By Hibben J.H.
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Additional resources for An Investigation of Intermediate Compound Formation by Means of the Raman Effect
The estimated values of βi , i = 1, . . , k, are known as the first k principal components of the data matrix m. A key point is that the principal components are an intrinsic property of the data, and do not depend of the particular specification they belong to. In other words, the maximum likelihood estimator of β1 in a specification with one component and in a specification with five are identical, so that β1 is always the first principal component, so long as the data remain the same. This implies that there is a natural order of importance among the principal components: The first is more important than the second, which is more important than third and so on.
Partially pooling countries by assuming similarity of coefficients in neighboring countries is an improvement, and serves to automate some aspects of the analysis. ) We extend this logic to combine partial pooling of neighboring countries, and consecutive time periods, simultaneously with partial pooling of adjacent age groups. The fact that 5-year-olds and 80-year-olds die of completely different causes and at very different rates would normally prevent pooling these groups. However, we also know that 5year-olds and 10-year-olds die at similar rates, as do 10-year-olds and 15-year-olds, and 15-year-olds and 20-year-olds, etc.
And it serves our goal of automating the analysis, since far less cross-section-specific tuning is required (and many fewer hyperparameter values need be set) when using this formulation. Since the mapping from the vector of expected mortality rates, on the scale of our prior, to the coefficients, on the scale of estimation, is a many-to-few transformation, it may seem less than obvious how to accomplish this. We have however developed a relatively straightforward procedure to do this (that we describe in Chapter 4).
An Investigation of Intermediate Compound Formation by Means of the Raman Effect by Hibben J.H.