# Gaussian Mixture Model-Based Ensemble Kalman Filtering for State and Parameter Estimation for a PMMA Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State Estimation Techniques for Nonlinear Systems

#### 2.1. Particle Filter (PF)

#### 2.2. Ensemble Kalman Filter (EnKF)

#### 2.3. Gaussian Mixture Model Based Ensemble Kalman Filter (EnKF-GMM)

#### 2.3.1. Expectation Maximization (EM) for Clustering of the Gaussian Mixture Model

**The pseudo-code for the EM algorithm is provided below**.

Algorithm 1: Expectation Maximization algorithm. Inputs are data set ${\{{x}_{i}\}}_{i=1,\mathrm{..},N}$, component number M and initial values $\{{\mathsf{\theta}}^{0}\}$ of ${\{{\mathsf{\pi}}_{j}\}}_{j=1,\dots ,M},{\{{\mathsf{\mu}}_{j}\}}_{j=1,\dots M},{\{{P}_{j}\}}_{j=1,\dots M}$, ${\mathsf{\theta}}^{k}={\mathsf{\theta}}^{0}$. |

EM[{x}$,M,\{{\mathsf{\theta}}^{k}\}$] |

// E step |

while $\mathsf{\epsilon}\le 1e-6$ |

for i = 1: N |

for j = 1:M |

$p[{({c}_{i})}_{j}|{x}_{i},{\mathsf{\theta}}^{k}]=p({x}_{i}|{({c}_{i})}_{j},{\mathsf{\theta}}^{k})p({({c}_{i})}_{j}|{\mathsf{\theta}}^{k})/p\left({x}_{i}\right)$ |

end for |

end for |

// M step |

for j = 1:M |

${\mathsf{\pi}}_{j}^{k+1}={\displaystyle \sum}_{i=1}^{N}p[{({c}_{i})}_{j}|{x}_{i},{\mathsf{\theta}}^{k}]/N$ |

${\mathsf{\mu}}_{j}^{k+1}={\displaystyle \sum}_{i=1}^{N}p[{({c}_{i})}_{j}|{x}_{i},{\mathsf{\theta}}^{k}]{x}_{i}/{\displaystyle \sum}_{i=1}^{N}p[{({c}_{i})}_{j}|{x}_{i},{\mathsf{\theta}}^{k}]$ |

${P}_{j}^{k+1}=\frac{{{\displaystyle \sum}}_{i=1}^{N}p[{({c}_{i})}_{j}|{x}_{i},{\mathsf{\theta}}^{k}]\left({x}_{i}-{\mathsf{\mu}}_{j}^{k+1}\right){\left({x}_{i}-{\mathsf{\mu}}_{j}^{k+1}\right)}^{T}+\mathsf{\lambda}{I}_{d}}{{{\displaystyle \sum}}_{i=1}^{N}p[{({c}_{i})}_{j}|{x}_{i},{\mathsf{\theta}}^{k}]+1}$ |

end for |

$\mathsf{\epsilon}={\mathsf{\mu}}^{k+1}-{\mathsf{\mu}}^{k}$ |

end while |

return ${\mathsf{\theta}}^{k+1}$ |

#### 2.3.2. EnKF-GMM Algorithm

**Forecast:**

- The first portion of the forecast step is to determine the number of components $M$ in the multimodal distribution. $M$ can be determined using the Bayesian or other information criteria [27,28], or using prior knowledge. For example, in reservoir models, petrophysical properties (such as porosity or permeability) are typically related to geological units (facies), and variables inside the facies are characterized by underlying multimodal distributions which are known beforehand [9]. In our work, this information can be considered as prior knowledge if we know the distribution of the process noise.
- With the knowledge of the process model and the number of components $M$, the prior ensemble ${\{{x}_{i}\}}_{i=1,\dots ,N}$ is propagated through the model to get the predicted values of the ensemble ${\{{x}_{i}^{f}\}}_{i=1,\dots ,N}$. These are the realizations of the predicted state space ${x}^{f}$.

**Update:**

- 3.
- For each component $j$ of the distribution, the Kalman gain matrix for each Gaussian component is computed by utilizing the membership probability matrix $W$.$$P{\left[j\right]}^{f}{H}^{T}={\displaystyle \sum}_{i=1}^{N}{w}_{i,j}({x}_{i}^{f}-{\mathsf{\mu}}_{j}){(H{x}_{i}^{f}-H{\mathsf{\mu}}_{j})}^{T}/{n}_{j}$$$$HP{\left[j\right]}^{f}{H}^{T}={\displaystyle \sum}_{i=1}^{N}{w}_{i,j}(H{x}_{i}^{f}-H{\mathsf{\mu}}_{j}){(H{x}_{i}^{f}-H{\mathsf{\mu}}_{j})}^{T}/{n}_{j}$$$$K\left[j\right]=P{\left[j\right]}^{f}{H}^{T}{(HP{\left[j\right]}^{f}{H}^{T}+R)}^{-1}$$
- 4.
- In the update step, assuming one Gaussian component $j$ claims the ownership of all the ensemble members, the Kalman update can be performed for each component member under component $j$. This gives us an ensemble size of $N\times M$.$${x}_{i}^{a,j}={x}_{i}^{f}+K\left[k\right](d-H{x}_{i}^{f}-{e}_{i})$$
- 5.
- The $N\times M$ ensemble members can be combined to form $N$ members by using the probability matrix. This gives us the final posterior ensemble ${\{{x}_{i}^{a}\}}_{i=1,\dots ,N}$.$${x}_{i}^{a}={\displaystyle \sum}_{j=1}^{M}{w}_{i,j}{x}_{i}^{a,j}$$$${\mathsf{\mu}}_{j}^{a}={\displaystyle \sum}_{i=1}^{N}{w}_{i,j}{x}_{i}^{a,j}/{n}_{j}$$$$P{\left[j\right]}^{a}={\displaystyle \sum}_{i=1}^{N}{w}_{i,j}({x}_{i}^{a,j}-{\mu}_{j}^{a}){({x}_{i}^{a,j}-{\mathsf{\mu}}_{j}^{a})}^{T}/{n}_{j}$$
- 6.
- The posterior weight of each component of the distribution can be computed based on the observed data $d$, which contains the measurements y.$${\mathsf{\tau}}_{j}^{a}=p\left({\mathsf{\mu}}_{j},{{{\displaystyle \sum}}^{\text{}}}_{j},R|d\right)=\frac{p(d|{\mu}_{j},{{{\displaystyle \sum}}^{\text{}}}_{j},R){n}_{j}}{{{\displaystyle \sum}}_{j=1}^{M}p(d|{\mu}_{j},{{{\displaystyle \sum}}^{\text{}}}_{j},R){n}_{j}}$$$$p(d|{\mathsf{\mu}}_{j},{{{\displaystyle \sum}}^{\text{}}}_{j},R)=\frac{exp[-\frac{1}{2}{\left(d-H{\mathsf{\mu}}_{j}\right)}^{T}\left(H{{{\displaystyle \sum}}^{\text{}}}_{j}{H}^{T}+R{)}^{-1}\left(d-H{\mathsf{\mu}}_{j}\right)\right]}{\sqrt{{(2\pi )}^{m}|H{{{\displaystyle \sum}}^{\text{}}}_{j}{H}^{T}+R|}}$$
- 7.
- The point estimate is given by:$${x}^{a}={{\displaystyle \sum}}_{j=1}^{M}{\mathsf{\tau}}_{j}^{a}{\mathsf{\mu}}_{j}^{a}$$

Algorithm 2: EnKF-GMM algorithm. Inputs include the initial distribution of x, the total number of the particles N, the components M, and the time steps T. Inputs and observations at each time step are ${u}_{n}$ and ${d}_{n}$. |

[${\{{x}_{i}^{a}\}}_{i=1}^{N}$, ${\{{\mu}_{j}^{a},{P}_{j}^{a}.{\tau}_{j}^{a}\}}_{j=1}^{M}$] = EnKF-GMM[${\{{\mathrm{x}}_{\mathrm{i}}\}}_{i=1}^{N}$, ${d}_{t}$] |

for n = 1:T |

for I = 1 : N |

Draw ${x}_{i}^{f}~f\left(I,{u}_{n-1},{v}_{n-1}^{i}\right)$ |

Calculate ${y}_{i}=H{\mathrm{x}}_{\mathrm{i}}^{\mathrm{f}}+{v}_{n}^{i}$ |

end for |

Apply the EM algorithm on ${\{{x}_{i}^{f}\}}_{i=1,\dots ,N}$ using algorithm 1: |

$\{{\mathsf{\tau}}_{j}^{f}$, ${\mathsf{\mu}}_{j}^{f}$, ${P}_{j}^{f}{\}}_{j=1}^{M}=EM[{\{{\mathrm{x}}_{\mathrm{i}}^{\mathrm{f}}\}}_{\mathrm{i}=1,\dots ,\mathrm{N}},M,\{{\mathsf{\theta}}^{k}\}]$ |

for j = 1 : M |

Calculate the Kalman gain of each component $K\left[j\right]$ using Equation (21) |

foI i = 1 : N |

Calculate the updated particles for each component ${\{{x}_{i}^{a,j}\}}_{i=1}^{N}$ using Equation (22) |

end for |

Combine ${\{{x}_{i}^{a,j}\}}_{i=1}^{N}$ to obtain the posterior particles ${\{{x}_{i}^{a}\}}_{i=1}^{N}$ using Equation (23) |

Calculate the parameters of the posterior distribution ${\mathsf{\mu}}_{j}^{a},{P}_{j}^{a}.{\mathsf{\tau}}_{j}^{a}$ using Equations (24)–(26). |

end for |

Calculate the point estimate ${x}^{a}$ using Equation (28) |

end for |

## 3. Results and Discussion

#### 3.1. Mathematical Model of the Methyl Methacrylate ( MMA) Polymerization Process

#### 3.2. Comparison of State Estimation with the EnKF-GMM, EnKF, and PF (Case Study 1)

^{−4}, 10

^{−4}, 1.2 × 10

^{−4}, 10

^{−5}, 2 × 10

^{−4}, 4 × 10

^{−4}], respectively, confirming the significance of the estimates. The PF performs better than the EnKF only for some states. Increasing the number of particles for each of the algorithms to 200 (results not shown) improves the performance of the PF slightly, but the same conclusions hold.

#### 3.3. Comparison of State and Parameter Estimation with the EnKF-GMM, EnKF and PF (Case Studies 3 and 4)

#### 3.3.1. State Estimation with Uncertain Parameter (Case Study 3)

#### 3.3.2. State and Parameter Estimation with Uncertain Parameter (Case Study 4)

#### 3.4. Alternate Point Estimates for the PF (Case Study 5)

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Comparison of the estimation performance of the ensemble Kalman filter (EnKF)-Gaussian mixture model (GMM), EnKF, and particle filter (PF) for the polymethyl methacrylate (PMMA) process with multimodal process noise (Case Study 1).

**Figure 2.**Comparison of the estimation performance of the EnKF-GMM, EnKF, and PF for the PMMA process with more significant multimodal process noise (Case Study 2).

**Figure 3.**Evolution of the multimodal posterior distributions of ${C}_{M}$ at time steps 1, 2, 4, and 9 (Case Study 2).

**Figure 4.**Evolution of the multimodal posterior distributions of ${T}_{j}$ at time steps 2, 6, 9, and 10 (Case Study 2).

**Figure 5.**Comparison of state estimation with the EnKF-GMM, EnKF, and PF for the PMMA process with uncertain parameter ${E}_{p}$ (Case Study 3).

**Figure 6.**Comparison of state estimation with the EnKF-GMM, EnKF, and PF for the PMMA process with uncertain parameters (Case Study 4).

$F=1.0{m}^{3}/h$ | ${M}_{m}=100.12kg/kgmol$ |

${F}_{I}=0.0032{m}^{3}/h$ | ${f}^{*}=0.58$ |

${F}_{cw}=0.1588{m}^{3}/h$ | $R=8.314kJ/kgmol\xb7K$ |

${C}_{min}=6.4678kgmol/{m}^{3}$ | $-\Delta H=57800kJ/kgmol$ |

${C}_{Iin}=8.0kgmol/{m}^{3}$ | ${E}_{p}=1.8283\times {10}^{4}$$kJ/kgmol$ |

${T}_{in}=350K$ | ${E}_{I}=1.2877\times {10}^{5}kJ/kgmol$ |

${T}_{w0}=293.2K$ | ${E}_{fm}=7.4478\times {10}^{4}kJ/kgmol$ |

$U=720kJ/h\xb7K\xb7{m}^{2}$ | ${E}_{tc}=2.9442\times {10}^{3}kJ/kgmol$ |

$A=2.0{m}^{2}$ | ${E}_{td}=2.9442\times {10}^{3}kJ/kgmol$ |

$V=0.1{m}^{3}$ | ${A}_{p}=1.77\times {10}^{9}{m}^{3}/kgmol\xb7h$ |

${V}_{0}=0.02{m}^{3}$ | ${A}_{I}=3.792\times {10}^{18}1/h$ |

$\mathsf{\rho}=866kg/{m}^{3}$ | ${A}_{fm}=1.0067\times {10}^{15}{m}^{3}/kgmol\xb7h$ |

${\mathsf{\rho}}_{w}=1000kg/{m}^{3}$ | ${A}_{tc}=3.8223\times {10}^{10}{m}^{3}/kgmol\xb7h$ |

${C}_{p}=2.0kJ/\left(kg\xb7K\right)$ | ${A}_{td}=3.1457\times {10}^{11}{m}^{3}/kgmol\xb7h$ |

${C}_{pw}=4.2kJ/\left(kg\xb7K\right)$ |

**Table 2.**RMSE of the Gaussian mixture model based ensemble Kalman filter (EnKF-GMM), ensemble Kalman filter (EnKF), and particle filter (PF) for the polymethyl methacrylate (PMMA) process with multimodal process noise (Case Study 1).

Variable | EnKF-GMM | EnKF | PF |
---|---|---|---|

${C}_{M},kg\xb7mol/{m}^{3}$ | 0.20 | 0.20 | 0.33 |

${C}_{I},kg\xb7mol/{m}^{3}$ | 0.24 | 0.20 | 0.33 |

$T,K$ | 4.3 | 4.4 | 3.1 |

${D}_{0},kg\xb7mol/{m}^{3}$ | 0.019 | 0.014 | 0.032 |

${D}_{1},kg/{m}^{3}$ | 11.85 | 11.53 | 10.44 |

${T}_{j},K$ | 2.3 | 2.2 | 1.4 |

NAMW | 209 | 338 | 357 |

**Table 3.**RMSE of the EnKF-GMM, EnKF, and PF for the PMMA process with more significant multimodal process noise (Case Study 2).

Variable | EnKF-GMM | EnKF | PF |
---|---|---|---|

${C}_{M},kg\xb7mol/{m}^{3}$ | 0.44 | 0.68 | 0.69 |

${C}_{I},kg\xb7mol/{m}^{3}$ | 0.37 | 0.14 | 0.17 |

$\mathrm{T},K$ | 5.8 | 11.8 | 14.4 |

${D}_{0},kg\xb7mol/{m}^{3}$ | 0.042 | 0.062 | 0.078 |

${D}_{1},kg/{m}^{3}$ | 9.73 | 36.13 | 51.38 |

${T}_{j},K$ | 5.1 | 8.2 | 9.2 |

NAMW | 559 | 1400 | 831 |

**Table 4.**Comparison of the estimation errors of the EnKF-GMM, EnKF, and PF for ${C}_{M}$ at time steps 1, 3, 4, and 9 (in $\text{kg}\xb7\text{mol}/{\mathrm{m}}^{3}$) (Case Study 2).

Estimator | Time Step 1 | Time Step 3 | Time Step 4 | Time Step 9 |
---|---|---|---|---|

EnKF-GMM | 0.23 | 0.14 | 0.40 | 0.04 |

EnKF | 2.06 | 1.06 | 0.80 | 0.10 |

PF | 3.60 | 2.20 | 1.65 | 0.22 |

**Table 5.**Comparison of the estimation errors of the EnKF-GMM, EnKF, and PF for ${T}_{j}$ at time steps 2, 6, 9, and 10 (in $K$) (Case Study 2).

Estimator | Time Step 1 | Time Step 3 | Time Step 4 | Time Step 9 |
---|---|---|---|---|

EnKF-GMM | 6.4 | 2.8 | 1.5 | 1.3 |

EnKF | 6.6 | 3.0 | 1.9 | 1.7 |

PF | 13.5 | 4.5 | 2.9 | 2.9 |

**Table 6.**RMSE of the EnKF-GMM, EnKF, and PF for state estimation in the case with uncertain parameter ${E}_{p}$ (Case Study 3).

Variable | EnKF-GMM | EnKF | PF |
---|---|---|---|

${C}_{M},kg\xb7mol/{m}^{3}$ | 0.29 | 0.26 | 0.32 |

${C}_{I},kg\xb7mol/{m}^{3}$ | 0.12 | 0.10 | 0.27 |

$\mathrm{T},K$ | 7.2 | 8.9 | 10.3 |

${D}_{0},kg\xb7mol/{m}^{3}$ | 0.111 | 0.092 | 0.144 |

${D}_{1},kg/{m}^{3}$ | 32.27 | 35.11 | 45.34 |

${T}_{j},K$ | 5.5 | 5.7 | 7.5 |

NAMW | 487 | 869 | 653 |

Variable | EnKF-GMM | EnKF | PF | PF-mode |
---|---|---|---|---|

${C}_{M},\text{kg}\xb7\text{mol}/{\mathrm{m}}^{3}$ | 0.44 | 0.68 | 0.68 | 0.85 |

${C}_{I},\text{kg}\xb7\text{mol}/{\mathrm{m}}^{3}$ | 0.37 | 0.14 | 0.17 | 0.55 |

$T,K$ | 5.8 | 11.8 | 14.4 | 8.31 |

${D}_{0},\text{kg}\xb7\text{mol}/{\mathrm{m}}^{3}$ | 0.042 | 0.062 | 0.078 | 0.047 |

${D}_{1},\text{kg}/{\mathrm{m}}^{3}$ | 9.73 | 36.13 | 51.38 | 13.05 |

${T}_{j},K$ | 5.1 | 8.2 | 9.2 | 7.9 |

NAMW | 559 | 1400 | 831 | 706 |

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**MDPI and ACS Style**

Li, R.; Prasad, V.; Huang, B.
Gaussian Mixture Model-Based Ensemble Kalman Filtering for State and Parameter Estimation for a PMMA Process. *Processes* **2016**, *4*, 9.
https://doi.org/10.3390/pr4020009

**AMA Style**

Li R, Prasad V, Huang B.
Gaussian Mixture Model-Based Ensemble Kalman Filtering for State and Parameter Estimation for a PMMA Process. *Processes*. 2016; 4(2):9.
https://doi.org/10.3390/pr4020009

**Chicago/Turabian Style**

Li, Ruoxia, Vinay Prasad, and Biao Huang.
2016. "Gaussian Mixture Model-Based Ensemble Kalman Filtering for State and Parameter Estimation for a PMMA Process" *Processes* 4, no. 2: 9.
https://doi.org/10.3390/pr4020009