these thresholds are examined to identify specific types of aberrations (gains ( $\ge {log}_{2}{\scriptstyle \frac{3}{2}}$) or losses ( $\le {log}_{2}{\scriptstyle \frac{1}{2}}$)). The optimal processing window length depends on CNV size which is not a priori known. Thus the data may be processed using windows of various length, each resulting in different SNR gains. In typical applications of matched filtering the filtered signal is used in all further analysis. In the absence of precise knowledge of the template waveform and length, longer processing windows may result in higher SNR but also in signals where the original waveform is distorted. Thus, there is always a trade-off between window length and signal resolution. However, in this application matched filtering is solely used to identify segments in pairs of sequences potentially containing CNVs, i.e., the filtered signal is not used in further analysis, and the issue of its waveform shape is not as relevant. Nevertheless, for different filter lengths it is important to examine segments adjacent to the one of interest for potentially spurious SNR changes not associated with signal similarity. This issue is addressed in simulations below.

Simulations were performed using the following signal model:

$$x(k)=\{\begin{array}{ll}\hfill {\sum}_{i=1}^{N}s{(k)}_{Ki}+n(k)\hfill & k\in [{k}_{0,i}:{k}_{0,i}+{K}_{i}]\hfill \\ \hfill n(k)\hfill & \text{otherwise},\hfill \end{array}$$

(1)

where *x*(*k*) is a linear superposition of Gaussian noise *n*(*k*) and the CNV signal *s*(*k*)_{Ki}, i.e., the log_{2} ratio that is non-zero in the interval [*k*_{0,}* _{i}*,

In this simplest case, one CNV (200 points long, in locations 2,001 to 2,200) was completely aligned in all sequences that contained it. The only variable parameter was noise level. An additional simulation with one large CNV (2,000 points long, in locations 22–2,021) was also performed. There are pathologies, such as cancer, where large genomic amplifications or deletions sometimes spanning an entire chromosome arm have been reported. The same window lengths were used as in all other simulations.

In real data a CNV may have somewhat different lengths in individual sequences, even if its starting location is exactly the same. Depending on pairwise differences in CNV length, SNR gains following match-filtering may be substantially different than SNR gains in sequence pairs that contain a CNV with the same length. In this simulation, CNV length was increased by 2 points in each sequence, i.e., in the first sequence, CNV length was 202 points and in the 100th sequence it was 400 points. Thus, different processing windows covered each CNV differently.

Both length and starting location were simultaneously varied. CNV length was progressively increased by 2 points as before and the starting location was progressively shifted to the left by 2 points. Thus, the first sequence started at location 2,001 − 2 = 1,999 and ended at location 2,202 (2,200 + 2 points), and the 100th sequence started at location 1,801 and ended at location 2400 (2,200+200 points).

This simulation addressed the issue of SNR gain in a processing window partially/completely covering CNVs in close proximity to each other. One hundred sequences included a gain in locations 2,001–2,200, a loss in locations 2,401–2,600, a gain in locations 2,801–2,900, and a loss at locations 3,201–3,450. Another 100 sequences included the same number of CNVs and at the same locations, but with gains and losses switched in comparison to the first 100 sequences.

A more realistic data scenario was also simulated, where a processing window length was selected based on the data structure. Two hundred sequences contained CNVs 10, 50, 100, 120, 200, 350 and 620 points long. A simple but imprecise detector was applied to one sequence with fairly low SNR (~ −5 dB) to identify segments of consecutive data (or non-consecutive but no more than 5 data points apart) with amplitudes $\ge {log}_{2}({\scriptstyle \frac{3}{2}})$ or $\le {log}_{2}({\scriptstyle \frac{1}{2}})$. Application of the detector to a sequence will low SNR was selected as a worst case scenario, as it led to imprecise estimates of CNV length and consequently a selection of processing window lengths that in some cases were not well correlated with CNV length. In practice it could be applied to any and all sequences in a dataset prior to the selection of processing windows. Based on this detector, we estimated CNV lengths 9, 27 (not close to any of the CNV lengths), 50, 94, 123, 203, 350 and 617 points. The data were analyzed using these as processing window lengths.

To compare the performance of the original and proposed SDMF methods, CNV frequency was also varied in data generated for simulation 5. A single processing window of moderate size (123 points) was selected, given that the effects of varying the window length were assessed in simulation 5. The goal of this simulation was to compare the two approaches using the same method parameters, including window length. Simulations 1–5 assumed that CNVs were in 200 of 300 sequences (frequency ~67 percent). The original SDMF is expected to perform relatively poorly for CNVs with high frequency, since the averaged SNR change for all comparisons of each sequence with all others may not meet the mismatch threshold. In this simulation an increasingly larger number of randomly selected sequences were replaced by pure noise, to progressively decrease the CNV frequency in the dataset from ~67 to ~33 percent, ~17 and 10 percent.

The proposed and original SDMF were applied to 429 normal sequences from TCGA, also analyzed in [26]. Data from 10 chromosomes were processed. Detected CNVs were also compared to these detected by CBS and by a HMM. Similarly to SDMF which compares local data patterns (waveforms) in pairs of sequences, the HMM approach explicitly models these local patterns and was, therefore, chosen as an additional relevant method for comparison to SDMF. Without an independent validation of identified CNVs, it is not possible to assess the sensitivity and specificity of any detection method when applied to real data, despite agreement with other computational approaches. To compare the relative performance of the two versions of SDMF, CBS and the HMM for real data, a subset of CNVs (gains and losses) in the database for genomic variants was selected, from studies with ≥250 samples. Numbers of gains and losses separately estimated for each chromosome, and CNV locations that our results were compared to can be found in http://www.hsph.harvard.edu/rebecca-betensky/software. Merged-level CNV coordinates are reported, i.e., sample-level CNVs with at least 70% spatial overlap were merged in the DGV. Note that in Stamoulis& Betensky (2011), CNVs were selected from the DGV using more subjective criteria for CNV selection and no coordinate merging. No criteria based on sample demographics were used to select CNVs in the DGV as corresponding demographic information for the TCGA samples were not available.

In all simulations true and false positive rates (TPR and FPR) were calculated for every segment covering a CNV partially or totally. Sensitivity (TPR) was calculated from 200 sequences containing a CNV, as the ratio of detected true positives over all true positives. Specificity (1-FPR) was calculated from 100 pure noise sequences as the ratio of detected true negatives over all true negatives. False negatives were calculated again from 200 sequences containing a CNV, and false positives were calculated from 100 noise sequences. Receiver operating characteristic (ROC) curves and areas under the curves (AUC) were also estimated, as the SNR gain threshold varied from 0 to ~90 dB at 1 dB increments. Fig. 2 shows SNR gain as a function of the original sequence SNR, for different template lengths in simulation 1. Each data point on the y-axis corresponds to an SNR change following matched filtering of one pair of sequences (with a total of $\left(\begin{array}{c}300\\ 2\end{array}\right)$ SNRs) at individual segments containing CNVs. Given that in these simulations the noise level is a priori known, the SNR of the original sequences are known and are plotted in the x-axis.

Case 1: SNR change following matched filtering as a function of the original sequence SNR. Each color corresponds to a window length: 50 (black), 100 (blue) 150 (cyan), 200 (green), 300 (yellow), and 400 (magenta) points. SNR changes for each sequence **...**

SNR changes were clustered: SNR gains >20 dB irrespective of the original signal SNR were estimated for both gains and losses (loss-related SNR gains were slightly higher) when pairs of sequences both containing the CNV were match-filtered. A wide range of SNR changes were estimated (from ~ − 40 dB to ~ + 40 dB) for sequences with very low SNR (< ~ − 5 dB). SNR gains were estimated for pairs of low SNR sequences that both contained the CNV, SNR changes < 0 dB were estimated for pairs of pure noise sequences or pairs of noise and low SNR sequences. Finally, another cluster of SNR changes, including lower than ~ 20 dB gains and <0 dB changes were identified for noisier sequences matched to each other or to pure noise. These clustered SNR changes suggest that a threshold for SNR gain may be estimated to maximize CNV detection.

The ROC curves for simulations 1–4 are shown in Fig. 3. In the case of one CNV with fixed length and location the optimum processing window corresponded to CNV length. However, even for a 400 points-long window covering the entire CNV (200 points) and a 200-point noise segment, the performance of the detector was still good. In the case of the 2,000-point CNV, AUCs were 0.91, 0.94, 0.95, 0.97. 0.94 and 0.79 for window lengths 50, 100, 150, 200, 300 and 400 points, respectively. For the largest window, the segment 2,000–2,400 points only included 22 points of signal (probes 2,000–2,021), and 380 points of pure noise. Consequently the AUC for this window was lower than all others. Overall SDMF performed very well for this CNV too, with comparable performance for window lengths 50–300 points. In simulations where CNVs varied either only in length or also in location, the detector performed poorly for small windows and substantially better for larger windows (300–400 points). Finally, in simulations of multiple CNVs, both smaller (50–100 points) and large windows (400 points) resulted in similar AUCs (Table 1). Note that SNR gain following matched filtering depends on several factors. First, it is proportional to the window length, as well as the amplitudes of the signals of interest. For small processing windows and/or low-amplitude signals, SNR changes following filtering may be small. In addition, noise levels in the signals being filtered also affects SNR changes. In these simulations noise levels varied widely, and 175 of 200 sequences with CNVs had noise levels greater than equal to the CNV amplitudes (gain or loss). Although such high noise levels may be substantially higher than those encountered in real genomic data, we have included them in the simulations to demonstrate the robust performance of SDMF even in high noise levels. Nevertheless, in cases of relatively small window lengths, high noise levels and relatively low or high SNR thresholds there may be an increasing number of false positives (noise segments identified as signals) or increased number of false negatives (signals identified as noise), respectively. In these cases, the ROC curves may fall below the diagonal, highlighting the need of larger processing windows for improved detection. Overall, the simulations suggest that a modestly large processing window may be more appropriate that a small window. However, a small CNV may not be easily detectable if the window is very large and thus there should be a trade-off between the two.

ROC curves for each simulation and each processing window length, from 50 to 400 points (color denotes window length). The scenario for CNV length and location simulated in each case is also schematically shown.

Fig. 4 shows the ROC curves for simulation 5. Both the overall sensitivity and specificity (averaged over all segments containing CNVs) is shown (top left panel), as well as CNV-specific sensitivity and specificity. Corresponding AUCs are summarized in Table 2.

ROC curves for each simulation and each processing window length, from 9 to 617 points (color denotes window length).

AUC as a Function of Window Length, Overall for All CNVs (Column 2) and for Individual CNVs (columns 3–9) Where the Locations of the Seven CNVs were 401–410 (First), 1001–1100 (Second), 1681–1800 (Third), 1901–1950 **...**

These results suggest that 1) small CNVs are difficult to detect even when using a small processing window, particularly when the noise level is high, and 2) medium size windows (here ~50–200 points) may be more appropriate for identifying segments potentially including CNVs and may yield similar results, and 3) in addition to window length, the way the window covers a CNV also affects SNR gain and consequently detection. For example, for some CNVs a window length of 94 points was worse than 50 and 123 points, in part due to differential CNV coverage.

AUCs for simulation 6 comparing the original and modified SDMF are summarized in Table 3. As expected, the proposed SDMF substantially outperformed the original SDMF for 67 percent CNV frequency (overall AUC was 0.9 compared to 0.42), demonstrating the shortcomings of the original method in the case of common CNVs with relatively high frequencies in a set of genomic sequences. It also outperformed the original SDMF for all CNVs >1% of the sequence length (six of seven CNVs). Interestingly, for the smallest CNV (0.25 percent of the sequence length) the AUC for the original SDMF was 0.75 compared to 0.42 AUC for the proposed SDMF, for 67 percent CNV frequency. This is the only case where the original method substantially outperformed the proposed SDMF. As previously noted, short CNVs are difficult to detect, particularly with larger processing windows, and SNR gains may be small and may not meet the thresholds. In contrast, sequence mismatch may be easier to detect in these cases. Therefore, if the goal of the detection is to identify primarily very short CNVs, it may be appropriate to apply both versions of SDMF and compare the findings. The performance of the original SDMF improved substantially with decreasing CNV frequency, though noise level in sequences with the CNV also affected the AUCs. Thus, at 33 percent frequency, the overall AUC was higher than at 17 percent frequency. However, given the random selection of sequences that are replaced by pure noise, these differences are probably due to the distribution of noise levels in the respective datasets, as AUCs for the proposed SDMF also differed by ~12 percent. Overall, the performance of the proposed SDMF varied less with decreasing CNV frequency, and since SNR changes are compared at the signal pair level, performance is independent of CNV frequency. For each CNV frequency the random selection of sequence subsets were repeated 5 times. For each CNV frequency the overall AUC varied ~10–15 percent in datasets with the same CNV frequency but different sets of selected sequences. In contrast, for the original SDMF, AUCs varied substantially in repeated simulations for the same CNV frequency by ~10–25 percent.

In simulations, we can estimate a reasonable threshold for SNR gain from plots like the one shown in Fig. 2. In real data, with no independent knowledge of noise level, we only have knowledge of the raw signal variance and SNR gain following matched filtering. A similar plot may, however, be used to select a reasonable SNR threshold. We selected two SNR gain thresholds, 5 and 10 dB. As suggested by Fig. 2, a higher SNR threshold may be required for larger windows, since lower thresholds for larger windows raise the issue of specificity. Table 4 summarizes the number of gains and losses in each chromosome in the DGV, the number of CNVs estimated by SDMF (original and proposed) using two processing window lengths, 200 and 400 data points long, and the number of discrete regions estimated by CBS and by the HMM. The reported CNVs estimated by SDMF are only those that have any overlap with those in the DGV. This overlap varied substantially among CNVs, and in some cases was <1% of the CNV length. Furthermore, given that SDMF uses a sequential segmentation approach for detection, probes that met the threshold for gain or loss in different segments, but based on their genomic coordinates were part of the same CNV in the DGV, were reported as a single CNV. The threshold for mismatch in the original SDMF was set at −1.5 dB. CBS detects CNVs in individual sequences using a two-sample t-statistic to compare the mean of a data segment to the mean of the sequence outside that segment. A change in the mean is assumed to be statistically significant if the p-value is less than a threshold of 0.01. Note that each genomic segment corresponding to a particular set of breakpoints estimated by CBS or by the HMM overlapped with at least one CNV in the DGV. This is because both CBS and HMM estimated a relatively small number of discrete genomic regions covering most of the chromosome and thus had limited spatial specificity compared to SDMF. To highlight this difference and limitation of CBS and HMM-based approaches, Table 4 reports the total number of regions, defined by CBS-based breakpoints and HMM-based transition points, that overlap with the DGV. For SDMF, each estimated CNV with any overlap with the DGV is included in the list.

CNVs in DGV (Partial), i.e., Subset with Lengths Detectable by the 244 Array (Column 2), CNVs Using the Original SDMF (Old) (Columns 3,6) CNVs Using the Proposed SDMF Approach (New) (Column 4,5,7,8) Both for 200- and 400-pointWindows, and Thresholds of **...**

Overall, the proposed SDMF performed better that the original SDMF, particularly when a 400-point window was used with a lower SNR gain threshold. The highest numbers of CNVs overlapping with the DGV were consistently estimated with the proposed SDMF, a 400-point window and 5 dB SNR gain threshold for 9 of 10 chromosomes. For chromosome 10, a higher number of CNVs overlapping with the DGV was estimated with the smaller 200-point window. As shown in simulations, SNR gain and consequently CNV detection also depends on how the CNVs are covered by the processing window. Thus for chromosome 10, possibly smaller CNVs were better covered by the smaller window. For the 200-point window, the original SDMF performed better than the proposed SDMF with a 10 dB gain threshold, and worse for a 5 dB threshold, again highlighting the fact that a high threshold may substantially lower the sensitivity of the SDMF algorithm. Note that the mismatch threshold for the original SDMF was kept constant at −1.5 dB, and only the threshold for the proposed SDMF was varied. For the 400-point window, the choice of threshold had a smaller effect on CNV detection, i.e., only slightly higher numbers of CNVs were detected with the a lower threshold. A threshold ≥10 dB may have had a more substantial effect on the sensitivity of SDMF with this window. In regard to the other methods, the HMM-based approach estimated a higher number of discrete regions than CBS, and for chromosomes 4, 5, and 7 this number was comparable, though consistently lower, to the number of CNVs estimated with SDMF.

In contrast to other algorithms that process individual sequences, SDMF compares sequence pairs, both in its original and modified forms. Thus, as the dataset size increases the computational time also increases. The length of the processing window also affects computational time. Ten sets of 400 sequences were simulated as before, varying the additive Gaussian noise level as previously described. Each set included an increasing number of CNVs from 1 to 10. All sequences contained the CNVs. The speed of the original and optimized SDMF are expected to be approximately equal, since both involve the same decomposition (first step) and convolution process (involved in matched filtering). Thus, only the optimized SDMF and CBS, because to its widespread use in genomic research, were applied to these data. The window length affects only the performance of SDMF. However, the number of CNVs in each sequence, and thus the non-random structure of the signal could affect the performance of both SDMF and CBS. Central processing unit (CPU) time was used to measure performance. Clock time was also recorded, and the two were approximately the same (±10–20 s from each other). All simulations were ran on a Dell Precision T7500 Four-processor workstation, using a Linux operating system, and the software Matlab. Fig. 5 shows CPU time for SDMF (black curves) and CBS (green curves) as a function of number of CNVs. Each panel corresponds to a different processing window length (20, 50, 100, 200, 400, 500 points).

CPU time (s) for SDMF (black curves) and CBS (green curves) as a function of number of CNVs for varying window lengths 20–500 points (top left to bottom right). All sequences were 4,000 points long.

For a 20-point window, CBS was faster than SDMF for all numbers of CNVs, although the CBS CPU time rapidly increased with increasing number of CNVs. As the window length increased, the SDMF CPU time decreased. For windows ≥100 points, SDMF was faster than CBS for sequences with ≥3–4 CNVs and substantially faster for sequences with ≥8 CNVs. Genomic sequences typically contain a large number of aberrations and thus have significant spatial structure. Thus, simulations with higher number of CNVs are more realistic. For datasets of 400 sequences containing ≥8 CNVs, SDMF was at least three times faster than CBS for windows ≥100 points. When the number of sequences was also varied from 20 to 400, SDMF was up to 100 times faster than CBS. SDMF was faster for windows 200–400 points than for longer windows. Thus, the relationship between window length and performance may not inversely proportional. These results support the selection of a medium-size window for efficient processing.

In this study we aimed to modify and improve the SDMF method and propose an approach for selecting its parameters for robust CNV detection. The original SDMF indirectly depends on CNV frequency as it uses averaged SNR changes due to mismatch as the relevant measure. As shown in simulations, its performance may decrease substantially with increasing CNV frequency. To improve the sensitivity of the method independently of CNV frequency, we have proposed a modification that uses non-averaged SNR gain as the relevant measure of sequence similarity, to identify segments of pairwise match that may contain CNVs. Overall, the proposed SDMF consistently outperformed the original method for all CNV frequencies, and is thus a more flexible approach for robust CNV detection. Our simulations have also shown that it performs well even in sequences contaminated by very high noise levels. The only case where the original SDMF may perform better than the proposed method is for very short CNVs and processing windows, where small SNR gains may be difficult to detect. Finally, when detection of pathological CNVs is of interest, the proposed SDMF is more appropriate for identifying subsets of disease-related CNVs that are common in samples from patients sharing the pathology, since it compares sequences for common genomic similarity (match) rather than relatively rare mismatch in normal samples.

The modified and original SDMFs, as well as CBS and an HMM-based approach were also applied to normal samples from the TCGA. To compare their performance we assessed the overlap of detected CNVs with a subset of normal CNVs reported in DGV. Overall, the proposed SDMF consistently outperformed the original SDMF, CBS and HMM, particularly with a larger processing window and lower SNR thresholds. CBS detected relatively long segments containing CNVs but lacked the spatial specificity of SDMF, which in part depends on the choice of the processing window. The HMM approach detected a higher number of discrete segments than CBS, but consistently lower than SDMF. Based on separate simulations for SDMF parameter selection and the assessment of the relative computational cost of SDMF and CBS, and the results of the real data analysis, a medium-size processing window relative to CNV length may be optimal. Although CNV lengths are not a priori known, an adequate window may be chosen based on approximate estimates of the data structure. In fact, another advantage of SDMF is that its parameters are not hardwired and can be easily changed according to the research question or dataset of interest.

Although developed and optimized for aCGH data, SDMF may be adapted and extended to other high-dimensional genomic data, such as NGS. By design, it directly compares pairs of genomes, e.g., a reference and a target genome sequenced using NGS, and may thus be used to rapidly identify and separate regions that may contain DNA aberrations, which may be further analyzed with smaller processing windows, and regions of no further interest, which may be selectively compressed for efficient processing. Thus, SDMF may become a valuable tool both for efficient preprocessing and CNV detection in very high-dimensional data.

**Funding:** This work was supported by NIH Grant R03 CA121884.

*Conflict of interest:* None declared.

**Software and Data**

The results published here are based on data generaged by The Cancer Genome Atlas established by the NCI and NHGRI [1]. Information from the Database of Genomic Variants [2] has also been used in this study. Software associated with various simulations and the SDMF method can be found at http://www.hsph.harvard.edu/betensky/.

Personal use is permitted, but republication/redistribution requires IEEE permission.

Catherine Stamoulis, Department of Radiology, Harvard Medical School and Boston Children’s Hospital, Boston, MA 02115.

Rebecca A. Betensky, Department of Biostatistics, Harvard School of Public Health, Boston, MA 02115.

1. The results published here are based on data generated by The Cancer Genome Atlas established by the NCI and NHGRI.

2. Information from the Database of Genomic Variants has been used, http://cancergenome.nih.gov/

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