A multi-class predictor based on a probabilistic model: application to gene expression profiling-based diagnosis of thyroid tumors
- Naoto Yukinawa^{1},
- Shigeyuki Oba^{1},
- Kikuya Kato^{2}Email author,
- Kazuya Taniguchi^{2},
- Kyoko Iwao-Koizumi^{2},
- Yasuhiro Tamaki^{3},
- Shinzaburo Noguchi^{3} and
- Shin Ishii^{1}
DOI: 10.1186/1471-2164-7-190
© Yukinawa et al; licensee BioMed Central Ltd. 2006
Received: 13 December 2005
Accepted: 27 July 2006
Published: 27 July 2006
Abstract
Background
Although microscopic diagnosis has been playing the decisive role in cancer diagnostics, there have been cases in which it does not satisfy the clinical need. Differential diagnosis of malignant and benign thyroid tissues is one such case, and supplementary diagnosis such as that by gene expression profile is expected.
Results
With four thyroid tissue types, i.e., papillary carcinoma, follicular carcinoma, follicular adenoma, and normal thyroid, we performed gene expression profiling with adaptor-tagged competitive PCR, a high-throughput RT-PCR technique. For differential diagnosis, we applied a novel multi-class predictor, introducing probabilistic outputs. Multi-class predictors were constructed using various combinations of binary classifiers. The learning set included 119 samples, and the predictors were evaluated by strict leave-one-out cross validation. Trials included classical combinations, i.e., one-to-one, one-to-the-rest, but the predictor using more combination exhibited the better prediction accuracy. This characteristic was consistent with other gene expression data sets. The performance of the selected predictor was then tested with an independent set consisting of 49 samples. The resulting test prediction accuracy was 85.7%.
Conclusion
Molecular diagnosis of thyroid tissues is feasible by gene expression profiling, and the current level is promising towards the automatic diagnostic tool to complement the present medical procedures. A multi-class predictor with an exhaustive combination of binary classifiers could achieve a higher prediction accuracy than those with classical combinations and other predictors such as multi-class SVM. The probabilistic outputs of the predictor offer more detailed information for each sample, which enables visualization of each sample in low-dimensional classification spaces. These new concepts should help to improve the multi-class classification including that of cancer tissues.
Background
Histopathological analysis is the traditional mainstay of cancer diagnostics, and is relied on heavily to discriminate between malignant and benign tissues. However, in spite of its long and successful history as a routine medical technique, it is plagued by a number of well-known problems. For example, histological diagnosis often depends upon judgment calls by individual pathologists, leading to variations in diagnosis [1, 2]. In addition, it may be difficult in some cases to differentiate between malignant and benign tissues. In such cases, microscopic observations have often been complemented by staining for molecular markers, such as expressed genes. This led directly to today's emerging approach, gene expression profiling, by which the expression levels of thousands of genes throughout the genome are measured by DNA microarrays or alternative techniques. Using this technique, diagnostic systems using multiple genes have been constructed [3]. Clinical status parameters such as prognosis or drug resistance are the popular targets, and the supervised learning theory is often applied.
In most examples of cancer classification, binary classifiers have been used, with multi-class predictors only rarely applied. However, because more than three tissue types are often differentiated during pathological diagnosis, a stable or optimized multi-class predictor is needed.
In this study, we developed a new multi-class predictor based on a probabilistic model. We examined various combinations of binary classifiers to perform optimized multi-class prediction, and applied our system to gene expression data from four tissue types of human thyroid origin. Thyroid cancer is a relatively common cancer, accounting for roughly 1% of total cancer incidence. There are two main types of thyroid cancer, papillary carcinoma (PC) and follicular carcinoma (FC). In addition to these malignant types, a benign tumor, follicular adenoma (FA), is also prevalent. The intriguing problem in thyroid cancer therapy is the preoperative differential diagnosis of follicular carcinoma vs. adenoma. The main diagnostic procedure is fine needle aspiration, but because the tissue structure is disrupted during the sampling process, the differential diagnosis is extremely difficult [4]. Papillary carcinoma is relatively easier to distinguish, but good molecular diagnostics would still be of potential clinical value. Thus, thyroid cancer represents a good example of a tumor type for which conventional histopathologic diagnosis faces limitations.
We performed gene expression profiling of the four thyroid tissues by adaptor-tagged competitive polymerase chain reaction (ATAC-PCR) [5, 6], a high-throughput RT-PCR technique. We first analyzed the global differences in gene expression among the four thyroid tissues, and confirmed that the data matrix contained information for class separation. Then, we applied our new multi-class predictor for differential diagnosis of the four tissue types. The multi-class predictor outperformed previous methods used in the field.
Results
Difference in gene expression among the four thyroid tissues
We then compared the CR spectra of all pairwise comparisons. The CR of each pair was calculated; genes were sorted by CR value order, and plotted (Figure 1b). Figure 1b shows the CR values of the top 250 genes: this figure represents the degree of gene expression difference of pairwise comparison, not indicating significance of gene expression difference (see below). Throughout this region, FA vs. N and FA vs. FC exhibited the smallest and the second smallest CR values. This result may reflect the difficulty in diagnosis: FA and N maintain microscopic follicular structures; FA and FC are the most difficult tissues to be differentially diagnosed. The other four comparisons, i.e., FC vs. N, FC vs. PC, FA vs. PC, N vs. PC, exhibited no substantial difference in CR in the high CR region (top 50 genes). In the middle CR region (the top 50–250 genes), the CRs of N vs. PC are close to those of FA vs. FC.
The CR spectra directly represent global differences in gene expression. We also evaluated these results by conventional measures, using p-values and q-values [8] of the standard t-test. When the cut-off p-value (significance level) is set at 0.0001, the numbers of genes and corresponding q-values are as follows: FA vs. N, 8 genes (q-value, 0.018); FA vs. FC, 7 (0.017); N vs. PC, 44 (0.0038); FC vs. N, 25 (0.0048); PC vs. FA, 66 (0.002); PC vs. FC, 41 (0.0026). These results parallel those from the CR spectrum analysis. Comparisons such as FA vs. N and FA vs. FC exhibited small numbers of significant genes and high q-values, and those such as PC vs. FA, and PC vs. FC exhibited large numbers of significant genes and low q-values. FC vs. N and N vs. PC are the middle of the above two.
Differential diagnosis by binary classifier
Strategy for construction of a multi-class predictor
In this study, we constructed a multi-class prediction system based on a combination of binary classifiers. Each unit binary classifier was constructed by the weighted-voting algorithm, where the diagnostic genes were selected by the criterion p ≤ 0.0001. In this criterion, all of the corresponding q-values were less than 0.05, maintaining the least inclusion of false positive genes. It should be noted that application of the following strategy is not limited to weighted-voting. We examined the following four types of construction methods using different numbers of binary classifiers.
1R: constructed using classifiers discriminating between one class and all of the other classes. 1R is conventionally referred to as "one-to-the-rest" [9, 10].
11: constructed using classifiers discriminating a pair of classes. 11 is conventionally referred to as "one-to-one" [11–13].
1A: constructed using classifiers discriminating between one class and any subset of other classes.
AA: constructed using classifiers discriminating between any pair of disjoint subsets of classes.
The conventional construction rule for multi-class prediction from outputs of each binary classifier was designed for 11 and 1R. Binary output functions are usually used (see the materials and methods section), and prediction is performed by simple summation of the outputs. It should be noted that outputs are null when a group of classes, i.e., ''rest'' of ''one-to-the-rest'', wins. This rule, however, is not applicable to AA, because AA includes classification between groups of classes.
From four patterns of classifier combination (1R, 11, 1A, AA) and two output summation methods (SIS and SHS), we generated seven types of multi-class predictors (AA with SIS is not applicable).
Classification of thyroid tissues
Prediction accuracies (%) of the learning sample set, evaluated by leave-one-out cross-validation.
SIS | SHS | |
---|---|---|
1R | 69.8 | 69.8 |
11 | 77.3 | 77.3 |
1A | 79.8 | 79.8 |
AA | N/A | 79.0 |
Our multi-class predictor was validated with an independent test set consisting of 49 samples. A multi-class predictor based on 1A-SIS, 1A-SHS, and AA-SHS was constructed using gene expression data from the learning set (119 samples), and was applied to the independent test set. The overall prediction accuracy was 85.7% with all of the three predictors. The confusion matrix of 1A-SHS is shown in Figure 5b.
Comparison with other methods
We compared our method with other multi-class predictors popular in the field of cancer classification using the same learning data set (119 samples) as well as the other three data sets. The additional data sets are as follows. All data sets and algorithms were evaluated by LOO. With any algorithms, limited numbers of trials in classifier conditions (e.g., number of selected genes) were performed to avoid excessive overlearning.
The esophageal cancer data set
This data set is also composed of original gene expression profiles obtained from esophageal cancer of Japanese patients by ATAC-PCR [15]. The task here is differential diagnosis of three histological types: poorly differentiated, 14; moderately differentiated, 97; well differentiated, 30.
The GCM data set
The original global cancer map data set consists of 14 cancer types [14]. Because the number of combinations of binary classifiers increases exponentially as the number of classes increases, we selected five cancer types: breast, 11 samples; prostate, 10; leukemia, 30; renal, 11; mesothelioma, 11.
The SRBCT data set
Gene expression profiles about small round blue cell tumors (SRBCTs) of childhood, which contain 83 samples and 2308 genes measured by cDNA microarrays [16]. SRBCTs, which include the Ewing family of tumors (EWS), rhabdomyosarcoma (RMS), Burkitt lymphoma (BL), and neuroblastoma (NB) [17]. The composition of the samples are 29 (EWS), 25 (RMS), 11 (BL), and 18 (NB).
Comparison of prediction accuracies of various algorithms. Each figure represents the best accuracy obtained by the gene-selection condition shown as the value in the parenthesis. The greatest accuracy of each data set is shown as bold letters. Values in parentheses are numbers of diagnostic genes selected by recursive feature elimination for MC-SVM, shrinkage parameters for SC, and threshold p-value for others.
11-SIS | 11-SHS | 1R-SIS | 1R-SHS | 1A-SIS | 1A-SHS | AA-SHS | MC-SVM | SC | |
---|---|---|---|---|---|---|---|---|---|
thyroid | 79.8 (10^{-3}) | 79.8 (10^{-3}) | 74.8 (10^{-3}) | 74.8 (10^{-3}) | 79.8 (10^{-4}) | 79.8 (10^{-4}) | 79.0 (10^{-4}) | 74.8 (2000) | 74.8 (0.5) |
GCM | 86.3 (10^{-3}) | 86.3 (10^{-3}) | 89.0 (10^{-5}) | 89.0 (10^{-5}) | 89.0 (10^{-6}) | 89.0 (10^{-6}) | 90.4 (10^{-6}) | 80.8 (32) | 74.0 (0) |
SRBCT | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 (32~2308) | 100 (1~2) |
esophageal | 72.3 (10^{-1}) | 72.3 (10^{-1}) | 71.6 (10^{-1}) | 71.6 (10^{-1}) | 73.1 (10^{-1}) | 73.8 (10^{-1}) | 73.8 (10^{-1}) | 71.7 (8) | 68.8 (≥ 1) |
Our predictors were compared with other multi-class prediction methods reported in the field. One is the combination of support vector machines (multi-class SVM, MC-SVM) described by Ramaswamy et al. [14], and the other is the shrunken centroid method (SC) [18]. The first method corresponds to 1R-SIS without probabilistic outputs by our definition, and linear-kernel SVM is used as a unit binary classifier instead of a weighted-voting algorithm. To select diagnostic genes, we performed recursive feature elimination [19]: prediction accuracies were determined with various numbers of genes, i.e., all genes, 2000, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. The greatest accuracies with respective gene number are shown in Table 2: the accuracies were less than those obtained by our method.
The shrunken centroid is an improved method of the simple nearest prototype (centroid) classifier. We evaluated this method by LOO with various shrinkage parameters from 0 to 6 with 0.5 as an interval, i.e., a total of 13 parameters. With the all four data sets, our method yielded results superior to the shrunken centroid.
Tissue sample display by class probability
Discussion
In spite of their integral role in cancer diagnosis, many histopathological diagnostic protocols are still dependent on judgment calls made by individual pathologists, and as a result are significantly prone to error. More objective measures are needed, such as computational interpretation of microscopic images or complementary diagnosis by gene expression. For diagnosis by gene expression profile, conventional binary classifiers are not sufficient because the number of tissue types to be differentially diagnosed is usually greater than two. In this study, we have described the differential diagnosis of four thyroid tissues by use of a novel multi-class predictor with a test prediction accuracy of as high as 79~85%. As a result, our diagnostic system represents a promising first step towards an automated pathological tool to complement current diagnostic procedures.
Our approach to multi-class prediction is a novel one characterized by increasing combination of binary classifiers and a probabilistic approach for outputs. Traditionally, "one-to-one" or "one-to-the-rest" approaches have been applied to integrate combinations of unit binary classifiers. As shown above, 1A and AA exhibited prediction accuracy better than conventional approaches. Our results suggest that conventional methods such as "one-to-one" and "one-to-the-rest" are not necessarily optimal, recommending use of more combinations of binary classifiers. The main disadvantage of AA and 1A is that the number of classifiers increases exponentially as the number of classes increases. However, this is not a significant disadvantage when the method is applied to cancer classification problems, because the number of classes for separation is nearly always less than seven.
Another characteristic of our predictor is the introduction of a probabilistic model to the outputs. Probabilistic output of the multi-class predictor, here designated as class probability, offers detailed information about each sample, which could not be obtained by binary outputs. In particular, it is useful to visualize the relationships between the classes using a two- or three-dimensional plot of class probabilities. This visualization may in fact be more effective than unsupervised approaches such as principal component analysis and multi-dimensional scaling [20], because the class probabilities were calculated from weighted sums of selected diagnostic genes. In particular, it is useful for gaining a deeper understanding of individual samples. For example, among the FC samples, some appear rather similar to FA, and a few seem separate from all of the classes. Detailed analysis of class probability may lead to discoveries of new tumor classes. It is also possible that some of the samples may have been initially misdiagnosed by the pathologist. Reexamination of these tissue sections may lead to more accurate diagnosis.
One may argue that our assumption that the choice of unit binary classifier is an independent process (see Materials and Methods) is too simplified. For example, because one class appears in different classifiers in different combinations, these classifiers may not be independent. A simple solution for this problem is introduction of prior probability in the choice of unit binary classifier. For example, our predictor is only modestly effective at separating FC from FA, but this shortcoming could be improved by the introduction of appropriate prior probabilities.
Another possible critique is our equal sharing of outputs of unit binary classifiers. The optimal ratio of output sharing can be obtained by taking account of correlations between similar targets, such as [c_{1}|c_{2}] and [c_{1},c_{2}|c_{3},c_{4}]. Our preliminary results using a method similar to that previously used by Hastie et al. in the pairwise case [12] suggested that the prediction accuracy was equal to that of SHS. We suspect that SHS is a good estimate for the sharing method deduced by this more detailed model. In any case, the optimum model of outputs should not decrease accuracy as increase in number of constituent binary classifiers. However, with the thyroid data set, the accuracy of 1A was higher than that of AA, suggesting that the current probabilistic model is incomplete. Our complete version of the output model, requiring a larger computer resource, will be described elsewhere (Yukinawa et al., submitted).
Multi-class classification techniques can be roughly divided into two types. One is the decomposition of multi-class problems into binary ones. ''one-to-the-rest'' methods [9, 10], pairwise comparisons, i.e., ''one-to-one'' methods [11–13], error-correcting output coding [21–23] belong to this type. There have been comprehensive studies centered on SVM [24, 25], but there have been no definitive methods in this type. Recent developments also include logitBoost [26] or genetic algorithm [27] as unit classifiers. The other type is binary classification algorithms that can be naturally extended to handle multi-class problems directly. SVM with multiclass objective functions [28, 29], discriminant analysis [30] and regression and decision trees are of this type. Recent developments include shrunken centroid [18], an advanced version of nearest centroid [31], and total principal component regression [32]. Our method is an extension of one of the former type approaches. For comparison with previous methods, we chose a ''one-to-the-rest'' method using a support vector machine from the first type [14], and the shrunken centroid method [18] from the latter type. With the four data sets, including two in the public domain, we established that our method was superior to these two methods.
Ramaswamy et al. [14] pointed out that the weighted-voting was generally not well-performed as MC-SVM. This may be because they selected genes in the order of signal-to-noise ratio, and used the gene number as the threshold. In contrast, we used the p-value as the threshold: the p-value threshold would be essential to obtain multi-class predictor with the weighed-voting algorithm, because using the number of genes instead may incorporate unnecessary genes or ignore necessary genes in some classifiers.
The shrunken centroid [18] is a simple modification of the nearest centroid method. The prediction accuracy obtained was higher than that of the combination of SVM, but still fell significantly short of our algorithm. Moreover, because the best results were obtained by comparison of different shrinkage parameters, the predictor may be overfitted. The discriminant hyperplanes resulting from the shrunken centroid were rather simple, and a more complex classification such as that for thyroid tissue may require more complicated discriminant surfaces, such those obtained by our method. One of the advantages of the shrunken centroid method is that each sample can be described by class probabilities calculated from discriminant scores. This feature is similar to our class probability, and enables visualization of samples as in our method.
The main shortcoming common to previous multi-class studies of cancer classification is the use of sample data matrices, which are easily diagnosed by conventional techniques. The solid tumors or small round blue cell tumors used have been of different tissues of origin, and could be differentiated easily by microscopic observations as well as by gene expression analysis. Because solving such problems is of little clinical value, such data matrices are not adequate for the benchmark test of predictors. Use of data matrices which model realistic problems is of pivotal importance for evaluation of techniques.
Conclusion
Among tasks in histopathology, differential diagnosis of thyroid tissues is a difficult one. We have developed a new multi-class predictor based on a probabilistic model using gene expression data. Tests on four types of thyroid tissues revealed its excellent performance in prediction. The novel approaches introduced in this study show promise as a means to differentiate between similar tumor types from the same tissue of origin. However, before we can draw a conclusion on their efficacy, a number of confirmatory experiments, including analysis of artificial data sets, are necessary. Nevertheless, we believe these algorithms will contribute to significant advances in the pathological diagnosis of cancer in the near future.
Methods
Patients and tumor samples
The learning set consisted of 41 FA, 20 FC, 30 PC and 28 N samples, and the test set consisted of 17 FA, 8 FC, 12 PC and 12 N samples. For the test set, we simply chose samples with the most recent surgery dates. The tumor samples were obtained from patients who underwent surgery (hemithyroidectomy or total thyroidectomy) during the period from February 1998 to December 2002. Histological diagnosis was performed by an experienced pathologist. Tumor samples obtained during surgery were snap frozen in liquid nitrogen and stored at -80°C until use. The study protocol was approved by the Institutional Review Board of Osaka University Medical School, and written informed consent was obtained from each patient.
ATAC-PCR assay and data processing
To select genes for ATAC-PCR, we first constructed four cDNA libraries; one from a mixture of five normal thyroid tissues, one from a mixture of five PC samples, one from a mixture of five FA samples and one from a mixture of four FC samples as described [33]. We then performed single pass sequencing of 6154 clones (1424 from follicular carcinoma, 1592 from papillary carcinoma, 1575 from follicular adenoma, and 1563 from normal thyroid). A total of 2383 genes were selected from this EST collection, prioritizing abundant genes. Then, 133 genes were added by experts in thyroid cancer, based on literature information. We designed PCR primers for ATAC-PCR reactions for these 2516 genes. The specificity of this gene selection provides an advantage over more universal gene sets, such as those selected from the UniGene database, which include genes not expressed in thyroid tissue. Total RNA was purified from thyroid tumors using Trizol reagent (GibcoBRL), and was subjected to gene-expression profiling by ATAC-PCR. The ATAC-PCR experimental procedure was performed as previously described [34]. The measurements of 2516 genes were repeated twice with ten tumor samples, and correlation coefficient of these duplicates was calculated. The correlation coefficient and corresponding p-value was 0.92 and 0.000163, showing the sufficient reproducibility of the measurement. The raw data describing gene expression levels were divided by the median of each sample. Because the median of each sample is expected to reflect the overall mRNA level, normalization by this value corrects for variation in mRNA level from sample to sample. Values less than 0.05 and more than 20 were converted to 0.05 and 20, respectively, and subsequently the entire data matrix was converted to a logarithmic scale. The detailed protocols for the ATAC-PCR experimental procedure are available on our web site [35].
For the analysis, missing values were estimated and filled in by a method based on Bayesian inference [36] (see the additional files 1 and 2). In the ATAC-PCR reaction, the fluorescence intensity of the products is correlated with the data quality. Genes were sorted by the order of the average of the fluorescence intensity, and we selected the top 2000 genes for the following analysis.
Statistical analysis
The correlation ratio R of gene i is defined by the following equation.
$R=\sqrt{\frac{{\displaystyle \sum _{c\in C}{n}_{c}{({\overline{x}}_{ci}-{\overline{x}}_{i})}^{2}}}{{\displaystyle \sum _{i}{\displaystyle \sum _{j}{({x}_{ij}-{\overline{x}}_{i})}^{2}}}}}$
where n_{ c }is the number of samples in a particular class c ∈ C; x_{ ij }is the expression level of gene i in sample j; $\overline{x}$_{ ci }is the average expression level of gene i for samples in a particular class c ∈ C; and $\overline{x}$_{ i }is the average expression level of gene i for all samples. Permutation p-value of a correlation ratio is defined by the proportion of trials where R of randomly permutated data exceeded that of original data, among 10,000 permutation trials.
p-values were calculated using t-statistics. q-values were calculated using software by Dr Storey [37].
Unit binary classifiers
The data set used in this study is defined as X = {x^{(n)},t^{(n)}}_{n = 1:N}; here, x^{(n)}∈ ℜ^{ D }and t^{(n)}∈ C ≡ {c_{1},...,c_{ M }} denote gene expression pattern and class label, respectively, of the n-th sample; D is the number of genes.
C is the set of whole classes, and 2^{ c }is the labels' power set, as follows.
2^{ c }≡ {{c_{1}}, {c_{2}},..., {c_{1}, c_{2}},..., {c_{1}, c_{2}, c_{3}},..., C, φ}.
We designated a pair of subsets to be classified as a "target". Consider a binary classifier trained on a binary target [l|m] ∈ B where indices l, m ∈ 2^{ C }- {C, φ} denote two disjoint (l ∩ m = φ) subsets of class labels and B is a certain set of targets. In this study, we used some selections of B such as B_{1R}, B_{11}, B_{1A}and B_{ AA }(Figure 3).
We employed the weighted-voting algorithm for the binary classification of each target. [l|m]. The discriminant function f_{[l|m]}(x) ∈ ℜ was defined using selected diagnostic genes with the criterion p ≤ 0.0001 for each gene-wise t-test.
Output function of the binary classifier
When a discriminant function f_{[l|m]}(x) is determined from a learning data set X_{[l|m]}= {(x^{(n)}, t^{(n)}) ∈ X|t^{(n)}∈ l ∪ m}, the binary output function of the classifier is described as follows.
${q}_{[l|m]}(x)=\{\begin{array}{cc}1& \text{if}{f}_{[l|m]}(x)\ge 0\\ 0& \text{if}{f}_{[l|m]}(x)0.\end{array}$
In this study, we took a probabilistic approach, where q_{[l|m]}(·) may be a real value of 0 ≤ q_{[l|m]}(·) ≤ 1 which corresponds to a posterior probability, Pr(t ∈ l|x, t ∈ l ∪ m, X_{[l|m]}). The following logistic function r(x; a_{[l|m]}, b_{[l|m]}) can represent a mapping from the discriminant function f_{[l|m]}(x) to the probability. The mapping is based on one-dimensional logistic regression,
$r(x;{a}_{[l|m]},{b}_{[l|m]})=\frac{1}{1+\mathrm{exp}(-{a}_{[l|m]}{f}_{[l|m]}(x)-{b}_{[l|m]}},$
where a_{[l|m]}and b_{[l|m]}are determined by maximum-likelihood estimation so that r(x^{(n)}; a_{[l|m]}, b_{[l|m]}) for all learning data x^{(n)}∈ X[_{[l|m]}best fits the corresponding labels t^{(n)}. The probabilistic output for the gene expression vector of a test sample x^{NEW},q_{[l|m]}(x^{NEW}), is determined as follows.
q_{[l|m]}(x^{NEW}) = r(x^{NEW}; a_{[l|m]}, b_{[l|m]}).
Multi-class prediction scheme
For each expression vector x, we considered a probabilistic process of selecting a single class c ∈ C as a prediction of its label.
1. Random selection of a binary target [l|m] in a set B, and calculation of q_{[l|m]}(x).
2. Stochastic selection of a class subset l or m, with probability q_{[l|m]}(x) for l and 1 - q_{[l|m]}(x) for m.
3. Random selection of a single class prediction c in the above selected subset l or m. According to this process, the probability of selecting a class c ∈ C, referred to as "class probability", becomes
${p}_{c}(x)=\frac{1}{Z}{\displaystyle {\sum}_{[l|m]\in B}\left\{I(c\in l){q}_{[l|m]}(x)/\#l+I(c\in m)(1-{q}_{[l|m]}(x))/\#m\right\}}.$
Here I(A) is equal to 1 if condition A holds, and zero otherwise; #l and #m denote the numbers of classes in the class subset l and m; Z is a normalization constant which guarantees $\sum _{c\in C}{p}_{c}(x)=1$. The above equation takes the summation of q_{[l|m]}(x) for all targets [l|m] where subset l or m contains class c, and where probability scores are shared equally to member classes of the subsets. Therefore, this is called shared summation (SHS) of probabilistic output. Subsequently, we can obtain a Bayes-optimal class prediction as $\underset{c\in C}{\mathrm{arg}\mathrm{max}}{p}_{c}(x)$.
In simple summation (SIS), we determined class probability p_{ c }(x) for each class c ∈ C as a summation of q_{[l|m]}(x) for all targets [l|m] such that l or m is {c} itself,
${p}_{c}(x)=\frac{1}{Z}{\displaystyle {\sum}_{[l|m]\in B}\left\{I(l=\{c\}){q}_{[l|m]}(x)+I(m=\{c\})(1-{q}_{[l|m]}(x))\right\}},$
where Z is a normalization constant. Because of the condition l = {c} or m = {c}, classifiers whose output has more than one label were not included in the above summation. As a result, AA and 1A are equivalent in the SIS case. Although SIS is a straightforward extension of conventional voting criterion to handle probabilistic guesses, it lacks a consistent model of decision process, such as that presents in the background of SHS.
Matlab programs to perform 1A/AA-SIS/SHS using the weighted-voting algorithm is available from NY.
Declarations
Acknowledgements
We thank Ms Noriko Ueno, Satoko Maki-Kinjo, Keiko Miyaoka-Ikegami and Mihoko Yoshino for their technical assistance. This work was supported by the Knowledge Cluster Initiative (the Keihanna Science City area) of the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Authors’ Affiliations
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