Path modeling in linear networks

In many areas of applied sciences and in industrial applications it is known how data blocks are related in time or sequence. It can be instructive to model the data according to this known sequence of the data blocks. There have been developed a large collection of methods to handle this situation. For a review of some introductory analysis see an example.


1. Introduction. Subdivision of data.
2. Graphic display of interconnections.
3. Measures of relationships
4. The H-method of estimation. One and two data blocks
5. Three data blocks
6. Case study
7. Correlation and prediction
8. Simple paths
9. Examples
10. Comparison of methods
11. Network of data blocks
12. Vectors computed (W, T, P, R)i
13. Regressions among data blocks
14. Several input (exogenous) and output (endogenous) data blocks
15. Multiple objectives
16. Comparisons with overall models
17. Confidence intervals
18. Detection of special features
19. Sensitivity analysis
20. Case study
21. Guidelines for presentation of results

There are two types of approach that are of importance. They are correlation and regression approaches. They are illustrated in the figure below. Consider the case of correlation. If there are two blocks, we are looking for latent variables h1=X1w1 and h2=X2w2 such that latent variables have maximal correlation. In the case of three data bocks, we are looking for three latent variables, h1, h2, h3, such that their correlation is as 'large' as possible.

In the case of regression approach the search is for a score matrix T1 such that the samples of T1 provide with as good prediction of samples of X2 as possible.

In the case of three data blocks, Xi, i=1,2 and 3, we are looking for a score matrix T1 and score matrix T2, such that if we know a new sample of X1, x10, we can estimate samples of X2 and X3, x20 and x30, as well as possible. For instance, such that a prediction of x30 is as precise as possible.