- Research
- Open Access
Secure Multi-pArty Computation Grid LOgistic REgression (SMAC-GLORE)
- Haoyi Shi^{1, 2},
- Chao Jiang^{1, 3},
- Wenrui Dai^{1},
- Xiaoqian Jiang^{1},
- Yuzhe Tang^{2},
- Lucila Ohno-Machado^{1} and
- Shuang Wang^{1}Email author
https://doi.org/10.1186/s12911-016-0316-1
© The Author(s). 2016
- Published: 25 July 2016
Abstract
Background
In biomedical research, data sharing and information exchange are very important for improving quality of care, accelerating discovery, and promoting the meaningful secondary use of clinical data. A big concern in biomedical data sharing is the protection of patient privacy because inappropriate information leakage can put patient privacy at risk.
Methods
In this study, we deployed a grid logistic regression framework based on Secure Multi-party Computation (SMAC-GLORE). Unlike our previous work in GLORE, SMAC-GLORE protects not only patient-level data, but also all the intermediary information exchanged during the model-learning phase.
Results
The experimental results demonstrate the feasibility of secure distributed logistic regression across multiple institutions without sharing patient-level data.
Conclusions
In this study, we developed a circuit-based SMAC-GLORE framework. The proposed framework provides a practical solution for secure distributed logistic regression model learning.
Keywords
- Matrix Multiplication
- Matrix Inversion
- Ordinary Logistic Regression
- Oblivious Transfer
- Garble Circuit
Background
Biomedical research can benefit from data sharing from distributed sources. For example, comparative effectiveness research requires data comparison among different data sources to determine which existing health care interventions work best for certain patients. This requires a large amount of data to be harmonized. Big biomedical data networks, such as the patient-centered SCAlable National Network for Effectiveness Research (pSCANNER) clinical data research network (CDRN) [1], the Scalable Architecture for Federated Translational Inquiries Network (SAFTINet) [2] and the Electronic Medical Records and Genomics (eMERGE) Network [3] have been established to enable cross-institutional biomedical studies. However, information exchange of biomedical data (e.g., genome sequences, diagnoses, medication information, etc.) can put patient privacy at risk, where the potential risks include, but are not limited to, denial of certain types of insurance [4]. As a result, research participants may lose trust in research institutions, which may have an adverse impact on biomedical research. Privacy risks in biomedical studies have been demonstrated in many recent studies. For example, Vaidya demonstrated the possibility to re-identifying individuals from the public query system of the Healthcare Cost and Utilization Project (HCUP) [5]. Sweeney’s study successfully identified participants of the Personal Genome Project (PGP) [6]. In addition, with some background information, an attacker can even identify sensitive information of a participant using “anonymized” biomedical data [7–9]. The study in [10] demonstrated that aggregated genome statistics (e.g., allele frequencies) can present privacy risks. Therefore, it is imperative to develop privacy-preserving techniques to facilitate biomedical research. For this purpose, many distributed model learning frameworks [11–18] have been proposed for building a global model involving multiple participants, but without sharing sensitive patient-level information.
In this paper, we consider the scenario of horizontally partitioned data, where different institutions possess data from different patients, but with the same variables. Our previous work, Grid binary LOgistic REgression model (GLORE [11]) was developed to allow sharing models without necessarily sharing patient data in a distributed manner. It leveraged the aggregation of non-sensitive decomposable intermediary results to build a shared model. Its Bayesian extension, EXPLORER [19], proposed online learning to update the model with incremental data. It enabled asynchronous communication to alleviate probable service breakdown when coordinating multiple participants. Recently, grid ordinal and multinomial logistic regressions [13] were developed to deal with multi-center modeling with multiple categorical values for response variables. Remarkably, distributed modeling learning can also be employed in Cox regression for survival analysis [20–22].
Existing privacy-preserving solutions to multi-site regression [23, 24] can guarantee the precision of model learning. However, patient information could leak in these solutions due to disclosure of the information matrix and score vectors during iterative model learning [25, 26]. To protect these exchanged data, many secure multi-party computation (SMC) methods [18, 27–33] have been developed for distributed model learning. Unfortunately, existing SMC-based methods would still suffer from inappropriate disclosure under certain conditions due to the secure sum protocol. Therefore, El-Eman et al. [15] proposed the SPARK protocol that utilized different secure blocks to build a secure distributed logistic regression, which aims to offer stronger privacy protection for patient data. Although, homomorphic encryption based systems [34–36] can protect secure outsourcing, they need to assign the same public keys in the case of multi-party computation, which may leak intermediary results during communications among participants.
In this paper, we propose a secret-sharing circuit-based secure multi-party computation framework for grid logistic regression (SMAC-GLORE). Inheriting the distributed model learning framework from GLORE, SMAC-GLORE protects not only patient-level data, but also all the intermediary information exchanged during the model learning phase. Introducing secure multi-party computation to build boolean circuits for private data in learning, the proposed framework prevents participants from interpreting arbitrary intermediary information, such as aggregation of summary statistics, and only releases the final learned model parameters.
Methods
To securely evaluate the logistic function, we introduced secret-sharing circuits-based Secure Multi-party Computation (SMC) into procedure of calculation. SMC provides a method for parties to jointly compute a function over their data while keeping the data private in semi-honest scenarios, where all the participants are those who are honest in running programs and algorithms correctly, but might be curious about the information transferred among entities.
Garbled circuits
In the case in which there is no trusted CRYSP, party A could, for example, serve as the CRYSP and party B could be CSP to avoid potential collusion risk between CSP and CRYSP. Moreover, one can choose more advanced encryption algorithms (other than the streaming cipher used in this example) to achieve a better protected OT protocol. The above example demonstrates how a secure integer comparison function can be achieved using garbled circuit-based SMC. In practice, advanced circuits are required to handle more complicated tasks, such as secure distributed logistic regression, where only the learned model parameters are allowed to be released as circuit outputs.
However, Yao’s garbled circuit is only secure in 2-party semi-honest scenarios, which is not sufficient for practical use. Usually, there are more than 2 parties or participants engaged in the same computing task. We based our model on the GMW project developed by Choi [44], who implemented the classical SMC protocols of Goldreich, Micali, and Wigderson (the GMW protocol) [45]. The GMW protocol uses secret-sharing rather than garbled truth tables to implement the secure computation, which enables the computation among more than 2 parties. In GMW, all the variables are represented as binary numbers and the protocol itself is able to protect against a semi-honest adversary with any number of corrupted parties. All the functions should be interpreted as boolean-circuits and each participant feeds the encrypted private data as input to the circuits. During the process of computing, none of the participants can interpret any temporary values except the final output. However, the GMW project only supports non-negative integers. We established our own encoding format, enabling the support for real number arithmetic, and built libraries for secure matrix operation primitives, which made it possible to solve the practical problems of building a secure distributed logistic regression model.
The proposed framework is based on the well-developed GMW protocol [44] for SMC. We overcame the limitations of GMW protocol and built several secure computation primitives to support SMAC-GLORE. As discussed in the original GMW implementation paper [44], the Naor-Pinkas OT was implemented as the encryption scheme to secure the computation. The mathematical definitions and proofs of security of the Naor-Pinkas OT protocol for GMW have been discussed in [46].
Platform preparation
In our project, all the floating values are represented by binary vectors in 28-bits fixed-point format, in which 11 bits are assigned for the fractional part and the other 17 bits for the integer part. The highest bit of the integer part is reserved as a sign of positive or negative value. The two’s complement [47] method is adopted to represent negative values. Thus, all possible values under the proposed fixed-point format are ranging from − (2^{28} − 1) × 2^{−11} to (2^{27} − 1) × 2^{−11}. The proposed platform extends the integer addition and multiplication to support floating number arithmetic. Here, we describe methods for implementing subtraction and multiplication.
Subtraction
When doing subtractions, we need to iteratively compare bits of minuend and subtrahend. If the bit in minuend is smaller than the corresponding bit in the subtrahend, we may need to borrow a bit leftwards. However, in circuits, it is very difficult and expensive to implement borrowing bits. Therefore, we choose to use two’s complement. To calculate the subtraction, we first calculate the two’s complement of the subtrahend, where we invert or flip all the bits of subtrahend and then add 1 to the least significant bit. After that, we add the calculated value to the minuend.
Multiplication
In multiplication, the result may, at most, double the number of required bits to represent the product. For example, the product of two n-bits numbers should have at most 2n bits. However, the GMW project requires the two factors involved in a multiplication to have the same number of bits, and allocates the same number of bits for the product. So, if we still use 28 bits for multiplication, we will suffer from significant precision loss or computing error. To solve this problem, we double the size of all the values when doing multiplication. In another word, we expand the 28-bits values to 56-bits values before multiplication and use the 56-bits values for the multiplication. As the calculated result will also have 56 bits, we need to drop 28 bits. Even though, in this procedure, we may waste a lot of bits and computing effort, we can secure the computing precision.
Based on the basic operations described above, we built several secure primitives in the GMW project for matrix operations, including matrix addition, matrix subtraction and matrix multiplication, which are elaborated in detail later in this section.
Problem definition
Equation (5) shows that each party can calculate its own intermediary results conditioned on its local data (X ^{ j }, y ^{ j }), and share them for the combined results. However, this method requires a trusted server [11] to exchange local statistics. In this paper, we will build a decentralized framework for logistic regression using secret-sharing circuits based on secure multi-party communication. The proposed framework protects the intermediary statistics (X ^{ j })^{ T }(W ^{ j })^{(t)} X ^{ j } and (X ^{ j })^{ T }(y ^{ j } − (μ ^{ j })^{(t)}) with a joint function for all the parties without disclosing any private information.
Secure Hessian matrix inversion
Equation (7) implies that each party p _{ j } can locally calculate its own part of a partial Hessian matrix \( {\tilde{\boldsymbol{H}}}_i=-{\left({X}^j\right)}^T{X}^j/4 \), and feed \( {\tilde{\boldsymbol{H}}}_j \) as part of the input to the circuit. While in the circuit, the approximated Hessian matrix is constructed by aggregating all the partial Hessian matrices as \( \tilde{\boldsymbol{H}}={\displaystyle {\sum}_{j=1}^h{\tilde{\boldsymbol{H}}}_j} \).
where \( {\boldsymbol{M}}_t={\boldsymbol{N}}_t\ \tilde{\boldsymbol{H}} \), and c is constant. After convergence (e.g., in ~10 to 15 iterations), N _{ t } will provide an accurate approximation to the inversion of \( \tilde{\boldsymbol{H}} \).
Matrix multiplication
We transfer the matrix inversion problem into an iterative procedure of matrix multiplication and addition. Therefore, optimizing the implementation of matrix multiplication can improve the efficiency of the proposed framework. In this subsection, we adopted the Strassen algorithm for matrix multiplication.
The matrices can be iteratively partitioned k times, when n = 2^{ k }. Thus, the Strassen algorithm can reduce the complexity of matrix multiplication from O(n ^{3}) to O(n ^{2.8}).
In this work, the Strassen algorithm is implemented for matrix inversion, which has a significant effect on computational complexity. For other ordinary matrix multiplications, we still use the standard method. However, it is also possible to extend the Strassen algorithm to employ it in ordinary matrix multiplication, e.g. multiplication between non-square matrices.
The first derivative of the maximum likelihood function
To simplify computation and avoid overflow, we set a filter to bound the exponential within the interval between −5 and 5. When the exponentials are greater than 5 or smaller than −5, the evaluation results of the logit function (i.e., 1/(1 + e ^{ x })) would be smaller than 6.7 × 10^{−2} according to Eq. (1). Thus, we will not lose much accuracy by using this bound.
We built a look-up table storing the inversion of integers from 1 to 15, to avoid divisions and to speed up the calculation. For the other divisions involved in the logistic function, we treat them as a matrix of size 1.
It is worth mentioning that all the computations in this section are carried out in a customized Boolean circuit, where all the inputs and intermediary information exchange are protected by the OT protocol and the circuits. The only outputs in plaintext are the learned model parameter β in the proposed SMAC-GLORE.
Results
In this section, we first describe computational performance evaluations for fundamental operations, including matrix addition, matrix multiplication and matrix inversion. We then describe accuracy evaluations over real datasets with three features, including the Edinburgh dataset, which contains T wave inversion, Sweating and Pain in right arm features, and three genome datasets [50], where the first two features are ethnicity groups and the third feature is a SNP. The last column for each dataset is the intercept.
Computational performance evaluation
We first evaluated the performance of matrix addition, matrix multiplication and matrix inversion under a 2-party setup. We varied the size of matrices from 1 × 1 to 20 × 20 for each party. We simulated both parties on a 64-bit Ubuntu 14.04 platform with an Intel Xeon CPU at 3.10GHz and 256GB RAM, under the 28-bit fixed-point encoding, where both parties were connected by a 1GB network.
Computational performance for different matrix sizes in terms of number of gates, OT and total time cost for matrix addition, matrix multiplication and matrix inversion in a 2-party setup
Matrix addition operation | ||||||
Matrix size | # of AND gates | # of total gates | OT time (s) | Total time (s) | ||
Party 1 | Party 2 | Party 1 | Party 2 | |||
1 × 1 | 27 | 250 | 0.346 | 0.194 | 0.354 | 0.202 |
2 × 2 | 108 | 994 | 0.348 | 0.194 | 0.357 | 0.20 |
4 × 4 | 432 | 2,850 | 0.343 | 0.202 | 0.353 | 0.212 |
10 × 10 | 2,700 | 24,802 | 0.369 | 0.230 | 0.387 | 0.247 |
Matrix multiplication operation | ||||||
Matrix size | # of AND gates | # of total gates | OT time (s) | Total time (s) | ||
Party 1 | Party 2 | Party 1 | Party 2 | |||
1 × 1 | 4,621 | 21,594 | 0.367 | 0.245 | 0.384 | 0.262 |
2 × 2 | 37,076 | 273,034 | 0.518 | 0.609 | 0.577 | 0.660 |
4 × 4 | 580,325 | 2,707,002 | 2.135 | 3.636 | 2.603 | 4.060 |
10 × 10 | 4,645,300 | 21,664,002 | 21.174 | 50.413 | 29.646 | 58.769 |
Matrix inversion operation (15 iterations) | ||||||
Matrix size | # of AND gates | # of total gates | OT time (s) | Total time (s) | ||
Party 1 | Party 2 | Party 1 | Party 2 | |||
2 × 2 | 1,030,869 | 4,872,479 | 4.864 | 10.908 | 6.519 | 12.472 |
4 × 4 | 8,027,793 | 37,694,207 | 36.503 | 85.848 | 49.619 | 98.314 |
6 × 6 | 26,847,253 | 125,771,967 | 121.266 | 296.780 | 170.634 | 349.398 |
8 × 8 | 63,345,729 | 296,412,479 | 281.653 | 676.747 | 405.865 | 810.214 |
10 × 10 | 123,379,701 | 576,922,463 | 528.062 | 1286.500 | 751.421 | 1519.897 |
Accuracy evaluation
In this section, we perform accuracy evaluation of the proposed framework. As we introduced several approximation schemes (e.g., fixed-point encoding format, Taylor expansion for exponential function, division-free matrix inversion, etc.), the accuracy evaluation intends to measure how the results of the proposed framework differ from those of ordinary methods.
We also evaluated the accuracy of learned coefficients β in secure distributed logistic regression using different data sets. Our experiments were carried out under two setups: a local setting (simulated on a single server, which is the same as the one used in the Computational performance evaluation section) and a remote setting (on three different machines). In the remote setting, we executed the program on three servers, including the server in the local setting and two other 64-bits Ubuntu servers with Intel Xeon CPUs at 2.40GHz and 96 GB RAM.
Model parameters β learned in SMAC-GLORE and ordinary logistic regression model
β | 2 parties | Ordinary logistic regression | Two-sample Z test | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Value | Wald test | Value | Wald test | Test statistic | p-value | |||||
SE | Z value | p-value | SE | Z value | p-value | |||||
β _{1} | −0.6182 | 0.7759 | −0.7968 | 0.4256 | −0.6274 | 0.7779 | −0.8065 | 0.4199 | 0.0084 | 0.9933 |
β _{2} | 2.5454 | 0.8461 | 3.0084 | 0.0026 | 2.5767 | 0.8511 | 3.0275 | 0.0025 | −0.0261 | 0.9792 |
β _{3} | 1.2246 | 1.1226 | 1.0909 | 0.2753 | 1.2407 | 1.1369 | 1.0913 | 0.2751 | −0.0101 | 0.9920 |
β _{4} | 0.6177 | 0.8283 | 0.7457 | 0.4558 | 0.6198 | 0.8319 | 0.7450 | 0.4562 | −0.0018 | 0.9986 |
Model parameters β learned in local 2-party, 3-party scenarios, and remote 4-party scenarios
β | 2-party | 3-party | 4-party (remote) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Value | Wald test | Value | Wald test | Value | Wald test | |||||||
SE | Z value | p-value | SE | Z value | p-value | SE | Z value | p-value | ||||
β _{1} | −0.6182 | 0.7759 | −0.7968 | 0.4256 | −0.6182 | 0.7759 | −0.7968 | 0.4256 | −0.6182 | 0.7759 | −0.7968 | 0.4256 |
β _{2} | 2.5454 | 0.8461 | 3.0084 | 0.0026 | 2.5454 | 0.8461 | 3.0084 | 0.0026 | 2.5454 | 0.8461 | 3.0084 | 0.0026 |
β _{3} | 1.2246 | 1.1226 | 1.0909 | 0.2753 | 1.2246 | 1.1226 | 1.0909 | 0.2753 | 1.2246 | 1.1226 | 1.0909 | 0.2753 |
β _{4} | 0.6177 | 0.8283 | 0.7457 | 0.4558 | 0.6177 | 0.8283 | 0.7457 | 0.4558 | 0.6177 | 0.8283 | 0.7457 | 0.4558 |
Computing performances in local 2-party, 3-party scenarios, and remote 4-party scenarios
2-party | 3-party | 4-party (remote) | |||||||
---|---|---|---|---|---|---|---|---|---|
Party 1 | Party 2 | Party 1 | Party 2 | Party 3 | Party 1 | Party 2 | Party 3 | Party 4 | |
OT time (s) | 1,785.51 | 4,021.83 | 5,075.79 | 5,192.17 | 5,168.24 | 7,225.62 | 15,927.10 | 10,051.50 | 9,929.10 |
Computing time (s) | 601.76 | 625.59 | 538.79 | 659.43 | 612.51 | 650.65 | 1288.07 | 1011.14 | 857.72 |
Total time (s) | 2,607.11 | 4,683.56 | 5,822.17 | 6,064.08 | 5,976.22 | 8123.55 | 17,556.40 | 11,412.60 | 11,026.5 |
# of AND gates | 355,074,784 | 355,075,216 | 355,075,648 |
Differences between models learned from SMAC-GLORE and Ordinary Logistic Regression (LR) for Datasets 1-5
β | Dataset 1 | Dataset 2 | Dataset 3 | Dataset 4 | ||||
---|---|---|---|---|---|---|---|---|
2-party SMAC-GLORE | Ordinary LR | 2-party SMAC-GLORE | Ordinary LR | 2-party SMAC-GLORE | Ordinary LR | 2-party SMAC-GLORE | Ordinary LR | |
β _{1} | 1.7632 | 1.7647 | −0.6592 | −0.6567 | −0.5093 | −0.5066 | −1.5126 | −1.5168 |
β _{2} | 0.3369 | 0.3374 | 0.3174 | 0.3179 | 0.5767 | 0.5777 | −0.3516 | −0.3488 |
β _{3} | 1.1885 | 1.1902 | −0.2212 | −0.2195 | 0.4102 | 0.4138 | 0.2822 | 0.2855 |
β _{4} | −1.6514 | −1.6500 | −1.3115 | −1.3098 | −1.8940 | −1.8939 | −1.4873 | −1.4873 |
Limitations and discussion
Our project, which is in part based on the GMW project, developed secure fixed-point algorithms to handle floating number computation and constructed pipelines for securely building a distributed logistic regression model. The experimental results demonstrated the feasibility of the proposed framework, but there are still some limitations.
First, all the variables are represented in a 28-bit fixed-point format, where we allocate 1 bit for the sign (i.e., positive or negative numbers) and 11 bits for fractional part of a floating number. Therefore, the proposed framework cannot handle a number that is larger than 2^{16} or has precision higher than \( \frac{1}{2^{11}} \). Although the range and precision of a floating number can be improved by adding more bits in the fixed-point format, it will result in a significant increase of circuit size. Another potential workaround is to replace the fixed-point format with a floating-point format, which will require further investigation. In addition, during fixed-point multiplication, results may end up taking twice the number of bits, where the intermediary product results have 56 bits. In order to maintain a consistent 28-bit fixed-point encoding format, we need to truncate the intermediary product results, which may also result in precision loss. Based on our simulation, we need at least an 80-bit fixed point representation to achieve precision of a single precision floating point encoding format.
Second, to handle secure distributed exponential and logit function evaluations, we implemented a secure Taylor expansion algorithm up to a 15 order and truncated the input range (i.e., −5 to 5). The output precision of both functions can be further improved by increasing the order of expansion and of the truncated input range, but this results in additional computational costs. Similarly, we also developed a secure matrix inversion protocol based on an iterative algorithm [41], which only requires multiplication and addition operations. This secure matrix inversion protocol may also introduce precision loss in the final results.
Third, the current implication of the secure multiplication primitive is partially based on the Strassen algorithm, which requires several levels of block-wise decompositions of the input matrix to achieve maximum performance gain. However, we only perform a one-level decomposition rather than repeating the procedure in the whole multiplication. As matrix multiplication is one of the most time-consuming operations, we may need to fully utilize the Strassen algorithm to improve the performance in our future work.
In addition, based on our experiments, we find that the OT phase contributes the most to the computational time, due to the limitations of the GMW project which uses only a single thread for each pair’s OT procedure. To reduce the time for OT and improve performance, we can resort to parallel computation in the OT procedure. The GMW project was developed for a 32-bit computing environment, which means that the total number of gates cannot exceed 2^{31}. Therefore, in our future work, we plan to extend GMW to support a 64-bit address space, in which we can handle larger data sets. Another limitation is that the GMW project needs to preload the entire circuit into memory during the computation, which requires a very large amount of memory for a complex circuit. In our future work, we will investigate the possibility of dynamically generating a part of a circuit that will be required for the next execution to reduce memory consumption.
Although the proposed framework can protect the entire model learning phase, there is still no protection of the final output of the learned model parameter. Differential privacy has emerged as one of the strongest privacy guarantees for statistical data release [51]. Roughly speaking, it ensures (to a pre-defined extent, and quantifiably) that no risk is incurred when data from individual patients are entered in a particular database. It will be useful to integrate an optional component to enable differentially private model learning [52] in our future work.
Conclusion
In this study, we developed a secret-sharing, circuit-based SMAC-GLORE framework. To overcome the limitation of GMW, which only supports integer operations, we designed a fixed-point encoding format to support floating number arithmetic in SMAC-GLORE. We also implemented several secure matrix-operation primitives and built a pipeline for secure distributed logistic regression calculation. SMAC-GLORE is able to build a shared model without sharing each party’s private data. To the best of the authors’ knowledge, the proposed SMAC-GLORE is the first attempt to enable secret-sharing circuit based secure distributed logistic regression model learning for biomedical research studies. The experimental results show that our framework is reliable and can be used to solve practical problems in secure distributed logistic regression model learning.
Abbreviations
CSP, computation service provider; CRYSP, crypto service provider; GMW, Goldreich, Micali, and Wigderson; MSE, mean squared error; OT, oblivious transfer; SMC, secure multi-party computation; SMAC-GLORE, secure multi-pArty computation grid logistic regression
Declarations
Declaration
Publication of this article has been funded by the NIH grants R00HG008175, K99HG008175, R01HG007078, R00LM011392, R21LM012060, and U54HL108460. This article has been published as part of BMC Medical Informatics and Decision Making Volume 16 Supplement 3, 2016. Selected articles from the 5th Translational Bioinformatics Conference (TBC 2015): medical genomics. The full contents of the supplement are available online https://bmcmedgenomics.biomedcentral.com/articles/supplements/volume-16-supplement-3.
Availability of data and materials
Authors’ contributions
The authors HS, CJ, and SW contributed the majority of the writing and conducted major parts of the methodology and experiments. WD wrote a part of this paper and provided detailed edits. YT and XJ provided helpful comments on both methods and presentation. LO-M provided the motivation for this work, detailed edits and critical suggestions. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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