# Distributed Global Function Model Finding for Wireless Sensor Network Data

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Data Mining in Wireless Sensor Networks (WSNs)

#### 2.2. Function Mining

## 3. Function Finding Algorithm by Using Gene Expression Programming for Wireless Sensor Networks Data

#### 3.1. Function Finding in Wireless Sensor Networks

#### 3.2. Coding of Gene Expression Programming (GEP)

**Definition 1.**Let string $G$ be defined as a triplet $G=<GHead,GTail,L>$, $F$ be basic elementary function set and $T$ be terminal set. Where $GHead$, $GTail$ and $L$ represent head, tail, length of the $G$ respectively. The elements of $GHead$ randomly generates from $F$ and $T$, the elements of $GTail$ randomly generates from $T$. Then string $G$ is called gene.

**Property 1.**Let the length of $GHead$ be $h$, the length of $GTail$ be $t$, maximum number of arguments of operator in the $GHead$ be $n$. Then, $h$ and $t$ follow the equation:

**Definition 2.**The string which is composed of one or more $G$ is called the chromosome, and denoted as $C$.

**Example 1.**Let function set be $F=\{+,-,\times ,Q\}$, terminal set be $T=\{a,b\}$, length of gene head be $h=5$, where “$Q$” represents the square root function. From function set $F$, we know that maximum number of arguments of all operators is 2. According to Equation (1), length of gene tail is 6. The randomly generated chromosome is shown in Figure 2.

_{1}and Sub-ET

_{2}is respectively performed. The result of decoding is linked by addition function and simplified by mathematica software. The final function model is $f\left(a,b\right)=3a+ab+\frac{1}{b}-\sqrt{b}$.

#### 3.3. Adaptive Population Generation Strategy Based on Collaborative Evolution of Sub-Population

Algorithm 1. APGS-CESP ($Pop$) |

Input: Pop, ${P}_{s}$, ${P}_{m}$, ${P}_{t}$, ${P}_{r}$, popSize; |

Output: new Population; |

Begin { |

1. for (int i = 1; i <= MaxGen; i++){ |

2. ${F}_{\mathrm{max}}[i]$ = MaxFitness (Pop [i]); |

3. $Sum{F}_{\mathrm{max}}$ = 0; $Sum{F}_{\mathrm{max}}+={F}_{\mathrm{max}}[i]$; |

4. if ($i\text{\hspace{0.17em}}\mathrm{mod}\text{\hspace{0.17em}}10==0$){ |

5. if ($sum{F}_{max}\text{\hspace{0.17em}}\mathrm{mod}\text{\hspace{0.17em}}10==0$) { |

6. $subPop\leftarrow RandomInitPop(subPopSize)$; |

7. $Pop\leftarrow Insert(subPop,Pop)$; |

8. $EvalueFitness(Pop)$; |

9. $Pop\leftarrow Select({P}_{s},Pop,popSize)$; |

10. $Pop\leftarrow Mutate({P}_{m},Pop)$; |

11. $Pop\leftarrow ISTransposition({P}_{t},Pop)$; |

12. $Pop\leftarrow RISTransposition({P}_{t},Pop)$; |

13. $Pop\leftarrow GeneTransposition({P}_{t},Pop)$; |

14. $Pop\leftarrow OnePointRecombination({P}_{r},Pop)$; |

15. $Pop\leftarrow TwoPointRecombination({P}_{r},Pop)$; |

16. $Pop\leftarrow GeneRecombination({P}_{r},Pop)$;}}} |

17. $newPopulation=Pop$; |

18. Return newPopulation;} |

#### 3.4. Description of Function Finding Algorithm Using Gene Expression Programming (FF-GEP)

Algorithm 2. FF-GEP |

Input: popSize, maxGen, maxFitness, ${P}_{s}$, ${P}_{m}$, ${P}_{t}$, ${P}_{r}$; |

Output: Best Function Expression; |

Begin { |

1. double $fitness=0.0$; int $i=0$; |

2. $Pop\leftarrow InitPop(popSize)$; |

3. $EvalueFitness(Pop)$; |

4. while (($fitness\le maxFitness$) or ($i\le maxGen$)) { |

5. $Pop\leftarrow Select({P}_{s},Pop,popSize)$; |

6. $Pop\leftarrow Mutate({P}_{m},Pop)$; |

7. $Pop\leftarrow ISTransposition({P}_{t},Pop)$; |

8. $Pop\leftarrow RISTransposition({P}_{t},Pop)$; |

9. $Pop\leftarrow GeneTransposition({P}_{t},Pop)$; |

10. $Pop\leftarrow OnePoint\mathrm{Re}combination({P}_{r},Pop)$; |

11. $Pop\leftarrow TwoPoint\mathrm{Re}combination({P}_{r},Pop)$; |

12. $Pop\leftarrow Gene\mathrm{Re}combination({P}_{r},Pop)$; |

13. $Pop\leftarrow APGS-SPAE(Pop)$; |

14. $EvalueFitness(Pop)$;} |

15. return BestFunctionExpression;} |

## 4. Distributed Global Function Model Finding for Wireless Sensor Networks Data

#### 4.1. Algorithm Idea

#### 4.2. Global Model Generation Algorithm Based on Unconstrained Nonlinear Least Squares

**Definition 3.**In WSNs, we propose the number of the sensor node and sink node are k and n, respectively. For each sink node, it contains a sensor data set $S=[{x}_{1},\mathrm{...},{x}_{m}\text{\hspace{0.17em}}{y}_{m+1}]$, where $S\in {R}^{m+1}$ and ${y}_{m+1}$ represents target value for each sensor data set. Then, the set of each sink node can be obtained yielding to GEP by employing the approach of function model mining, such that ${y}_{i}({x}_{m+1})={f}_{i}({x}_{1},{x}_{2},\mathrm{...},{x}_{m}),i\in [1,n]$. Hence, ${y}_{i}({x}_{m+1})={f}_{i}({x}_{1},{x}_{2},\mathrm{...},{x}_{m})$ is the local function model with m-dimension of the i-th sink node.

**Definition 4.**Suppose that there exist n sink nodes and ${f}_{i}(X),i\in [1,n]$, where $X=({x}_{1},{x}_{2},\mathrm{...},{x}_{m})$. There exists a set of constants ${a}_{i}\ne 0,i\in [1,n]$ such that $f({x}_{1},{x}_{2},\mathrm{...},{x}_{m})={\displaystyle \sum _{i=1}^{n}{a}_{i}{f}_{i}(X)}$. Thus, $f({x}_{1},{x}_{2},\mathrm{...},{x}_{m})$ is called global function model.

**Lemma 1.**

**Proof:**

Algorithm 3. GMG-UNLS |

Input: local Function Model ${f}_{i}\left(X\right),i\in \left[1,n\right]$, k sample data; |

Output: global Function Model $f(X)$; |

Begin { |

1. double ${a}_{1},{a}_{2},\mathrm{...},{a}_{n}$;//Defining n real variables. |

2. Set $f(X)={\displaystyle \sum _{i=1}^{n}{a}_{i}{f}_{i}(X)}$. //Building global function equation. |

3. Set $Q({a}_{1},{a}_{2},\mathrm{...},{a}_{n})={\displaystyle \sum _{i=1}^{k}{({y}_{i}-{\displaystyle \sum _{j=1}^{n}{a}_{j}{f}_{j}({X}_{i})})}^{2}}$;//Building function model, where ${y}_{i},i\in [1,k]$ is target value for k sample data. |

4. $k\text{\hspace{0.17em}sample data}\to Q({a}_{1},{a}_{2},\mathrm{...},{a}_{n})$; // Substituting k sample data into $Q({a}_{1},{a}_{2},\mathrm{...},{a}_{n})$. |

5. $\frac{\partial Q}{\partial {a}_{i}}=0$, $i\in [1,n]$; // $({a}_{1},{a}_{2},\mathrm{...},{a}_{n})$ is obtained by solving non homogeneous linear equations. |

6. return $f(X)$;}// Substituting $({a}_{1},{a}_{2},\mathrm{...},{a}_{n})$ into $\sum _{i=1}^{n}{a}_{i}{f}_{i}(X)$ and returning $f(X)$. |

#### 4.3. Description of DGFMF-WSND

Algorithm 4. DGFMF-WSND |

Input: GEPGSH, popSize; maxGen; maxFitness; ${P}_{s}$, ${P}_{m}$, ${P}_{t}$, ${P}_{r}$; |

Output: Global Function; |

Begin { |

Server: |

1. ReceivePara (T, GEPParas, i, GEPGSH); // Parameters are received from ith client according to GEPGSH. |

2. InitPop (Pop S); //Initializing population of FF-GEP. |

3. LocalFunction (i) = FF-GEP (popSize; maxGen; maxFitness; ${P}_{s}$, ${P}_{m}$, ${P}_{t}$, ${P}_{r}$); |

Client: |

4. T = Read (SampleData); |

5. int gridcodes = SelectGridCodes (); |

6. for (int i = 0; i < gridcodes; i++) { |

7. TransPara (T, GEPParas, i, GEPGSH); //Transmitting parameters to server. |

8. TransService (LocalFunction [i]);} |

9. GlobalFunction = GMG-UNLS (LocalFunction); |

10. return Global Function;} |

## 5. Experimental Section

#### 5.1. Experimental Environment

**Figure 4.**Distributed data mining on grid computing platform using web services with gene expression programming (GEP) and function finding algorithm using gene expression programming (FF-GEP) and global model generation algorithm based on unconstrained nonlinear least squares (GMG-UNLS).

#### 5.2. Data Resources

Datasets | Number of Attributes | Number of Instances |
---|---|---|

Leaf optical properties experiment 93 (LOPEX93) | 9 | 1938 |

European Network of Excellence on Intelligent Technologies for Smart Adaptive Systems (EUNITE) | 3 | 730 |

Gas Sensor Array Drift Dataset (GSADD) | 128 | 13,910 |

Dodgers Loop Sensor (DLS) | 3 | 50,400 |

## 6. Comparative Analysis

**Definition 5.**Let ${\widehat{y}}_{i}$, ${y}_{i}$ and ${\overline{y}}_{i}$ be predicted value, real value and mean value of the i-th original data, respectively. Let $SSR={\displaystyle \sum _{i=1}^{n}{({\widehat{y}}_{i}-{\overline{y}}_{i})}^{2}}$ be sum of squares for regression, $SST={\displaystyle \sum _{i=1}^{n}{({y}_{i}-{\overline{y}}_{i})}^{2}}$ be sum of squares for total. Then ${R}^{2}=\frac{SSR}{SST}$ is called coefficient of determination.

**Definition 6.**Let ${F}_{R-max}$ be real maximum fitness value, ${F}_{M-max}$ be model-based maximum fitness value. If $\frac{{F}_{R-max}-{F}_{M-max}}{{F}_{R-max}}\le \mathsf{\delta}$, then the corresponding algorithm is convergent.

**Definition 7.**Let $Stime$ be time-consumption of DGFMF-WSND in a single machine environment, $Ctime$ be time-consumption of DGFMF-WSND in parallel computing environment. Then ${S}_{peedup}=\frac{Stime}{Ctime}\times 100\%$ is called speed-up ratio.

**Definition 8.**Let $m\cdot dataT$ be time-consumption to perform dataset with an increase of m times on a cluster with an increase of m times, $dataT$ be time-consumption of the original dataset. Then ${S}_{caleup}=\frac{m\cdot dataT}{dataT}\times 100\%$ is called scale-up ratio.

**Definition 9.**Suppose that the algorithm runs N times independently, and $\frac{{F}_{R-max}-{F}_{M-max}\left[i\right]}{{F}_{R-max}}\le \mathsf{\delta},i\in \left[1,N\right]$, where ${F}_{M-max}\left[i\right]$ be the i-th model-based maximum fitness value. Then, by Definition 6, it is clear that the i-th run of the algorithm is convergent. Thus, the sum of the number of algorithm convergence $K,K\le N$ is called number of convergence of the algorithm.

**Definition 10.**Suppose that the algorithm runs N times independently, $K[i],i\le N$ represents the corresponding number of generation when the algorithm is convergent under the condition of the i-th run. Thus, $\frac{{\displaystyle \sum _{i=1}^{N}K\left[i\right]}}{N}$ is called average number of convergence generation.

**Example 1:**To compare the performance of ACO (Ant Colony Optimization) [50], SA (Simulated Annealing) [51], GP [16], GA [17], GEP [20] and FF-GEP, for four datasets in Table 1, the four algorithms run 50 times independently, and the maximum number of generation of four algorithms is 5000. By Definition 6, Figure 5 shows comparison of number of convergence for GP, GA, GEP and FF-GEP. Comparison of average generation of convergence for GP, GA, GEP and FF-GEP are shown in Figure 6. Meanwhile, Table 2 shows comparison of value of ${R}^{2}$ for four test datasets in Table 1 based on the four algorithms. Degree of fitting between model value and real value of four test datasets in Table 1 based on FF-GEP is shown Figure 7 without taking into account the time-consumption.

**Figure 5.**Comparison of number of convergence for ACO (Ant Colony Optimization), SA(Simulated Annealing), genetic programming (GP), genetic algorithm (GA), gene expression programming (GEP) and function finding algorithm using gene expression programming (FF-GEP).

**Figure 6.**Comparison of average number of convergence generation for ACO (Ant Colony Optimization), SA(Simulated Annealing), genetic programming (GP), genetic algorithm (GA), gene expression programming (GEP) and function finding algorithm using gene expression programming (FF-GEP).

**Table 2.**Comparison of value of ${R}^{2}$ for four test datasets based on ACO (Ant Colony Optimization), SA(Simulated Annealing), genetic programming (GP), genetic algorithm (GA), gene expression programming (GEP) and function finding algorithm using gene expression programming (FF-GEP).

Datasets | Algorithms | |||||
---|---|---|---|---|---|---|

ACO | SA | GP | GA | GEP | FF-GEP | |

LOPEX93 | 0.8355 | 0.8291 | 0.8700 | 0.8490 | 0.9118 | 0.9381 |

EUNITE | 0.8914 | 0.8901 | 0.9187 | 0.8926 | 0.9263 | 0.9575 |

GSADD | 0.7012 | 0.6997 | 0.7393 | 0.7035 | 0.8321 | 0.8686 |

DLS | 0.7801 | 0.7729 | 0.7903 | 0.7844 | 0.88 | 0.9097 |

**Example 2:**In order to better evaluate performance of algorithm, Example 2 focuses on comparison of average time-consumption and fitting degree between real value and model value. Figure 8 shows average time-consumption of ACO, SA, GP, GA, GEP and FF-GEP. Average time-consumption of DGFMF-WSND with the increase of number of computing nodes is shown in Figure 9. Comparison of value of ${R}^{2}$ for LOPEX 93, EUNITE, GSADD and DLS datasets with the increase of number of computing nodes is shown in Figure 10. Figure 11 shows fitting degree between model value and real value of four test datasets in Table 1 based on DGFMF-WSND on six computing nodes.

**Figure 7.**Comparison between model value and real value of four test datasets in Table 1 using gene expression programming (GEP) and function finding algorithm using gene expression programming (FF-GEP). (

**a**) Comparison between model value and real value of LOPEX93 datasets using GEP and FF-GEP; (

**b**) comparison between model value and real value of EUNITE datasets using GEP and FF-GEP; (

**c**) comparison between model value and real value of GSADD datasets using GEP and FF-GEP; and (

**d**) comparison between model value and real value of DLS datasets using GEP and FF-GEP.

**Figure 8.**Comparison of average time-consumption of ACO (Ant Colony Optimization), SA(Simulated Annealing), genetic programming (GP), genetic algorithm (GA), gene expression programming (GEP) and function finding algorithm using gene expression programming (FF-GEP) for four datasets.

**Figure 9.**Comparison of average time-consumption of DGFMF-WSND for four datasets with the increase of number of computing nodes.

**Figure 10.**Comparison of value of ${R}^{2}$ for four test datasets in Table 1 based on distributed global function model finding for wireless sensor networks data (DGFMF-WSND) with the increase of number of computing nodes.

**Example 3:**To reflect the parallel performance of DGFMF-WSND, LOPEX93 and EUNITE datasets in Table 1 are expanded 1000, 2000, 4000 and 8000 times to respectively form four new datasets. Comparison of speed-up ratio of DGFMF-WSND for the four new datasets with the increase of number of computing nodes is shown in Figure 12. Figure 13 shows comparison of scale-up ratio of DGFMF-WSND for LOPEX93 and EUNITE datasets with the increase of number of computing nodes.

**Figure 11.**Comparison between model value and real value of four test datasets in Table 1 based on DGFMF-WSND. (

**a**) Comparison between model value and real value of LOPEX93 datasets based on DGFMF-WSND; (

**b**) comparison between model value and real value of EUNITE datasets based on DGFMF-WSND ; (

**c**) comparison between model value and real value of GSADD datasets based on DGFMF-WSND; and (

**d**) comparison between model value and real value of DLS datasets based on DGFMF-WSND.

**Figure 12.**Comparison of speed-up ratio of DGFMF-WSND for two datasets with the increase of number of computing nodes. (

**a**) Comparison of speed-up ratio of DGFMF-WSND for LOPEX93 datasets with the increase of number of computing nodes; and (

**b**) comparison of speed-up ratio of DGFMF-WSND for EUNITE datasets with the increase of number of computing nodes.

**Figure 13.**Comparison of scale-up ratio of DGFMF-WSND for LOPEX93 and EUNITE datasets with the increase of number of computing nodes.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Deng, S.; Yang, L.-C.; Yue, D.; Fu, X.; Ma, Z.
Distributed Global Function Model Finding for Wireless Sensor Network Data. *Appl. Sci.* **2016**, *6*, 37.
https://doi.org/10.3390/app6020037

**AMA Style**

Deng S, Yang L-C, Yue D, Fu X, Ma Z.
Distributed Global Function Model Finding for Wireless Sensor Network Data. *Applied Sciences*. 2016; 6(2):37.
https://doi.org/10.3390/app6020037

**Chicago/Turabian Style**

Deng, Song, Le-Chan Yang, Dong Yue, Xiong Fu, and Zhuo Ma.
2016. "Distributed Global Function Model Finding for Wireless Sensor Network Data" *Applied Sciences* 6, no. 2: 37.
https://doi.org/10.3390/app6020037