**Causal analysis in linear networks**

In many areas of applied sciences
it is natural to setup a network of data blocks that reflect the measurements of
a large interconnected systems. In these models it is often useful to study
which variables cause what. The direction of causality is supposed to be given
by the person. The methods presented here are designed to simplify the structure
of the loading matrix matrix **P** according to a given theory or derived
from the data.

*Topics*

1. | Introduction. Ideas and methods. |

2. | One data block. Latent structure. |

3. | The H-method of estimation |

4. | Measures of causality |

5. | Plots of latent structure |

6. | Three data blocks |

7. | Estimation procedures |

8. | Cross-validation |

9. | Network of data blocks |

10. | Comparison of methods |

11. | Examples |

12. | Information measures |

13. | Correlation and prediction |

14. | Several input (exogenous) and output (endogenous) data blocks |

15. | Numerical methods |

16. | Increasing networks |

17. | Confidence intervals |

18. | Detection of special features |

19. | Sensitivity analysis |

20. | Useful measures of causality |

21. | Guidelines for presentation of results |

The type of approach chosen for these methods are briefly presented here.

The given data matrix **X** (N times
K) is decomposed by some method into a product of a score matrix **T** (N
times A) and a loading matrix **P** (K times A). Mathematically the
decomposition can be written as

**X** = **T** **P**^{T}
+ **X**_{0}

Here **X**_{0} is the part of the **X**
matrix that was not used in the analysis. The interpretation of the score matrix
**T** is that the rows represents the 'scores' that the individuals have
obtained. By simplifying the structure of **P** one will be able to explain
better how the individuals obtain their scores. There are many
algorithms that can be used, see an example of an approach.

In a network of data blocks each data matrix, **
X**_{i}, is decomposed in a similar way. And their structure is
simplified in similar ways as in the case of one data block. The interpretation of
the structure will then depend on the method chosen and its role in the network.
An example of a network of data block is given in the figure below. It is given
by 5 data blocks. The first two, **X**_{1} and **X**_{2},
are the input (exogenous) data blocks.

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