1. Recovering the time-delayed varying causal coefficients:

function [A G p_val] = Tdepent_FCM_delayed(Data, p)

model type:
- linear model (equation 5 in the paper), and only consider time-delayed causal effects
in this code we assume all time-dependent coefficients share the same kernel width
we apply a trick to make the computation much more efficient
INPUT:
- Data : TxN matrix of samples(T: number of samples; N: number of variables)
- p: time lag
OUTPUT:
- A: the estimated posterior mean of time-delayed causal coefficients
  - A{i}(j,k,:): the ith time-lagged causal coefficients from Xk to Xj(Xk ->Xj)
- G: the estimated posterior mean of confounder term
  - G(i,:): the confounder term for Xi
- p_val: p values derived from the independence test between estimated noise terms

EXAMPLE:

see example1.m

2. Recovering the instantaneous varying causal coefficients:

function [B,p_val] = Tdepent_FCM_ins(Data,causal_ordering)

model type:
- linear model, and only consider the time-dependent instantaneous causal effect
- the hypothetcal causal ordering needs to be assigned in advance
- in this code we assume all time-dependent coefficients share the same kernel width
- we apply a trick to make the computation much more efficient
INPUT:
- Data : TxN matrix of samples(T: number of samples; N: number of variables)
- causal_ordering: 1xN vector. The root node is labelled as 1, and the sink node is labelled as N
  - for example: if x1->x2->x3, and Data = [x1,x2,x3], then causal ordering = [1,2,3]; if x3->x2->x1, and Data = [x1,x2,x3], then causal ordering = [3,2,1];
OUTPUT:
- B: the estimated posterior mean of the instantaneous causal coefficients
  - ```
  B(i,j,:): means the causal coefficicents from Xj to Xi (Xj -> Xi)
```
- p_val: p values derived from the independence test between estimated noise term and hypothetical causes

EXAMPLE

see example2.m

3. Recovering both time-delayed and instantaneous varying causal coefficients, as well as cofounder terms:

function [A, G, B, p_val] = Tdepent_FCM_delayIns(Data, causal_ordering,p);

model type:
- linear model (equation 2 in the paper), and consider time-delayed and instantaneous causal effects simultaneously. We estimate them in one step.
- In this code we assume all time-dependent coefficients share the same kernel hypermeters

INPUT:

Data : TxN matrix of samples(T: number of samples; N: number of variables)
p: time lag

causal_ordering: the instantaneous causal ordering, a 1xN vector.

  The root node is labelled as 1, and the sink node is labelled as N

  for example: if x1->x2->x3, and Data = [x1,x2,x3], then causal ordering = [1,2,3];

              if x3->x2->x1, and Data = [x1,x2,x3], then causal ordering = [3,2,1];

OUTPUT:
- A: the estimated posterior mean of time-delayed causal coefficients
  - ```
  A{i}(j,k,:): the ith time-lagged causal coefficients from Xk to Xj(Xk ->Xj)
```
- G: the estimated posterior mean of confounder term
  - ```
  G(i,:): the confounder term for Xi
```
- B: the estimated posterior mean of the instantaneous causal coefficients
  - ```
   B(i,j,:): means the causal coefficicents from Xj to Xi (Xj -> Xi)
```
- p_val: p values derived from the independence test between estimated noise term and hypothetical causes

EXAMPLE:

see example3.m

CITATION

Biwei Huang ,Kun Zhang , Bernhard Scholkopf, Causal discovery from nonstationary/heterogeneous data: skeleton estimation and orientation determination, IJCAI, 2015.

Biwei-Huang/Identification-of-Time-Dependent-Functional-Causal-Model

IMPORTANT FUNCTIONS

1. Recovering the time-delayed varying causal coefficients:

EXAMPLE:

2. Recovering the instantaneous varying causal coefficients:

EXAMPLE

3. Recovering both time-delayed and instantaneous varying causal coefficients, as well as cofounder terms:

EXAMPLE:

CITATION