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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)
I have a dataframe which contains three more or less significant correlations between target column and other columns ( LinarRegressionModel.coef_ from sklearn shows 57, 97 and 79). And I don't know what exact model to choose: should I use only most correlated column for regression or use regression with all three predictors. Is there any way to compare models effectiveness? Sorry, I'm very new to data analysis, I couldn't google any tools for this task
Well first at all, you must know that when we are choosing the best model to apply to new data, we are going to choose the best model to fit out of sample data, which is the kind of samples that might are not present in the training process, after all, you want to predict new probabilities or cases. In your case, predict a new number.
So, how can we do this? Well, the best is to use metrics which can help us to choose which model is better for our dataset.
There are so many kinds of metrics for regression:
MAE: Mean absolute error is the mean of the absolute value of the errors. This is the easiest of the metrics to understand since it’s just the average error.
MSE: Mean squared error is the mean of the squared error. It’s more popular than a mean absolute error because the focus is geared more towards large errors.
RMSE: Root means the squared error is the square root of the mean squared error. This is one of the most popular of the evaluation metrics because root means the squared error is interpretable in the same units as the response vector or y units, making it easy to relate its information.
RAE: Relative absolute error, also known as the residual sum of a square, where y bar is a mean value of y, takes the total absolute error and normalizes it by dividing by the total absolute error of the simple predictor.
You can work with any of these, but I highly recommend to use MSE and RMSE.
I am given a data that consists of N sequences of variable lengths of hidden variables and their corresponding observed variables (i.e., I have both the hidden variables and the observed variables for each sequence).
Is there a way to find the order K of the "best" HMM model for this data, without exhaustive search? (justified heuristics are also legitimate).
I think there may be a confusion about the word "order":
A first-order HMM is an HMM which transition matrix depends only on the previous state. A 2nd-order HMM is an HMM which transition matrix depends only on the 2 previous states, and so on. As the order increases, the theory gets "thicker" (i.e., the equations) and very few implementations of such complex models are implemented in mainstream libraries.
A search on your favorite browser with the keywords "second-order HMM" will bring you to meaningful readings about these models.
If by order you mean the number of states, and with the assumptions that you use single distributions assigned to each state (i.e., you do not use HMMs with mixtures of distributions) then, indeed the only hyperparameter you need to tune is the number of states.
You can estimate the optimal number of states using criteria such as the Bayesian Information Criterion, the Akaike Information Criterion, or the Minimum Message Length Criterion which are based on model's likelihood computations. Usually, the use of these criteria necessitates training multiple models in order to be able to compute some meaningful likelihood results to compare.
If you just want to get a blur idea of a good K value that may not be optimal, a k-means clustering combined with the percentage of variance explained can do the trick: if X clusters explain more than, let say, 90% of the variance of the observations in your training set then, going with an X-state HMM is a good start. The 3 first criteria are interesting because they include a penalty term that goes with the number of parameters of the model and can therefore prevent some overfitting.
These criteria can also be applied when one uses mixture-based HMMs, in which case there are more hyperparameters to tune (i.e., the number of states and the number of component of the mixture models).
I'm analysing longitudinal panel data, in which individuals transition between different states in a Markov chain. I'm modelling the transition rates between states using a series of multinomial logistic regressions. This means that I end up with a very large number of regression slopes.
For each regression slope, I obtain a posterior distribution (using WinBUGS). From the posterior distribution, we get the mean, standard deviation, and 95% credible interval associated with the slope in question.
The value I am ultimately interested in is the expected first passage time ('hitting time') through the Markov chain. This is a function of all the different predictor variables, and so is built from the many regression slopes produced by the multinomial logistic regressions.
A simple approach would be to take the mean of each posterior distribution as a point-estimate for each regression slope, and solve for the expected first passage time at a series of different values of the predictor variables. I have now done this, but it is potentially misleading because it doesn't show the uncertainty around the predicted values of expected first passage time.
My question is: how can I calculate a credible interval for the expected first passage time?
My first thought was to approximate the error via simulation, by sampling individual values for the regression slopes from each posterior distribution, obtaining the expected first passage time given those values, and then plotting the standard deviation of all these simulated values. However, I feel like (a) this would make a statistician scream and (b) it doesn't take into account the fact that different posterior distributions will be correlated (it samples from each one independently).
In WinBUGS, you can actually obtain the correlations between the posterior distributions. So if the simulation idea is appropriate, I could in theory simulate the regression slope coefficients incorporating these correlations.
Is there a more direct and less approximate way to find the uncertainty? Could I, for instance, use WinBUGS to find the posterior distribution of the expected first passage time for a given set of values of the predictor variables? Rather like the answer to this question: define a new node and monitor it. I would imagine defining a series of new nodes, where each one is for a different set of actual predictor values, and monitoring each one. Does this make good statistical sense?
Any thoughts about this would be really appreciated!
I am now implementing an email filtering application using the Naive Bayes algorithm. My application uses the Spambase Data Set from the UCI Machine Learning Repository. Since the attributes are continuous, I calculate the probability using the Probability Density Function (PDF). However, when I evaluate the data using the k-fold cross validation, a training set may contain only 0 for one of its attributes. For this reason, I got a 0 standard deviation and the PDF returns NaN and it leads to a huge number of spams are not correctly classified with that training set. What should I do to fix the problem?
You could use a discrete PDF, which will always be bounded.
Alternatively, simply ignore any attribute with zero variance. There is no point in including distributions with zero variance, because they won't actually do anything. For example, you want to know how old I am, and then I tell you that I live on planet Earth. That shouldn't change your estimate, because every single piece of data you have is for people on planet Earth.