Figure 1 lists the average testing accuracies of the gene selection methods with classifiers NMSC, SVM, NBC, and RF. Again, the performances of NBC-MMC, NMSC-MMC, NBC-MSC, and NMSC-MSC are close to one another; therefore, the average testing accuracies of the gene selection methods NBC-MMC, NMSC-MMC, and NBC-MSC are not listed in the figures. It indicates that the average testing accuracy of NMSC-MSC is the best, followed by GLGS, LOOCSFS, and SVM-RFE. SFS-LS bound, SFFS-LS bound, and T-TEST did not perform well. Figure 1 also demonstrates that, spanning several data sets and learning classifiers, the performance and stabilization of the gene selection method of NMSC-MSC is the best.

Table 1 provides the mean values and standard errors of the testing accuracies ms_hr(i), (i = 1, 2,..., 20) and the highest testing accuracies hs_hr(i), (i = 1, 2, ..., 20) under the highest training classification, respectively. After applying each classifier to each data set, the highest mean value of the ten gene selection methods is shaded. In each data set, the highest mean value in the shade is in bold. With the use of the four learning classifiers, under the best training, RFA, GLGS, LOOCSFS, SVMRFE, SFS-LSBOUND, SFFS-LSBOUND, and T-test respectively achieve the highest testing accuracies (HS_HR), 99.9%, 99.6%, 99.3%, 98.0%, 97.4%, 97.3%, and 96.8% for the leukemia data set; 99.5%, 98.6%, 93.0%, 99.2%, 95.1%, 96.1%, and 94.4% for lymphoma; 96.9%, 96.1%, 95.2%, 95.7%, 93.4%, 92.7%, and 94.0% for prostate; 91.1%, 90.5%, 86.8%, 86.8%, 87.1%, 86.0%, and 85.5% for colon; 94.0%, 91.1%, 85.0%, 85.1%, 76.2%, 76.2%, and 77.4% for CNS; and 85.9%, 83.7%, 80.3%, 80.4%, 81.5%, 81.3%, and 77.6% for the breast cancer data set. In applying the ten gene selection methods to the six benchmark data sets, all the highest testing accuracies are obtained from the gene set chosen by RFA.

Table 2 lists the number of occurrences for each gene selection method that achieved the best testing accuracy. Table 2 shows that 61 out of 67 highest mean values were obtained by MMC- or MSC-based methods; GLGS, LOOCSFS, and SVMRFE obtained the best twice, three times, and once, respectively; LSBOUND and T-TEST never got the best value. Results indicate that RFA outperforms other gene selection methods.

On the other side, to see whether the new methods are superior to others, regression models were built based on average testing accuracy (ms_hr) and highest testing accuracy (hs_hr), respectively, with data set (six benchmark microarray data set), gene selection method (four new methods and six other methods) and classifier (four classification methods) as independent variables. After adjusting data set effect and classifier effect, the main effects for the new feature selection methods (NBC-MMC, NMSC-MMC, NBC-MSC, and NMSC-MSC) and others (GLGS, LOOCSFS, SVMRFE, SFS-LSBOUND, SFFS-LSBOUND, and T-test) are 91.86%, 91.67%, 92.47%, 92.27%, 90.65%, 86.96%, 88.89%, 84.70%, 85.38%, and 83.93% for the highest testing accuracy, and 86.38%, 86.15%, 87.30%, 86.97%, 85.48%, 82.76%, 83.96%, 79.45%, 80.58%, and 79.36% for the average testing accuracy, respectively. Table 3 gives the p-values of testing superiority of each new method to other six methods, which are calculated based on one-tailed t-test from the output of the regression models. From the p-values, the performances of our new methods are statistically significantly better than all other methods (most p-values are <0.0001) except for GLGS. From Table 3 MSC-based methods (NBC-MSC, NMSC-MSC) are significantly better than GLGS based on both highest testing accuracy and average testing accuracy at a significance level of 0.05. Although the p-values for NBC-MMC and NMSC-MMC to GLGS are not small enough due to the small sample size (only six testing data sets) and therefore lower power, we would expect that the differences will be detected at lower significance levels if more data sets are used. To see whether the four new gene methods perform differently, we also test each pair of the four methods and calculate the p-values based on two-tailed t-test from the output of the regression models. All the p-values are bigger than 0.2, so the four new methods perform equally well.

Table 4 lists the mean values of the differences between the testing values (denoted as S_LPPO) by applying NMSC, SVM, NBC, and RF to LPPO and ms_hr. This table shows that, on average, LPPO is superior to the random strategy under the best training accuracies. In summary, spanning the six benchmark data sets, in comparison with ms_hr, LPPO improves the testing accuracy by 0.8% for NMSC, 0.7% for SVM, 0.4% for NBC, and 0.9% for RF on average.

Figure 2 gives the boxplots of the testing values with the use of learning classifier random forest for the feature sets from LPPO with RFA and varSelRF. The gene selection methods are NBC-MMC, NMSC-MMC, NBC-MSC, NMSC-MSC, and varSelRF from left to right in each subfigure. Figure 2 indicates that the testing accuracies by applying random forest to the feature sets of LPPO with RFA are better than those of varSelRF. In comparison with varSelRF, LPPO with RFA increases the average testing accuracy by about 5% for the leukemia data set, 9% for lymphoma, 3% for colon and prostate, 10% for CNS, and 14% for the breast cancer data set.

In microarray data analysis, generally, the number of features in the final feature set is far smaller than the total variables. Suppose the number of total variables is n, the number of features of the final feature set is m (m <<n). In forward feature selection, with the use of some learning classifier, the computational time is F(s, d) for a s×d feature matrix, here s is the number of data samples (s << n) and d is the feature dimensionality at each sample. Without losing the generality, if d₁<d₂, F(s, d₁) <F(s, d₂). The computational cost of our feature selection algorithm is analyzed as follows.

Let T₁ denote the total computational time for supervised learning

T 1 =   n * F s , 1   + n - 1 * F s , 2   +   …   +   n - m + 1 * F s , m ≤ n + ( n - 1 ) + . . . + ( n - m + 1 ) * F ( s , m ) = m * ( 2 * n - m + 1 ) 2 * F ( s , m )

Let T₂ denote the computational time for similarity calculation among the candidates and chosen genes, the calculation time between two single- variant vectors with s samples is C(s), then

T 2 ≤ ( n - 1 ) * C ( s ) + 2 * ( n - 2 ) * C ( s ) + . . . + m * ( n - m ) * C ( s ) = C ( s ) * 1 2 n m ( m + 1 ) - ∑ i = 1 m i 2

Due to the fact of m << n and s << n with microarray data, the computational cost of our feature selection is obtained by

T = T 1 + T 2 ~ O ( n )

Our method of RFA uses supervised learning to achieve the highest level of training accuracy and statistical similarity measures to choose the next variable with the least dependence on or correlation to the already identified variables as follows:

1. Insignificant genes are removed according to their statistical insignificance. Specifically, a gene with a high p-value is usually not differently expressed and therefore has little contribution in distinguishing normal tissues from tumor tissues or in classifying different types of tissues. To reduce the computational load, those genes should be removed. The filtered gene data is then normalized. Here we use the standard normalization method, MANORM, which is available from MATLAB bioinformatics toolbox.

2. Each individual gene is selected by supervised learning. A gene with highest classification accuracy is chosen as the most important feature and the first element of the feature set. If multiple genes achieve the same highest classification accuracy, the one with the lowest p-value measured by test-statistics (e.g., score test), is the target of the first element. At this point the chosen feature set, G₁, contains just one element, g₁, corresponding to the feature dimension one.

3. The (N+1)^st dimension feature set, G_N+1 = {g_1, g₂, ..., g_N, g_N+1} is obtained by adding g_N+1 to the N^th dimension feature set, G_N = {g_1, g₂, ..., g_N}. The choice of g_N+1 is described as follows:

Add each gene g_i (g_i ∉ G_N) into G_N and obtain the classification accuracy of the feature set G_N ∪{g_i}. The g_i (g_i ∉ G_N) associated with the group, G_N ∪{g_i} that obtains the highest classification accuracy, is the candidate for g_N+1 (not yet g_N+1). Considering the large number of variables, it is highly possible that multiple features correspond to the same highest classification accuracy. These multiple candidates are placed into the set C, but only one candidate from C will be identified as g_N+1. How to make the selection is described next.

To find the most informative (or least redundant) next feature g_N+1, two formulas may be designed by measuring the statistical similarity between the chosen feature set and each candidate. Here we use, say, Pearson's correlation coefficient 28 between chosen features g_n (g_n ∈ G_N , n = 1, 2, ..., N) and candidate g_c (g_c ∈ C) to measure the similarity.

In the first formula, the sum of the square of the correlation, SC, is calculated to measure the similarity and is defined as follows:

SC g c = ∑ n = 1 N co r 2 g c ,   g n n = 1 , 2 . . . N

Where, g_c ∈ C, g_n ∈ G_N.

Then selection of g_N+1 can be based on the Minimum Sum of the square of the Correlation (MSC), that is,

g N + 1 ← { g c | SC ( g c ) = min ( SC ) . g c ∈ C }

In the second formula, the maximum value of the square of the correlation, MC, is calculated:

MC ( g c ) = max ( co r 2 ( g c , g n ) ) , n = 1 , 2 , . . . , N

Where, g_c ∈ C, g_n ∈ G_N.

The selection of g_N+1 follows the criterion that the MC value is the minimum, which we call Minimum of Maximum value of the square of the Correlation (MMC).

g N + 1 ← { g c | MC ( g c ) = min ( MC ) . g c ∈ C }

In the methods mentioned above, a feature is recursively added to the chosen feature set based on supervised learning and the similarity measures. With the use of a classifier XXX, we call the first gene selection method XXX-MSC and the second one XXX-MMC. For example, if the classifier is Naive Bayes Classifier (NBC), we call the two strategies NBC-MSC and NBC-MMC, respectively.

We want to find a combination of features (genes) that yields the best performance on breaking down solvents. Normally, with the recursive addition for the next feature, the training accuracy will increase and reach a peak classification performance at some point, and then may maintain it with subsequent feature additions; but after that the training accuracy may decrease. Generally speaking, all strategies for determining the final feature set should be based on the best training classification. In high-volume data analysis, it is common that the best training accuracy corresponds to different feature sets; that is, multiple feature sets achieve the same highest training accuracy. However, although all these feature sets are associated with the same highest training accuracy, the testing accuracy of these feature sets may be different. Among these highest training feature sets, the one having the best testing accuracy is called the optimal feature set, which is highly complicated to characterize when a sample size is small. Either applying different gene methods to the same training samples, or applying the same gene selection method to different training samples, or applying different learning classifiers to the same training samples, will produce a different optimization of the feature set. Pochet et al. 29 presented a method of determining the optimal number of genes by means of a cross-validation procedure; the drawback of this method is that it actually utilizes whole data information, including training samples and testing samples.

How do we choose the optimal feature set? If there are multiple best training classifications, a random choice, called random strategy, works for best training classification. In the recursive addition of the features, for training samples, a classification model is one of the best methods. But for testing samples, at this point, the classification model may not be optimal because of the difference between the training samples and the testing samples; the optimal classification model will lag in appearance (see Figure 1). Based on this observation, we propose the following algorithm for optimization.

Under feature dimension j, the training accuracy of the i^th experiment is r(i, j). If the feature set G_k, corresponding to feature dimension k, has the best training accuracy in the trainings from the feature set G₁ to G_D, corresponding to the feature dimensions from 1 to D, let HR denote the set that contains all the combinations of G_k, corresponding to all the feature set having the highest classification accuracy under feature dimension 1 to D.

H R = { G k | r ( i , k ) = max ( r ( i , ∙ ) ) , 1 ≤ k ≤ D }

In general, the best classification model for testing samples will lag in appearance behind the initial best training model. We will exclude the elements of HR that correspond to the initial best training. The remaining elements in HR constitute the candidate set HRC for optimization.

Each element in HRC is associated with the best training accuracy. We set a peephole for each element and choose the element associated with the optimal peephole. The details are described as follows:

a. For each element G_k ∈ HRC, the peephole over G_k with length of 2l+1 covers the feature sets G_k-l, G_k-l+1, ..., G_k , ..., G_k+l-1, G_k+l, corresponding to the training accuracy r(i, k-l), r(i, k-l+1), ..., r(i, k), ..., r(i, k+l-1), r(i, k+l). The mean training value of the peephole is denoted by mp_r(i, k).

m p _ r ( i , k ) = ( 1 ∕ ( 2 l + 1 ) ) ∑ m = k - l m = k + l r ( i , m )

The peephole with the best classification of mp_r is then chosen as the optimal one.

b. If there are multiple optimal peepholes, then we apply random forest to these peepholes and check the mean values of the Out-of-Bag (OOB) error rates 242530. The feature sets G_k-l, G_k-l+1, ..., G_k,, ..., G_k+l-1, G_k+l correspond to the OOB errors, oob_e(i, k-l), oob_e(i, k-l+1), ..., oob_e(i, k), ..., oob_e(i, k+l-1), oob_e(i, k+l). The mean value of the OOB errors is denoted by mp_oob_e(i, k)

m p _ o o b _ e ( i , k ) = ( 1 ∕ ( 2 l + 1 ) ) ∑ m = k - l m = k + l o o b _ e ( i , m )

The peephole with minimum mp_oob_e is the optimal one.

c. If there are multiple peepholes corresponding to the best mp_r and minimum mp_oob_e, then set l +1 → l, and repeat 'a' to 'c', until a unique optimal peephole is determined.

d. The feature set located at the center of the final optimal peephole is chosen as the final optimal feature set.

This optimization of RFA is called Lagging Prediction Peephole Optimization (LPPO). Figure 3 briefly outlines the LPPO on the prostate data set, which was studied by Singh et al. 31.

The following six benchmark microarray data sets have been extensively studied and used in our experiments to compare the performances of our methods with others. Data sources that are not specified are available at: http://www.broad.mit.edu/cgi-bin/cancer/datasets.cgi.

1) The LEUKEMIA data set consists of two types of acute leukemia: 48 acute lymphoblastic leukemia (ALL) samples and 25 acute myeloblastic leukemia (AML) samples with over 7129 probes from 6817 human genes. It was studied by Golub et al. 32.

2) The LYMPHOMA data set consists of 58 diffuse large B-cell lymphoma (DLBCL) samples and 19 follicular lymphoma (FL) samples. It was studied by Shipp et al. 33. The data file, lymphoma_8_lbc_fscc2_rn.res, and the class label file, lymphoma_8_lbc_fscc2.cls were used in our experiments for identifying DLBCL and FL.

3) The PROSTATE data set used by Singh et al. 31 contains 52 prostate tumor samples and 50 non-tumor prostate samples.

4) The COLON cancer data set used by Alon et al. 34 contains 62 samples collected from colon-cancer patients. Among them, 40 tumor biopsies are from tumors, and 22 normal biopsies are from healthy parts of the colons of the same patients. Based on the confidence in the measured expression levels, 2000 genes were selected. The data source is available at: http://microarray.princeton.edu/oncology/affydata/index.html.

5) The Central Nervous System (CNS) embryonal tumor data set that was originally studied by Pomeroy et al. 35 contains 60 patient samples. Among them, 21 are survivors who are alive after treatment, and 39 are failures who succumbed to their diseases. There are 7129 genes.

6) The Breast cancer data set studied by Van et al. 36 contains 97 patient samples, 46 of which are relapse patients who had developed distance metastases within 5 years, and 51 patients who are non-relapsed who remained healthy for at least 5 years from the distance after their initial diagnosis. This data source is available at: http://www.rii.com/publications/2002/vantveer.htm.

Our experiments are designed as follows:

1. The data sets are first divided randomly into training samples and testing samples. The ratio of training samples to testing samples is approximately 1:1 in each class.

2. Recursive feature additions with Naive Bayes Classifier (NBC) and Nearest Mean Scaled Classifier (NMSC) for gene selection (NBC-MSC, NBC-MMC, NMSC-MSC, and NMSC-MMC) were applied to the training samples for gene selection. Different feature sets of the gene expression data are produced under feature dimensions 1 to 100. We compared the above proposed methods to several recently developed and published gene selection methods: LOOCSFS, GLGS, SVMRFE, SFFS-LS bound, SFS-LS bound, and also T-TEST.

3. To compare different gene selection methods, the learning classifiers including NBC, NMSC, SVM 3738, and Random Forest are applied to the testing samples.

4. The experiments were performed in 20 runs, and the average testing accuracies were compared to evaluate performance.