Hi everyone,
I'm trying to fit certain data using nonlinear least squares which is of the following form
y(t) = a1*X1(t)+a2*X2(t)+…an*Xn(t)+b1*Y1(t-1)+..+bk*Y(t-k)
Where a1,..an and b1,..bk are the parameters, which I want to find during training the data. I have around 5000 sample points (X,Y), which varies over time. The curve fits very well during training. My R-square value is 98%, but the major issue is with validation. I used the same data, which I used in training to validate the model. I knew this is not the right process to validate the model, but I did to make sure that I can reproduce the same R-square with same data. I observed that my R-square drops drastically at least 20% for the same data. The reason I found is while validating the error from past outputs, y(t-1) is taken in to account at y(t) and also during training, the feedback output is free from error . The error keeps on growing and results in a drop of R-square value. In other words, training the data is more like an open loop, but during Validation, it turns in to real closed loop. Is there some way to accommodate this effect in optimisation function during training the data? so that I can have the same R-square value when I train and validate with the same data set
Thanks!
Maybe you can alernatively consider the recursive least squares algorithm (RLS). RLS is the recursive application of least squares (LS) regression, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For RLS with forgetting factor (RLS-FF), acquired data is weighted according to its age, with increased weight given to the most recent data. No prior 'training phase' is required.
Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS algorithm with forgetting factor (RLS-FF) to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter (pilot plant). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):
Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...
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