Accurate wisdom of the crowd from unsupervised dimension reduction

Wisdom of the crowd, the collective intelligence from responses of multiple human or machine individuals to the same questions, can be more accurate than each individual and improve social decision-making and prediction accuracy. Crowd wisdom estimates each individual’s error level and minimizes the overall error in the crowd consensus. However, with problem-specific models mostly concerning binary (yes/no) predictions, crowd wisdom remains overlooked in biomedical disciplines. Here we show, in real-world examples of transcription factor target prediction and skin cancer diagnosis, and with simulated data, that the crowd wisdom problem is analogous to one-dimensional unsupervised dimension reduction in machine learning. This provides a natural class of generalized, accurate and mature crowd wisdom solutions, such as PCA and Isomap, that can handle binary and also continuous responses, like confidence levels. They even outperform supervised-learning-based collective intelligence that is calibrated on historical performance of individuals, e.g. random forest. This study unifies crowd wisdom and unsupervised dimension reduction, and extends its applications to continuous data. As the scales of data acquisition and processing rapidly increase, especially in high-throughput sequencing and imaging, crowd wisdom can provide accurate predictions by combining multiple datasets and/or analytical methods.

However, previous crowd wisdom classification studies have focused predominantly on binary responses and problem-specific models ( [9,7,4,10]). The confusion matrices of individuals and the binary true classes are fit in turn to maximize the model's likelihood with expectation-maximization.
Where available, continuous individual predictions such as confidence levels are thresholded and mostly lost, potentially limiting the classification accuracy and generalizability, whilst the proper choice of threshold can also be difficult.
To resolve this issue and to link crowd wisdom with machine learning, we consider continuous rather than binary variables for individual responses. Due to a lack of complete information to perfectly determine the true class, we introduce an unknown intermediate layer representing the probability of the true class (class probability, Figure 1B). In the simplest scenario, individual responses are then independent continuous estimations of the class probability. More generally, individuals can also characterize and estimate classification confidence with any other continuous scores, which are assumed to be equivalent in ranking with the class probability. Binary responses can also be treated as numerical 0s and 1s.
The continuous crowd wisdom classification problem can then be solved by unsupervised dimension reduction. Unsupervised dimension reduction infers the latent lower dimensions by which the input data are assumed to be parameterized. In crowd wisdom ( Figure 1B), each individual independently estimates, and is effectively parameterized by, the class probability alone. Therefore, the class probability may be recovered as the first and only dimension ( Figure 1C, subjecting to a monotonic transformation). This makes dimension reduction the natural crowd wisdom for classification problems with continuous information. Which dimension reduction method is the best then depends on various aspects of the problem, such as nonlinearity. As a brief demonstration with the DREAM2 BCL6 Transcription Factor Prediction challenge dataset, containing the confidence scores of 200 genes as potential targets of BCL6 (i.e. questions) submitted by 8 teams (i.e. individuals) ( [11,12,13]), the first principal component (PC1) direction of the gene-by-individual matrix gave an accurate representation of the class probability ranking ( Figure 1D) and the performance of each individual ( Figure   1E).
To first evaluate dimension reduction methods on binary responses, we envisioned an algorithmassisted diagnostic committee of 24 dermatologists whose skin cancer classifications are known for 111 dermoscopy images ( [14]). As a comparison, we applied principal component analysis (PCA), factor analysis (FA), multi-dimensional scaling (MDS), locally linear embedding (LLE), Hessian LLE, local tangent space alignment (LTSA), Isomap, and spectral embedding to estimate the class probability ranking from the individual classifications (Methods). PCA and FA were superior to most dermatologists and were among the top crowd wisdoms. Nearest neighbor based methods were not significantly more accurate than PCA, but instead converged towards PCA at large numbers of neighbors, suggesting no significant nonlinearity ( Figure 2E). PCA and FA offered continuous confidence levels which reduced to state-of-the-art binary crowd wisdom solutions from SML ( [4]) and CUBAM ( [15]) at certain thresholds ( Figure 2AB, Figure S1AB). Interestingly, more than 15 crowd wisdoms had better classification performance than a deep neural network trained on 130k clinical images ( Figure   2ABE, Table S1, [14]). This demonstrates the cutting-edge efficacy from dimension reduction on the binary crowd wisdom task.
To test whether continuous confidence information can improve accuracy, we applied the same dimension reduction methods on the DREAM2 dataset, as well as on their perfectly binarized yes/no responses (Methods). PCA on continuous confidence levels was more accurate than SML and CUBAM on binarized responses ( Figure 2CD, Figure S2). Performance differences between crowd wisdoms were in agreement with the skin cancer classification data, except that mean and median -often the default crowd wisdom method for continuous data ( [3]) -could not account for worse-than-random individuals ( Figure 2CDF, Figure S1CD, Figure S3). Many dimension reduction methods, including PCA and Isomap, outperformed every team. Dimension reduction provided reliable and superior crowd wisdom from confidence information without knowing the true class distribution.
Knowledge of the ground-truth for a subset of questions may help calibrating response aggregations for the remaining questions. For instance, in daily life we trust people and favor programs that were more accurate historically. To compare calibrated response aggregations against ground-truth-ignorant crowd wisdoms, we cross-validated crowd wisdoms and 8 popular supervised classifiers [including linear, logistic, lasso, and elasticnet regression, linear discriminant analysis (LDA), support vector machine (SVM), kNN, and random forest] that were trained on randomly selected question subsets (Methods). Surprisingly, crowd wisdom had equal or better performance than supervised classifiers for both the DREAM2 and the skin cancer datasets in terms of AUROC and AUPR ( Figure 3, Figure S4, Figure S5). Supervised classifiers could only reach crowd wisdom's performance with 50% of training data or more ( Figure S4, Figure S5). Considering that the true answers in practical research questions are largely unknown, unsupervised crowd wisdom outperformed supervised learning by integrating the test dataset to better estimate individual accuracies.
We further interrogated crowd wisdoms in controlled simulations. With 2000 replicated simulations for each parameter set, we found SML to highly correlate with and converge to thresholded PCA as the number of individuals increases ( Figure 4AB, Methods). SML was consequently less sensitive than PCA due to the loss of information, even in perfect binarizations of confidence levels ( Figure   4C, Student's t-tests P < 10 −160 , Figure S6, Methods). CUBAM was also less sensitive after binarization than PCA. In single simulations (Methods), PCA, FA, Isomap, and LLE converged to perfect class probability predictions as the number of individuals increased ( Figure 4D, Figure   S7, Figure S8B-I), but LLE based methods were unreliable on noisy datasets ( Figure S8JK, [16]). Single simulations also reaffirmed our existing conclusions. PCA, FA, and Isomap continued to lead the performances ( Figure S9, Figure S8) and crowd wisdom remained superior to supervised classifiers ( Figure S9B, Figure S10). Mean and median were again hindered by worse-than-random individuals ( Figure S9A, Figure S8). Overall, PCA and Isomap are more reliable and accurate than other dimension reduction methods and previous wisdom of the crowd methods.
By embedding wisdom of the crowd in unsupervised dimension reduction, we have found that

DREAM2 BCL6 Transcription Factor Prediction challenge dataset
The DREAM2 BCL6 Transcription Factor Prediction Challenge is an open crowd challenge to infer BCL6 gene's transcriptional targets ( [11,12,13]). Participating teams inferred BCL6 targets from gene expression microarray and optional external data, and submited confidence scores for 200 potential target genes. Submissions were evaluated against the gold standard derived from ChIP-on-chip and perturbation experiments, containing 53 BCL6 targets. We had access to submissions from 11 teams, in which 8 were full (without missing predictions) and were used for crowd wisdom.

Skin cancer classification dataset
Deep neural networks outperformed an average dermatologist in the classification of skin cancer from dermoscopy images ( [14]). Based on dermoscopy images alone, dermatologists were asked whether to biopsy/treat the lesion or to reassure the patient. We obtained 24 dermatologists' responses to 111 biopsy-proven dermoscopy images in which 71 were malignant. We also obtained the predicted confidence scores for these images from the deep neural network in [14].

Simulated datasets
A simulated dataset of n binary (yes/no) questions contains their true classes, the (posterior) class probabilities given all the relevant data for each question as P i (Y es | data), and the responses from k individuals to all n questions as matrix R = {r ji }, for i = 1, . . . , n, j = 1, . . . , k. Given the desired occurrence frequency of class yes as P (Y es), the dataset needs to contain nP (Y es) questions in class yes and n(1 − P (Y es)) in class no. We simulated the true classes, class probabilities, and individual responses ( Figure 1B) according to the following steps: 1. Simulate class probabilities P (Y es | data) ∼ B(β, β), where B is the Beta distribution and β characterizes the question difficulty given all the data. For each question, set the true class to yes with probability P (Y es | data) and no otherwise. Only the first nP (Y es) questions in yes class and the first n(1 − P (Y es)) questions in no class were retained, merged, and shuffled to form the full list of questions i = 1, . . . , n. Their class probability P i (Y es | data) and true classes were recorded.

Perfect binarization
To transform confidence level responses to binary (yes/no) responses, we chose the ideal scenario for existing binary crowd wisdom methods, by assuming that each individual knows the true total number of yes responses. Consequently, each individual will select that same number of their most confident predictions as yes, and the rest as no. Ties at the yes/no boundary are selected at random.

Normalization
We normalized raw answers from multiple individuals to multiple questions before applying crowd wisdom or supervised learning (in cross validation). For continuous datasets, we first converted raw answers into rankings, separately for each individual and with ties averaged. Then, for all datasets, we shifted the raw or rank-converted values to zero mean and scaled them to unit variance, separately for each individual.

Dimension reduction as wisdom of the crowd
From the python package scikit-learn, we applied the following dimension reduction methods for crowd wisdom: TruncatedSVD (as PCA) and FactorAnalysis in sklearn.decomposition, and Local-lyLinearEmbedding (with methods standard, hessian, and ltsa), Isomap, and SpectralEmbedding in sklearn.manifold. Nearest-neighbor based methods took 5, 7, 10, 15, 25, 40, 60, and 90 neighbors. We also included mean and median as simple statistics for crowd wisdom.

Evaluation metrics
We used the Receiver Operating Characteristic (ROC) and Precision-Recall curves, as well as their areas under the curves (AUROC and AUPR) as evaluation metrics. To tackle the sign indeterminacy from dimension reduction, we always computed these metrics twice, on the original output and on its negative, and selected the one with a larger area under the curve for comparison. For fair comparison, the same procedure was applied on supervised learning methods. In practice, sign indeterminancy can be solved by assuming more than half of the individuals have better-than-random responses, and then aligning crowd wisdom with the majority of the crowd.

Method comparison in cross validation
To compare crowd wisdom and supervised classifiers, we randomly split each dataset into a training set (containing 10, 20, 25, 40, 60, 80, or 90 percent of all samples) and a test set (for the rest), using sklearn.model selection.StratifiedShuffleSplit and requiring the number of questions to be larger than that of individuals in the training set. Supervised classifiers were trained on individual predictions against ground-truths in the training set, and then predicted for the test set. For crowd wisdom, we performed crowd wisdom on the full data (not using ground-truth) and then extracted predictions for the test set. Evaluation metrics were computed for every random split. The random split was repeated 200 times per split ratio per dataset.    See external file: supa3.pdf Figure S4: Crowd wisdom outperformed supervised learning in cross-validation. Empirical distributions and medians of AUROC (left) and AUPR (right) of all crowd wisdom and supervised learning methods in 200 cross-validations with 10%, 20%, 25%, 40%, 60%, 80%, or 90% (A to G) random partitions of training data are shown for the DREAM2 dataset. Method names include the numbers of nearest neighbors in brackets, and are italicized for supervised classifiers. Numbers next to the frames represent rankings of the methods in terms of median AUROC or AUPR. Colors reflect methods' relative rankings in performance.

Method comparison on binarized data
See external file: supa7.pdf Figure S5: Crowd wisdom outperformed supervised learning in cross-validation. Empirical distributions and medians of AUROC (left) and AUPR (right) of all crowd wisdom and supervised learning methods in 200 cross-validations with 25%, 40%, 60%, 80%, or 90% (A to E) random partitions of training data are shown for the skin cancer classification dataset. Method names include the numbers of nearest neighbors in brackets, and are italicized for supervised classifiers. Numbers next to the frames represent rankings of the methods in terms of median AUROC or AUPR. Colors reflect methods' relative rankings in performance.   (Table S2). (B) Comparison of dimension reduction and supervised learning in cross validation at 25% training data (cf Figure 3) for simulation 1. Color reflects relative ranking. (C, D, E, F) ROC (C, E) and Precision-Recall (D, F) curves for dimension reductions, existing crowd wisdoms, the class probability, and individual predictions of simulation 1. In C, D, the best parameter (in Figure 4A) was selected according to AUROC (C) or AUPR (D) for each parametric dimension reduction and PCA was selected for non-parametric dimension reduction. All methods are shown in E, F. Existing crowd wisdoms were performed on binarized input data. See external file: supsima5.pdf Figure S10: Crowd wisdom outperformed supervised learning in cross-validation in simulated dataset. Empirical distributions and medians of AUROC (left) and AUPR (right) of all crowd wisdom and supervised learning methods in 200 cross-validations with 10%, 20%, 25%, 40%, 60%, 80%, or 90% (A to G) random partitions of training data are shown for simulation 1.