I am trying to minimize the difference of a function with a data point over different time points. So the objective function is the sum of the squares of the difference between the model (my function) and the data points over different times.
My model has analytical first and second order derivatives. How can I provide these derivatives to Gekko Python?
There are several examples in the APMonitor webpage regarding parameter estimation. Please check the link below. It also provides the data and model that you can use for practice.
TCLab C - Parameter Estimation
You can also get the idea how to implement the higher order differential equations in GEKKO in the link below. You basically want to introduce additional variable which links the first derivative variable to the 2nd derivative variable. That way, you can collapse the higer order DE down into the multiple 1st order DEs.
Solve 2nd Order Differential Equation
Related
I am new to the field of optimization and I need help in the following optimization problem. I have tried to solve it using normal coding to make sure that I got he correct results. However, the results I got are different and I am not sure my way of analysis is correct or not. This is a short description of the problem:
The objective function shown in the picture is used to find the optimal temperature of the insulating system that minimizes the total cost over a given horizon.
[This image provides the mathematical description of the objective function and the constraints] (https://i.stack.imgur.com/yidrO.png)
The data of the problems are as follow:
1-
Problem data:
A=1.07×10^8
h=1
T_ref=87.5
N=20
p1=0.001;
p2=0.0037;
This is the curve I want to obtain
2- Optimization variable:
u_t
3- Model type:
The model is a nonlinear cost function with non-linear constraints and it is solved using non-linear solver SNOPT.
4-The meaning of the symbols in the objective and constrained functions
The optimization is performed over a prediction horizon of N years.
T_ref is The reference temperature.
Represent the degree of polymerization in the kth year.
X_DP Represents the temperature of the insulating system in the kth year.
h is the time step (1 year) of the discrete-time model.
R is the ratio of the load loss at the rated load to the no-load loss.
E is the activation energy.
A is the pre-exponential constant.
beta is a linear coefficient representing the cost due to the decrement of the temperature.
I have developed the source code in MATLAB, this code is used to check if my analysis is correct or not.
I have tried to initialize the Ut value in its increasing or decreasing states so that I can have the curves similar to the original one. [This is the curve I obtained] (https://i.stack.imgur.com/KVv2q.png)
I have tried to simulate the problem using conventional coding without optimization and I got the figure shown above.
close all; clear all;
h=1;
N=20;
a=250;
R=8.314;
A=1.07*10^8;
E=111000;
Tref=87.5;
p1=0.0019;
p2=0.0037;
p3=0.0037;
Utt=[80,80.7894736842105,81.5789473684211,82.3684210526316,83.1578947368421,... % The value of Utt given here represent the temperature increament over a predictive horizon.
83.9473684210526,84.7368421052632,85.5263157894737,86.3157894736842,...
87.1052631578947,87.8947368421053,88.6842105263158,89.4736842105263,...
90.2631578947369,91.0526315789474,91.8421052631579,92.6315789473684,...
93.4210526315790,94.2105263157895,95];
Utt1 = [95,94.2105263157895,93.4210526315790,92.6315789473684,91.8421052631579,... % The value of Utt1 given here represent the temperature decreament over a predictive horizon.
91.0526315789474,90.2631578947369,89.4736842105263,88.6842105263158,...
87.8947368421053,87.1052631578947,86.3157894736842,85.5263157894737,...
84.7368421052632,83.9473684210526,83.1578947368421,82.3684210526316,...
81.5789473684211,80.7894736842105,80];
Ut1=zeros(1,N);
Ut2=zeros(1,N);
Xdp =zeros(N,N);
Xdp(1,1)=1000;
Xdp1 =zeros(N,N);
Xdp1(1,1)=1000;
for L=1:N-1
for k=1:N-1
%vt(k+L)=Ut(k-L+1);
Xdq(k+1,L) =(1/Xdp(k,L))+A*exp((-1*E)/(R*(Utt(k)+273)))*24*365*h;
Xdp(k+1,L)=1/(Xdq(k+1,L));
Xdp(k,L+1)=1/(Xdq(k+1,L));
Xdq1(k+1,L) =(1/Xdp1(k,L))+A*exp((-1*E)/(R*(Utt1(k)+273)))*24*365*h;
Xdp1(k+1,L)=1/(Xdq1(k+1,L));
Xdp1(k,L+1)=1/(Xdq1(k+1,L));
end
end
% MATLAB code
for j =1:N-1
Ut1(j)= -p1*(Utt(j)-Tref);
Ut2(j)= -p2*(Utt1(j)-Tref);
end
sum00=sum(Ut1);
sum01=sum(Ut2);
X1=1./Xdp(:,1);
Xf=1./Xdp(:,20);
Total= table(X1,Xf);
Tdiff =a*(Total.Xf-Total.X1);
X22=1./Xdp1(:,1);
X2f=1./Xdp1(:,20);
Total22= table(X22,X2f);
Tdiff22 =a*(Total22.X2f-Total22.X22);
obj=(sum00+(Tdiff));
ob1 = min(obj);
obj2=sum01+Tdiff22;
ob2 = min(obj2);
plot(Utt,obj,'-o');
hold on
plot(Utt1,obj)
My problem is described in this picture(It's like a Pyramid structure):
The objective function is below:
In this problem, D is known, A is the object that I want to get. It is a layered structure, each block in the upper layer is divided into four sub-blocks in the layer below. And the value of the upper layer node is equal to the sum of the four child nodes of the lower layer. In above example, I used only 2 layers.
What I want to do is simulate the distribution of D with A, so in the objective function is the ratio of two adjacent squares in each row in A compared to the value in D. I do this comparison on each layer and sum them. Then it is all of my objective function. But in the finest layer, the value in A has a constrain A<=1, the value in A can be a number between 0 and 1. I have tried to solve it using Quadratic programming in python library CVXPY. However, it seems the speed is slow.
So I want to solve it in another way, because this is a convex optimization problem, which can guarantee the global optimal solution. What I think is whether it is possible to use the method of derivation. There are two unknown variables in each item, that is, the two items with A in the formula. Partial derivatives are obtained for them, and the restriction of A<=1 is added, then solve using gradient descent method. Is this mathematically feasible, because I don't know much about optimization, and if it is possible, how should I do it? If not possible, what other methods can I use?
I am having a non-linear optimization model with several variables and a certain function between three of them should be defined as a constraint. (Let us say, that the efficiency of a machine is dependent on the inlet and outlet temperatures). I have calculated some values in a table to visualize the dependency for T_inlets and T_outlets. It gives back a pretty ugly surface. A good fit would be something like a 5 degree polynomial equation if I wanted to define a function directly, but I do not think that would boost my computation speed... So instead I am considering simply having the created table and use it as a lookup table. Is a non-linear solver able to interpret this? I am using ipopt in Pyomo environment.
Another idea would be to limit my feasible temperature range and simplify the connection...maybe with using peace-wise linearization. Is it doable with 3d surfaces?
Thanks in advance!
I have a multiclass model with 4 classes. I have already implemented a callback able to calculate the precision/recall for each class and their macro average. But for some technical reason, I have to calculate them using the metrics mechanism.
I'm using TensorFlow 2 and Keras 2.3.0. I have already used the tensorflow.keras.metrics.Recall/Precision to get the class-wise metrics:
metrics_list = ['accuracy']
metrics_list.extend([Recall(class_id=i, name="recall_{}".format(label_names[i])) for i in range(n_category)])
metrics_list.extend([Precision(class_id=i, name="precision_{}".format(label_names[i])) for i in range(n_category)])
model = Model(...)
model.compile(...metrics=metrics_list)
However, this solution is not satisfying:
firstly, tensorflow.keras.metrics.Recall/Precision uses a threshold to define the affiliation to a class, while it should use argmax to define the most probable class, if class_id is defined
Secondly, I have to create 2 new metrics that would calculate the average over all classes, which itself requires to calculate the class-wise metrics. This is inelegant and inefficient to calculate twice the same thing.
Is there a way to create a class or a function that would calculate directly the class-wise and the average predicion/recall using the TensorFlow/Keras metrics logic?
Apparently I can easily obtain the confusion matrix using tf.math.confusion_matrix(). However, I do not see how to inject a list of scalar at once, instead of returning a single scalar.
Any comment is welcomed!
It occurs that in my very specific case, I can simply use CategoricalAccuracy() as unique metric because i'm using a batch_size=1. It this case, accuracy=recall=precision={1.|0.} for a batch. That only partially solve the problem. The best solution would be to update the confusion matrix using argmax at each batch end, then calculate the Precision/Recall based on that. I don't known how it is possible to do that yet, but it should be doable.
I am using the code in this website http://blog.chrislowis.co.uk/2008/11/24/ruby-gsl-pearson.html to implement a Pearson Correlation given two time series data like so:
require 'gsl'
pearson_correlation = GSL::Stats::correlation(
GSL::Vector.alloc(first_metrics),GSL::Vector.alloc(second_metrics)
)
This returns a number such as -0.2352461593569471.
I'm currently using the highcharts library and am feeding it two sets of timeseries data. Given that I have a finite time series for both sets, can I do something with this number (-0.2352461593569471) to create a third time series showing the slope of this curve? If anyone can point me in the right direction I'd really appreciate it!
No, correlation doesn't tell you anything about the slope of the line of best fit. It just tells you approximately how much of the variability in one variable (or one time series, in this case) can be explained by the other. There is a reasonably good description here: http://www.graphpad.com/support/faqid/1141/.
How you deal with the data in your specific case is highly dependent on what you're trying to achieve. Are you trying to show that variable X causes variable Y? If so, you could start by dropping the time-series-ness, and just treat the data as paired values, and use linear regression. If you're trying to find a model of how X and Y vary together over time, you could look at multivariate linear regression (I'm not very familiar with this, though).