How to set up GEKKO for parameter estimation from multiple independent sets of data? - gekko

Question

I am learning how to use GEKKO for kinetic parameter estimation based on laboratory batch reactor data, which essentially consists of the concentration profiles of three species A, C, and P. For the purposes of my question, I am using a model that I previously featured in a question related to parameter estimation from a single data set.
My ultimate goal is to be able to use multiple experimental runs for parameter estimation, leveraging data that may be collected at different temperatures, species concentrations, etc. Due to the independent nature of individual batch reactor experiments, each data set features samples collected at different time points. These different time points (and in the future, different temperatures for instance) are difficult for me to implement into a GEKKO model, as I previosly used the experimental data collection time points as the m.time parameter for the GEKKO model. (See end of post for code) I have solved problems like this in the past with gPROMS and Athena Visual Studio.
To illustrate my problem, I generated an artificial data set of 'experimental' data from my original model by introducing noise to the species concentration profiles, and shifting the experimental time points slightly. I then combined all data sets of the same experimental species into new arrays featuring multiple columns. My thought process here was that GEKKO would carry out the parameter estimation by using the experimental data of each corresponding column of the arrays, so that times_comb[:,0] would be related to A_comb[:,0] while times_comb[:,1] would be related to A_comb[:,1].
When I attempt to run the GEKKO model, the system does obtain a solution for the parameter estimation, but it is unclear to me if the problem solution is reasonable, as I notice that the GEKKO Variables A, B, C, and P are 34 element vectors, which is double the elements in each of the experimental data sets. I presume GEKKO is somehow combining both columns of the time and Parameter vectors during model setup that leads to those 34 element variables? I am also concerned that during this combination of the columns of each input parameter, that the relationship between a certain time point and the collected species information is lost.
How could I improve the use of multiple data sets that GEKKO can simultaneously use for parameter estimation, with the consideration that the time points of each data set may be different? I looked on the GEKKO documentation examples as well as the APMonitor website, but I could not find examples featuring multiple data sets that I could use for guidance, as I am fairly new to the GEKKO package.
Thank you for your time reading my question and for any help/ideas you may have.
Code below:
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO
#Experimental data
times = np.array([0.0, 0.071875, 0.143750, 0.215625, 0.287500, 0.359375, 0.431250,
0.503125, 0.575000, 0.646875, 0.718750, 0.790625, 0.862500,
0.934375, 1.006250, 1.078125, 1.150000])
A_obs = np.array([1.0, 0.552208, 0.300598, 0.196879, 0.101175, 0.065684, 0.045096,
0.028880, 0.018433, 0.011509, 0.006215, 0.004278, 0.002698,
0.001944, 0.001116, 0.000732, 0.000426])
C_obs = np.array([0.0, 0.187768, 0.262406, 0.350412, 0.325110, 0.367181, 0.348264,
0.325085, 0.355673, 0.361805, 0.363117, 0.327266, 0.330211,
0.385798, 0.358132, 0.380497, 0.383051])
P_obs = np.array([0.0, 0.117684, 0.175074, 0.236679, 0.234442, 0.270303, 0.272637,
0.274075, 0.278981, 0.297151, 0.297797, 0.298722, 0.326645,
0.303198, 0.277822, 0.284194, 0.301471])
#Generate second set of 'experimental data'
times_new = times + np.random.uniform(0.0,0.01)
P_obs_noisy = P_obs+np.random.normal(0,0.05,P_obs.shape)
A_obs_noisy = A_obs+np.random.normal(0,0.05,A_obs.shape)
C_obs_noisy = A_obs+np.random.normal(0,0.05,C_obs.shape)
#Combine two data sets into multi-column arrays
times_comb = np.array([times, times_new]).T
P_comb = np.array([P_obs, P_obs_noisy]).T
A_comb = np.array([A_obs, A_obs_noisy]).T
C_comb = np.array([C_obs, C_obs_noisy]).T
m = GEKKO(remote=False)
t = m.time = times_comb #using two column time array
Am = m.Param(value=A_comb) #Using the two column data as observed parameter
Cm = m.Param(value=C_comb)
Pm = m.Param(value=P_comb)
A = m.Var(1, lb = 0)
B = m.Var(0, lb = 0)
C = m.Var(0, lb = 0)
P = m.Var(0, lb = 0)
k = m.Array(m.FV,6,value=1,lb=0)
for ki in k:
ki.STATUS = 1
k1,k2,k3,k4,k5,k6 = k
r1 = m.Var(0, lb = 0)
r2 = m.Var(0, lb = 0)
r3 = m.Var(0, lb = 0)
r4 = m.Var(0, lb = 0)
r5 = m.Var(0, lb = 0)
r6 = m.Var(0, lb = 0)
m.Equation(r1 == k1 * A)
m.Equation(r2 == k2 * A * B)
m.Equation(r3 == k3 * C * B)
m.Equation(r4 == k4 * A)
m.Equation(r5 == k5 * A)
m.Equation(r6 == k6 * A * B)
#mass balance diff eqs, function calls rxn function
m.Equation(A.dt() == - r1 - r2 - r4 - r5 - r6)
m.Equation(B.dt() == r1 - r2 - r3 - r6)
m.Equation(C.dt() == r2 - r3 + r4)
m.Equation(P.dt() == r3 + r5 + r6)
m.Minimize((A-Am)**2)
m.Minimize((P-Pm)**2)
m.Minimize((C-Cm)**2)
m.options.IMODE = 5
m.options.SOLVER = 3 #IPOPT optimizer
m.options.NODES = 6
m.solve()
k_opt = []
for ki in k:
k_opt.append(ki.value[0])
print(k_opt)
plt.plot(t,A)
plt.plot(t,C)
plt.plot(t,P)
plt.plot(t,B)
plt.plot(times,A_obs,'bo')
plt.plot(times,C_obs,'gx')
plt.plot(times,P_obs,'rs')
plt.plot(times_new, A_obs_noisy,'b*')
plt.plot(times_new, C_obs_noisy,'g*')
plt.plot(times_new, P_obs_noisy,'r*')
plt.show()

Answer 1

To have multiple data sets with different times and data points, you can join the data sets as a pandas dataframe. Here is a simple example:
# data set 1
t_data1 = [0.0, 0.1, 0.2, 0.4, 0.8, 1.00]
x_data1 = [2.0, 1.6, 1.2, 0.7, 0.3, 0.15]
# data set 2
t_data2 = [0.0, 0.15, 0.25, 0.45, 0.85, 0.95]
x_data2 = [3.6, 2.25, 1.75, 1.00, 0.35, 0.20]
The merged data has NaN where the data is missing:
x1 x2
Time
0.00 2.0 3.60
0.10 1.6 NaN
0.15 NaN 2.25
0.20 1.2 NaN
0.25 NaN 1.75
Take note of where the data is missing with a 1=measured and 0=not measured.
# indicate which points are measured
z1 = (data['x1']==data['x1']).astype(int) # 0 if NaN
z2 = (data['x2']==data['x2']).astype(int) # 1 if number
The final step is to set up Gekko variables, equations, and objective to accommodate the data sets.
xm = m.Array(m.Param,2)
zm = m.Array(m.Param,2)
for i in range(2):
m.Equation(x[i].dt()== -k * x[i]) # differential equations
m.Minimize(zm[i]*(x[i]-xm[i])**2) # objectives
You can also calculate the initial condition with m.free_initial(x[i]). This gives an optimal solution for one parameter value (k) over the 2 data sets. This approach can be expanded to multiple variables or multiple data sets with different times.
from gekko import GEKKO
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# data set 1
t_data1 = [0.0, 0.1, 0.2, 0.4, 0.8, 1.00]
x_data1 = [2.0, 1.6, 1.2, 0.7, 0.3, 0.15]
# data set 2
t_data2 = [0.0, 0.15, 0.25, 0.45, 0.85, 0.95]
x_data2 = [3.6, 2.25, 1.75, 1.00, 0.35, 0.20]
# combine with dataframe join
data1 = pd.DataFrame({'Time':t_data1,'x1':x_data1})
data2 = pd.DataFrame({'Time':t_data2,'x2':x_data2})
data1.set_index('Time', inplace=True)
data2.set_index('Time', inplace=True)
data = data1.join(data2,how='outer')
print(data.head())
# indicate which points are measured
z1 = (data['x1']==data['x1']).astype(int) # 0 if NaN
z2 = (data['x2']==data['x2']).astype(int) # 1 if number
# replace NaN with any number (0)
data.fillna(0,inplace=True)
m = GEKKO(remote=False)
# measurements
xm = m.Array(m.Param,2)
xm[0].value = data['x1'].values
xm[1].value = data['x2'].values
# index for objective (0=not measured, 1=measured)
zm = m.Array(m.Param,2)
zm[0].value=z1
zm[1].value=z2
m.time = data.index
x = m.Array(m.Var,2) # fit to measurement
x[0].value=x_data1[0]; x[1].value=x_data2[0]
k = m.FV(); k.STATUS = 1 # adjustable parameter
for i in range(2):
m.free_initial(x[i]) # calculate initial condition
m.Equation(x[i].dt()== -k * x[i]) # differential equations
m.Minimize(zm[i]*(x[i]-xm[i])**2) # objectives
m.options.IMODE = 5 # dynamic estimation
m.options.NODES = 2 # collocation nodes
m.solve(disp=True) # solve
k = k.value[0]
print('k = '+str(k))
# plot solution
plt.plot(m.time,x[0].value,'b.--',label='Predicted 1')
plt.plot(m.time,x[1].value,'r.--',label='Predicted 2')
plt.plot(t_data1,x_data1,'bx',label='Measured 1')
plt.plot(t_data2,x_data2,'rx',label='Measured 2')
plt.legend(); plt.xlabel('Time'); plt.ylabel('Value')
plt.xlabel('Time');
plt.show()

Answer 2

Including my updated code (not fully cleaned up to minimize number of variables) incorporating the selected answer to my question for reference. The model does a regression of 3 measured species in two separate 'datasets.'
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from gekko import GEKKO
#Experimental data
times = np.array([0.0, 0.071875, 0.143750, 0.215625, 0.287500, 0.359375, 0.431250,
0.503125, 0.575000, 0.646875, 0.718750, 0.790625, 0.862500,
0.934375, 1.006250, 1.078125, 1.150000])
A_obs = np.array([1.0, 0.552208, 0.300598, 0.196879, 0.101175, 0.065684, 0.045096,
0.028880, 0.018433, 0.011509, 0.006215, 0.004278, 0.002698,
0.001944, 0.001116, 0.000732, 0.000426])
C_obs = np.array([0.0, 0.187768, 0.262406, 0.350412, 0.325110, 0.367181, 0.348264,
0.325085, 0.355673, 0.361805, 0.363117, 0.327266, 0.330211,
0.385798, 0.358132, 0.380497, 0.383051])
P_obs = np.array([0.0, 0.117684, 0.175074, 0.236679, 0.234442, 0.270303, 0.272637,
0.274075, 0.278981, 0.297151, 0.297797, 0.298722, 0.326645,
0.303198, 0.277822, 0.284194, 0.301471])
#Generate second set of 'experimental data'
times_new = times + np.random.uniform(0.0,0.01)
P_obs_noisy = (P_obs+ np.random.normal(0,0.05,P_obs.shape))
A_obs_noisy = (A_obs+np.random.normal(0,0.05,A_obs.shape))
C_obs_noisy = (C_obs+np.random.normal(0,0.05,C_obs.shape))
#Combine two data sets into multi-column arrays using pandas DataFrames
#Set dataframe index to be combined time discretization of both data sets
exp1 = pd.DataFrame({'Time':times,'A':A_obs,'C':C_obs,'P':P_obs})
exp2 = pd.DataFrame({'Time':times_new,'A':A_obs_noisy,'C':C_obs_noisy,'P':P_obs_noisy})
exp1.set_index('Time',inplace=True)
exp2.set_index('Time',inplace=True)
exps = exp1.join(exp2, how ='outer',lsuffix = '_1',rsuffix = '_2')
#print(exps.head())
#Combine both data sets into a single data frame
meas_data = pd.DataFrame().reindex_like(exps)
#define measurement locations for each data set, with NaN written for time points
#not common in both data sets
for cols in exps:
meas_data[cols] = (exps[cols]==exps[cols]).astype(int)
exps.fillna(0,inplace = True) #replace NaN with 0
m = GEKKO(remote=False)
t = m.time = exps.index #set GEKKO time domain to use experimental time points
#Generate two-column GEKKO arrays to store observed values of each species, A, C and P
Am = m.Array(m.Param,2)
Cm = m.Array(m.Param,2)
Pm = m.Array(m.Param,2)
Am[0].value = exps['A_1'].values
Am[1].value = exps['A_2'].values
Cm[0].value = exps['C_1'].values
Cm[1].value = exps['C_2'].values
Pm[0].value = exps['P_1'].values
Pm[1].value = exps['P_2'].values
#Define GEKKO variables that determine if time point contatins data to be used in regression
#If time point contains species data, meas_ variable = 1, else = 0
meas_A = m.Array(m.Param,2)
meas_C = m.Array(m.Param,2)
meas_P = m.Array(m.Param,2)
meas_A[0].value = meas_data['A_1'].values
meas_A[1].value = meas_data['A_2'].values
meas_C[0].value = meas_data['C_1'].values
meas_C[1].value = meas_data['C_2'].values
meas_P[0].value = meas_data['P_1'].values
meas_P[1].value = meas_data['P_2'].values
#Define Variables for differential equations A, B, C, P, with initial conditions set by experimental observation at first time point
A = m.Array(m.Var,2, lb = 0)
B = m.Array(m.Var,2, lb = 0)
C = m.Array(m.Var,2, lb = 0)
P = m.Array(m.Var,2, lb = 0)
A[0].value = exps['A_1'][0] ; A[1].value = exps['A_2'][0]
B[0].value = 0 ; B[1].value = 0
C[0].value = exps['C_1'][0] ; C[1].value = exps['C_2'][0]
P[0].value = exps['P_1'][0] ; P[1].value = exps['P_2'][0]
#Define kinetic coefficients, k1-k6 as regression FV's
k = m.Array(m.FV,6,value=1,lb=0,ub = 20)
for ki in k:
ki.STATUS = 1
k1,k2,k3,k4,k5,k6 = k
#If doing paramrter estimation, enable free_initial condition, else not include them in model to reduce DOFs (for simulation, for example)
if k1.STATUS == 1:
for i in range(2):
m.free_initial(A[i])
m.free_initial(B[i])
m.free_initial(C[i])
m.free_initial(P[i])
#Define reaction rate variables
r1 = m.Array(m.Var,2, value = 1, lb = 0)
r2 = m.Array(m.Var,2, value = 1, lb = 0)
r3 = m.Array(m.Var,2, value = 1, lb = 0)
r4 = m.Array(m.Var,2, value = 1, lb = 0)
r5 = m.Array(m.Var,2, value = 1, lb = 0)
r6 = m.Array(m.Var,2, value = 1, lb = 0)
#Model Equations
for i in range(2):
#Rate equations
m.Equation(r1[i] == k1 * A[i])
m.Equation(r2[i] == k2 * A[i] * B[i])
m.Equation(r3[i] == k3 * C[i] * B[i])
m.Equation(r4[i] == k4 * A[i])
m.Equation(r5[i] == k5 * A[i])
m.Equation(r6[i] == k6 * A[i] * B[i])
#Differential species balances
m.Equation(A[i].dt() == - r1[i] - r2[i] - r4[i] - r5[i] - r6[i])
m.Equation(B[i].dt() == r1[i] - r2[i] - r3[i] - r6[i])
m.Equation(C[i].dt() == r2[i] - r3[i] + r4[i])
m.Equation(P[i].dt() == r3[i] + r5[i] + r6[i])
#Minimization objective functions
m.Obj(meas_A[i]*(A[i]-Am[i])**2)
m.Obj(meas_P[i]*(P[i]-Pm[i])**2)
m.Obj(meas_C[i]*(C[i]-Cm[i])**2)
#Solver options
m.options.IMODE = 5
m.options.SOLVER = 3 #APOPT optimizer
m.options.NODES = 6
m.solve()
k_opt = []
for ki in k:
k_opt.append(ki.value[0])
print(k_opt)
plt.plot(t,A[0],'b-')
plt.plot(t,A[1],'b--')
plt.plot(t,C[0],'g-')
plt.plot(t,C[1],'g--')
plt.plot(t,P[0],'r-')
plt.plot(t,P[1],'r--')
plt.plot(times,A_obs,'bo')
plt.plot(times,C_obs,'gx')
plt.plot(times,P_obs,'rs')
plt.plot(times_new, A_obs_noisy,'b*')
plt.plot(times_new, C_obs_noisy,'g*')
plt.plot(times_new, P_obs_noisy,'r*')
plt.show()