 Research
 Open Access
Hadamard Kernel SVM with applications for breast cancer outcome predictions
 Hao Jiang^{1},
 WaiKi Ching^{2},
 WaiShun Cheung^{2},
 Wenpin Hou^{2} and
 Hong Yin^{1}Email author
https://doi.org/10.1186/s1291801705141
© The Author(s) 2017
 Published: 21 December 2017
Abstract
Background
Breast cancer is one of the leading causes of deaths for women. It is of great necessity to develop effective methods for breast cancer detection and diagnosis. Recent studies have focused on genebased signatures for outcome predictions. Kernel SVM for its discriminative power in dealing with small sample pattern recognition problems has attracted a lot attention. But how to select or construct an appropriate kernel for a specified problem still needs further investigation.
Results
Here we propose a novel kernel (Hadamard Kernel) in conjunction with Support Vector Machines (SVMs) to address the problem of breast cancer outcome prediction using gene expression data. Hadamard Kernel outperform the classical kernels and correlation kernel in terms of Area under the ROC Curve (AUC) values where a number of realworld data sets are adopted to test the performance of different methods.
Conclusions
Hadamard Kernel SVM is effective for breast cancer predictions, either in terms of prognosis or diagnosis. It may benefit patients by guiding therapeutic options. Apart from that, it would be a valuable addition to the current SVM kernel families. We hope it will contribute to the wider biology and related communities.
Keywords
 SVM
 Hadamard Kernel
 Breast Cancer
Background
It is known that 13% of deaths all over the world are caused by cancer [1]. For women, breast cancer is a leading cause of deaths worldwide. In the U.S. alone, it is estimated that 246,660 new patients will be diagnosed with breast cancer, and 40,450 deaths associated with malignancy are estimated [2]. Early detection and identification of breast cancer is necessary for reducing the sideeffects of the disease. On the other hand, cancer prognosis can assist in designing treatment protocol which is also of great importance. Cancer prognosis can be interpreted as estimating survival probability within a certain period of time. A 10year prognosis of 60% represents the probability of surviving 10 years after surgery or diagnosis is 60%. Here we formulate the prognosis problem as a classification one where label information can be retrieved from the survival information beyond the prognosis period. For example, patients who died before the considered prognosis period are labeled negative and vice versa.
In cancer research, cDNA Microarrays and high density oligonucleotide chips are increasingly used and in the meantime they raise numerous excellent and challenging research problems in fields. By monitoring expression levels in cells for tens of thousands of genes simultaneously, microarray experiments may lead to a better understanding of the molecular variations among tumors and hence to a more informative classification [3]. Over the last few years, substantial efforts [4–7] have been made on gene expression profile based classifiers for predicting patient outcomes in breast cancer.
Maglogiannis et al. [8] proposed Support Vector Machines (SVMs) based classifier for the prognosis and diagnosis of breast cancer disease. It was compared with Bayesian classifiers and Artificial Neural Networks (ANNs). Delen et al. [9] compared three algorithms for predicting breast cancer survivability where they used SEER data for evaluation. Endo et al. [10] proposed optimal model for 5year prognosis of breast cancer. They compared seven algorithms (Logistic Regression model, ANN, Naive Bayes, Bayes Net, Decision Trees with Naive Bayes, Decision Trees (ID3) and Decision Trees (J48)) on SEER data and results show that decision tree J48 showed the highest sensitivity, ANN had the highest specificity. We note that the data used for model comparisons in [9, 10] is very large in samples (over 30,000) but relatively small in attributes. When the data sets involved small number of samples, SVM based algorithms can usually outperform other considered algorithms. Vikas et al. [11] compared Naive Bayes, SVMRBF kernel, RBF neural networks, Decision Trees (J48) and Classification And Regression Tree (CART) to find the best classifier for the breast cancer data sets. Experimental on 286 samples show that SVMRBF kernel is more accurate. Aruna et al. [12] compared SVM, Decision Tree, and RBF Neural Networks in prediction of Wisconsin Breast Cancer Dataset (there are 699 samples). Results show that SVMRBF kernel is the best among the considered methods. Asri et al. [13] compared SVM, Decision Tree (C4.5), Naive Bayes, KNearest Neighbors (KNN) on the Wisconsin Breast Cancer Datasets to assess the efficiency and effectiveness of algorithms. Experimental results show that SVM yields the highest accuracy.
In the current perspective, SVM demonstrates as a benchmark for various disciplines in particular for dealing with small sample problems. The effectiveness of SVMs depends on the choice of kernels. In [14], we proposed a novel kernel based on correlation matrix for cancer diagnosis purpose. Experiments on 5 realworld cancer data sets with gene expression profiles showed that correlation based kernel outperformed other classical kernels.
In this paper, we propose a parsimonious kernel named Hadamard Kernel for breast cancer outcome predictions. The remainder of this paper is structured as follows. In “Method” section, we propose the parsimonious positive semidefinite kernel. Theoretical proof on the positive semidefinite property of the kernel is provided. In “Results” section, publicly available data sets are utilized to check the performance of the proposed method. Finally, concluding remarks are given in “Conclusions” section.
Method

Preliminaries
The basic SVM considers binary classification problem through building an appropriate model representing data points, mapping them so as to best separate different categories. In a formal setting, if we assume a data set of n data instances with corresponding class annotations:where x _{ i }∈R ^{ p },y _{ i }∈{−1,1}. SVM constructs a hyperplane to ensure good separation having largest distance from it to the nearest data points in each class category [15]. The optimization problem can be formulated as follows:$$\left\{\left(\mathbf{x}_{1},y_{1}\right),\cdots,\left(\mathbf{x}_{n},y_{n}\right)\right\} $$$$ \left\{ \begin{array}{l} \text{Minimize} \ \frac{1}{2}{\\mathbf{w}\}^{2} \\ \text{subject to} \ y_{i}\left(\mathbf{w} \cdot \mathbf{x}_{i} b\right)\geq 1 \\ \text{for any} \ i \in \{1,2,\ldots,n\} \end{array} \right. $$(1)The dual form of the primal optimization problem is given by:$$ \left\{ \begin{array}{l} \text{Maximize} \sum_{i=1}^{n}\alpha_{i} \frac{1}{2}{\alpha}^{T}\mathbf{H} {\alpha} \\ \text{subject to} \ \alpha_{i} \geq 0 \\ \text{for any} \ i \in \{1,2,\ldots,n\} \\ \sum_{i=1}^{n}\alpha_{i} y_{i}=0 \end{array} \right. $$(2)where α=[α _{1},α _{2},…,α _{ n }],$$\mathbf{H}=\left(\begin{array}{cccc} y_{1}^{2}\mathbf{x}_{1}^{T}\mathbf{x}_{1} & y_{1}y_{2}\mathbf{x}_{1}^{T}\mathbf{x}_{2} & \ldots & y_{1}y_{n}\mathbf{x}_{1}^{T}\mathbf{x}_{n} \\ y_{2}y_{1}\mathbf{x}_{2}^{T}\mathbf{x}_{1} & y_{2}y_{3}\mathbf{x}_{2}^{T}\mathbf{x}_{3} & \ldots & y_{2}y_{n}\mathbf{x}_{2}^{T}\mathbf{x}_{n} \\ \vdots & \vdots & \ddots & \vdots \\ y_{n}y_{1}\mathbf{x}_{n}^{T}\mathbf{x}_{1} & \ldots & \ldots & y_{n}^{2}\mathbf{x}_{n}^{T}\mathbf{x}_{n} \\ \end{array} \right) $$When the data sets are nonlinearly separable, one can construct a nonlinear mapping for input vectors into feature space of higher dimensionality [16]. Different from previous setting based on inner product of input vectors, kernel matrix is constructed in terms of similarity measure through pairwise comparisons. Given n data instances X={x _{1},x _{2},…,x _{ n }}, kernel matrix K is a n×n matrix which is symmetric, i.e.,for any x,x ^{′}∈X.$$K(\mathbf{x},\mathbf{x}') = K(\mathbf{x}',\mathbf{x}) $$There are a number of popular kernels, the most straightforward one is:
Linear Kernel.which is an inner product of x and x ^{′} in R ^{ p }.$$K(\mathbf{x},\mathbf{x}') = \mathbf{x}^{T}\mathbf{x}', $$Another popularly used kernel matrix is polynomial kernel that is expressed as$$K(\mathbf{x},\mathbf{x}') = \left(\mathbf{x}^{T}\mathbf{x}'+1\right)^{d}, $$Gaussian Radial Basis Function (RBF) kernel is defined aswhere d is parameter. If the distance between x and x ^{′} is small, the kernel value would be large; on the contrary, if x is far away from x ^{′} in terms of Euclidean distance, the kernel value would be small. Hence this kernel provides a similarity measure between data points.$$K(\mathbf{x},\mathbf{x}')=\text{exp}\left({d\\mathbf{x}\mathbf{x}'\^{2}}\right) $$ 
Hadamard Kernel
Kernel trick is useful in the sense that there is no need to calculate ϕ(x) explicitly as long as constructing appropriate kernel matrix. The Positive SemiDefinite (PSD) property [17] of a kernel matrix is required to ensure the existence of a Reproducing Kernel Hilbert Space (RKHS) where a convex optimization formulation can be deduced to yield an optimal solution.
We propose Hadamard Kernel in this way:Here α≠0 is a flexible parameter within the kernel matrix. For some k, if x _{ ik }=0, then \(\frac {x_{ik}^{\alpha } x_{jk}^{\alpha }}{(x_{ik}^{\alpha }+x_{jk}^{\alpha })}\) is defined as 0.$$K_{\alpha}\left(\mathbf{x}_{i},\mathbf{x}_{j}\right)=\sum_{k=1}^{p}\frac{x_{ik}^{\alpha} x_{jk}^{\alpha}}{2\left(x_{ik}^{\alpha}+x_{jk}^{\alpha}\right)}, i,j = 1,2,\ldots, n. $$
Theorem
Kernel K _{ α } is positive semidefinite for all data matrix X.
The proposed Hadamard Kernel with varying parameter α constitute to a broad range of kernel families which can fit all kinds of data matrix if the theorem holds.
This kernel is not generally positive semidefinite. Let’s consider the following example.
Example
∘
However, in our particular case, where all the gene expression values are positive valued, the kernel here is positive semidefinite. We give the proof in the subsequent statement.
Theorem
Kernel K _{0} is positive semidefinite when that data matrix X is positive valued.
Proof
For a positive matrix A=(a _{ ij }), we define the Hadamard inverse of A by \( A^{\circ (1)}=\left (\frac 1{a_{ij}}\right). \) First proved by Bapat [18] and reformulated by Reams [19], we have the following proposition. □
Proposition
If A is a positive symmetric matrix with only one positive eigenvalue, then A ^{∘(−1)} is positive semidefinite.
Note that w _{ i } e+(w _{ i } e)^{ T } is a positive symmetric matrix of rank 2 and it is not positive semidefinite (the determinant of any principal 2×2 submatrix is negative), hence it has exactly one positive eigenvalue. Therefore by the result of Reams, \(\phantom {\dot {i}\!}V_{\mathbf {x}_{i}e}\) is positive semidefinite.
We can generalize the result to any nonnegative matrix X as well.
Theorem
Kernel K _{0} is positive semidefinite when data matrix X is nonnegative.
Proof
Then \(\phantom {\dot {i}\!}U_{\mathbf {X}}=U_{\mathbf {w}_{1}e}+\cdots U_{\mathbf {w}_{p}e}\). To show that U _{ X } is positive semidefinite, we only need to show that U _{ w e } is positive semidefinite for any nonnegative column vector w.
Suppose w has zero entries. Without loss of generality, write w=(y , 0)^{ T } where y>0, then \(U_{\mathbf {w}e}=\begin {pmatrix}U_{\mathbf {y}e}&0\\0&0\end {pmatrix}\) which is positive semidefinite if and only if U _{ y e } is positive semidefinite. Hence it suffices to show that U _{ y e } is positive semidefinite for any positive column vector y.
Take y ^{′} to be the Hadamard inverse of y, then U _{ y e }=(y ^{′} e+(y ^{′} e)^{ T })^{∘(−1)}. Note that y ^{′} e+(y ^{′} e)^{ T } is a positive symmetric matrix of rank 2 and it is not positive semidefinite (the determinant of any principal 2×2 submatrix is negative), hence it has exactly one positive eigenvalue. Therefore by the result of Reams [19], U _{ w e } is positive semidefinite. □
We now proceed to prove the first theorem.
Proof

Models for comparison
In this paper, we consider breast cancer outcome predictions based on high dimensional gene expression profiles. Hence the number of samples is relatively small. In the literature it is shown that SVM is statistically better than other machine learning algorithms. We therefore confine our research in the framework of SVMs and exclude other algorithms from our scope of research. Other kernels for a comparison are listed below.

SVM Linear Kernel$$K(\mathbf{x},\mathbf{x}') = \mathbf{x}^{T}\mathbf{x}', $$

SVM Quadratic Kernel$$K(\mathbf{x},\mathbf{x}') = \left(\mathbf{x}^{T}\mathbf{x}'+1\right)^{2}, $$

SVM RBF Kernel$$K(\mathbf{x},\mathbf{x}')=\text{exp}\left({\frac{\\mathbf{x}\mathbf{x}'\^{2}}{\sigma^{2}}}\right) $$

SVM Correlation Kernel
This kernel construction can be decomposed into three steps.
1. Based on the correlation matrix, we first construct a preliminary kernel.2. We do eigenvalue decomposition for the matrix K _{ CB } where V is the matrix composed of eigenvectors, P is the diagonal matrix where diagonal entries are eigenvalues.$$ K_{CB} = 1  e^{{\text{corr}}(\mathbf{X})} $$3. Denoising strategy. If we denote$$K_{CB}=V^{T}PV $$The denoising strategy is to transform the diagonal matrix P to another diagonal matrix \(\tilde {P}\),$$ P = \left(\begin{array}{cccc} p_{1} & 0 & \cdots & 0 \\ 0 & p_{2} & \cdots & 0 \\ \vdots& \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & p_{n} \\ \end{array} \right). $$where$$ \tilde{P} = \left(\begin{array}{cccc} \tilde{p}_{1} & 0 & \cdots & 0 \\ 0 & \tilde{p}_{2} & \cdots & 0 \\ \vdots& \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \tilde{p}_{n} \\ \end{array} \right) $$Finally, the kernel matrix becomes$$\tilde{p}_{i} = \left\{ \begin{array}{ll} 0, & \text{\(p_{i} < 0\);} \\ p_{i}, & \text{\(p_{i} \geq 0\).} \end{array} \right., \ i = 1, 2, \ldots, n. $$$$ K_{DCB}=V^{T}\tilde{P}V. $$
Results
Materials
We obtained a number of realworld data sets from National Center for Biotechnology Information [20]. The first data set is derived from a NmethylNnitrosoureainduced breast cancer model. It has 35 samples in total, of which 11 are normal. The number of attributes used to describe a sample is 15923. Expression profiles were obtained through Affymetrix Rat Expression 230A Array. The annotation ID for this data set is GSE1872.
Estrogen ReceptorPositive (ER+) and ER breast cancers tend to show different patterns of metastasis. In this data set where the access number is GSE32394, glycan structure analyse by Custom Affymetrix Glyco v4 GeneChip was conducted to compare the two types of breast cancer. There are 19 samples in total, of which 9 are ER+, the number of attributes is 1259.
The third data set is used to differentiate noninvasive breast cancer and invasive breast cancer, the access number is GSE59246. mRNA, miRNA and DNA copy number profiles are generated to measure the expression of different samples. Arrays consist of 3 normal controls, 46 ductal carcinoma in situ lesions, and 56 small invasive breast cancers. We discard the 3 normal controls, so we have 102 samples in total. In this data set, the number of attributes is 62976.
Studies show that circulating miRNAs have the potential to become biomarkers. This data set involves 78 samples in total, 1205 circulating miRNAs for measurements. 26 of the 78 samples are negative. Identification number for this data set in NCBI is GSE59993.
One more data set is related to breast cancer prognosis, GSE25055 is the identification number. A total number of 310 breast cancer patients is involved. The number of attributes is 22283. This study is conducted with Affymetrix Human Genome U133A Array. It is a neoadjuvant study of HER2negative breast cancer cases treated with taxaneanthracycline chemotherapy preoperatively and endocrine therapy if ERpositive. Response was assessed at the end of neoadjuvant treatment. Using 5 years as a cutoff, we conduct the outcome prediction.
The last data set contains 60 patients with ERpositive primary breast cancer and treated with tamoxifen monotherapy for 5 years [21], the identification number in NCBI is GSE1379. This study was conducted using expression profiling by array, with the number of attributes 22575. We build models to predict the 5year recurrence outcome for the considered patients. There were 28 patients who showed recurrence symptoms.
Performance evaluation
5fold cross validation
Cross validation is a standard way to evaluate the supervised learning model. The kfold cross validation is performed as follows: first of all, the training data set \(\mathcal {M}\) is randomly divided into k subsets \(\mathcal {M}_{1},\cdots,\mathcal {M}_{k}\) of approximately equal size. The prediction model is trained on k−1 subsets and the remaining subset is treated as the test set. Repeating this process k times such that each subset is tested once, all the prediction results are recorded for the computation of prediction accuracy. In our case, we conduct 5fold cross validation for model evaluations.
Area under the receiver operating characteristic (ROC) curve
Definitions for True/False Positive/Negatives
Result  

True  P ^{′}  N ^{′} 
P  True positive (tp)  False negative (fn) 
N  False positive (fp)  True negative (tn) 
The area under the ROC curve (AUC) [22, 23] is a widely adopted statistics for assessing the discriminatory capacity of models. It can be interpreted as a measure of aggregated classification performance, and also the tradeoff between specificity and sensitivity [24].
Experimental results
In this section, we will show the performance of the Hadamard Kernel in conjunction with SVM and the other 4 kernels for breast cancer outcome predictions as tested on the five data sets. We employed the AUC measured by 5fold crossvalidation run 10 times to evaluate the performance. All the experiments are conducted using Matlab R2012 under Window 7 Operations System.
Averaged AUC values for determining optimal σ in RBF kernel
σ  

Datasets  σ=0.01  σ=0.1  σ=1  σ=10  σ=100  σ=1000 
GSE1872  0.2379 ± 0.0538  0.2379 ± 0.0538  0.2379 ± 0.0538  0.2379 ± 0.0538  0.2379 ± 0.0538  0.2379 ± 0.0538 
GSE32394  0.1811 ± 0.0707  0.1811 ± 0.0707  0.2044 ± 0.0845  0.6767 ± 0.1125  0.9456 ± 0.0133  0.9456 ± 0.0122 
GSE59246  0.4408 ± 0.0446  0.4408 ± 0.0446  0.4408 ± 0.0446  0.4408 ± 0.0446  0.8424 ± 0.0379  0.8658 ± 0.0110 
GSE59993  0.3542 ± 0.0283  0.3542 ± 0.0283  0.4305 ± 0.0355  0.8392 ± 0.0235  0.6937 ± 0.0340  0.6940 ± 0.0342 
GSE25055  0.3651 ± 0.0182  0.3651 ± 0.0182  0.3651 ± 0.0182  0.3651 ± 0.0182  0.8092 ± 0.0156  0.7259 ± 0.0127 
GSE1379  0.3952 ± 0.0478  0.3952 ± 0.0478  0.3982 ± 0.0468  0.3970 ± 0.0468  0.6712 ± 0.0294  0.6276 ± 0.0374 
For hadamard kernel, we would like to see the performance of Hadamard Kernel in relation with parameter α. Figures S1 to S6 (attached in Additional file 1) record the performance of Hadamard Kernel in relation with parameter α from (0,5) with step size 0.1. Optimal α in Hadamard Kernel varies in different data sets. For example, one can see a steady decrement in performance when α>1.3 in GSE1872 and when α>2.8 in GSE59246. There is no obvious pattern detected in Additional file 1: Figure S2 in GSE32394, the performance is unstable with respect to α. But we can see a tendency of decrement in an overall manner. For GSE59993, the performance is firstly increasing, achieving the best for α=0.5. The performance is then decreasing steadily. In GSE25055, the performance of hadamard kernel stays in a stable range when α<2.8, it then decreases drastically. For GSE1379, the performance of hadamard kernel gradually increases when α>2. It can be seen that different datasets may fit for different best α in Hadamard Kerne, the optimal α determination becomes an interesting problem.
Figures S7 to S12 (attached in Additional file 1) depict the AUC values of the 5 considered methods in each 5fold cross validations. Dark blue refers to Hadamard Kernel, Linear Kernel is marked in blue, green represents Quadratic Kernel, and orange stands for RBF Kernel, brown stands for Correlation Kernel. In the xaxis, 1 represents the first 5fold crossvalidation. The corresponding values in y axis are the AUC values for the considered 5 methods. For example, in Additional file 1: Figure S7 for GSE1872, the best performance is shown in Hadamard Kernel and Correlation Kernel in the first round, achieving 100% in accuracy. The performances of Linear Kernel, RBF Kernel and Quadratic Kernel are not satisfactory. RBF Kernel shows the worst performance, the AUC values are below 30%. Similar patterns can be detected in the remaining 9 round 5fold crossvalidations. In summary, Hadamard Kernel and Correlation Kernel show the best performance regarding the 10 runs 5fold crossvalidations.
Additional file 1: Figure S8 shows the performance of different models for data set GSE32394. The best performance is shown in Hadamard Kernel, it is slightly better than Linear Kernel. RBF Kernel and Correlation Kernel show comparable performance, and the worst performance is shown in the Quadratic Kernel.
Additional file 1: Figure S9 depicts the result for GSE59246 breast cancer outcome prediction. Hadamard Kernel still demonstrates the best performance, the second best performance is shown in Linear Kernel. Overall, RBF Kernel is better than Correlation Kernel. They rank the third the fourth place this time. Quadratic Kernel can only get 50% in AUC values on average.
In GSE59993, Hadamard Kernel is better than the other 4 kernels as shown in Additional file 1: Figure S10. RBF Kernel shows the second best in this context. Linear Kernel ranks the third place and Quadratic Kernel shows the worst performance.
Additional file 1: Figure S11 shows the result for GSE25055 breast cancer outcome prediction. GSE25055 is a data set related to breast cancer prognosis. We formulate the problem into a classification one by labeling patients who survive within 5 years after diagnosis as positive classes. Hadamard Kernel and Linear Kernel reach the top places, yielding around 84% on average in AUC values. The performance of RBF kernel is also acceptable, achieving around 81% in Averaged AUC Values.
Additional file 1: Figure S12 reports the result for GSE1379, a data set related to ERPositive breast cancer recurrence status prediction. It can be clearly shown that hadamard kernel shows the best performance, RBF kernel ranks the second best, and Quadratic kernel ranks the worst.
Averaged AUC values for comparison of different methods
Methods  

Datasets  Linear kernel  Quadratic kernel  RBF kernel  Hadamard kernel  Correlation kernel 
GSE1872  0.3788 ± 0.1019  0.3686 ± 0.1136  0.2117 ± 0.0584  1.000 ± 0.000  0.9989 ± 0.0018 
GSE32394  0.9456 ± 0.0312  0.5544 ± 0.1248  0.9344 ± 0.0254  0.9589 ± 0.0166  0.9233 ± 0.0294 
GSE59246  0.8977 ± 0.0172  0.5386 ± 0.0579  0.8431 ± 0.0379  0.9022 ± 0.0145  0.8562 ± 0.0113 
GSE59993  0.8283 ± 0.0226  0.5935 ± 0.0694  0.8347 ± 0.0182  0.8855 ± 0.0088  0.7869 ± 0.0144 
GSE25055  0.8575 ± 0.0182  0.4743 ± 0.0393  0.8196 ± 0.0203  0.8653 ± 0.0171  0.7654 ± 0.0152 
GSE1379  0.6205 ± 0.0481  0.5237 ± 0.0701  0.6743 ± 0.0427  0.7300 ± 0.0375  0.6419 ± 0.0453 
To sum up, Hadamard Kernel is effective and robust in predicting breast cancer outcomes. There is no dominant algorithm for the other 4 considered kernels. Quadratic Kernel always shows the worst performance, implying that Quadratic Kernel may not be a good choice in breast cancer outcome predictions.
Discussions
Comparison of Hadamard kernel on raw data and different methods on normalized data
Methods  

Datasets  Linear  Quadratic  RBF  Correlation  Hadamard 
kernel  kernel  kernel  kernel  kernel  
GSE1872  1  1  0.2367  0.9962  1 
GSE32394  0.9556  0.9556  0.8500  0.9444  0.9833 
GSE59246  0.8546  0.8546  0.8061  0.8626  0.8849 
GSE59993  0.8521  0.8476  0.7977  0.8277  0.8913 
GSE25055  0.8619  0.8615  0.7914  0.7715  0.8590 
GSE1379  0.7009  0.5017  0.7411  0.6797  0.7623 
Comparison of Hadamard kernel on raw data and different methods on normalized data(RNA)
Methods  

Datasets  Linear  Quadratic  RBF  Correlation  Hadamard 
kernel  kernel  kernel  kernel  kernel  
GSE87517  0.6022  0.4562  0.5459  0.7189  0.9524 
GSE47462  0.7422  0.5322  0.4029  0.7506  0.8949 
GSE48213  0.9990  0.9982  0.3375  0.9993  0.9996 
One of the test data sets is also obtained from NCBI GEO database, the accession number is GSE47462. Raw counts lncRNAs are used to measure the expression levels. We have 72 samples in total, of which 24 are normal, 25 early neoplasia, 9 carcinoma in situ, and 14 invasive cancer. The number of attributes is 2173. We focus on differentiating normal samples from breast tumor samples. The best σ in RBF kernel and best α in Hadamard kernel are shown to be 1000 and 0.5 respectively where details are attached in Additional file 2 (Table S2, Figure S14). We further compare on Hadamard kernel with other kernel methods through 10 runs 5fold crossvalidations. Averaged AUC values are calculated and the results are reported in Table 5. It can be shown that Hadamard kernel is robust and can demonstrate satisfactory performance compared to other kernels even with data normalization. The averaged AUC value in Hadamard kernel is 0.8949 while in linear kernel 0.7422. The performance in RBF kernel is not satisfactory, achieving only 0.4029 in averaged AUC value.
The third data set is under accession number GSE48213. 56 breast cancer cell lines were profiled to identify patterns of gene expression associated with subtype and response to therapeutic compounds using RNAseq technology. There are 4 unknown cell lines, with 27 samples related to Luminal, 14 samples related to Basal like breast cancer, 5 normal samples and 6 samples of Claudinlow subtype. Subtype Luminal constitutes the majority of all the considered subtypes, hence we try to differentiate Luminal from others by removing the 4 unknown samples. Hadamard kernel on raw data can yield 0.9996 in averaged AUC value. The performance in other kernels after data normalization is also comparable except in RBF kernel.
In Additional file 2: Table S4, we also record the performance of the 4 compared kernels on considered RNAseq data sets without data normalization.
In a word, we can see that Hadamard kernel is robust for dealing with expression data in general.
Conclusions
In this paper, we proposed Hadamard Kernel for breast cancer outcome predictions. It is a valid and effective kernel for dealing with high dimensional gene expression data when they are positive valued. In particular, we have given theoretical verification on the positive semidefiniteness for all kinds of data. Through comparison with classical kernels in SVM and correlation kernel that is good at cancer predictions, we show the superiority of Hadamard Kernel. The hadamard kernel is flexible in varying the parameter α, the determination of optimal α can be devoted to our future work. We hope Hadamard kernel as a novel class of kernels can enrich kernel communities in SVM and contribute to the wider biological problems.
Declarations
Acknowledgements
Authors would like to thank the referees and the editors for their helpful comments and suggestions.
Funding
This research is supported in part by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China, National Natural Science Foundation of China Grant Nos. 11626229, 10971075, 61472428, 11671158 and S201201009985 and Research Grants Council of Hong Kong under Grant No. 15210815. The publication costs are funded by National Natural Science Foundation of China Grant No.61472428.
Availability of data and materials
All the datasets are publicly accessible through National Center for Biotechnology Information Gene Expression Omnibus, where the accession number are GSE1872, GSE32394, GSE59246, GSE59993, GSE25055, GSE1379, GSE87517, GSE47462, and GSE48213.
About this supplement
This article has been published as part of BMC Systems Biology Volume 11 Supplement 7, 2017: 16th International Conference on Bioinformatics (InCoB 2017): Systems Biology. The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume11supplement6.
Authors’ contributions
JH designed the research. JH, WKC and CWS proposed the methods and did theoretical analysis. HWP, YH collected the data. JH, HWP and YH conducted the experiments and analyze the results. JH, WKC, CWS, HWP and YH wrote the manuscript. All authors have read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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