# Triclustering Discovery Using the δ-Trimax Method on Microarray Gene Expression Data

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basis

#### 2.1. Perfect Shifting Clustering

#### 2.2. Mean Squared Residual

#### 2.3. Triclustering Quality Index (TQI)

## 3. Methodology

#### 3.1. Multiple Node Deletion Algorithm

- When $S\ge \delta $ then proceed to step 2, otherwise the process is not continued and gives $M(I,J,K)$ as the final result of this algorithm.
- Delete the ith gene if it satisfies the following inequality:$\frac{1}{\left|J\right|\left|K\right|}{\sum}_{j\in J,k\in K}{({m}_{ijk}-{m}_{iJK}-{m}_{IJk}+2{m}_{IJK})}^{2}>\lambda S$.
- Recalculate:${m}_{iJK},\forall i\in I;{m}_{IjK},\forall j\in J;{m}_{IJk},\forall k\in K;{m}_{IJK}$ and S.
- Delete the jth condition if it satisfies the following inequality:$\frac{1}{\left|I\right|\left|J\right|}{\sum}_{i\in I,j\in K}{({m}_{ijk}-{m}_{iJK}-{m}_{IJk}+2{m}_{IJK})}^{2}>\lambda S$.
- Recalculate:${m}_{iJK},\forall i\in I;{m}_{IjK},\forall j\in J;{m}_{IJk},\forall k\in K;{m}_{IJK}$ and S.
- Repeat step 2 to step 7. If there are no genes, conditions and times are deleted then the iteration stops.

#### 3.2. Single Node Deletion Algorithm

- Detect the gene, condition, and time that has the highest residual score in the following way:
- The residual score for the ith gene, $\forall i\in I$:$\mu \left(i\right)=\frac{1}{\left|J\right|\left|K\right|}{\sum}_{j\in J,k\in K}{({m}_{ijk}-{m}_{iJK}-{m}_{IjK}-{m}_{IJk}+2{m}_{IJK})}^{2}$
- The residual score for the jth condition, $\forall j\in J$:$\mu \left(j\right)=\frac{1}{\left|I\right|\left|K\right|}{\sum}_{i\in I,k\in}{({m}_{ijk}-{m}_{iJK}-{m}_{IjK}-{m}_{IJk}+2{m}_{IJK})}^{2}$
- The residual score for the kth time, $\forall k\in K$:$\mu \left(k\right)=\frac{1}{\left|I\right|\left|K\right|}{\sum}_{i\in I,k\in K}{({m}_{ijk}-{m}_{iJK}-{m}_{IjK}-{m}_{IJk}+2{m}_{IJK})}^{2}$

- Delete the gene, condition or time that has the highest score.
- Recalculate ${m}_{iJK}\forall i\in I,{m}_{IjK}\forall j\in J,{m}_{IJk}\forall k\in K,{m}_{IJK}$ and S.
- Repeat step 1 to step 3. If value $S\le \delta $ then iteration stops.The final result of the single node deletion algorithm is subspace $M({I}^{\prime},{J}^{\prime},{K}^{\prime})$ which has a value of $S\le \delta $, where ${I}^{\prime}\subseteq I,{J}^{\prime}\subseteq J$ and ${K}^{\prime}\subseteq K$.

#### 3.3. Node Addition Algorithm

- Add genes $i\notin I$ that satisfy$\frac{1}{\left|J\right|\left|K\right|}{\sum}_{j\in J,k\in K}{({m}_{ijk}-{m}_{iJK}-{m}_{IjK}-{m}_{IJk}+2{m}_{IJK})}^{2}\le S$
- Recalculate ${m}_{IjK}\forall j,{m}_{IJk}\forall k$ and S.
- Add conditions $j\notin J$ that satisfy$\frac{1}{\left|I\right|\left|K\right|}{\sum}_{i\in I,k\in K}{({m}_{ijk}-{m}_{iJK}-{m}_{IjK}-{m}_{IJk}+2{m}_{IJK})}^{2}\le S$.
- Recalculate ${m}_{IjK}\forall j,{m}_{IJk}\forall k$ and S.
- Add times $k\notin K$ that satisfy$\frac{1}{\left|I\right|\left|J\right|}{\sum}_{(}iinI,j\in J){({m}_{ijk}-{m}_{iJK}-{m}_{IjK}-{m}_{IJk}+2{m}_{IJK})}^{2}\le S$.
- Recalculate ${m}_{IjK}\forall j,{m}_{IJk}\forall k$ and S.
- Repeat step 1 to step 6. If there are no more nodes added to the gene, condition and time then the iteration stops.

#### 3.4. Algorithm Simulation

#### 3.4.1. Precomputing

#### 3.4.2. Single Node Deletion

- Iteration 1The single node deletion algorithm computes residual score on each gene, condition and time step using residual data in Table 2.
- Residual score at $gen{e}_{i},\forall i\in I$.$$\begin{array}{cc}\hfill \mu \left(1\right)& =\frac{1}{\left|J\right|\left|K\right|}\sum _{j\in J,k\in K}{r}_{1jk}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{12}[{\left(0.14\right)}^{2}+{(-6.21)}^{2}+{\left(8.34\right)}^{2}+{\left(10.68\right)}^{2}+...+{(-10.3)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =54.37\hfill \\ \hfill \mu \left(2\right)& =\frac{1}{\left|J\right|\left|K\right|}\sum _{j\in J,k\in K}{r}_{2jk}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{12}[{(-1.94)}^{2}+{(-1.29)}^{2}+{\left(11.26\right)}^{2}+{(-3.4)}^{2}+...+{(-1.66)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =17.48\hfill \\ \hfill \mu \left(3\right)& =\frac{1}{\left|J\right|\left|K\right|}\sum _{j\in J,k\in K}{r}_{3jk}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(3\right)\left(4\right)}[{(-4.11)}^{2}+{(-1.46)}^{2}+{(-6.91)}^{2}+{(-2.57)}^{2}+...+{(-2.49)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =23.14\hfill \\ \hfill \mu \left(4\right)& =\frac{1}{\left|J\right|\left|K\right|}\sum _{j\in J,k\in K}{r}_{4jk}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(3\right)\left(4\right)}[{(-3.19)}^{2}+{(-5.54)}^{2}+{\left(5.01\right)}^{2}+{(-2.65)}^{2}+...+{(-4.59)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =30.89\hfill \\ \hfill \mu \left(5\right)& =\frac{1}{\left|J\right|\left|K\right|}\sum _{j\in J,k\in K}{r}_{5jk}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(3\right)\left(4\right)}[{\left(8.64\right)}^{2}+{\left(2.29\right)}^{2}+{(-5.16)}^{2}+{\left(11.18\right)}^{2}+...+{(-4.76)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =43.54\hfill \end{array}$$Based on the calculation above, we’ve got ${g}_{1}$ as the highest gene residual score which yield $53.47$.
- Residual score calculation where $conditio{n}_{j},\forall j\in J$:$$\begin{array}{cc}\hfill \mu \left(1\right)& =\frac{1}{\left|I\right|\left|K\right|}\sum _{i\in I,k\in K}{r}_{i1j}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(5\right)\left(4\right)}[{\left(0.14\right)}^{2}+{\left(10.68\right)}^{2}+{\left(3.14\right)}^{2}+{(-1.46)}^{2}+{(-1.94)}^{2}+...+{\left(3.04\right)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =21.17\hfill \\ \hfill \mu \left(2\right)& =\frac{1}{\left|I\right|\left|K\right|}\sum _{i\in I,k\in K}{r}_{i2j}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(5\right)\left(4\right)}[{(-6.21)}^{2}+{\left(4.33\right)}^{2}+{(-2.21)}^{2}+{\left(10.19\right)}^{2}+{(-1.29)}^{2}+...+{(-6.31)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =24.49\hfill \\ \hfill \mu \left(3\right)& =\frac{1}{\left|I\right|\left|K\right|}\sum _{i\in I,k\in K}{r}_{i3j}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(5\right)\left(4\right)}[{\left(8.34\right)}^{2}+{(-13.12)}^{2}+{(-3.66)}^{2}+{(-10.3)}^{2}+{(-11.26)}^{2}+...+{(-4.76)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =49.99\hfill \end{array}$$Based on the calculation above, the residual score obtained at $conditio{n}_{j}=3$ in the amount of $49.99$.
- Residual score calculation for $time\_ste{p}_{k},\forall k\in K$:$$\begin{array}{cc}\hfill \mu \left(1\right)& =\frac{1}{\left|I\right|\left|J\right|}\sum _{i\in I,k\in K}{r}_{ij1}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(5\right)\left(3\right)}[{\left(0.14\right)}^{2}+{(-6.21)}^{2}+{\left(8.34\right)}^{2}+{(-1.94)}^{2}+...+{(-5.16)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =31.97\hfill \\ \hfill \mu \left(2\right)& =\frac{1}{\left|I\right|\left|J\right|}\sum _{i\in I,k\in K}{r}_{ij2}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(5\right)\left(3\right)}[{\left(10.68\right)}^{2}+{\left(4.33\right)}^{2}+{(-13.12)}^{2}+{(-3.4)}^{2}+...+{(-2.62)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =41\hfill \\ \hfill \mu \left(3\right)& =\frac{1}{\left|I\right|\left|J\right|}\sum _{i\in I,k\in K}{r}_{ij1}^{3}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(5\right)\left(3\right)}[{\left(3.14\right)}^{2}+{(-2.21)}^{2}+{(-3.66)}^{2}+{(-2.94)}^{2}+...+{(-5.16)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =32.97\hfill \\ \hfill \mu \left(4\right)& =\frac{1}{\left|I\right|\left|J\right|}\sum _{i\in I,k\in K}{r}_{ij4}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(5\right)\left(3\right)}[{(-1.46)}^{2}+{\left(10.19\right)}^{2}+{(-10.3)}^{2}+{\left(4.46\right)}^{2}+...+{(-4.76)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =31.97\hfill \end{array}$$

Table 3 summarizes residual scores on genes, conditions, and time steps for iteration 1.Based on Table 3 on all columns, we’ve got 54.37 as the highest score. This highest score is obtained on $gen{e}_{1}$, so that makes $gen{e}_{1}$ erased. The sets condition after the first iteration are $I=\{2,3,4,5\}$, $J=\{1,2,3\}$, and $K=\{1,2,3,4\}$. After the deletion is done, check again whether the S value on sub-space $M(I,J,K)$ is smaller than $\delta $. In order to do that, we need to recalculate ${m}_{iJK},{m}_{IjK},{m}_{IJk},{m}_{IJK}$ and S on the current sets.Using the same way as before, we have got the average value for each gene, condition and time step as follows: - Iteration 2Table 6 summarizes residual scores on genes, conditions, and time steps for iteration 2.The maximum score based on Table 6 is 50.43, so the $gen{e}_{5}$ removed from the data. The new data obtained after node deletion consists of $I=2,3,4,J=1,2,3$ and $K=1,2,3,4$. We compute ${m}_{iJK}\forall i\in I,{m}_{IjK}\forall j\in J,{m}_{IJk}\forall k\in K,{m}_{IJK}$ and S of the current data.
- Iteration 3Table 9 summarizes residual scores on gene $i\in I$, condition $j\in J$, and time step $k\in K$ for iteration 3. The maximum score obtained at $conditio{n}_{3}$ which equal to 39.19. Therefore, we remove $conditio{n}_{3}$ on the dataset and recalculate ${m}_{iJK}\forall i\in I,{m}_{IjK}\forall j\in J,{m}_{IJk}\forall k\in K$ and S. The current dataset are $I=(2,3,4),J=(1,2)$, and $K=(1,2,3,4)$. The same calculation steps are carried out and we got $S=5.48$ which is bigger than $\delta =2.7$. Therefore the calculation is repeated again on iteration 4.
- Iteration 4According to Table 10, we have got $S=1.43$. Because $S<\delta =2.7$, iteration for single node deletion is terminated. The next step is to perform a node addition algorithm using the data in Table 11. Table 12 summarizes residual scores on gene $i\in I$, condition $j\in J$, and time step $k\in K$ for iteration 3.

#### 3.4.3. Node Addition Algorithm

- Gene addition to $i\notin I$Calculate ${m}_{iJK}$ for $i\notin I$:$$\begin{array}{cc}\hfill {m}_{1JK}& =\frac{1}{\left|J\right|\left|K\right|}\sum _{j\in J,k\in K}{m}_{1jk}=\frac{1}{\left(2\right)\left(3\right)}\times (13+7+22+16+14+9)=13.5\hfill \\ \hfill {m}_{5JK}& =\frac{1}{\left|J\right|\left|K\right|}\sum _{j\in J,k\in K}{m}_{5jk}=\frac{1}{\left(2\right)\left(3\right)}\times (20+14+21+7+1+20)=13.83\hfill \end{array}$$Residual score for gene $i\in I$ calculation:$$\begin{array}{cc}\hfill \mu \left(1\right)& =\frac{1}{\left|J\right|\left|K\right|}\sum _{j\in J,k\in K}{({m}_{1jk}-{m}_{1JK}-{m}_{IjK}-{m}_{IJk}+2{m}_{IJK})}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(2\right)\left(3\right)}\times [{(13-13.5-2.56-3.67+2\times 2.83)}^{2}+...+{(9-13.5-3.11-2.83+2\times 2.83)}^{2}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =30.7\hfill \\ \hfill \mu \left(5\right)& =55.4\hfill \end{array}$$According to Table 13, we obtained $\mu \left(1\right)=30.7$ and $\mu \left(5\right)=55.4$, these values bigger than $S=1.43$, then no new datum is added to the dataset.
- Condition addition to $j\notin J$Calculates ${m}_{IjK}$ for $j\notin J$:$$\begin{array}{cc}\hfill {m}_{I3K}& =\frac{1}{\left|I\right|\left|K\right|}\sum _{i\in I,k\in K}{m}_{i1k}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(3\right)\left(3\right)}\times (21+6+4+2+11+16+17+20+21)=13.11\hfill \end{array}$$Residual score for condition $j\notin J$:$$\mu \left(3\right)=47.67$$According to Table 14 $\mu \left(3\right)=47.67$, then no new datum is added to the dataset.
- Time step addition for $k\notin K$Calculates ${m}_{IJk}$ for $k\notin K$:$$\begin{array}{cc}\hfill {m}_{IJ4}& =\frac{1}{\left|I\right|\left|J\right|}{m}_{ij4}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\left(3\right)\left(2\right)}\times (12+9+1+15+8+14)=9.83\hfill \end{array}$$Residual score for condition $j\in J$:$$\mu \left(4\right)=19.48$$According to Table 15, we obtained, $\mu \left(4\right)=19.48$, then no new time steps are added to the data in Table 11. From the addition node algorithm results, it turns out that there is no addition to genes, conditions, and time, so the algorithm is stopped and not continued to the next iteration. Because there are no additional nodes, the data for this algorithm’s final result is the same as the initial data used (as given in Table 11).The final result produced by addition node algorithm is a tricluster obtained from the first iteration of the $\delta $-Trimax method. Therefore, the data in Table 11 is a tricluster that has sub-space $M(I,J,K)$ with $I=\{2,3,4\}$, $J=\{1,2\}$, and $K=\{1,2,3\}$.

#### 3.4.4. Masking

## 4. Implementation Results

#### 4.1. The Dataset

#### 4.2. Determination of Threshold $\delta $ and $\lambda $

- In each condition, gene clustering was performed against time using the K-Means method.
- For each cluster generated from stage 1, time clustering was carried out on genes using the K-Means method.
- Each cluster produced in stage 2 is calculated the mean square residual (S) value. The smallest S is used as $\delta $.

#### 4.3. Simulation Comparison

## 5. Discussion

#### 5.1. HiPSC Triclustering Performance

#### 5.2. HIV-1 Triclustering Performance

#### 5.3. DNA/RNA Sequences

## 6. Conclusions

- From several simulations using different $\delta $ and $\lambda $, the best simulation is obtained when using $\delta =0.0068$ and $\lambda =1.2$ for HiPSC, $\delta =0.0046$ and $\lambda =1.25$ for HIV-1.
- The best five tricluster based on the smallest TQI for HiPSC data. This group of gene expression within the five tricluster is thought to be a feature of heart disease. Therefore, this gene group can be used by medical experts in providing further treatment, such as making the genes in this tricluster a therapeutic target or as a drug development.
- Three biomarkers for HIV-1 disease were obtained from the 10 selected tricluster. Biomarkers consist of genes AGFG1, EGR1, and HLA-C.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison graph of Triclustering Quality Index (TQI) and computation time in each simulation.

Datum | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ | ${\mathit{t}}_{4}$ | Total | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | ||

${g}_{1}$ | 13 | 7 | 25 | 22 | 16 | 2 | 14 | 9 | 11 | 13 | 25 | 8 | 165 |

${g}_{2}$ | 4 | 5 | 21 | 1 | 1 | 6 | 1 | 5 | 4 | 12 | 9 | 13 | 82 |

${g}_{3}$ | 1 | 4 | 2 | 1 | 2 | 11 | 4 | 2 | 16 | 1 | 15 | 13 | 72 |

${g}_{4}$ | 5 | 3 | 17 | 4 | 3 | 20 | 2 | 3 | 21 | 8 | 14 | 9 | 109 |

${g}_{5}$ | 20 | 14 | 10 | 21 | 7 | 11 | 1 | 20 | 8 | 16 | 7 | 12 | 147 |

Total | 43 | 33 | 75 | 49 | 29 | 50 | 22 | 39 | 60 | 50 | 70 | 55 | 575 |

Datum | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ | ${\mathit{t}}_{4}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | |

${g}_{1}$ | 0.14 | −6.21 | 8.34 | 1068 | 4.33 | −13.12 | 3.14 | −2.21 | −3.66 | −1.46 | 10.19 | −10.3 |

${g}_{2}$ | −1.94 | −1.29 | 1126 | −3.4 | −3.8 | −2.2 | −2.94 | 0.71 | −3.74 | 4.46 | 1.11 | 1.66 |

${g}_{3}$ | −4.11 | −1.46 | −6.91 | −2.57 | −1.9 | 3.63 | 0.89 | −1.46 | 9.09 | −5.71 | 7.94 | 2.49 |

${g}_{4}$ | −3.19 | −5.54 | 5.01 | −2.65 | −4 | 9.55 | −4.19 | −3.54 | 11.01 | −1.79 | 3.86 | −4.59 |

${g}_{5}$ | 8.64 | 2.29 | −5.16 | 11.18 | −3.2 | −2.62 | −8.36 | 10.29 | −5.16 | 3.04 | −6.31 | −4.76 |

Residual Gene Score | Residual Condition Score | Residual Time Step Score |
---|---|---|

$\mu \left(1\right)=54.37$ | $\mu \left(1\right)=27.17$ | $\mu \left(1\right)=31.97$ |

$\mu \left(2\right)=17.48$ | $\mu \left(2\right)=24.49$ | $\mu \left(2\right)=41$ |

$\mu \left(3\right)=23.14$ | $\mu \left(3\right)=49.99$ | $\mu \left(3\right)=32.7$ |

$\mu \left(4\right)=30.89$ | $\mu \left(4\right)=29.85$ | |

$\mu \left(5\right)=43.54$ |

${\mathit{m}}_{\mathit{iJK}}$ | ${\mathit{m}}_{\mathit{IjK}}$ | ${\mathit{m}}_{\mathit{IJk}}$ | ${\mathit{m}}_{\mathit{IJK}}$ |
---|---|---|---|

${m}_{2JK}=6.833$ | ${m}_{I1K}=6.38$ | ${m}_{IJ1}=8.83$ | 8.54 |

${m}_{3JK}=6$ | ${m}_{I2K}=7.13$ | ${m}_{IJ2}=7.73$ | |

${m}_{4JK}=9.08$ | ${m}_{I3K}=12.13$ | ${m}_{IJ3}=7.25$ | |

${m}_{5JK}=12.25$ | ${m}_{IJ4}=10.75$ |

Datum | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ | ${\mathit{t}}_{4}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | |

${g}_{2}$ | 4 | 5 | 21 | 1 | 1 | 6 | 1 | 5 | 4 | 12 | 9 | 13 |

${g}_{3}$ | 1 | 4 | 2 | 1 | 2 | 11 | 4 | 2 | 16 | 1 | 15 | 13 |

${g}_{4}$ | 5 | 3 | 17 | 4 | 3 | 20 | 2 | 3 | 21 | 8 | 14 | 9 |

${g}_{5}$ | 20 | 14 | 10 | 21 | 7 | 11 | 1 | 20 | 8 | 16 | 7 | 12 |

Residual Gene Score | Residual Condition Score | Residual Time Step Score |
---|---|---|

$\mu \left(2\right)=16.24$ | $\mu \left(1\right)=25.95$ | $\mu \left(1\right)=30.34$ |

$\mu \left(3\right)=20.12$ | $\mu \left(2\right)=19.45$ | $\mu \left(2\right)=24.41$ |

$\mu \left(4\right)=25.30$ | $\mu \left(3\right)=38.68$ | $\mu \left(3\right)=35.49$ |

$\mu \left(5\right)=50.43$ | $\mu \left(4\right)=21.84$ |

${\mathit{m}}_{\mathit{iJK}}$ | ${\mathit{m}}_{\mathit{IjK}}$ | ${\mathit{m}}_{\mathit{IJk}}$ | ${\mathit{m}}_{\mathit{IJK}}$ |
---|---|---|---|

${m}_{2JK}=6.833$ | ${m}_{I1K}=3.67$ | ${m}_{IJ1}=6.89$ | 7.31 |

${m}_{3JK}=6$ | ${m}_{I2K}=5.5$ | ${m}_{IJ2}=5.44$ | |

${m}_{4JK}=9.08$ | ${m}_{I3K}=12.75$ | ${m}_{IJ3}=6.44$ | |

${m}_{IJ4}=10.44$ |

Datum | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ | ${\mathit{t}}_{4}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | |

${g}_{2}$ | 4 | 5 | 21 | 1 | 1 | 6 | 1 | 5 | 4 | 12 | 9 | 13 |

${g}_{3}$ | 1 | 4 | 2 | 1 | 2 | 11 | 4 | 2 | 16 | 1 | 15 | 13 |

${g}_{4}$ | 5 | 3 | 17 | 4 | 3 | 20 | 2 | 3 | 21 | 8 | 14 | 9 |

Residual Gene Score | Residual Condition Score | Residual Time Step Score |
---|---|---|

$\mu \left(1\right)=54.37$ | $\mu \left(1\right)=27.17$ | $\mu \left(1\right)=31.97$ |

$\mu \left(2\right)=17.48$ | $\mu \left(2\right)=24.49$ | $\mu \left(2\right)=41$ |

$\mu \left(3\right)=23.14$ | $\mu \left(3\right)=49.99$ | $\mu \left(3\right)=32.7$ |

$\mu \left(4\right)=30.89$ | $\mu \left(4\right)=29.85$ | |

$\mu \left(5\right)=43.54$ |

${\mathit{m}}_{\mathit{iJK}}$ | ${\mathit{m}}_{\mathit{IjK}}$ | ${\mathit{m}}_{\mathit{IJk}}$ | ${\mathit{m}}_{\mathit{IJK}}$ |
---|---|---|---|

${m}_{2JK}=2.83$ | ${m}_{I1K}=2.56$ | ${m}_{IJ1}=3.67$ | 2.83 |

${m}_{3JK}=2.33$ | ${m}_{I2K}=3.11$ | ${m}_{IJ2}=2$ | |

${m}_{4JK}=3.33$ | ${m}_{IJ3}=2.83$ |

Datum | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ | |||
---|---|---|---|---|---|---|

c1 | c2 | c1 | c2 | c1 | c2 | |

g2 | 4 | 5 | 1 | 1 | 1 | 5 |

g3 | 1 | 4 | 1 | 2 | 4 | 2 |

g4 | 5 | 3 | 4 | 3 | 2 | 3 |

Residual Gene Score | Residual Condition Score | Residual Time Step Score |
---|---|---|

$\mu \left(2\right)=17.13$ | $\mu \left(1\right)=5.86$ | $\mu \left(1\right)=21.21$ |

$\mu \left(3\right)=16.98$ | $\mu \left(2\right)=9.35$ | $\mu \left(2\right)=9.61$ |

$\mu \left(4\right)=20.29$ | $\mu \left(3\right)=39.19$ | $\mu \left(3\right)=18.55$ |

$\mu \left(4\right)=23.16$ |

Datum | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ | |||
---|---|---|---|---|---|---|

c1 | c2 | c1 | c2 | c1 | c2 | |

${g}_{1}$ | 13 | 7 | 22 | 16 | 14 | 9 |

${g}_{5}$ | 20 | 14 | 21 | 7 | 1 | 20 |

Datum | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ |
---|---|---|---|

c3 | c3 | c3 | |

${g}_{2}$ | 21 | 6 | 4 |

${g}_{3}$ | 2 | 11 | 16 |

${g}_{4}$ | 17 | 20 | 21 |

Datum | ${\mathit{t}}_{4}$ | |
---|---|---|

c1 | c2 | |

g2 | 12 | 9 |

g3 | 1 | 15 |

g4 | 8 | 14 |

Datum | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ | ${\mathit{t}}_{4}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | c1 | c2 | c3 | |

${g}_{1}$ | 13 | 7 | 25 | 22 | 16 | 2 | 14 | 9 | 11 | 13 | 25 | 8 |

${g}_{2}$ | 22 | 15 | 21 | 2 | 19 | 6 | 20 | 17 | 4 | 12 | 9 | 13 |

${g}_{3}$ | 13 | 1 | 2 | 7 | 3 | 11 | 15 | 21 | 16 | 1 | 15 | 13 |

${g}_{4}$ | 7 | 23 | 17 | 2 | 15 | 20 | 5 | 8 | 21 | 8 | 14 | 9 |

${g}_{5}$ | 20 | 14 | 10 | 21 | 7 | 11 | 1 | 20 | 8 | 16 | 7 | 12 |

No. | TQI | Dimension | Time Point | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |||

1 | 2.8 $\times \text{}{10}^{-8}$ | 6735 × 3 × 12 | X | X | X | X | X | X | X | X | X | X | X | X |

2 | 4.5 $\times \text{}{10}^{-8}$ | 4384 × 3 × 10 | X | X | X | X | X | X | X | X | X | X | ||

3 | 4.58 $\times \text{}{10}^{-8}$ | 7783 × 2 × 7 | X | X | X | X | X | X | X | |||||

4 | 4.66 $\times \text{}{10}^{-8}$ | 4562 × 3 × 9 | X | X | X | X | X | X | X | X | X | |||

5 | 4.67 $\times \text{}{10}^{-8}$ | 5817 × 2 × 10 | X | X | X | X | X | X | X | X | X | X | ||

6 | 4.7 $\times \text{}{10}^{-8}$ | 7567 × 2 × 7 | X | X | X | X | X | X | X | |||||

7 | 4.71 $\times \text{}{10}^{-8}$ | 4169 × 3 × 10 | X | X | X | X | X | X | X | X | X | X | ||

8 | 4.72 $\times \text{}{10}^{-8}$ | 11027 × 2 × 4 | X | X | X | X | ||||||||

9 | 4.76 $\times \text{}{10}^{-8}$ | 5509 × 3 × 7 | X | X | X | X | X | X | X | |||||

10 | 4.8 $\times \text{}{10}^{-8}$ | 9395 × 2 × 5 | X | X | X | X | X |

Tricluster ID | TQI | Dimension | The Name of the Gene Associated with HIV-1 |
---|---|---|---|

158 | 8.43 $\times \text{}{10}^{-8}$ | 3602 × 4 × 2 | AGFG1,EGR1,HLA-C |

1 | 1.15 $\times \text{}{10}^{-8}$ | 1211 × 8 × 4 | AGFG1 |

2 | 1.24 $\times \text{}{10}^{-8}$ | 1332 × 6 × 4 | AGFG1 |

184 | 1.37 $\times \text{}{10}^{-8}$ | 2334 × 4 × 2 | AGFG1, EGR1,HLA-C |

76 | 1.38 $\times \text{}{10}^{-8}$ | 1899 × 4 × 3 | AGFG1 |

55 | 1.41 $\times \text{}{10}^{-8}$ | 1559 × 5 × 3 | AGFG1 |

35 | 1.45 $\times \text{}{10}^{-8}$ | 1306 × 6 × 3 | HLA-C |

7 | 1.46 $\times \text{}{10}^{-8}$ | 1201 × 5 × 4 | AGFG1 |

8 | 1.49 $\times \text{}{10}^{-8}$ | 1249 × 6 × 3 | - |

12 | 1.52 $\times \text{}{10}^{-8}$ | 1233 × 8 × 2 | AGFG1, EGR1 |

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## Share and Cite

**MDPI and ACS Style**

Siswantining, T.; Saputra, N.; Sarwinda, D.; Al-Ash, H.S.
Triclustering Discovery Using the *δ*-Trimax Method on Microarray Gene Expression Data. *Symmetry* **2021**, *13*, 437.
https://doi.org/10.3390/sym13030437

**AMA Style**

Siswantining T, Saputra N, Sarwinda D, Al-Ash HS.
Triclustering Discovery Using the *δ*-Trimax Method on Microarray Gene Expression Data. *Symmetry*. 2021; 13(3):437.
https://doi.org/10.3390/sym13030437

**Chicago/Turabian Style**

Siswantining, Titin, Noval Saputra, Devvi Sarwinda, and Herley Shaori Al-Ash.
2021. "Triclustering Discovery Using the *δ*-Trimax Method on Microarray Gene Expression Data" *Symmetry* 13, no. 3: 437.
https://doi.org/10.3390/sym13030437