I am working in the area of electronic waste. E-waste was ground to fine sizes. The cumulative mass distribution in different sizes is represented by Rosin-Rammler model. Similarly, the differential metal percent distribution in different sizes is represented by the Gaussian distribution model. I have fitting equations for both. Now, I want to get a model that represents the cumulative distribution of metal percent by combining the previous two models.
I found a way to do so, which is shown in the attached image file.
I hope the method followed is correct. Can anyone please suggest me scientific references for similar work.
Rahul.
May you post a dataset of 20 to 50 random values for your main two variables to work them? What does term "mass" mean? What distribution of the two mentioned variables do you want to study?
In general I believe that the asumption of normality is not a good a road. It is more important the data you have measured. OK, emilio
I agree with Emilio in that the normal model is not a good starting point. A proportion is restricted to the interval (0,1) or --(0,100) if working in percent--. The normal distribution is supported on the whole real line, including negative values. In the best case the normal distribution can be an acceptable approximation but frequently it fails and introduce probabilities for the negative values, and is unable to model the frequent skewness of these distributions.
I have no experience on the Rosin-Rammler model, but I think your data are compositional (proportions percents...) and that the model does not take into account this circumstance. If I understand well, you are trying to predict concentrations of different metals as a function of the grain size. Is it? This corresponds to a simplicial regression (response variable is a composition made of concentrations; explanatory variable can be grain size and/or its logarithm). The trick is to transform the composition into coordinates (to make it real spanning the whole R^n) preferably using ilr (isometric log-ratio transformation). Then you can regress these coordinates on the grain size and/or its logarithm or other covariates.
For simplicial regression have a look at
Egozcue, J. J., Daunis-i-Estadella, J., Pawlowsky-Glahn V., Hron K. and Filzmoser, P.: Simplicial regression. The normal model.
Journal of Applied Probability and Statistics (JAPS),
6, (1& 2), 87--108, 2012
You will find an introduction to compositional data analysis in
Pawlowsky-Glahn, V. and Buccianti A. (Eds.)
Compositional Data Analysis: Theory and Applications,
ISBN-10: 0-470-71135-3, Wiley, Chichester UK, 2011
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