Machine learning approach for pooled DNA sample calibration
- Andrew D Hellicar^{1}Email author,
- Ashfaqur Rahman^{1},
- Daniel V Smith^{1} and
- John M Henshall^{2}
Received: 23 July 2014
Accepted: 23 April 2015
Published: 9 July 2015
Abstract
Background
Despite ongoing reduction in genotyping costs, genomic studies involving large numbers of species with low economic value (such as Black Tiger prawns) remain cost prohibitive. In this scenario DNA pooling is an attractive option to reduce genotyping costs. However, genotyping of pooled samples comprising DNA from many individuals is challenging due to the presence of errors that exceed the allele frequency quantisation size and therefore cannot be simply corrected by clustering techniques. The solution to the calibration problem is a correction to the allele frequency to mitigate errors incurred in the measurement process. We highlight the limitations of the existing calibration solutions such as the fact they impose assumptions on the variation between allele frequencies 0, 0.5, and 1.0, and address a limited set of error types. We propose a novel machine learning method to address the limitations identified.
Results
The approach is tested on SNPs genotyped with the Sequenom iPLEX platform and compared to existing state of the art calibration methods. The new method is capable of reducing the mean square error in allele frequency to half that achievable with existing approaches. Furthermore for the first time we demonstrate the importance of carefully considering the choice of training data when using calibration approaches built from pooled data.
Conclusion
This paper demonstrates that improvements in pooled allele frequency estimates result if the genotyping platform is characterised at allele frequencies other than the homozygous and heterozygous cases. Techniques capable of incorporating such information are described along with aspects of implementation.
Keywords
Background
Recently the Illumina HiSeq X Ten [1] achieved a new low in per genome sequencing cost, continuing the ongoing reduction in cost per genome since 2001 [2]. These cost reductions now make it practical to genotype individuals in large association studies of humans. However, this is not the case for studies involving large populations of low economic value species where contemporary genotyping technology is cost prohibitive. The cost benefits achieved in [1] have not been realised on platforms based on alternative technology, such as Sequenom, and therefore pooling is still required in this scenario. This is evidenced by the ongoing use of DNA pooling in studies on low economic value species, specifically to reduce cost [3,4]. DNA pooling has been shown to provide a cost benefit over individual genotyping [5] and allows access to a broader community to enable genetic association studies.
Pooling techniques date back to 1943 when blood from soldiers was pooled for testing of disease [6] and pooling of DNA was first proposed in 1985 [7]. The field advanced rapidly and in 2002 a broad review of the approach (applied to SNP data) was published [8]. DNA pooling combines DNA from multiple individuals into a single sample which can be genotyped once, as opposed to genotyping each individual. This reduces the cost of genotyping by a factor equal to the number of individuals in the pooled sample. In general pooling strategies are more complex and involve the multiple genotyping of duplicate pools, the effiiency of pooling approaches is given in [8]. The general pooling approach changes the measurement from detecting whether or not a substance is present, to measuring the concentration of the substance. In the case of DNA pooling, the ‘substances’ are the discrete SNP genotypes A A,A B,B B with corresponding A-allele frequencies 1,1/2,0 and the ‘concentration’ is equivalent to the real valued A-allele frequency within the range [0, 1].
The most significant drawback of the pooling approach is the error incurred in the process of measuring the pool’s allele frequency. The impact of this error is illustrated in the context of a bi-allelic quantitative trait linkage study.
Three main factors contribute to the allele frequency variation including: sampling error E _{ s } (due to the limited pool size), sample construction error: E _{ p } (due to non ideal pool constructing resulting from the unequal contributions of individuals to the pool sample) and allele frequency measurement error: E _{ m } (due to chemistry and detection errors in the genotyping process). If the true sub-population allele frequency is p, then these errors result in a measured allele frequency f=p+E _{ s }+E _{ p }+E _{ m }. The variance introduced in f by approximating sub-population with N individuals is the expectation of the square error: \(\overline {{E_{s}^{2}}}=p(1-p)/2N\). Similarly the unequal contributions to a pool from individual samples contribute to a variance component \(\overline {{E_{p}^{2}}}=\tau p(1-p)/2N\) [9] where τ is the standard deviation in the fractions of the pool contributed by the individuals. A thorough analyses of these errors under different sampling conditions is given in [10]. Both these variance contributions can be reduced by increasing the pool size. Measurement error; however, is independent of pool size. Reducing measurement error requires averaging over multiple measurements, which reduces cost effectiveness of the pooling strategy. To resolve this issue, a range of calibration techniques have been proposed for E _{ m } reduction. Three example approaches are k-correction [11], linear interpolation [12] and the polynomial-based probe specific correction (PPC) method [13].
Despite the fact that these methods were developed for different platforms, they all contain a number of similarities which allow them to be applied to data generated by the Sequenom platform. All existing calibration techniques have a mapping which takes as input the raw allele frequency resulting from the platform’s response to each of the two alleles present for a SNP. The Sequenom data is also available in this format. Furthermore the SNP specific corrections are based on the platform’s allele responses to multiple individuals for the SNP being corrected. Sequenom data can also be generated by multiple individuals to provide such a data set. To explain these techniques the following notation is adopted:
Given a SNP requiring calibration, and a set of AA homozygous individuals in the SNP, define \(\overline {AA}=(\overline {H_{A}(AA)},\overline {H_{B}(AA)}\)) where \(\overline {H_{A}(AA)}\) and \(\overline {H_{B}(AA)}\) are the average value for H _{ A } and H _{ B } over the AA homozygous set of individuals. Similarly \(\overline {AB}\) and \(\overline {BB}\) are average values defined for heterozygous AB and homozygous BB sets of individuals respectively. The measured allele frequency f, corresponding to points \(\overline {AA}\), \(\overline {AB}\), and \(\overline {BB}\), are f _{ AA }, f _{ AB }, and f _{ BB } respectively. The calibration techniques all map f _{ AA } and f _{ BB } into A-allele frequencies 1 and 0 respectively with calibration specific approaches between these values to map f _{ AB } into A-allele frequency 0.5. How they achieve this varies between the methods.
k-correction approach can be applied to the Sequenom data without modification.
Minor changes include the fact that the ratio in Equation (2) is used in calculating allele frequency, as opposed to the normalised angle (2/p i)a t a n(H _{ A }/H _{ B }). Dividing by H _{ A } + H _{ B } in Eq. 4 introduces a normalising factor, and enforcing the homozygous values to 0, 1 and heterozygous cluster centre to 0.5 is equivalent to the rotation and shear transformation. However the translation transformation is not implemented as it requires estimating the intercept of the asymptotes of the AA and BB clusters. However the majority of the approach is captured in the expression above.
where \(E = \frac {1}{2}-f^{1}_{\textit {AB}}\), is the error for the heterozygous case \(\left (\,f^{1}_{\textit {AB}} = f^{1}\left (\,f_{\textit {AB}}\right)\right)\).
The expression for k-correction includes both H _{ A } and H _{ B } terms; however, these can be eliminated by solving (2) and (3) for H _{ B }, equating and cancelling H _{ A }. Furthermore after correcting homozygous cases using Eq. 6, the allele frequency for the heterozygous case is \(f^{1}_{\textit {AB}} = K/(1+K)\).
The expression in (11) is more complicated than those in (9) and (10). If distortion in the reciprocal of allele frequency is used (11) becomes a first order distortion correction of the form in (6); however, for consistency between three methods we have expressed all in terms of allele frequency.
- 1.
Impose assumptions on variation between 0, 0.5, 1.0,
- 2.
address a limited set of error types.
To highlight limitation 1 we show that when testing with allele frequencies between 0, 1/2 and 1, the performance of each interpolation method varies significantly between SNPs. We then outline a machine learning technique [14] that samples across the full allele frequency range. The technique can model non-linear distortions to correct the broad range of errors that occur in the chemistry/detection processes across different genotyping platforms. Therefore they resolve the drawbacks of existing approaches. Furthermore the technique substantially reduces the measurement error. After learning the calibration the approach can be used to calibrate pooled samples measured on the same platform without further training. Finally we demonstrate the training requirements for machine learning approaches by training and testing on sets containing individuals, pools, and a combination of both.
Method
Experimental data
Finally amongst the 850 individuals 41 individuals were genotyped twice, after data quality control a total of 1621 duplicate measurements of (H _{ A },H _{ B }) were available. These duplicate samples were used to estimate the underlying variability in the measurement process.
Measurement error calculation
To test our methods we introduce three testing regimes: individuals, pools and combined. The regimes evaluate the performance of the calibration methods by testing with samples that are either all individuals, all pools, or a combination of individual and pool samples. The combined set is constructed such that the error incurred on the combined set contains equal contributions from pools and individuals. Although the presented methods are developed to be applied to data sets containing pools, we include a data set comprising individuals only. The intent is not to provide results indicative of application of the methods, but to demonstrate the performance of the methods at detector values typical of homozygous and heterozygous samples. The individual regime highlights the contrast in performance with the existing methods developed with individual data in addition to allowing error to be decomposed into bias and variance components due to the presence of multiple calibrations at the same allele frequency.
Polynomial calibration
Three existing techniques were applied: linear interpolation, k-correction and 2nd order Lagrange interpolation. In addition we also implemented three variations of Hermite interpolation to explore whether alternative interpolating functions could achieve better corrections on the Sequenom platform. The techniques are equivalent to the existing methods in mapping the homozygous cases as in (6), with a distortion specific term D satisfying conditions in (8). Piecewise Hermite interpolation implements two Hermite polynomials over sub-domains [0,f _{ AB }] and [f _{ AB },1] and enforces a derivative at f=0, f = f _{ AB }, and f = 1. We enforce zero derivative in our implementation. An equal domain version creates a symmetric function either side of f _{ AB }, finally a fixed point variation of the equal domain version enforces the derivative to be unit valued at f _{ AB }. To highlight the differences in calibration polynomials the functions are plotted in Figure 1 for the case of correcting an erroneous heterozygous allele frequency measurement f _{ AB }=0.3.
The MSE in allele frequency was calculated by calibrating under the three regimes described previously: individuals, pools and combined.
Machine learning approaches
The new approach outlined in this paper utilises machine learning techniques to learn functions that correct and estimate allele frequency. Three approaches were implemented including linear regression (LR), multilayer perceptron (MLP), and support vector regression (SVR). WEKA implementations of the LR and MLP algorithms were used [17] and libSVM [18] used for the SVR. Each method learns a function that maps f into a calibrated allele frequency output f ^{′}. The training data set (which includes samples of f and known ground truth allele frequencies f∗) is used to learn the mapping function. The methods find different solutions to the function due to the fact the methods impose differing constraints on the solution and optimise different objectives. LR finds the linear representation that minimises the least squares error over the training set and requires no additional parameters to define the approach. Both MLP and SVR learn non-linear mappings and require a number of parameters to define both the type of function representation, and the optimisation approach.
The MLP [19] is implemented as a cascaded series of matrix-vector multiplications. The ’vector’ input to the first matrix-vector product is the uncalibrated allele frequency value. A non-linear operation is applied to the output of each matrix-vector multiplication, the result is then multiplied by the next matrix in the series. The output of the final matrix-vector product is the calibrated allele frequency value.
Parameters describing machine learning approaches
Multi-layer Perceptron (MLP) | |||
---|---|---|---|
Parameter | Best | Range minimum | Range maximum |
num layers | 2 | 1 | 3 |
nodes per layer | 2 | 1 | 6 |
learning rate | 0.11 | 0.01 | 1.0 |
momentum | 0.15 | 0.01 | 1.0 |
Non-linearity | Sigmoid in all hidden layers | ||
Support Vector Regression (Lib SVM nu-SVR) | |||
nu | 0.092 | 0.01 | 1.0 |
C | 0.027 | 0.01 | 1.0 |
Kernel | Gaussian |
Support vector regression builds a function based on the training data itself. The function is represented as a sum of non-linear basis functions (called kernels) centred at each training sample. Parameters are required to describe the choice of kernel, the cost function and the optimisation approach. A common choice of basis function are Gaussians with a specified standard deviation in the input domain. The SVR cost function has no cost for small errors, this allowable error can be explicitly provided as in the ε-SVR, or implicitly provided via a parameter ν in the nu-SVR which finds a balance between regularisation and ν. Here the nu-SVR [20] is used with the parameters provided in Table 1. Where parameters are not explicitly stated, default parameters provided by WEKA and lib-SVM were used.
Whereas the polynomial calibration used cluster centres for determining the calibration polynomials, machine learning directly use the sample values as training data. The question arises as to what proportion of pooled data should be used versus the individual data. To examine this we introduce three training regimes individuals, combined and pools, which train the models using data from the respective data sources. The intent of the machine learning approaches is to provide samples away from the homozygous and heterozygous sample cases, to improve the calibration in these regions; however, we provide the individuals training set to allow comparison with existing methods which rely on samples from individuals only, and also provide the ability to decompose errors into variance and bias components in the resulting f ^{′}.
An additional requirement on the data sets to ensure valid results for the machine learning approach is there is no intersection between the data used for training the models and the data used for testing the models. To achieve this the data sets are further refined. Specifically we use a cross-validation approach: the original data set containing all the data is partitioned into 10 blocks. One block is removed for testing and the remaining 9 blocks used for training. Consequently we create two pool data sets P _{ train } and P _{ test } which partition P _{ all }, and individual sets I _{ train } and I _{ test } which partition I _{ all }. Similar to the combined testing regime, we create data sets C _{ train } and C _{ test }, resampling from P _{ train } and P _{ test } to ensure equal representation by pooled samples in the combined data sets. The process for generating the data sets is shown in Figure 4.
Sets used for machine learning under different regimes in format: (training sets ; testing sets)
Test regime | |||
---|---|---|---|
Train regime | Individuals | Combined | Pools |
individuals | (I _{ train } ; I _{ test }) | (I _{ train } ; I _{ test } + P _{ all }) | (I _{ all } ; P _{ all }) |
combined | (I _{ train } + P _{ all } ; I _{ test }) | (C _{ train } ; C _{ test }) | (I _{ all } + P _{ train }; P _{ test }) |
pools | (P _{ all } ; I _{ all }) | (P _{ train }; P _{ test } + I _{ all }) | (P _{ train }; P _{ test }) |
Results and discussion
The pairs of duplicate measurements were used to calculate the underlying variation in the measurement process which cannot be removed by calibration. The difference in duplicated measurements d is a random variable with twice variance of the allele frequency measurement. Given m duplicate measurements the variance is \(\sum d^{2} / 2m\). After data cleaning m=1621 duplicate samples remained. The measurement process was found to contribute a variance component of 1.91×10^{−3} to \(\overline {{E_{N}^{2}}}\).
Allele frequency MSE ( 10 ^{ 3 } ) obtained by calibration polynomial methods
Method | Individuals | Combined | Pools | |
---|---|---|---|---|
E ^{2} | \(\overline {{E_{N}^{2}}}\) | |||
None | 8.83 | 3.27 | 12.32 | 15.80 |
k-correction | 4.26 | 3.74 | 8.35 | 12.44 |
Piecewise linear | 4.07 | 3.72 | 8.23 | 12.38 |
2nd order Lagrange | 4.21 | 4.01 | 8.28 | 12.34 |
Piecewise Hermite | 2.68 | 2.40 | 8.73 | 14.77 |
Piecewise Hermite equal deriv. | 7.62 | 7.54 | 14.16 | 20.69 |
Piecewise Hermite equal domain | 3.54 | 3.45 | 10.27 | 16.99 |
Best approach applied per SNP | 2.58 | 2.45 | 7.57 | 11.34 |
Percentage of SNPs where given method obtains best performance
Method | Individuals | Combined | Pools |
---|---|---|---|
None | 0 | 14.6 | 35.4 |
k-correction | 4.2 | 27.1 | 29.2 |
Piecewise linear | 0 | 14.6 | 10.4 |
2nd order Lagrange | 0 | 8.3 | 14.6 |
Piecewise Hermite | 85.4 | 22.9 | 4.2 |
Piecewise Hermite equal deriv. | 0 | 2.1 | 2.1 |
Piecewise Hermite equal domain | 10.4 | 10.4 | 4.2 |
Machine learning allele frequency MSE’s ( 10 ^{ 3 } )
Method | Training set | Individuals (0.6) | Combined (0.7) | Pools (2.2) | |
---|---|---|---|---|---|
E ^{ 2 } | \(\boldsymbol {\overline {{E_{N}^{2}}}}\) | ||||
individuals | 4.56 | 3.12 | 7.58 | 10.58 | |
LR | combined | 6.10 | 2.72 | 6.66 | 7.44 |
pools | 32.93 | 1.15 | 19.03 | 5.28 | |
individuals | 2.58 | 2.01 | 8.90 | 15.10 | |
MLP | combined | 4.96 | 2.42 | 6.34 | 8.00 |
pools | 16.35 | 2.17 | 10.92 | 5.91 | |
individuals | 4.22 | 2.68 | 6.78 | 9.29 | |
SVM | combined | 6.64 | 2.55 | 6.55 | 6.54 |
pools | 10.05 | 2.37 | 8.40 | 7.05 |
The reason for generating testing and training sets including mixtures of individuals and pools is evident in the results. Examination of just one testing set can lead to erroneous conclusions on performance. For example piecewise Hermite polynomials achieved the best results in Table 3 for minimising variance in individuals. However, this is a result of the zero derivatives enforced at 0, 0.5 and 1, which tend to compress the results towards the correct allele frequencies. The disadvantage of this is seen with the larger errors incurred when testing with pools. A similar, overfitting effect, occurs for learning models trained on individuals which result in flattening of the mapping in the AA, AB, BB cluster regions. The non-linear MLP and SVR methods can achieve this flattening, whereas LR cannot. Consequently MLP and SVR trained on individuals achieved poor results when tested on pools in contrast to LR.
The effect of changing the number of pools and individuals was also explored. The linear regression approach was applied to two scenarios, using individuals for training and testing, and using pools for training and testing. The results showed the existing method’s MSE was improved upon if at least 10 individuals and 8 pools were included in the respective training sets. Improvement stopped after 225 individuals were included. The available pools data set was not large enough to see performance stop improving.
In summary, whereas existing calibration approaches are trained using individual samples, machine learning approaches should not, and pooled samples are required. There is an advantage in including calibration pools when building calibration models. However care must be taken to avoid learning near the pool allele frequency values only. Models that achieved the best results (when tested on pools) were those trained only on the calibration pools and were not accurate elsewhere over the full allele frequency range. This is highlighted by the larger errors committed by all methods when trained on pools and tested on individuals. A typical experiment will involve calibration pools (with known ground truth allele frequency) and phenotype pools to be corrected. The spread of the calibration pool allele frequencies is determined by the allele frequency of the population the pool is taken, and the size of the pool. However, for phenotype specific pools being calibrated there is no guarantee a SNPs allele frequency lies within this spread, particularly if there exists a relationship between the SNP and phenotype. Therefore ideally a calibration function should be accurate over the full range of allele frequencies [0,1], or alternatively be only applied within the spread of allele frequencies on which the model was built. One alternative is to use smaller number of samples in constructing calibration pools, to increase spread. Another solution is to include a mixture of pools and individuals in the training of the algorithm such as the combined data set.
Ratio of the best machine learning approach MSE to the best existing technique MSE for each training and testing set combination
Testing set | |||
---|---|---|---|
Training set | Individuals | Combined | Pools |
individuals | 0.63 | 0.82 | 0.75 |
combined | 1.22 | 0.77 | 0.53 |
pools | 2.47 | 1.02 | 0.43 |
Conclusion
This is the first study of a machine learning approach to calibration of pooled SNP samples which has demonstrated the importance of training sample location on performance. The approach was tested on data generated by a Sequenom iPLEX SNP panel providing results for 61 SNPs on Tiger prawn individual and pooled samples. We showed that SNP to SNP variation is significant between the allele frequencies and different calibration polynomials are suitable for different SNPs. We introduced a machine learning technique to model each SNP separately and included data between the discrete allele frequencies of individuals by incorporating calibration pools into the model. The machine learning approach achieves significantly less error, by reducing error by a factor of 2 and improves study test statistic by 72% as a consequence of reduction in allele frequency variance.
An additional advantage of the machine learning technique is the ability to calibration functions on higher dimensional inputs. The use of additional input information can allow errors which previously created variance in f, to become predictable in the additional dimension. In this situation variance causing error is converted to a bias error which can be corrected by calibration with a resulting reduction in variance. Here we have limited access to auxiliary data from the experiment and using allele frequency alone has allowed comparison of the techniques with the same input data.
Declarations
Acknowledgements
We would like to acknowledge Gold Coast Marine Aquaculture for their contribution towards the development of the Black Tiger Prawn SNP assay used in this study, and for the tissue samples used in evaluating the methods. We are grateful to Leanne Dierens and Melony Sellars who undertook sample collection and DNA extractions.
Authors’ Affiliations
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