I would like to ask is there any rule to determine the optimal number of PLS components i.e. above which it will overfit and below which it underfits. Is it like one third of the number of total samples?
Tons of literature are available on that subject. References can be found on: http://www.chemometry.com/Site%20map.html
Mr. Faber, the owner of the website, is one of the specialists in this topic.
The simple answer to your question is: if your system has N components you will need N-1 variables. That is when there is no interaction between the components, Beer's law applies, your sample set is well designed, your measurements are nicely linear and so on.
In practice the optimal number is found by applying some cross validation (jack-knife, or other). The measurement set is divided in multiple sub-sets. The PLS models with increasing numbers of variables are developed on one set and tested on the other(s). This way an seemingly optimal number of variables is found in terms of prediction error.
What is crucial is validating the developed model on an independent dataset which was set aside before starting the model. The prediction on that set is the actual prediction error of your developed model. If it turns out that refinement is needed a new validation set must be measured/collected.
Good luck
After validation of your model you could check the optimum number of factors (rank) looking at the root mean square error of prediction (RMSEP) or of the cross validation (RMSECV). Plotting the RMSEP or RMSECV against the rank (number of factors that you would use in the model) a minimum can be observed. The minimum indicate the optimum rank or the number of factors that minimize the error of prediction.
But if the RMSEP's minima is rank 15, should we use 15 factors? what is the factor amount limit?
How should we choose the optimal number of components from validation plot or by using PRESS statistic?
Also how could we choose the number of components if I observe the global minimum if one of my validation curves goes beyond that minimal point
Hair, Joseph F., Jr.; Hult, G. Tomas M.; Ringle, Christian M.; & Sarstedt, Marko (2014). A primer on partial least squares structural equation modeling (PLS-SEM). Thousand Oaks, CA: Sage Publications.
statistical software such as spss and xlstat automatically determines the number of components
https://blogs.sas.com/content/iml/2017/08/02/retain-principal-components.html
According to cross-validation procedure, we find out the Root Mean Square (Error of Prediction) (RMS(EP)) for different factors and try to take the number of factors with least RMS(EP) whereas we also try to minimize the number of factors for a stable solution. We kind of balance error and minimum number of factors.
You can identify the optimum number of components in PLS by considering R2 and Mean Squared Error (MSE) for the developed model
The cross-validation method can be used:
1-Calculate the concentrations of the compounds, in each sample using the calibration spectra.
2- Compare these values with the known concentrations of the compounds in this reference sample.
3- Calculate the prediction error sum of squares PRESS.
PRESS=SUM(Ci(added) - Ci(found) )2
4- Add a new factor to the PLS and PCR models each time and calculate the PRESS in the same manner.
5-The optimum number of factor would be that number which yielded the minimum PRESS
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