# An Improved Fuzzy Based Missing Value Estimation in DNA Microarray Validated by Gene Ranking

Authors: Sujay Saha; Anupam Ghosh; Dibyendu Bikash Seal; Kashi Nath Dey
Journal: Advances in Fuzzy Systems
 (2016)

---

## Abstract

Most of the gene expression data analysis algorithms require the entire gene expression matrix without any missing values. Hence, it is necessary to devise methods which would impute missing data values accurately. There exist a number of imputation algorithms to estimate those missing values. This work starts with a microarray dataset containing multiple missing values. We first apply the modified version of the fuzzy theory based existing method LRFDVImpute to impute multiple missing values of time series gene expression data and then validate the result of imputation by genetic algorithm (GA) based gene ranking methodology along with some regular statistical validation techniques, like RMSE method. Gene ranking, as far as our knowledge, has not been used yet to validate the result of missing value estimation. Firstly, the proposed method has been tested on the very popular Spellman dataset and results show that error margins have been drastically reduced compared to some previous works, which indirectly validates the statistical significance of the proposed method. Then it has been applied on four other 2-class benchmark datasets, like Colorectal Cancer tumours dataset (GDS4382), Breast Cancer dataset (GSE349-350), Prostate Cancer dataset, and DLBCL-FL (Leukaemia) for both missing value estimation and ranking the genes, and the results show that the proposed method can reach 100% classification accuracy with very few dominant genes, which indirectly validates the biological significance of the proposed method.


---

## Body

## 1. Introduction

Microarray expression analysis is a widely used technique for profiling mRNA expression. The mRNA carries genetic information from DNA to the ribosome, where they specify the amino acid sequence of the protein products of gene expression. Microarray datasets often contain missing values which may occur due to various reasons including imperfections in data preparation steps (e.g., poor hybridization and chip contamination by dust and scratches) that create erroneous and low-quality values, which are usually discarded and referred to as missing. It is common for gene expression data to contain at least 5% missing values [1]. Most of the microarray data analysis algorithms, such as gene clustering, disease (experiment) classification, and gene network design, require the complete information, that is, the entire gene expression matrix without any missing values. Hence, different imputation techniques should be used which would accurately impute multiple missing data values. Numerous imputation algorithms have been proposed to estimate the missing values. At first, we have applied modified version of our existing imputation technique LRFDVImpute [2] that first finds a subset of similar genes using the fuzzy difference vector (FDV) algorithm used in [3] where gene expression profiles have been considered as continuous time series curves and then use linear regression on the subset to estimate the missing value. We have considered estimating only those genes with one, two, or three missing values since these genes constitute 5–10% of the entire dataset. Absolute error has been calculated from the difference between the original value and the estimated value. Root Mean Square Error (RMSE) of those absolute errors is then determined.The workflow for the first phase has been shown in Figure1.Figure 1
Workflow of the proposed missing value estimation technique.After that we rank those genes to find the top ranked genes [4]. We have used a hypothesis test, Wilcoxon rank sum test [5], to sort the features (genes) and rank them in order of their p values and select top n genes from them, thereby reducing the dimensionality, where n is the population size that has been used later for GA. The reduced set of genes has then been ranked by our GA method. The two ranks, one by Wilcoxon method and the other by our GA method, are then compared. The top m genes (value of m defined by the user) selected by our method are then used for classification using support vector machine (SVM) classifiers. The performance of classification justifies the efficiency of the ranking method used. Figure 2 shows the workflow for this phase.Figure 2
Workflow of the proposed gene ranking technique.Once this is done, we then forcibly make some cells missing in the top ranked genes and again estimate them using the same missing value estimation technique. Finally, we rank them once more to find the top ranked genes. Results show that most of the top ranked genes remain the same, which validates the proposed missing value estimation technique biologically as far as the estimation is concerned.


## 2. Present State of the Art

As discussed earlier, various statistical and analytical methods used for gene expression analysis are not robust to missing values and require the complete gene expression matrix for providing accurate results. Hence, it is necessary to devise accurate methods which would impute data values when they are missing. Many imputation methods have been proposed. The earliest method, named as row averaging or filling with zeroes, used to fill in the gaps for the missing values in gene dataset with zeroes or with the row average.KNNImpute method proposed in [1] selects genes with expression profiles similar to the gene of interest to impute missing values. After experimenting with a number of metrics to calculate the gene similarity, such as Pearson correlation, Euclidian distance, and variance minimization, it was found that Euclidian distance was a sufficiently accurate norm.The SVDImpute method, proposed in [1], uses Singular Value Decomposition of matrices to estimate the missing values of a DNA microarray. This method works by decomposing the gene data matrix into a set of mutually orthogonal expression patterns that can be linearly combined to approximate the expression of all genes in the dataset. These patterns, which in this case are identical to the principle components of the gene expression matrix, are further referred to as eigengenes [6, 7].Another method named as LLSImpute [8] represents a target gene with missing values as a linear combination of similar genes. The similar genes are chosen by k-nearest neighbours or k coherent genes that have large absolute values of correlation coefficients followed by least square regression and estimation.BPCAImpute method, proposed in [9], uses a Bayesian estimation algorithm to predict missing values. BPCA suggests using the number of samples minus 1 as the number of principal axes. Since BPCA uses an EM-like repetitive algorithm to estimate missing values, it needs intensive computations to impute missing values.Another algorithm for time series gene expression analysis is presented in [10] that permits the principled estimation of unobserved time points, clustering, and dataset alignment. Each expression profile is modelled as a cubic spline (piecewise polynomial) that is estimated from the observed data and every time point influences the overall smooth expression curve. The alignment algorithm uses the same spline representation of continuous time series gene expression profiles.FDVImpute method, proposed in [11], incorporates some fuzziness to estimate the missing value of a DNA microarray. The first step selects nearest (most similar) genes of the target gene (whose some component is missing) using fuzzy difference vector algorithm. Then the missing cell is estimated by using least square fit on the selected genes in the second step.FDVSplineImpute, presented in [3], takes into account the time series nature of gene expression data and permits the estimation of missing observations using B-splines of similar genes from fuzzy difference vectors.Another method, LRFDVImpute, proposed in [2], estimates multiple missing observations by first finding the most similar genes of the target gene and then applying the linear regression on those similar genes. This approach works in two stages. At the first stage, it estimates the real missing cells of SPELLMAN_COMBINED dataset and at the later stage, it makes some cells miss forcefully of the same dataset and then using the estimated results from the first step, this approach estimates those missed cells using the same approach used earlier. Absolute error has been calculated from the difference between the original value and the estimated value. Root Mean Square Error (RMSE) of those absolute errors is then determined.Extracting relevant information from microarray data is also difficult because of the inherent characteristics of the datasets, where there are the thousands of variables (genes) and very few numbers of samples. Finding out the set of significant genes or, in other words, the most differentially expressed genes, by studying data from tissues affected or unaffected by cancer cells, is an important task. This problem can be termed as gene selection. Several techniques have been used to rank genes and find out the most significant ones.In [12], the algorithm used discriminant partial least squares (DPLS) and fuzzy clustering methods to interpret the gene expression patterns of acute leukemia and identify leukemia subtypes.In [13], the proposed method used Mann-Whitney test and k-sample Kruskal-Wallis ANOVA test to rank genes. Dimension reduction was done using k-means clustering and PCA and classification performed using ANN trained during 8-fold cross-validation with recursive feature elimination (RFE) and leave-one-out testing.In [14], the algorithm proposed a gene selection method based on Wilcoxon rank sum test and SVM. Wilcoxon rank sum test was used to select a subset of genes and then each selected gene is trained and tested using SVM classifier with linear kernel separately, and genes with high testing accuracy rates were chosen to form the final reduced gene subset. Classification was performed on two datasets: Breast Cancer [15] and ALL/AML Leukemia [16] using leave-one-out cross-validation (LOOCV).A hybrid GA/SVM approach is proposed for gene selection in [17], where a fuzzy logic based preprocessing tool is used to reduce dimensionality, GA for finding out the most frequent genes, and a SVM classifier used for classification. Experiments were performed on two well-known cancer datasets, Leukemia [16] and Colon [18], and results were compared with six other methods.A multiobjective genetic approach is proposed in [19] for simultaneous clustering and gene ranking where a method to simultaneously optimize the feature ranking and clustering has been used. NSGA-II (Nondominated Sorting Genetic Algorithm-II) [20] has been used as a multiobjective evolutionary algorithm to optimize the chromosomes.In [21], the proposed algorithm uses feature selection method based on genetic algorithms (GAs) and classification methods focusing on constructive neural networks (CNNs), C-Mantec. Several comparison results on six public cancer databases are provided using other feature selection strategy (Stepwise Forward Selection method) and different classification techniques (LDA, SVM, and Naive Bayes).A PSO based graph theoretic approach, proposed in [22], is used for identifying the nonredundant gene markers from microarray gene expression data. The microarray data is first converted into a weighted undirected complete feature graph where the nodes represent the genes having gene’s relevance as node weights and the edges are weighted in order of correlation among the genes. The densest subgraph having minimum average edge weight (similarity) and maximum average node weight (relevance) is then identified from the original feature graph. Binary particle swarm optimization is then applied for minimizing the average edge weight (correlation) and maximizing the average node weight (gene relevance) through a single objective function.A web based tool DWFS, proposed in [23], is used to select significant features for a variety of problems efficiently. The search strategy is implemented using Parallel Genetic Algorithm. DWFS also applies various filtering methods as a preprocessing step in the feature selection process. It also uses three classifiers, like KNN classifier, Naive Bayes Classifier, and the combination of these two. Experiments using datasets taken from different biomedical applications show the efficiency of DWFS and lead to a significant reduction of the number of features without sacrificing performance as compared to several widely used existing methods.


## 3. Proposed Method

### 3.1. Missing Value Estimation Using Linear Regression

This phase of the work modifies an existing method LRFDVImpute for estimating missing values present in the microarray dataset using linear regression. Earlier version of LRFDVImpute inserts the newly estimated gene into the training data after estimation of each target gene. In this way, the newly estimated gene is taken into consideration while estimating the next target gene. This process has the risk of increasing the error while estimating the subsequent genes since the error term is cumulatively multiplied. To overcome this problem, modified LRFDVImpute does not add the target gene to the training data after it has been estimated. This way, the training gene set size remains constant and with increasing membership values ofθ, the size of training data reduces. The effects of modifications have been studied and results are shown in the experimental results section. In our problem, the genes with missing values in the (p
×
q) (p is the number of genes and q is the number of samples) dataset are to be estimated. The method of finding a similar gene as used in [3] using fuzzy difference vector (FDV) algorithm is described below.Target Row/Testing Data. The row whose missing value is being estimated: a target row may have multiple missing values but in a single run, a single value is estimated.Similar Rows/Training Data. The rows that are similar to the target row: in this case only those rows are selected that have no missing values. Before applying the similarity measures all the columns from the complete matrix are removed that correspond to missing values in target row.Letg
1
,
g
2
,
…
,
g
p be the set of genes in the dataset. Let ith g
i be the target gene, that is, the gene with m missing values. We remove the columns having missing values from the entire dataset. Let the resultant matrix contain (q
-
m) columns. Each target gene i is compared with each of the similar rows in the dataset. For the ith gene g
i, the difference vector V
i of g
i is calculated as follows:(1)
D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
i
,
j
=
g
i
j
-
g
i
j
+
1
,
1
≤
j
≤
q
-
m
-
1
.Once the difference vectors are calculated for each of the target rows and the similar rows, sayD
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 (for target row) and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 (for similar row), we then calculate M
e
m
b
e
r
s
h
i
p
(
i
) to obtain the number of matches between difference vectors D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 for each target gene g
i. A match in the jth component of the vectors D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 is determined by whether the signs of D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1
(
i
,
j
) and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2
(
i
,
j
) are the same or not. M
e
m
b
e
r
s
h
i
p
(
i
) defines the degree of match between the distribution of the target gene and the similar gene. We then define a membership grade for g
i as follows:(2)
m
e
m
g
r
a
d
e
i
=
M
e
m
b
e
r
s
h
i
p
i
q
-
m
-
1
.The genes in the training data that have a membership value greater than a chosen membership grade θ are considered to be a part of the similar genes.The steps for estimation can be summarized below:(1)
Load the dataset with missing values.
(2)
Calculate the missing number of columns for each gene and start with the first row with the least number of missing values (for our dataset it is 1).
(3)
Compute the corresponding membership grade for the target gene from the training data using the FDV algorithm as shown above.
(4)
Estimate the missing value using linear regression.
(5)
Obtain coefficients of the regression from the linear model object lmObj.
(6)
Add a bias of 1 at the beginning of the target row to allow for the bias parameter.
(7)
Perform a vector multiplication between the modified target row and the coefficients of regression and add the obtained vector’s elements together to get the estimated value.
(8)
Replace the missing value with the estimated value.
(9)
Go to step(
2
) and repeat the above steps to fill the missing values unless the mentioned “least number of missing values” in step (
2
) is less than or equal to 3.Although we mentioned here that we go on filling the missing value till a point, it is not true. In between we stop this filling in process to do assessment of our algorithm.After we have filled in all the missing values corresponding to rows with single missing values we select a particular collection of row-column positions corresponding to rows that did not have missing values initially and deliberately treat the values at these positions as missing and use the exact same process to estimate the values.The same collection of row-column positions are again used when the algorithm has filled up all the rows up to two missing vales and then when it has filled up missing values existing in rows with up to three missing values.


### 3.2. Gene Ranking Using Genetic Algorithm

In phase 2 of the proposed work, the result of the missing value estimation procedure carried out in phase 1 is biologically validated by ranking the genes using GA. Since a characteristic of gene expression microarray data is that the number of variables (genes) far exceeds the number of samplesn, we must reduce its dimension. Executing GA on the original dataset is quite impractical and time consuming. As a preprocessing step, we have reduced the dimension using Wilcoxon rank sum test.


#### 3.2.1. Dimension Reduction Using Wilcoxon Rank Sum Test (WRST)

The inputs to the Wilcoxon rank sum test function are the two gene sets, the diseased set and the normal set, both of which have individually undergone the missing value estimation procedure (if there was any missing value). The two gene sets may have different number of samples. Let us consider that the diseased set is a (p
×
q
1) sized gene expression data, where p is the number of genes and q
1 is the number of samples, and the normal set has a size (p
×
q
2), where q
2 is the number of samples. The Wilcoxon rank sum function processes the two datasets in order to find out for which genes the null hypothesis is accepted or rejected. It returns two values, p value and h-value, as discussed earlier. The null hypothesis for our problem is that the genes are not differentially expressed; that is, either all the samples have come from diseased patients or they have come from normal patients. The alternative hypothesis can be that genes are differentially expressed. We record the p values and h-values for each gene.In the next step, we consider only those genes for which the alternative hypothesis holds (h
=
1) at the significance level alpha and sort the genes according to the p values thereby ranking the genes. We then select the topmost k genes, where k is the population size that has been used for GA later. Thus, we have two reduced populations, one representing diseased and the other representing normal tissues. Let (
X
i
j
)
p
×
q
1 be the diseased set, where p is the reduced set of genes and q
1 is the number of samples, respectively, and let (
Y
i
j
)
p
×
q
2 be the normal set, where q
2 is the number of samples.


#### 3.2.2. Chromosome Representation and Initial Population for GA

The reduced gene sets(
X
i
j
)
p
×
q
1and (
Y
i
j
)
p
×
q
2 serve as the initial population for the genetic algorithm step. They contain pop_size number of genes which is preselected by the user. We use real value encoding to represent each chromosome; that is, X
i
j and Y
i
j are the measurements recorded for the ith gene and jth sample for each population, respectively.


#### 3.2.3. Fitness Calculation

The fitness for each gene in the reduced gene sets is again calculated by a method similar to that used in [14] where gene expression profiles have been considered as continuous time series curves.In our problem, we have two populations, one for the diseased tissues and the other for the normal tissues. The two populations contain the same number of genesp but may have different number of samples. In that case, we consider the minimum of the two and extract the same number of samples from each set.Letg
1
,
g
2
,
…
,
g
p be the reduced set of genes in each population. If q
=
min
⁡
q
1
,
q
2, then for each population, the difference vector V
i of g
i is calculated using (1). Once the difference vectors are calculated for each of the two populations, say D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 (for diseased) and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 (for normal), the number of matches between the difference vectors M
e
m
b
e
r
s
h
i
p
(
i
) and the membership grade for g
i is computed using (2).The fitness of geneg
i is the reciprocal of m
e
m
g
r
a
d
e
(
i
) and is calculated as(3)
f
i
t
i
=
1
m
e
m
g
r
a
d
e
i
.This signifies that the more similar the distributions of gene g
i in the two populations are, the less differentially expressed the gene is, and vice versa. Thus, a fitter gene will have different distributions in the two populations. We then rank the genes in order of their fitness.


#### 3.2.4. Elitism

We have used an elitist version of GA where the best chromosomes are carried forward to the next generation unchanged; that is, the crossover and mutation operators are not applied on the best chromosomes. This technique ensures faster convergence of the process by keeping track of the best solutions.


#### 3.2.5. Selection

For selection, we have used a roulette wheel technique where genes are selected based on their relative fitness values. The better the chromosomes are, the more chances to be selected they have. Let count be the number of elite children. We construct a roulette wheel as follows [22]:(i)
Calculate the fitness valuef
i
t
(
i
) for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p.
(ii)
Find the total fitness of the populationF = ∑
i
=
c
o
u
n
t
+
1
p
o
p
_
s
i
z
e
f
i
t
(
g
i
).
(iii)
Calculate the probability of selectionp
i for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p:(4)
p
i
=
f
i
t
g
i
F
.
(iv)
Calculate a cumulative probabilityc
i for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p:(5)
c
i
=
∑
j
=
1
i
p
j
.We now spin the wheel (pop_size − count) times and select a single chromosome as follows:(i)
Generate a random number (float)r between 0 and 1.
(ii)
Ifr
<
c
1, we select the first chromosome g
1; otherwise, select the ith chromosome g
i (2
≤
i
≤
pop_size
-
count) such that c
i
-
1
<
r
≤
c
i.Some chromosomes get selected more than once. According toSchema Theorem [24], the best chromosomes get more copies, the average stay even, and the worst die off.


#### 3.2.6. Crossover

For crossover, we proceed as follows.For each chromosomeg
i in the population,(i)
generate a random number (float)r between 0 and 1,
(ii)
ifr
<
p
c
r
o
s
s (crossover probability), we select the given chromosome for crossover.We have used single point crossover where the crossover site is also generated randomly in the range [
1
⋯
q
-
1
], where q is the number of samples. Thus after crossover, a pair of parent chromosomes generates a pair of offspring chromosomes [25]. The new population obtained after crossover contains the new generation produced by crossover as well as the elite children that did not undergo crossover. This new population is used in the mutation process.


#### 3.2.7. Mutation

A nonuniform mutation operator as proposed in literature [25] has been used here. The new operator is defined as follows:(i)
A random experiment is carried out which produces an outcome which is either 0 or 1.
(ii)
Another random number pos is generated in the range[
1
⋯
q
-
1
], where q is the number of samples, to select the mutation site.
(iii)
Letg
i
t
=
[
g
i
(
1
)
,
g
i
(
2
)
,
…
,
g
i
(
j
)
,
…
,
g
i
(
q
)
],  1
≤
j
≤
q, be the chromosome, and let g
i
(
j
) be selected for mutation. Domain of g
i is [
l
b
,
u
b
]; the resultant vector g
i
t
+
1
=
[
g
i
(
1
)
,
g
i
(
2
)
,
…
,
g
i
j
′
,
…
,
g
i
(
q
)
]:
(6)
g
i
j
′
=
g
i
j
+
Δ
t
,
ub
-
g
i
j
,
if  outcome  is
0
g
i
j
-
Δ
t
,
g
i
j
-
lb
,
if  outcome  is
1
,where t is the generation number and the function Δ
(
t
,
y
) returns a value in the range [
0
⋯
y
] such that the probability of Δ
(
t
,
y
) being close to 0 increases as t increases. This property causes this operator to search the space uniformly initially (when t is small) and very locally at later stages.Δ
(
t
,
y
) is calculated as(7)
Δ
t
,
y
=
y
1
-
r
1
-
t
/
T
β
,where r is a random number in the range [
0
⋯
1
], T is the maximum number of generations preselected by the user, and β is a system parameter determining the degree of uniformity. We have used β
=
2 for our experiment.The entire genetic transformation has been performed on one population with respect to the other. We made the diseased gene set to undergo genetic transformation while fitness evaluation has been made with respect to the normal gene set. The opposite transformation will produce similar results.Once the genetic transformations are done, we obtain a final population set (here, genetically transformed diseased gene set) which have been ranked in order of their fitness. We compare the two ranks, one by the Wilcoxon method and the other by our GA method. A threshold of±2 has been considered while comparing the two ranks. Results show that there is a good percentage of matches in the two ranks. Moreover, we find out the top ranked genes produced by both methods and the significant genes produced by the two methods are also similar. This also validates the result of the missing value estimation method carried out in phase 1.


### 3.3. Gene Classification Using SVM

In order to prove the significance of ranking by our GA method, we perform classification. The top rankedn genes,n’
{
5,10,15,20,25
}, ranked by our GA method are used for the purpose. We use k-fold LOO cross-validations, where k is varied from one dataset to another depending on the number of samples. For cross-validation, we have divided our dataset into two sets, a training set and a testing set, in 80 : 20 ratio. The reason behind taking this ratio is that 80 : 20 is a commonly occurring ratio, which is often referred to as Pareto Principle. So, if there are n samples in the training set and m
-
n samples in the test set, where m is the total number of samples, the training set is divided into k equal sized subsets. Of the k subsets, one subset is retained for validation and the remaining k
-
1 subsets are used as training data. Thus, k SVM classifiers with linear kernel are trained using the n training subsets. The classification accuracy rates are recorded and the classifier with the best accuracy rate is used to test the m
-
n samples.


## 3.1. Missing Value Estimation Using Linear Regression

This phase of the work modifies an existing method LRFDVImpute for estimating missing values present in the microarray dataset using linear regression. Earlier version of LRFDVImpute inserts the newly estimated gene into the training data after estimation of each target gene. In this way, the newly estimated gene is taken into consideration while estimating the next target gene. This process has the risk of increasing the error while estimating the subsequent genes since the error term is cumulatively multiplied. To overcome this problem, modified LRFDVImpute does not add the target gene to the training data after it has been estimated. This way, the training gene set size remains constant and with increasing membership values ofθ, the size of training data reduces. The effects of modifications have been studied and results are shown in the experimental results section. In our problem, the genes with missing values in the (p
×
q) (p is the number of genes and q is the number of samples) dataset are to be estimated. The method of finding a similar gene as used in [3] using fuzzy difference vector (FDV) algorithm is described below.Target Row/Testing Data. The row whose missing value is being estimated: a target row may have multiple missing values but in a single run, a single value is estimated.Similar Rows/Training Data. The rows that are similar to the target row: in this case only those rows are selected that have no missing values. Before applying the similarity measures all the columns from the complete matrix are removed that correspond to missing values in target row.Letg
1
,
g
2
,
…
,
g
p be the set of genes in the dataset. Let ith g
i be the target gene, that is, the gene with m missing values. We remove the columns having missing values from the entire dataset. Let the resultant matrix contain (q
-
m) columns. Each target gene i is compared with each of the similar rows in the dataset. For the ith gene g
i, the difference vector V
i of g
i is calculated as follows:(1)
D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
i
,
j
=
g
i
j
-
g
i
j
+
1
,
1
≤
j
≤
q
-
m
-
1
.Once the difference vectors are calculated for each of the target rows and the similar rows, sayD
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 (for target row) and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 (for similar row), we then calculate M
e
m
b
e
r
s
h
i
p
(
i
) to obtain the number of matches between difference vectors D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 for each target gene g
i. A match in the jth component of the vectors D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 is determined by whether the signs of D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1
(
i
,
j
) and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2
(
i
,
j
) are the same or not. M
e
m
b
e
r
s
h
i
p
(
i
) defines the degree of match between the distribution of the target gene and the similar gene. We then define a membership grade for g
i as follows:(2)
m
e
m
g
r
a
d
e
i
=
M
e
m
b
e
r
s
h
i
p
i
q
-
m
-
1
.The genes in the training data that have a membership value greater than a chosen membership grade θ are considered to be a part of the similar genes.The steps for estimation can be summarized below:(1)
Load the dataset with missing values.
(2)
Calculate the missing number of columns for each gene and start with the first row with the least number of missing values (for our dataset it is 1).
(3)
Compute the corresponding membership grade for the target gene from the training data using the FDV algorithm as shown above.
(4)
Estimate the missing value using linear regression.
(5)
Obtain coefficients of the regression from the linear model object lmObj.
(6)
Add a bias of 1 at the beginning of the target row to allow for the bias parameter.
(7)
Perform a vector multiplication between the modified target row and the coefficients of regression and add the obtained vector’s elements together to get the estimated value.
(8)
Replace the missing value with the estimated value.
(9)
Go to step(
2
) and repeat the above steps to fill the missing values unless the mentioned “least number of missing values” in step (
2
) is less than or equal to 3.Although we mentioned here that we go on filling the missing value till a point, it is not true. In between we stop this filling in process to do assessment of our algorithm.After we have filled in all the missing values corresponding to rows with single missing values we select a particular collection of row-column positions corresponding to rows that did not have missing values initially and deliberately treat the values at these positions as missing and use the exact same process to estimate the values.The same collection of row-column positions are again used when the algorithm has filled up all the rows up to two missing vales and then when it has filled up missing values existing in rows with up to three missing values.


## 3.2. Gene Ranking Using Genetic Algorithm

In phase 2 of the proposed work, the result of the missing value estimation procedure carried out in phase 1 is biologically validated by ranking the genes using GA. Since a characteristic of gene expression microarray data is that the number of variables (genes) far exceeds the number of samplesn, we must reduce its dimension. Executing GA on the original dataset is quite impractical and time consuming. As a preprocessing step, we have reduced the dimension using Wilcoxon rank sum test.


### 3.2.1. Dimension Reduction Using Wilcoxon Rank Sum Test (WRST)

The inputs to the Wilcoxon rank sum test function are the two gene sets, the diseased set and the normal set, both of which have individually undergone the missing value estimation procedure (if there was any missing value). The two gene sets may have different number of samples. Let us consider that the diseased set is a (p
×
q
1) sized gene expression data, where p is the number of genes and q
1 is the number of samples, and the normal set has a size (p
×
q
2), where q
2 is the number of samples. The Wilcoxon rank sum function processes the two datasets in order to find out for which genes the null hypothesis is accepted or rejected. It returns two values, p value and h-value, as discussed earlier. The null hypothesis for our problem is that the genes are not differentially expressed; that is, either all the samples have come from diseased patients or they have come from normal patients. The alternative hypothesis can be that genes are differentially expressed. We record the p values and h-values for each gene.In the next step, we consider only those genes for which the alternative hypothesis holds (h
=
1) at the significance level alpha and sort the genes according to the p values thereby ranking the genes. We then select the topmost k genes, where k is the population size that has been used for GA later. Thus, we have two reduced populations, one representing diseased and the other representing normal tissues. Let (
X
i
j
)
p
×
q
1 be the diseased set, where p is the reduced set of genes and q
1 is the number of samples, respectively, and let (
Y
i
j
)
p
×
q
2 be the normal set, where q
2 is the number of samples.


### 3.2.2. Chromosome Representation and Initial Population for GA

The reduced gene sets(
X
i
j
)
p
×
q
1and (
Y
i
j
)
p
×
q
2 serve as the initial population for the genetic algorithm step. They contain pop_size number of genes which is preselected by the user. We use real value encoding to represent each chromosome; that is, X
i
j and Y
i
j are the measurements recorded for the ith gene and jth sample for each population, respectively.


### 3.2.3. Fitness Calculation

The fitness for each gene in the reduced gene sets is again calculated by a method similar to that used in [14] where gene expression profiles have been considered as continuous time series curves.In our problem, we have two populations, one for the diseased tissues and the other for the normal tissues. The two populations contain the same number of genesp but may have different number of samples. In that case, we consider the minimum of the two and extract the same number of samples from each set.Letg
1
,
g
2
,
…
,
g
p be the reduced set of genes in each population. If q
=
min
⁡
q
1
,
q
2, then for each population, the difference vector V
i of g
i is calculated using (1). Once the difference vectors are calculated for each of the two populations, say D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 (for diseased) and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 (for normal), the number of matches between the difference vectors M
e
m
b
e
r
s
h
i
p
(
i
) and the membership grade for g
i is computed using (2).The fitness of geneg
i is the reciprocal of m
e
m
g
r
a
d
e
(
i
) and is calculated as(3)
f
i
t
i
=
1
m
e
m
g
r
a
d
e
i
.This signifies that the more similar the distributions of gene g
i in the two populations are, the less differentially expressed the gene is, and vice versa. Thus, a fitter gene will have different distributions in the two populations. We then rank the genes in order of their fitness.


### 3.2.4. Elitism

We have used an elitist version of GA where the best chromosomes are carried forward to the next generation unchanged; that is, the crossover and mutation operators are not applied on the best chromosomes. This technique ensures faster convergence of the process by keeping track of the best solutions.


### 3.2.5. Selection

For selection, we have used a roulette wheel technique where genes are selected based on their relative fitness values. The better the chromosomes are, the more chances to be selected they have. Let count be the number of elite children. We construct a roulette wheel as follows [22]:(i)
Calculate the fitness valuef
i
t
(
i
) for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p.
(ii)
Find the total fitness of the populationF = ∑
i
=
c
o
u
n
t
+
1
p
o
p
_
s
i
z
e
f
i
t
(
g
i
).
(iii)
Calculate the probability of selectionp
i for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p:(4)
p
i
=
f
i
t
g
i
F
.
(iv)
Calculate a cumulative probabilityc
i for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p:(5)
c
i
=
∑
j
=
1
i
p
j
.We now spin the wheel (pop_size − count) times and select a single chromosome as follows:(i)
Generate a random number (float)r between 0 and 1.
(ii)
Ifr
<
c
1, we select the first chromosome g
1; otherwise, select the ith chromosome g
i (2
≤
i
≤
pop_size
-
count) such that c
i
-
1
<
r
≤
c
i.Some chromosomes get selected more than once. According toSchema Theorem [24], the best chromosomes get more copies, the average stay even, and the worst die off.


### 3.2.6. Crossover

For crossover, we proceed as follows.For each chromosomeg
i in the population,(i)
generate a random number (float)r between 0 and 1,
(ii)
ifr
<
p
c
r
o
s
s (crossover probability), we select the given chromosome for crossover.We have used single point crossover where the crossover site is also generated randomly in the range [
1
⋯
q
-
1
], where q is the number of samples. Thus after crossover, a pair of parent chromosomes generates a pair of offspring chromosomes [25]. The new population obtained after crossover contains the new generation produced by crossover as well as the elite children that did not undergo crossover. This new population is used in the mutation process.


### 3.2.7. Mutation

A nonuniform mutation operator as proposed in literature [25] has been used here. The new operator is defined as follows:(i)
A random experiment is carried out which produces an outcome which is either 0 or 1.
(ii)
Another random number pos is generated in the range[
1
⋯
q
-
1
], where q is the number of samples, to select the mutation site.
(iii)
Letg
i
t
=
[
g
i
(
1
)
,
g
i
(
2
)
,
…
,
g
i
(
j
)
,
…
,
g
i
(
q
)
],  1
≤
j
≤
q, be the chromosome, and let g
i
(
j
) be selected for mutation. Domain of g
i is [
l
b
,
u
b
]; the resultant vector g
i
t
+
1
=
[
g
i
(
1
)
,
g
i
(
2
)
,
…
,
g
i
j
′
,
…
,
g
i
(
q
)
]:
(6)
g
i
j
′
=
g
i
j
+
Δ
t
,
ub
-
g
i
j
,
if  outcome  is
0
g
i
j
-
Δ
t
,
g
i
j
-
lb
,
if  outcome  is
1
,where t is the generation number and the function Δ
(
t
,
y
) returns a value in the range [
0
⋯
y
] such that the probability of Δ
(
t
,
y
) being close to 0 increases as t increases. This property causes this operator to search the space uniformly initially (when t is small) and very locally at later stages.Δ
(
t
,
y
) is calculated as(7)
Δ
t
,
y
=
y
1
-
r
1
-
t
/
T
β
,where r is a random number in the range [
0
⋯
1
], T is the maximum number of generations preselected by the user, and β is a system parameter determining the degree of uniformity. We have used β
=
2 for our experiment.The entire genetic transformation has been performed on one population with respect to the other. We made the diseased gene set to undergo genetic transformation while fitness evaluation has been made with respect to the normal gene set. The opposite transformation will produce similar results.Once the genetic transformations are done, we obtain a final population set (here, genetically transformed diseased gene set) which have been ranked in order of their fitness. We compare the two ranks, one by the Wilcoxon method and the other by our GA method. A threshold of±2 has been considered while comparing the two ranks. Results show that there is a good percentage of matches in the two ranks. Moreover, we find out the top ranked genes produced by both methods and the significant genes produced by the two methods are also similar. This also validates the result of the missing value estimation method carried out in phase 1.


## 3.2.1. Dimension Reduction Using Wilcoxon Rank Sum Test (WRST)

The inputs to the Wilcoxon rank sum test function are the two gene sets, the diseased set and the normal set, both of which have individually undergone the missing value estimation procedure (if there was any missing value). The two gene sets may have different number of samples. Let us consider that the diseased set is a (p
×
q
1) sized gene expression data, where p is the number of genes and q
1 is the number of samples, and the normal set has a size (p
×
q
2), where q
2 is the number of samples. The Wilcoxon rank sum function processes the two datasets in order to find out for which genes the null hypothesis is accepted or rejected. It returns two values, p value and h-value, as discussed earlier. The null hypothesis for our problem is that the genes are not differentially expressed; that is, either all the samples have come from diseased patients or they have come from normal patients. The alternative hypothesis can be that genes are differentially expressed. We record the p values and h-values for each gene.In the next step, we consider only those genes for which the alternative hypothesis holds (h
=
1) at the significance level alpha and sort the genes according to the p values thereby ranking the genes. We then select the topmost k genes, where k is the population size that has been used for GA later. Thus, we have two reduced populations, one representing diseased and the other representing normal tissues. Let (
X
i
j
)
p
×
q
1 be the diseased set, where p is the reduced set of genes and q
1 is the number of samples, respectively, and let (
Y
i
j
)
p
×
q
2 be the normal set, where q
2 is the number of samples.


## 3.2.2. Chromosome Representation and Initial Population for GA

The reduced gene sets(
X
i
j
)
p
×
q
1and (
Y
i
j
)
p
×
q
2 serve as the initial population for the genetic algorithm step. They contain pop_size number of genes which is preselected by the user. We use real value encoding to represent each chromosome; that is, X
i
j and Y
i
j are the measurements recorded for the ith gene and jth sample for each population, respectively.


## 3.2.3. Fitness Calculation

The fitness for each gene in the reduced gene sets is again calculated by a method similar to that used in [14] where gene expression profiles have been considered as continuous time series curves.In our problem, we have two populations, one for the diseased tissues and the other for the normal tissues. The two populations contain the same number of genesp but may have different number of samples. In that case, we consider the minimum of the two and extract the same number of samples from each set.Letg
1
,
g
2
,
…
,
g
p be the reduced set of genes in each population. If q
=
min
⁡
q
1
,
q
2, then for each population, the difference vector V
i of g
i is calculated using (1). Once the difference vectors are calculated for each of the two populations, say D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
1 (for diseased) and D
i
f
f
e
r
e
n
c
e
T
a
b
l
e
2 (for normal), the number of matches between the difference vectors M
e
m
b
e
r
s
h
i
p
(
i
) and the membership grade for g
i is computed using (2).The fitness of geneg
i is the reciprocal of m
e
m
g
r
a
d
e
(
i
) and is calculated as(3)
f
i
t
i
=
1
m
e
m
g
r
a
d
e
i
.This signifies that the more similar the distributions of gene g
i in the two populations are, the less differentially expressed the gene is, and vice versa. Thus, a fitter gene will have different distributions in the two populations. We then rank the genes in order of their fitness.


## 3.2.4. Elitism

We have used an elitist version of GA where the best chromosomes are carried forward to the next generation unchanged; that is, the crossover and mutation operators are not applied on the best chromosomes. This technique ensures faster convergence of the process by keeping track of the best solutions.


## 3.2.5. Selection

For selection, we have used a roulette wheel technique where genes are selected based on their relative fitness values. The better the chromosomes are, the more chances to be selected they have. Let count be the number of elite children. We construct a roulette wheel as follows [22]:(i)
Calculate the fitness valuef
i
t
(
i
) for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p.
(ii)
Find the total fitness of the populationF = ∑
i
=
c
o
u
n
t
+
1
p
o
p
_
s
i
z
e
f
i
t
(
g
i
).
(iii)
Calculate the probability of selectionp
i for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p:(4)
p
i
=
f
i
t
g
i
F
.
(iv)
Calculate a cumulative probabilityc
i for each chromosome g
i, c
o
u
n
t
+
1
≤
i
≤
p:(5)
c
i
=
∑
j
=
1
i
p
j
.We now spin the wheel (pop_size − count) times and select a single chromosome as follows:(i)
Generate a random number (float)r between 0 and 1.
(ii)
Ifr
<
c
1, we select the first chromosome g
1; otherwise, select the ith chromosome g
i (2
≤
i
≤
pop_size
-
count) such that c
i
-
1
<
r
≤
c
i.Some chromosomes get selected more than once. According toSchema Theorem [24], the best chromosomes get more copies, the average stay even, and the worst die off.


## 3.2.6. Crossover

For crossover, we proceed as follows.For each chromosomeg
i in the population,(i)
generate a random number (float)r between 0 and 1,
(ii)
ifr
<
p
c
r
o
s
s (crossover probability), we select the given chromosome for crossover.We have used single point crossover where the crossover site is also generated randomly in the range [
1
⋯
q
-
1
], where q is the number of samples. Thus after crossover, a pair of parent chromosomes generates a pair of offspring chromosomes [25]. The new population obtained after crossover contains the new generation produced by crossover as well as the elite children that did not undergo crossover. This new population is used in the mutation process.


## 3.2.7. Mutation

A nonuniform mutation operator as proposed in literature [25] has been used here. The new operator is defined as follows:(i)
A random experiment is carried out which produces an outcome which is either 0 or 1.
(ii)
Another random number pos is generated in the range[
1
⋯
q
-
1
], where q is the number of samples, to select the mutation site.
(iii)
Letg
i
t
=
[
g
i
(
1
)
,
g
i
(
2
)
,
…
,
g
i
(
j
)
,
…
,
g
i
(
q
)
],  1
≤
j
≤
q, be the chromosome, and let g
i
(
j
) be selected for mutation. Domain of g
i is [
l
b
,
u
b
]; the resultant vector g
i
t
+
1
=
[
g
i
(
1
)
,
g
i
(
2
)
,
…
,
g
i
j
′
,
…
,
g
i
(
q
)
]:
(6)
g
i
j
′
=
g
i
j
+
Δ
t
,
ub
-
g
i
j
,
if  outcome  is
0
g
i
j
-
Δ
t
,
g
i
j
-
lb
,
if  outcome  is
1
,where t is the generation number and the function Δ
(
t
,
y
) returns a value in the range [
0
⋯
y
] such that the probability of Δ
(
t
,
y
) being close to 0 increases as t increases. This property causes this operator to search the space uniformly initially (when t is small) and very locally at later stages.Δ
(
t
,
y
) is calculated as(7)
Δ
t
,
y
=
y
1
-
r
1
-
t
/
T
β
,where r is a random number in the range [
0
⋯
1
], T is the maximum number of generations preselected by the user, and β is a system parameter determining the degree of uniformity. We have used β
=
2 for our experiment.The entire genetic transformation has been performed on one population with respect to the other. We made the diseased gene set to undergo genetic transformation while fitness evaluation has been made with respect to the normal gene set. The opposite transformation will produce similar results.Once the genetic transformations are done, we obtain a final population set (here, genetically transformed diseased gene set) which have been ranked in order of their fitness. We compare the two ranks, one by the Wilcoxon method and the other by our GA method. A threshold of±2 has been considered while comparing the two ranks. Results show that there is a good percentage of matches in the two ranks. Moreover, we find out the top ranked genes produced by both methods and the significant genes produced by the two methods are also similar. This also validates the result of the missing value estimation method carried out in phase 1.


## 3.3. Gene Classification Using SVM

In order to prove the significance of ranking by our GA method, we perform classification. The top rankedn genes,n’
{
5,10,15,20,25
}, ranked by our GA method are used for the purpose. We use k-fold LOO cross-validations, where k is varied from one dataset to another depending on the number of samples. For cross-validation, we have divided our dataset into two sets, a training set and a testing set, in 80 : 20 ratio. The reason behind taking this ratio is that 80 : 20 is a commonly occurring ratio, which is often referred to as Pareto Principle. So, if there are n samples in the training set and m
-
n samples in the test set, where m is the total number of samples, the training set is divided into k equal sized subsets. Of the k subsets, one subset is retained for validation and the remaining k
-
1 subsets are used as training data. Thus, k SVM classifiers with linear kernel are trained using the n training subsets. The classification accuracy rates are recorded and the classifier with the best accuracy rate is used to test the m
-
n samples.


## 4. Experimental Results

### 4.1. Datasets Used

The missing value estimation part of the proposed modified LRFDVImpute technique has been evaluated on the publicly available yeast cell cycle time series dataset from Spellman et al. [26] described in Table 1.Table 1
Characteristic of Spellman dataset.
Dataset
Start
End
Sampling
Complete genes
alpha
0 m
119 m
Every 7 m
4489
cdc15
10 m
290 m
Every 20 m for 1 hr,10 m for 3 hr, and20 m for final hr
4381
cdc28
0 m
160 m
Every 10 m
1383
elu
0 m
390 m
Every 30 m
5766
YeastSaccharomyces cerevisiae dataset of Spellman et al. [26].
Source:http://genome-www.stanford.edu/cellcycle/data/rawdata/combined.txt.
Organism: yeast.After the experiments on Spellman dataset are done, the combined gene ranking and classification portion of the proposed method are evaluated on four publicly available datasets: Colorectal Cancer tumours dataset (GDS4382), Breast Cancer dataset (GSE349-350), Prostate Cancer dataset, and Leukaemia Cancer dataset (DLBCL-FL).


### 4.2. Platform Used

All algorithms have been implemented using MATLAB R2013a in Windows 8.1.


### 4.3. Results

#### 4.3.1. Results of Missing Value Estimation Part

We perform the initial estimation using modified version of LRFDVImpute with a membership gradeθ
=
0.55. After the initial estimation is over, we forcibly treat cells at specified locations as missing and estimate them using different membership values of θ and both earlier and modified versions. This has been carried out only once, after estimating rows with single missing values and the corresponding RMSE values have been recorded. We have performed our experiments only on alpha, cdc15, and elu data of Spellman dataset. The number of missing values is too large for cdc28; that is why we ignore that segment. The results for the alpha, cdc15, and elu datasets using both methods are shown in Tables 2–4. Figures 3–5 show the corresponding plots of RMSE versus membership grade θ for each of the four datasets.Table 2
Results for missing value estimation algorithm (Spellman, alpha).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0.012405344
0.012488181
0.012562782
0.012562782
0.012690904
0.012197374
0.012638865
Modified LRFDVImpute
0.012439936
0.012439366
0.012389872
0.012389872
0.012645466
0.011988263
0.013268721Table 3
Results for missing value estimation algorithm (Spellman, cdc15).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0.016832094
0.016760968
0.016706119
0.016682418
0.016768837
0.016733642
0.049482242
Modified LRFDVImpute
0.016781257
0.016710318
0.016736613
0.016723349
0.016637753
0.017023671
0.057225615Table 4
Results for missing value estimation algorithm (Spellman, elu).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0
0
0
0
0
0
0
Modified LRFDVImpute
0
0
0
0
0
0
0Figure 3
Plot of RMSE versus membership gradeθ for alpha dataset.Figure 4
Plot of RMSE versus membership gradeθ for cdc15 dataset.Figure 5
Plot of RMSE versus membership gradeθ for elu dataset.Table5 compares the performance of both versions of LRFDVImpute method to that of some other existing methods, like SVDImpute, LLSImpute, FDVLLSImpute, FDVSPLINEImpute, and so forth, and the results show that modified version of LRFDVImpute outperforms the other existing methods as far as RMSE value is concerned.Table 5
Performance comparison with other existing methods.
Dataset
SVDImpute
LLSImpute
FDVLLSImpute
FDVSPLINEImpute
FDVLRImpute withθ
=
0.55
Original LRFDVImpute
Modified LRFDVImpute
alpha
0.03395
0.07853
0.096
0.063
0.012562782
0.012389872
cdc15
0.05055
0.1208
0.258
0.127
0.016682418
0.016723349
elu
0.01585
0.0033
0.044
.019
0
0


#### 4.3.2. Combined Results

We test the significance of our proposed missing value estimation technique using the gene ranking method. We have not found any state-of-the-art work on gene ranking so far where Spellman dataset is used. That is why we use four more publicly available real-life gene expression datasets, like Colorectal Cancer dataset (GDS4382), Breast Cancer dataset (GSE349-350), Prostate Cancer dataset, and Leukaemia Cancer dataset (DLBCL-FL) [4, 27–32], to perform steps such as missing values estimation and gene ranking and analyze the results. We start with the microarray dataset containing missing values and apply our proposed missing value estimation technique to estimate the genes with missing values (if any). We rank them using proposed gene ranking method and find the top ranked genes. We then forcibly insert missing values in the top ranked genes and again estimate them using the same missing value estimation technique. Finally, we rank them once more to find the top ranked genes. Results show that most of the top ranked genes remain the same, which implies that the proposed missing value estimation technique has been accurate in estimating the unknown values. We have normalized most of the datasets using z-score normalization method in order to bring the data values to a common scale.Tables6, 8, 10, and 13 show the estimated values for the four datasets, Tables 7, 9, 11, and 14 show the common gene indices before and after the estimation, and Tables 12 and 15 compare the performance of the proposed approach with two state-of-the-art methods [22, 23] for Prostate and Leukaemia dataset on the basis of accuracy, sensitivity, specificity, F
1-score, and  G-mean metrics. We have found that Prostate and Leukaemia are the common dataset on which both the existing methods have done their experiments. The results show that the proposed gene ranking approach performs far better compared to those existing approaches, where one is a PSO based graph theoretic approach [22] and the other is a web based tool DWFS, which uses KNN and NBC classifiers [23] as far as those metrics are concerned.Table 6
Estimated values for Colorectal Cancer GDS4382.
Colorectal Cancer GDS4382
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
714
12
0.677252397
0.703080766
714
12
0.140108496
−0.163286572
1245
5
1.56686079
1.755118642
1245
5
1.212949296
1.148663192
1578
3
0.787510895
1.003134829
1578
3
0.256998387
0.22381735
1763
11
1.024714768
0.737414862
1763
11
0.20681001
0.056394235
2792
9
−0.861395162
−0.865043164
2792
9
−1.064597763
−1.005338727
4025
1
0.781326343
1.137532185
4025
1
−0.31562626
−0.297767551
4134
15
0.892338308
0.958542974
4134
15
0.329755342
0.247541595
5082
2
−0.006360431
0.32763543
5082
2
−0.480000425
−0.370717546
8426
13
1.210288879
0.790345743
8426
13
−0.082823022
0.157094029
9979
6
2.932068401
2.953449691
9979
6
2.387826603
2.246418246
10083
11
1.369142269
1.258410425
10083
11
1.915272483
1.809541605
10145
10
1.940467541
1.852338203
10145
10
2.750210569
2.834220655
10208
3
1.868505436
1.778084682
10208
3
2.224446282
2.122262802
10280
6
1.424951861
1.521769713
10280
6
0.98680627
1.139738659
10323
1
0.895845032
0.963530648
10323
1
0.378278185
0.376819137
10725
14
0.562534293
0.903674258
10725
14
1.263242627
1.285858157
10789
4
1.582210172
1.834961289
10789
4
0.507667085
0.540477481
10855
10
3.17769843
3.162360832
10855
10
2.562212194
2.546399932
11050
3
2.676634486
2.657786027
11050
3
1.872943693
1.894214081
11055
8
2.112261939
2.105431748
11055
8
1.680697142
1.713521495
11100
16
2.522783399
2.455429314
11100
16
3.208462284
3.064140274
11465
1
2.454481056
2.18211664
11465
1
1.232460868
1.356633277
11485
12
−0.701537989
−0.49772711
11485
12
0.609015246
0.316407836
11650
6
1.470662615
1.35686403
11650
6
1.940624638
2.047625027
11677
4
2.591635801
2.602068714
11677
4
2.903830552
2.934334312Table 7
Top 25 gene indices before and after estimation for GDS4382.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
714
714
2
1245
1245
3
1578
1578
4
1763
1763
5
2792
2792
6
4025
4025
7
4134
4134
8
5082
5082
9
8426
8426
10
9979
9979
11
10083
10083
12
10145
10145
13
10208
10208
14
10280
10280
15
10323
10323
16
10725
10725
17
10789
10789
18
10855
10855
19
11050
11050
20
11055
11055
21
11100
11100
22
11465
11465
23
11485
11485
24
11650
11650
25
11677
11677
Number of common genes in top 25 positions = 25
% of common genes = 100Table 8
Estimated values for Breast Cancer dataset.
Breast Cancer
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
272
12
−0.354039943
−0.419709522
272
8
−0.407057103
−0.396601672
329
5
−0.176687651
−0.295980517
329
2
−0.021584122
0.092981006
491
3
0.222486126
0.30260015
491
10
0.140530363
0.067353238
869
11
0.79646566
0.852100043
869
5
0.345360946
0.279331888
1143
9
0.017956445
0.256958128
1143
4
0.054582833
−0.145618319
1937
1
0.22672464
0.33943153
1937
7
−0.128257731
0.05338273
2825
8
7.552441375
8.108907291
2825
3
6.080319682
5.969522121
3004
2
0.128475064
0.04404869
3004
6
−0.124353092
−0.113116385
4911
4
−0.550612392
−0.372472073
4911
9
−0.090276567
−0.152051028
5328
6
10.57228289
9.556257324
5328
5
8.316072021
7.887609502
6184
11
−0.493537621
−0.504575199
6184
8
0.035382884
−0.132574115
7941
10
0.233974208
0.14984931
7941
6
0.077199916
0.101298927
8452
14
−0.312716903
−0.32992237
8452
3
−0.071344506
−0.092371778
9076
6
0.708996621
0.154489296
9076
1
0.060554625
0.036078623
9267
13
−0.270092957
−0.33842598
9267
10
−0.034956721
0.30142636
9574
7
0.093118778
0.074772718
9574
7
−0.227882015
−0.159726559
9723
4
−0.018883445
−0.279787171
9723
5
0.407076237
0.815509208
9753
10
−0.23265185
−0.264909557
9753
8
−0.228475796
−0.265794544
9905
3
−0.31537376
−0.417539213
9905
3
−0.286431974
−0.199211435
10319
8
−0.049692673
−0.038561161
10319
9
−0.218073382
−0.242557019
10614
9
−0.511734814
−0.430792872
10614
2
−0.268390734
−0.1922339
11377
1
−0.055809992
0.083662727
11377
7
−0.268295765
−0.144527083
11737
12
−0.34275829
−0.269422135
11737
4
−0.387083296
−0.093221982
11976
6
0.030992978
−0.133189906
11976
9
−0.270598103
−0.272899078
12053
4
−0.374640827
−0.328380483
12053
10
−0.360077886
−0.244255658Table 9
Top 25 gene indices before and after estimation for Breast Cancer dataset.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
3004
1143
2
7941
3004
3
10319
7941
4
869
10319
5
5328
11737
6
9723
491
7
491
9753
8
9574
869
9
9905
5328
10
11737
9723
11
2825
12053
12
4911
2825
13
8452
9574
14
272
9905
15
329
4911
16
6184
8452
17
9076
9076
18
9267
329
19
9753
6184
20
10614
9267
21
11377
11976
22
11976
2218
23
12053
2459
24
1143
2995
25
1937
4200
Number of common genes in top 25 positions = 21
% of common genes = 84Table 10
Estimated values for Prostate Cancer dataset.
Prostate Cancer
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
205
45
7.890085103
9.434976702
205
45
5.168380524
5.416011322
2839
5
−0.176203136
0.004447962
2839
5
−0.173585261
−0.183852307
3649
10
0.296457522
0.046654036
3649
10
0.329958724
0.617420849
3794
8
−0.241993865
−0.216817937
3794
8
−0.106322335
−0.109294537
4365
17
0.059874409
2.443242853
4365
17
−0.122967971
−0.099879376
5757
14
0.090277183
0.70723886
5757
14
−0.196702226
−0.067191045
5944
22
−0.180239863
−0.186977867
5944
22
−0.137411352
−0.217659599
6185
36
1.557345019
1.548681099
6185
36
−0.010802226
−0.181261903
6462
28
−0.16997265
−0.175762613
6462
28
0.202288499
0.191383577
7247
32
0.859643204
0.956551639
7247
32
0.932414697
2.21258573
7520
18
0.270631813
0.179429504
7520
18
0.064953651
0.155169008
7557
11
0.393712194
0.375941042
7557
11
1.450975133
1.084180805
7768
3
0.042090452
0.11563321
7768
3
0.59875936
0.576608917
8123
47
0.216968028
0.283330867
8123
47
0.128003038
0.165883246
8554
4
0.015161421
0.083268762
8554
4
0.40771258
−0.051156112
8768
26
−0.135452978
−0.04501778
8768
26
−0.138712043
−0.188517852
8850
17
2.708006948
1.916254815
8850
17
−0.364936796
−0.228740529
9034
29
−0.190217605
−0.178815842
9034
29
−0.194249889
−0.047431811
9050
34
0.351634344
0.174924563
9050
34
0.284389403
0.469260339
9172
42
2.277059356
2.347704101
9172
42
1.727648712
1.562259352
9850
50
−0.046243874
−0.290856086
9850
50
1.371476261
2.134924482
10138
14
0.599428228
0.628396963
10138
14
0.765017083
1.001711406
10494
22
−0.189181177
−0.122575727
10494
22
0.220022109
0.365466432
10537
48
0.139373755
0.406906442
10537
48
0.035059438
0.011805144
10956
7
0.337716243
0.155279221
10956
7
0.146282156
0.169249764Table 11
Top 25 gene indices before and after estimation for Prostate Cancer dataset.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
6185
6185
2
10494
10494
3
9850
4365
4
4365
9850
5
10138
9034
6
9172
10138
7
9034
5944
8
5944
9172
9
3649
3649
10
8554
2839
11
2839
7557
12
7557
10956
13
205
9050
14
3794
7520
15
10956
3794
16
8850
205
17
7520
8850
18
9050
10537
19
10537
5757
20
5757
8554
21
8123
8768
22
6462
8123
23
8768
6462
24
7247
7247
25
7768
9093
Number of common genes in top 25 positions = 24
% of common genes = 96Table 12
Performance comparison for Prostate dataset.
Dataset name
Prostate
Number of genes
Number of samples
Number of samples in Cancer dataset
Number of samples in normal dataset
Number of samples in training dataset
Number of samples in testing dataset
SVM kernel used
Number of folds for LOOCV
12600
102
52
50
82 (42 cancer, 40 normal)
20 (10 cancer, 10 normal)
Linear
41
Algorithm
% accuracy
Sensitivity/recall
Specificity
F1-score
G-mean
Proposed approach
100 (with top 5 genes)
1
1
1
1
PSO based graph theoretic approach
91
0.91
0.92
0.91
0.91
DWFS using KNN classifier
86
0.87
0.85
0.86
0.86
DWFS using NBC classifier
80
0.76
0.85
0.78
0.80Table 13
Estimated values for DLBCL-FL.
Leukaemia (DLBCL-FL)
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
28
50
−0.188603483
−0.203477278
28
2
3.886811534
0.84848513
447
45
0.103798973
0.304241026
447
18
−0.146274929
0.009235547
546
8
1.055798216
1.246420135
546
8
0.140697423
0.320415944
640
52
0.052220359
−0.311252744
640
7
0.12552933
0.364757746
913
5
0.15759485
0.180411915
913
2
−0.135855848
−0.165390479
1129
4
1.051658222
0.598949762
1129
10
0.04903928
0.096097027
1142
31
−0.249901255
−0.311776993
1142
1
−0.249625598
−0.196558516
1293
34
1.91037315
2.330011528
1293
3
0.27829269
0.224522592
1553
48
−0.218067263
−0.107003318
1553
9
−0.00577268
0.602953445
1731
26
0.555312243
0.39207635
1731
7
0.448099319
0.524633075
2062
22
2.366047008
4.45272374
2062
13
0.201613611
0.849245039
2929
17
0.072886107
0.125837311
2929
11
−0.007410482
0.004016071
3965
56
0.137340738
−0.905551574
3965
16
−0.133906259
−0.083702381
3969
36
2.328255836
2.60971631
3969
6
0.520731724
0.584352541
4124
17
0.077170048
0.183068221
4124
5
−0.157303536
−0.036843328
4135
10
−0.35554626
−0.339641516
4135
10
−0.232750407
−0.161598404
4143
14
0.185986189
−0.024609518
4143
7
−0.032712551
0.024485269
4233
18
3.990305253
1.582890447
4233
8
0.770838584
0.706726475
4313
11
0.453186971
0.126025314
4313
15
−0.129336617
−0.079142717
4510
22
−0.167456263
−0.135667934
4510
17
−0.046588554
−0.100490086
5327
46
−0.415159893
−0.324486992
5327
12
−0.146934163
−0.22684469
6120
3
4.432535499
4.497409395
6120
3
0.919305283
0.723423487
6417
7
−0.253214481
−0.312493286
6417
10
−0.141004928
−0.007636336
6434
42
−0.220525385
−0.208472395
6434
14
−0.001659417
0.063668162
6756
55
2.0189449
1.307321937
6756
4
−0.009878439
0.570901233Table 14
Top 25 gene indices before and after estimation for DLBCL-FL.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
447
447
2
913
4135
3
4135
913
4
546
640
5
2929
2929
6
640
1142
7
4510
546
8
3969
3969
9
5327
4510
10
6756
4233
11
4233
5327
12
4313
4313
13
6120
6756
14
1142
6120
15
1129
1553
16
1731
1129
17
4124
6417
18
3965
4124
19
1293
1731
20
6434
1293
21
28
6434
22
4143
28
23
2062
2062
24
1553
4094
25
6417
1984
Number of common genes in top 25 positions = 23
% of common genes = 92Table 15
Performance comparison for DLBCL-FL.
Dataset name
DLBCL-FL
Number of genes
Number of samples
Number of samples in Cancer dataset
Number of samples in normal dataset
Number of samples in training dataset
Number of samples in testing dataset
SVM kernel used
Number of folds for LOOCV
7070
77
58
19
62 (47 DLBCL, 15 FL)
15 (11 DLBCL, 4 FL)
Linear
31
Algorithm
% accuracy
Sensitivity/recall
Specificity
F1-score
G-mean
Proposed approach
100 (with top 5, 10, 15, and 20 genes)
1
1
1
1
PSO based graph theoretic approach
94
0.95
0.94
0.89
0.94
DWFS using KNN classifier
91
0.97
0.85
0.94
0.9
DWFS using NBC classifier
96
1
0.9
0.98
0.94


## 4.1. Datasets Used

The missing value estimation part of the proposed modified LRFDVImpute technique has been evaluated on the publicly available yeast cell cycle time series dataset from Spellman et al. [26] described in Table 1.Table 1
Characteristic of Spellman dataset.
Dataset
Start
End
Sampling
Complete genes
alpha
0 m
119 m
Every 7 m
4489
cdc15
10 m
290 m
Every 20 m for 1 hr,10 m for 3 hr, and20 m for final hr
4381
cdc28
0 m
160 m
Every 10 m
1383
elu
0 m
390 m
Every 30 m
5766
YeastSaccharomyces cerevisiae dataset of Spellman et al. [26].
Source:http://genome-www.stanford.edu/cellcycle/data/rawdata/combined.txt.
Organism: yeast.After the experiments on Spellman dataset are done, the combined gene ranking and classification portion of the proposed method are evaluated on four publicly available datasets: Colorectal Cancer tumours dataset (GDS4382), Breast Cancer dataset (GSE349-350), Prostate Cancer dataset, and Leukaemia Cancer dataset (DLBCL-FL).


## 4.2. Platform Used

All algorithms have been implemented using MATLAB R2013a in Windows 8.1.


## 4.3. Results

### 4.3.1. Results of Missing Value Estimation Part

We perform the initial estimation using modified version of LRFDVImpute with a membership gradeθ
=
0.55. After the initial estimation is over, we forcibly treat cells at specified locations as missing and estimate them using different membership values of θ and both earlier and modified versions. This has been carried out only once, after estimating rows with single missing values and the corresponding RMSE values have been recorded. We have performed our experiments only on alpha, cdc15, and elu data of Spellman dataset. The number of missing values is too large for cdc28; that is why we ignore that segment. The results for the alpha, cdc15, and elu datasets using both methods are shown in Tables 2–4. Figures 3–5 show the corresponding plots of RMSE versus membership grade θ for each of the four datasets.Table 2
Results for missing value estimation algorithm (Spellman, alpha).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0.012405344
0.012488181
0.012562782
0.012562782
0.012690904
0.012197374
0.012638865
Modified LRFDVImpute
0.012439936
0.012439366
0.012389872
0.012389872
0.012645466
0.011988263
0.013268721Table 3
Results for missing value estimation algorithm (Spellman, cdc15).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0.016832094
0.016760968
0.016706119
0.016682418
0.016768837
0.016733642
0.049482242
Modified LRFDVImpute
0.016781257
0.016710318
0.016736613
0.016723349
0.016637753
0.017023671
0.057225615Table 4
Results for missing value estimation algorithm (Spellman, elu).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0
0
0
0
0
0
0
Modified LRFDVImpute
0
0
0
0
0
0
0Figure 3
Plot of RMSE versus membership gradeθ for alpha dataset.Figure 4
Plot of RMSE versus membership gradeθ for cdc15 dataset.Figure 5
Plot of RMSE versus membership gradeθ for elu dataset.Table5 compares the performance of both versions of LRFDVImpute method to that of some other existing methods, like SVDImpute, LLSImpute, FDVLLSImpute, FDVSPLINEImpute, and so forth, and the results show that modified version of LRFDVImpute outperforms the other existing methods as far as RMSE value is concerned.Table 5
Performance comparison with other existing methods.
Dataset
SVDImpute
LLSImpute
FDVLLSImpute
FDVSPLINEImpute
FDVLRImpute withθ
=
0.55
Original LRFDVImpute
Modified LRFDVImpute
alpha
0.03395
0.07853
0.096
0.063
0.012562782
0.012389872
cdc15
0.05055
0.1208
0.258
0.127
0.016682418
0.016723349
elu
0.01585
0.0033
0.044
.019
0
0


### 4.3.2. Combined Results

We test the significance of our proposed missing value estimation technique using the gene ranking method. We have not found any state-of-the-art work on gene ranking so far where Spellman dataset is used. That is why we use four more publicly available real-life gene expression datasets, like Colorectal Cancer dataset (GDS4382), Breast Cancer dataset (GSE349-350), Prostate Cancer dataset, and Leukaemia Cancer dataset (DLBCL-FL) [4, 27–32], to perform steps such as missing values estimation and gene ranking and analyze the results. We start with the microarray dataset containing missing values and apply our proposed missing value estimation technique to estimate the genes with missing values (if any). We rank them using proposed gene ranking method and find the top ranked genes. We then forcibly insert missing values in the top ranked genes and again estimate them using the same missing value estimation technique. Finally, we rank them once more to find the top ranked genes. Results show that most of the top ranked genes remain the same, which implies that the proposed missing value estimation technique has been accurate in estimating the unknown values. We have normalized most of the datasets using z-score normalization method in order to bring the data values to a common scale.Tables6, 8, 10, and 13 show the estimated values for the four datasets, Tables 7, 9, 11, and 14 show the common gene indices before and after the estimation, and Tables 12 and 15 compare the performance of the proposed approach with two state-of-the-art methods [22, 23] for Prostate and Leukaemia dataset on the basis of accuracy, sensitivity, specificity, F
1-score, and  G-mean metrics. We have found that Prostate and Leukaemia are the common dataset on which both the existing methods have done their experiments. The results show that the proposed gene ranking approach performs far better compared to those existing approaches, where one is a PSO based graph theoretic approach [22] and the other is a web based tool DWFS, which uses KNN and NBC classifiers [23] as far as those metrics are concerned.Table 6
Estimated values for Colorectal Cancer GDS4382.
Colorectal Cancer GDS4382
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
714
12
0.677252397
0.703080766
714
12
0.140108496
−0.163286572
1245
5
1.56686079
1.755118642
1245
5
1.212949296
1.148663192
1578
3
0.787510895
1.003134829
1578
3
0.256998387
0.22381735
1763
11
1.024714768
0.737414862
1763
11
0.20681001
0.056394235
2792
9
−0.861395162
−0.865043164
2792
9
−1.064597763
−1.005338727
4025
1
0.781326343
1.137532185
4025
1
−0.31562626
−0.297767551
4134
15
0.892338308
0.958542974
4134
15
0.329755342
0.247541595
5082
2
−0.006360431
0.32763543
5082
2
−0.480000425
−0.370717546
8426
13
1.210288879
0.790345743
8426
13
−0.082823022
0.157094029
9979
6
2.932068401
2.953449691
9979
6
2.387826603
2.246418246
10083
11
1.369142269
1.258410425
10083
11
1.915272483
1.809541605
10145
10
1.940467541
1.852338203
10145
10
2.750210569
2.834220655
10208
3
1.868505436
1.778084682
10208
3
2.224446282
2.122262802
10280
6
1.424951861
1.521769713
10280
6
0.98680627
1.139738659
10323
1
0.895845032
0.963530648
10323
1
0.378278185
0.376819137
10725
14
0.562534293
0.903674258
10725
14
1.263242627
1.285858157
10789
4
1.582210172
1.834961289
10789
4
0.507667085
0.540477481
10855
10
3.17769843
3.162360832
10855
10
2.562212194
2.546399932
11050
3
2.676634486
2.657786027
11050
3
1.872943693
1.894214081
11055
8
2.112261939
2.105431748
11055
8
1.680697142
1.713521495
11100
16
2.522783399
2.455429314
11100
16
3.208462284
3.064140274
11465
1
2.454481056
2.18211664
11465
1
1.232460868
1.356633277
11485
12
−0.701537989
−0.49772711
11485
12
0.609015246
0.316407836
11650
6
1.470662615
1.35686403
11650
6
1.940624638
2.047625027
11677
4
2.591635801
2.602068714
11677
4
2.903830552
2.934334312Table 7
Top 25 gene indices before and after estimation for GDS4382.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
714
714
2
1245
1245
3
1578
1578
4
1763
1763
5
2792
2792
6
4025
4025
7
4134
4134
8
5082
5082
9
8426
8426
10
9979
9979
11
10083
10083
12
10145
10145
13
10208
10208
14
10280
10280
15
10323
10323
16
10725
10725
17
10789
10789
18
10855
10855
19
11050
11050
20
11055
11055
21
11100
11100
22
11465
11465
23
11485
11485
24
11650
11650
25
11677
11677
Number of common genes in top 25 positions = 25
% of common genes = 100Table 8
Estimated values for Breast Cancer dataset.
Breast Cancer
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
272
12
−0.354039943
−0.419709522
272
8
−0.407057103
−0.396601672
329
5
−0.176687651
−0.295980517
329
2
−0.021584122
0.092981006
491
3
0.222486126
0.30260015
491
10
0.140530363
0.067353238
869
11
0.79646566
0.852100043
869
5
0.345360946
0.279331888
1143
9
0.017956445
0.256958128
1143
4
0.054582833
−0.145618319
1937
1
0.22672464
0.33943153
1937
7
−0.128257731
0.05338273
2825
8
7.552441375
8.108907291
2825
3
6.080319682
5.969522121
3004
2
0.128475064
0.04404869
3004
6
−0.124353092
−0.113116385
4911
4
−0.550612392
−0.372472073
4911
9
−0.090276567
−0.152051028
5328
6
10.57228289
9.556257324
5328
5
8.316072021
7.887609502
6184
11
−0.493537621
−0.504575199
6184
8
0.035382884
−0.132574115
7941
10
0.233974208
0.14984931
7941
6
0.077199916
0.101298927
8452
14
−0.312716903
−0.32992237
8452
3
−0.071344506
−0.092371778
9076
6
0.708996621
0.154489296
9076
1
0.060554625
0.036078623
9267
13
−0.270092957
−0.33842598
9267
10
−0.034956721
0.30142636
9574
7
0.093118778
0.074772718
9574
7
−0.227882015
−0.159726559
9723
4
−0.018883445
−0.279787171
9723
5
0.407076237
0.815509208
9753
10
−0.23265185
−0.264909557
9753
8
−0.228475796
−0.265794544
9905
3
−0.31537376
−0.417539213
9905
3
−0.286431974
−0.199211435
10319
8
−0.049692673
−0.038561161
10319
9
−0.218073382
−0.242557019
10614
9
−0.511734814
−0.430792872
10614
2
−0.268390734
−0.1922339
11377
1
−0.055809992
0.083662727
11377
7
−0.268295765
−0.144527083
11737
12
−0.34275829
−0.269422135
11737
4
−0.387083296
−0.093221982
11976
6
0.030992978
−0.133189906
11976
9
−0.270598103
−0.272899078
12053
4
−0.374640827
−0.328380483
12053
10
−0.360077886
−0.244255658Table 9
Top 25 gene indices before and after estimation for Breast Cancer dataset.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
3004
1143
2
7941
3004
3
10319
7941
4
869
10319
5
5328
11737
6
9723
491
7
491
9753
8
9574
869
9
9905
5328
10
11737
9723
11
2825
12053
12
4911
2825
13
8452
9574
14
272
9905
15
329
4911
16
6184
8452
17
9076
9076
18
9267
329
19
9753
6184
20
10614
9267
21
11377
11976
22
11976
2218
23
12053
2459
24
1143
2995
25
1937
4200
Number of common genes in top 25 positions = 21
% of common genes = 84Table 10
Estimated values for Prostate Cancer dataset.
Prostate Cancer
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
205
45
7.890085103
9.434976702
205
45
5.168380524
5.416011322
2839
5
−0.176203136
0.004447962
2839
5
−0.173585261
−0.183852307
3649
10
0.296457522
0.046654036
3649
10
0.329958724
0.617420849
3794
8
−0.241993865
−0.216817937
3794
8
−0.106322335
−0.109294537
4365
17
0.059874409
2.443242853
4365
17
−0.122967971
−0.099879376
5757
14
0.090277183
0.70723886
5757
14
−0.196702226
−0.067191045
5944
22
−0.180239863
−0.186977867
5944
22
−0.137411352
−0.217659599
6185
36
1.557345019
1.548681099
6185
36
−0.010802226
−0.181261903
6462
28
−0.16997265
−0.175762613
6462
28
0.202288499
0.191383577
7247
32
0.859643204
0.956551639
7247
32
0.932414697
2.21258573
7520
18
0.270631813
0.179429504
7520
18
0.064953651
0.155169008
7557
11
0.393712194
0.375941042
7557
11
1.450975133
1.084180805
7768
3
0.042090452
0.11563321
7768
3
0.59875936
0.576608917
8123
47
0.216968028
0.283330867
8123
47
0.128003038
0.165883246
8554
4
0.015161421
0.083268762
8554
4
0.40771258
−0.051156112
8768
26
−0.135452978
−0.04501778
8768
26
−0.138712043
−0.188517852
8850
17
2.708006948
1.916254815
8850
17
−0.364936796
−0.228740529
9034
29
−0.190217605
−0.178815842
9034
29
−0.194249889
−0.047431811
9050
34
0.351634344
0.174924563
9050
34
0.284389403
0.469260339
9172
42
2.277059356
2.347704101
9172
42
1.727648712
1.562259352
9850
50
−0.046243874
−0.290856086
9850
50
1.371476261
2.134924482
10138
14
0.599428228
0.628396963
10138
14
0.765017083
1.001711406
10494
22
−0.189181177
−0.122575727
10494
22
0.220022109
0.365466432
10537
48
0.139373755
0.406906442
10537
48
0.035059438
0.011805144
10956
7
0.337716243
0.155279221
10956
7
0.146282156
0.169249764Table 11
Top 25 gene indices before and after estimation for Prostate Cancer dataset.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
6185
6185
2
10494
10494
3
9850
4365
4
4365
9850
5
10138
9034
6
9172
10138
7
9034
5944
8
5944
9172
9
3649
3649
10
8554
2839
11
2839
7557
12
7557
10956
13
205
9050
14
3794
7520
15
10956
3794
16
8850
205
17
7520
8850
18
9050
10537
19
10537
5757
20
5757
8554
21
8123
8768
22
6462
8123
23
8768
6462
24
7247
7247
25
7768
9093
Number of common genes in top 25 positions = 24
% of common genes = 96Table 12
Performance comparison for Prostate dataset.
Dataset name
Prostate
Number of genes
Number of samples
Number of samples in Cancer dataset
Number of samples in normal dataset
Number of samples in training dataset
Number of samples in testing dataset
SVM kernel used
Number of folds for LOOCV
12600
102
52
50
82 (42 cancer, 40 normal)
20 (10 cancer, 10 normal)
Linear
41
Algorithm
% accuracy
Sensitivity/recall
Specificity
F1-score
G-mean
Proposed approach
100 (with top 5 genes)
1
1
1
1
PSO based graph theoretic approach
91
0.91
0.92
0.91
0.91
DWFS using KNN classifier
86
0.87
0.85
0.86
0.86
DWFS using NBC classifier
80
0.76
0.85
0.78
0.80Table 13
Estimated values for DLBCL-FL.
Leukaemia (DLBCL-FL)
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
28
50
−0.188603483
−0.203477278
28
2
3.886811534
0.84848513
447
45
0.103798973
0.304241026
447
18
−0.146274929
0.009235547
546
8
1.055798216
1.246420135
546
8
0.140697423
0.320415944
640
52
0.052220359
−0.311252744
640
7
0.12552933
0.364757746
913
5
0.15759485
0.180411915
913
2
−0.135855848
−0.165390479
1129
4
1.051658222
0.598949762
1129
10
0.04903928
0.096097027
1142
31
−0.249901255
−0.311776993
1142
1
−0.249625598
−0.196558516
1293
34
1.91037315
2.330011528
1293
3
0.27829269
0.224522592
1553
48
−0.218067263
−0.107003318
1553
9
−0.00577268
0.602953445
1731
26
0.555312243
0.39207635
1731
7
0.448099319
0.524633075
2062
22
2.366047008
4.45272374
2062
13
0.201613611
0.849245039
2929
17
0.072886107
0.125837311
2929
11
−0.007410482
0.004016071
3965
56
0.137340738
−0.905551574
3965
16
−0.133906259
−0.083702381
3969
36
2.328255836
2.60971631
3969
6
0.520731724
0.584352541
4124
17
0.077170048
0.183068221
4124
5
−0.157303536
−0.036843328
4135
10
−0.35554626
−0.339641516
4135
10
−0.232750407
−0.161598404
4143
14
0.185986189
−0.024609518
4143
7
−0.032712551
0.024485269
4233
18
3.990305253
1.582890447
4233
8
0.770838584
0.706726475
4313
11
0.453186971
0.126025314
4313
15
−0.129336617
−0.079142717
4510
22
−0.167456263
−0.135667934
4510
17
−0.046588554
−0.100490086
5327
46
−0.415159893
−0.324486992
5327
12
−0.146934163
−0.22684469
6120
3
4.432535499
4.497409395
6120
3
0.919305283
0.723423487
6417
7
−0.253214481
−0.312493286
6417
10
−0.141004928
−0.007636336
6434
42
−0.220525385
−0.208472395
6434
14
−0.001659417
0.063668162
6756
55
2.0189449
1.307321937
6756
4
−0.009878439
0.570901233Table 14
Top 25 gene indices before and after estimation for DLBCL-FL.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
447
447
2
913
4135
3
4135
913
4
546
640
5
2929
2929
6
640
1142
7
4510
546
8
3969
3969
9
5327
4510
10
6756
4233
11
4233
5327
12
4313
4313
13
6120
6756
14
1142
6120
15
1129
1553
16
1731
1129
17
4124
6417
18
3965
4124
19
1293
1731
20
6434
1293
21
28
6434
22
4143
28
23
2062
2062
24
1553
4094
25
6417
1984
Number of common genes in top 25 positions = 23
% of common genes = 92Table 15
Performance comparison for DLBCL-FL.
Dataset name
DLBCL-FL
Number of genes
Number of samples
Number of samples in Cancer dataset
Number of samples in normal dataset
Number of samples in training dataset
Number of samples in testing dataset
SVM kernel used
Number of folds for LOOCV
7070
77
58
19
62 (47 DLBCL, 15 FL)
15 (11 DLBCL, 4 FL)
Linear
31
Algorithm
% accuracy
Sensitivity/recall
Specificity
F1-score
G-mean
Proposed approach
100 (with top 5, 10, 15, and 20 genes)
1
1
1
1
PSO based graph theoretic approach
94
0.95
0.94
0.89
0.94
DWFS using KNN classifier
91
0.97
0.85
0.94
0.9
DWFS using NBC classifier
96
1
0.9
0.98
0.94


## 4.3.1. Results of Missing Value Estimation Part

We perform the initial estimation using modified version of LRFDVImpute with a membership gradeθ
=
0.55. After the initial estimation is over, we forcibly treat cells at specified locations as missing and estimate them using different membership values of θ and both earlier and modified versions. This has been carried out only once, after estimating rows with single missing values and the corresponding RMSE values have been recorded. We have performed our experiments only on alpha, cdc15, and elu data of Spellman dataset. The number of missing values is too large for cdc28; that is why we ignore that segment. The results for the alpha, cdc15, and elu datasets using both methods are shown in Tables 2–4. Figures 3–5 show the corresponding plots of RMSE versus membership grade θ for each of the four datasets.Table 2
Results for missing value estimation algorithm (Spellman, alpha).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0.012405344
0.012488181
0.012562782
0.012562782
0.012690904
0.012197374
0.012638865
Modified LRFDVImpute
0.012439936
0.012439366
0.012389872
0.012389872
0.012645466
0.011988263
0.013268721Table 3
Results for missing value estimation algorithm (Spellman, cdc15).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0.016832094
0.016760968
0.016706119
0.016682418
0.016768837
0.016733642
0.049482242
Modified LRFDVImpute
0.016781257
0.016710318
0.016736613
0.016723349
0.016637753
0.017023671
0.057225615Table 4
Results for missing value estimation algorithm (Spellman, elu).
RMSE∖
θ
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Original LRFDVImpute
0
0
0
0
0
0
0
Modified LRFDVImpute
0
0
0
0
0
0
0Figure 3
Plot of RMSE versus membership gradeθ for alpha dataset.Figure 4
Plot of RMSE versus membership gradeθ for cdc15 dataset.Figure 5
Plot of RMSE versus membership gradeθ for elu dataset.Table5 compares the performance of both versions of LRFDVImpute method to that of some other existing methods, like SVDImpute, LLSImpute, FDVLLSImpute, FDVSPLINEImpute, and so forth, and the results show that modified version of LRFDVImpute outperforms the other existing methods as far as RMSE value is concerned.Table 5
Performance comparison with other existing methods.
Dataset
SVDImpute
LLSImpute
FDVLLSImpute
FDVSPLINEImpute
FDVLRImpute withθ
=
0.55
Original LRFDVImpute
Modified LRFDVImpute
alpha
0.03395
0.07853
0.096
0.063
0.012562782
0.012389872
cdc15
0.05055
0.1208
0.258
0.127
0.016682418
0.016723349
elu
0.01585
0.0033
0.044
.019
0
0


## 4.3.2. Combined Results

We test the significance of our proposed missing value estimation technique using the gene ranking method. We have not found any state-of-the-art work on gene ranking so far where Spellman dataset is used. That is why we use four more publicly available real-life gene expression datasets, like Colorectal Cancer dataset (GDS4382), Breast Cancer dataset (GSE349-350), Prostate Cancer dataset, and Leukaemia Cancer dataset (DLBCL-FL) [4, 27–32], to perform steps such as missing values estimation and gene ranking and analyze the results. We start with the microarray dataset containing missing values and apply our proposed missing value estimation technique to estimate the genes with missing values (if any). We rank them using proposed gene ranking method and find the top ranked genes. We then forcibly insert missing values in the top ranked genes and again estimate them using the same missing value estimation technique. Finally, we rank them once more to find the top ranked genes. Results show that most of the top ranked genes remain the same, which implies that the proposed missing value estimation technique has been accurate in estimating the unknown values. We have normalized most of the datasets using z-score normalization method in order to bring the data values to a common scale.Tables6, 8, 10, and 13 show the estimated values for the four datasets, Tables 7, 9, 11, and 14 show the common gene indices before and after the estimation, and Tables 12 and 15 compare the performance of the proposed approach with two state-of-the-art methods [22, 23] for Prostate and Leukaemia dataset on the basis of accuracy, sensitivity, specificity, F
1-score, and  G-mean metrics. We have found that Prostate and Leukaemia are the common dataset on which both the existing methods have done their experiments. The results show that the proposed gene ranking approach performs far better compared to those existing approaches, where one is a PSO based graph theoretic approach [22] and the other is a web based tool DWFS, which uses KNN and NBC classifiers [23] as far as those metrics are concerned.Table 6
Estimated values for Colorectal Cancer GDS4382.
Colorectal Cancer GDS4382
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
714
12
0.677252397
0.703080766
714
12
0.140108496
−0.163286572
1245
5
1.56686079
1.755118642
1245
5
1.212949296
1.148663192
1578
3
0.787510895
1.003134829
1578
3
0.256998387
0.22381735
1763
11
1.024714768
0.737414862
1763
11
0.20681001
0.056394235
2792
9
−0.861395162
−0.865043164
2792
9
−1.064597763
−1.005338727
4025
1
0.781326343
1.137532185
4025
1
−0.31562626
−0.297767551
4134
15
0.892338308
0.958542974
4134
15
0.329755342
0.247541595
5082
2
−0.006360431
0.32763543
5082
2
−0.480000425
−0.370717546
8426
13
1.210288879
0.790345743
8426
13
−0.082823022
0.157094029
9979
6
2.932068401
2.953449691
9979
6
2.387826603
2.246418246
10083
11
1.369142269
1.258410425
10083
11
1.915272483
1.809541605
10145
10
1.940467541
1.852338203
10145
10
2.750210569
2.834220655
10208
3
1.868505436
1.778084682
10208
3
2.224446282
2.122262802
10280
6
1.424951861
1.521769713
10280
6
0.98680627
1.139738659
10323
1
0.895845032
0.963530648
10323
1
0.378278185
0.376819137
10725
14
0.562534293
0.903674258
10725
14
1.263242627
1.285858157
10789
4
1.582210172
1.834961289
10789
4
0.507667085
0.540477481
10855
10
3.17769843
3.162360832
10855
10
2.562212194
2.546399932
11050
3
2.676634486
2.657786027
11050
3
1.872943693
1.894214081
11055
8
2.112261939
2.105431748
11055
8
1.680697142
1.713521495
11100
16
2.522783399
2.455429314
11100
16
3.208462284
3.064140274
11465
1
2.454481056
2.18211664
11465
1
1.232460868
1.356633277
11485
12
−0.701537989
−0.49772711
11485
12
0.609015246
0.316407836
11650
6
1.470662615
1.35686403
11650
6
1.940624638
2.047625027
11677
4
2.591635801
2.602068714
11677
4
2.903830552
2.934334312Table 7
Top 25 gene indices before and after estimation for GDS4382.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
714
714
2
1245
1245
3
1578
1578
4
1763
1763
5
2792
2792
6
4025
4025
7
4134
4134
8
5082
5082
9
8426
8426
10
9979
9979
11
10083
10083
12
10145
10145
13
10208
10208
14
10280
10280
15
10323
10323
16
10725
10725
17
10789
10789
18
10855
10855
19
11050
11050
20
11055
11055
21
11100
11100
22
11465
11465
23
11485
11485
24
11650
11650
25
11677
11677
Number of common genes in top 25 positions = 25
% of common genes = 100Table 8
Estimated values for Breast Cancer dataset.
Breast Cancer
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
272
12
−0.354039943
−0.419709522
272
8
−0.407057103
−0.396601672
329
5
−0.176687651
−0.295980517
329
2
−0.021584122
0.092981006
491
3
0.222486126
0.30260015
491
10
0.140530363
0.067353238
869
11
0.79646566
0.852100043
869
5
0.345360946
0.279331888
1143
9
0.017956445
0.256958128
1143
4
0.054582833
−0.145618319
1937
1
0.22672464
0.33943153
1937
7
−0.128257731
0.05338273
2825
8
7.552441375
8.108907291
2825
3
6.080319682
5.969522121
3004
2
0.128475064
0.04404869
3004
6
−0.124353092
−0.113116385
4911
4
−0.550612392
−0.372472073
4911
9
−0.090276567
−0.152051028
5328
6
10.57228289
9.556257324
5328
5
8.316072021
7.887609502
6184
11
−0.493537621
−0.504575199
6184
8
0.035382884
−0.132574115
7941
10
0.233974208
0.14984931
7941
6
0.077199916
0.101298927
8452
14
−0.312716903
−0.32992237
8452
3
−0.071344506
−0.092371778
9076
6
0.708996621
0.154489296
9076
1
0.060554625
0.036078623
9267
13
−0.270092957
−0.33842598
9267
10
−0.034956721
0.30142636
9574
7
0.093118778
0.074772718
9574
7
−0.227882015
−0.159726559
9723
4
−0.018883445
−0.279787171
9723
5
0.407076237
0.815509208
9753
10
−0.23265185
−0.264909557
9753
8
−0.228475796
−0.265794544
9905
3
−0.31537376
−0.417539213
9905
3
−0.286431974
−0.199211435
10319
8
−0.049692673
−0.038561161
10319
9
−0.218073382
−0.242557019
10614
9
−0.511734814
−0.430792872
10614
2
−0.268390734
−0.1922339
11377
1
−0.055809992
0.083662727
11377
7
−0.268295765
−0.144527083
11737
12
−0.34275829
−0.269422135
11737
4
−0.387083296
−0.093221982
11976
6
0.030992978
−0.133189906
11976
9
−0.270598103
−0.272899078
12053
4
−0.374640827
−0.328380483
12053
10
−0.360077886
−0.244255658Table 9
Top 25 gene indices before and after estimation for Breast Cancer dataset.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
3004
1143
2
7941
3004
3
10319
7941
4
869
10319
5
5328
11737
6
9723
491
7
491
9753
8
9574
869
9
9905
5328
10
11737
9723
11
2825
12053
12
4911
2825
13
8452
9574
14
272
9905
15
329
4911
16
6184
8452
17
9076
9076
18
9267
329
19
9753
6184
20
10614
9267
21
11377
11976
22
11976
2218
23
12053
2459
24
1143
2995
25
1937
4200
Number of common genes in top 25 positions = 21
% of common genes = 84Table 10
Estimated values for Prostate Cancer dataset.
Prostate Cancer
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
205
45
7.890085103
9.434976702
205
45
5.168380524
5.416011322
2839
5
−0.176203136
0.004447962
2839
5
−0.173585261
−0.183852307
3649
10
0.296457522
0.046654036
3649
10
0.329958724
0.617420849
3794
8
−0.241993865
−0.216817937
3794
8
−0.106322335
−0.109294537
4365
17
0.059874409
2.443242853
4365
17
−0.122967971
−0.099879376
5757
14
0.090277183
0.70723886
5757
14
−0.196702226
−0.067191045
5944
22
−0.180239863
−0.186977867
5944
22
−0.137411352
−0.217659599
6185
36
1.557345019
1.548681099
6185
36
−0.010802226
−0.181261903
6462
28
−0.16997265
−0.175762613
6462
28
0.202288499
0.191383577
7247
32
0.859643204
0.956551639
7247
32
0.932414697
2.21258573
7520
18
0.270631813
0.179429504
7520
18
0.064953651
0.155169008
7557
11
0.393712194
0.375941042
7557
11
1.450975133
1.084180805
7768
3
0.042090452
0.11563321
7768
3
0.59875936
0.576608917
8123
47
0.216968028
0.283330867
8123
47
0.128003038
0.165883246
8554
4
0.015161421
0.083268762
8554
4
0.40771258
−0.051156112
8768
26
−0.135452978
−0.04501778
8768
26
−0.138712043
−0.188517852
8850
17
2.708006948
1.916254815
8850
17
−0.364936796
−0.228740529
9034
29
−0.190217605
−0.178815842
9034
29
−0.194249889
−0.047431811
9050
34
0.351634344
0.174924563
9050
34
0.284389403
0.469260339
9172
42
2.277059356
2.347704101
9172
42
1.727648712
1.562259352
9850
50
−0.046243874
−0.290856086
9850
50
1.371476261
2.134924482
10138
14
0.599428228
0.628396963
10138
14
0.765017083
1.001711406
10494
22
−0.189181177
−0.122575727
10494
22
0.220022109
0.365466432
10537
48
0.139373755
0.406906442
10537
48
0.035059438
0.011805144
10956
7
0.337716243
0.155279221
10956
7
0.146282156
0.169249764Table 11
Top 25 gene indices before and after estimation for Prostate Cancer dataset.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
6185
6185
2
10494
10494
3
9850
4365
4
4365
9850
5
10138
9034
6
9172
10138
7
9034
5944
8
5944
9172
9
3649
3649
10
8554
2839
11
2839
7557
12
7557
10956
13
205
9050
14
3794
7520
15
10956
3794
16
8850
205
17
7520
8850
18
9050
10537
19
10537
5757
20
5757
8554
21
8123
8768
22
6462
8123
23
8768
6462
24
7247
7247
25
7768
9093
Number of common genes in top 25 positions = 24
% of common genes = 96Table 12
Performance comparison for Prostate dataset.
Dataset name
Prostate
Number of genes
Number of samples
Number of samples in Cancer dataset
Number of samples in normal dataset
Number of samples in training dataset
Number of samples in testing dataset
SVM kernel used
Number of folds for LOOCV
12600
102
52
50
82 (42 cancer, 40 normal)
20 (10 cancer, 10 normal)
Linear
41
Algorithm
% accuracy
Sensitivity/recall
Specificity
F1-score
G-mean
Proposed approach
100 (with top 5 genes)
1
1
1
1
PSO based graph theoretic approach
91
0.91
0.92
0.91
0.91
DWFS using KNN classifier
86
0.87
0.85
0.86
0.86
DWFS using NBC classifier
80
0.76
0.85
0.78
0.80Table 13
Estimated values for DLBCL-FL.
Leukaemia (DLBCL-FL)
Cancer
Normal
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
⁢Missing values inserted
Old_value
Estimated value with mem = 0.55
At row
At column
At row
At column
28
50
−0.188603483
−0.203477278
28
2
3.886811534
0.84848513
447
45
0.103798973
0.304241026
447
18
−0.146274929
0.009235547
546
8
1.055798216
1.246420135
546
8
0.140697423
0.320415944
640
52
0.052220359
−0.311252744
640
7
0.12552933
0.364757746
913
5
0.15759485
0.180411915
913
2
−0.135855848
−0.165390479
1129
4
1.051658222
0.598949762
1129
10
0.04903928
0.096097027
1142
31
−0.249901255
−0.311776993
1142
1
−0.249625598
−0.196558516
1293
34
1.91037315
2.330011528
1293
3
0.27829269
0.224522592
1553
48
−0.218067263
−0.107003318
1553
9
−0.00577268
0.602953445
1731
26
0.555312243
0.39207635
1731
7
0.448099319
0.524633075
2062
22
2.366047008
4.45272374
2062
13
0.201613611
0.849245039
2929
17
0.072886107
0.125837311
2929
11
−0.007410482
0.004016071
3965
56
0.137340738
−0.905551574
3965
16
−0.133906259
−0.083702381
3969
36
2.328255836
2.60971631
3969
6
0.520731724
0.584352541
4124
17
0.077170048
0.183068221
4124
5
−0.157303536
−0.036843328
4135
10
−0.35554626
−0.339641516
4135
10
−0.232750407
−0.161598404
4143
14
0.185986189
−0.024609518
4143
7
−0.032712551
0.024485269
4233
18
3.990305253
1.582890447
4233
8
0.770838584
0.706726475
4313
11
0.453186971
0.126025314
4313
15
−0.129336617
−0.079142717
4510
22
−0.167456263
−0.135667934
4510
17
−0.046588554
−0.100490086
5327
46
−0.415159893
−0.324486992
5327
12
−0.146934163
−0.22684469
6120
3
4.432535499
4.497409395
6120
3
0.919305283
0.723423487
6417
7
−0.253214481
−0.312493286
6417
10
−0.141004928
−0.007636336
6434
42
−0.220525385
−0.208472395
6434
14
−0.001659417
0.063668162
6756
55
2.0189449
1.307321937
6756
4
−0.009878439
0.570901233Table 14
Top 25 gene indices before and after estimation for DLBCL-FL.
Ranking
Rank
Gene indices prior to missing value insertion
Gene indices after missing value insertion
1
447
447
2
913
4135
3
4135
913
4
546
640
5
2929
2929
6
640
1142
7
4510
546
8
3969
3969
9
5327
4510
10
6756
4233
11
4233
5327
12
4313
4313
13
6120
6756
14
1142
6120
15
1129
1553
16
1731
1129
17
4124
6417
18
3965
4124
19
1293
1731
20
6434
1293
21
28
6434
22
4143
28
23
2062
2062
24
1553
4094
25
6417
1984
Number of common genes in top 25 positions = 23
% of common genes = 92Table 15
Performance comparison for DLBCL-FL.
Dataset name
DLBCL-FL
Number of genes
Number of samples
Number of samples in Cancer dataset
Number of samples in normal dataset
Number of samples in training dataset
Number of samples in testing dataset
SVM kernel used
Number of folds for LOOCV
7070
77
58
19
62 (47 DLBCL, 15 FL)
15 (11 DLBCL, 4 FL)
Linear
31
Algorithm
% accuracy
Sensitivity/recall
Specificity
F1-score
G-mean
Proposed approach
100 (with top 5, 10, 15, and 20 genes)
1
1
1
1
PSO based graph theoretic approach
94
0.95
0.94
0.89
0.94
DWFS using KNN classifier
91
0.97
0.85
0.94
0.9
DWFS using NBC classifier
96
1
0.9
0.98
0.94


## 5. Conclusion and Future Scope

The proposed modified version of LRFDVImpute technique has been tested on the dataset from Spellman et al. [26] and has shown impressive results. It outperforms some state-of-the-art methods. The plots of RMSE versus membership grade θ show that modified version is equivalent to or better than earlier version for the alpha and cdc15 datasets. However, for the cdc28 dataset, earlier version has shown better results. For the elu datasets, both have reached 0 error margin. For both versions, a membership grade between 0.55 and 0.65 produces minimum error and any value in this range can be considered as a threshold to be used for fresh experiments.The validation of the missing value estimation shows that most of the top ranked genes remain the same, before and after imputation, which implies that the proposed modified LRFDVImpute technique has been accurate in estimating the unknown values.As a future scope, we would like to analyze the effects of using quadratic regression for estimation of missing values and the use of data cleaning techniques before imputation which may remove outliers if any and may further reduce the error margin. For gene ranking, we wish to analyze the effects of different parameter settings for GA and observe the ranking and classification results using SVM with other kernels and also compare results with the ones mentioned in literature. We would also wish to modify our algorithms so as to make this ranking more efficient and find out the most significant genes that would correctly identify the subtypes of a particular type of cancer. For the Leukemia dataset [16], this could be identifying the B-cell and T-cell lineages for the acute lymphoblastic leukemia (ALL) samples.



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*Source: 6134736-2016-07-18.xml*