Causal Structure Learning Algorithm Based on Partial Rank Correlation under Additive Noise Model

Figures & data

Table

Notation	Mathematical meanings
G V X, Y, Z $X_i,Y_i$ $x_i$, $y_i$ PN(i) k$K_{\alpha }$	Directed acyclic graph Variable set in G A variable in V i-th N-dimensional feature vextor Value assignments for X and Y The candidate neighbors for the i-th feature Partial rank correlation threshold Threshold of partial rank correlation hypothesis test

Table 1. PRCB algorithm framework

Algorithm:
Input: Data set D =$\left\{D_1,D_2,\dots ,D_i,\dots ,D_n\right\}$; a given threshold k
Output: Bayesian Network Graph between Nodes G
1./* In the constraint phase, obtain the skeleton of the primary Bayesian network*/
Select a set of candidate neighbor node sets $\mathrm{P}\mathrm{N}(X_j)$ for each variable $X_i\in X$
2./* In the search stage, the final Bayesian network graph is obtained*/
In the first step, get the skeleton of the Bayesian network by performing a greedy hill climbing search to find the network (G) with the best score
3./* Return the Bayesian network graph*/
Output: G

Table 2. PRCB constraint phase algorithm framework

Algorithm:
Input: Data set D =$\left\{D_1,D_2,\dots ,D_i,\dots ,D_n\right\}$; a given threshold k
Output: Set of candidate neighbor nodes $\mathrm{P}\mathrm{N}(X_j)$ for each variable $X_j$; Candidate neighbor matrix PNM
1. $PN(X_j)=\varphi$, ($X_j\in \mathbf{V}$, j = 1 to n) PNM(i, j) = 0 (i = 1 to n, j = 1 to n)
2. For $X_j\in \mathbf{V}$, j = 1 to n do
3. For $X_i\in \mathbf{V}$, i = 1 to n, $i\ne j$, $\mathbf{Z}\subseteq \mathbf{V}\mathrm{\setminus }\left\{X_i,X_j\right\}$ do
4. Calculation ${\rho }_r\left(X_i,X_j|\mathbf{Z}\right)$
5. If abs(${\rho }_r\left(X_i,X_j|\mathbf{Z}\right)$) $\ge k$
6. $\mathrm{P}\mathrm{N}(X_j)=\mathrm{P}\mathrm{N}(X_j){\cup }^{X_i},\mathrm{P}\mathrm{N}\mathrm{M}\left(\mathrm{i},\mathrm{j}\right)=1$
7. else
8. $\mathrm{P}\mathrm{N}\mathrm{M}\left(\mathrm{i},\mathrm{j}\right)=0$
9. Output $\mathrm{P}\mathrm{N}(X_j)$ and PNM

Table 3. PRCS algorithm framework

Algorithm:
Input: Data set D =$\left\{D_1,D_2,\dots ,D_i,\dots ,D_n\right\}$; Hypothesis testing threshold $K_{\alpha }$
Output: Bayesian Network G
1.Initialization $\mathrm{P}\mathrm{N}(X_j)=\mathrm{\varnothing }$, ($X_j\in \mathbf{V}$, j = 1 to n)
PNM(i,j) = 0 (i = 1 to n, j = 1 to n)
2./* In the constraint phase, obtain the skeleton of the primary Bayesian network*/
For $X_j\in \mathbf{V}$, j = 1 to n do
For $X_i\in \mathbf{V}$, i = 1 to n, $i\ne j$, $\mathbf{Z}\subseteq \mathbf{V}\mathrm{\setminus }\left\{X_i,X_j\right\}$ do
Calculation ${\rho }_r\left(X_i,X_j|\mathbf{Z}\right)$
Calculation p-value $\left(X_i,X_j\right)$
If p-value $\left(X_i,X_j\right)<K_{\alpha }$
$PN(X_j)=PN(X_j){\cup }^{X_i},\mathrm{P}\mathrm{N}\mathrm{M}\left(\mathrm{i},\mathrm{j}\right)=1$
else
$\mathrm{P}\mathrm{N}\mathrm{M}\left(\mathrm{i},\mathrm{j}\right)=0$
3./* In the search stage, the final Bayesian network graph is obtained*/
G = DAGsearch(D,PNM)
Output G

Table 4. Network information

Network	Number of nodes	Number of sides
1 alarm	37	46
2 carpo	61	74
3 factors	27	68
4 insurance	27	52
5 mildew	35	46
6 water	32	66
7 chain	7	6

Figure 1. For ANM<1>, in different networks and data sets, the structural error of the PRCB algorithm at different thresholds.

Figure 2. For ANM<1>, the structural errors and running time of the nine algorithms in different networks and data sets.

Figure 3. For ANM<2>, the structural error and running time of the five algorithms in different networks and data sets.

Figure 4. For ANM<3>, the structural error and running time of the five algorithms in different networks and data sets.

Figure 5. For ANM<6>, the structural error and running time of the five algorithms in different networks and data sets.

Table 5. Network information

Network	Number of nodes	Number of sides
1 alarm10	370	570
2 hailfinder10	560	1017
3 gene	801	972

Figure 6. For ANM<1>, the structural error and running time of the two algorithms on the high-dimensional network large sample data set.

Figure 7. For ANM<2>, the structural error and running time of the two algorithms on the high-dimensional network large sample data set.

Figure 8. For ANM<6>, the structural error and running time of the two algorithms on the high-dimensional network large sample data set.

Table 6. Performance comparison of PRCS and RCS on high-dimensional large sample data

Algorithm	Functional relationship	Structural error (Win)	Time performance (Win)
PRCS/RCS	Linear function	3/0	3/0
	Sin function	3/0	3/0
	Power function	3/0	3/0
	Cos function	3/0	3/0
	Exponential function	3/0	3/0
	Mixed function	3/0	3/0

Table 7. Some sensor measuring point information of a power plant

Measuring point number	Measuring point	Unit
1	Unit load	MW
2	A side steam turbine inlet steam temperature	°C
3	Inlet steam temperature of A side medium pressure main valve	°C
4	A side cold re-steam temperature	°C
5	First extraction steam temperature	°C
6	Second extraction steam temperature	°C
7	Six extraction steam temperature	°C
8	Low pressure exhaust steam temperature (motor end)	°C
9	Low pressure exhaust steam temperature (valve end)	°C
10	1# bearing metal temperature (right side)	°C
11	2# bearing metal temperature (left side)	°C
12	Speed 1#	r/min
13	Rotor eccentric	%
14	Axial displacement 1	um
15	Differential expansion	um
16	2#Bearing vibration	um
17	Six-stage extraction steam pressure	MPa
18	Main steam flow	t/h

Figure 9. Flow chart of causal fault detection system based on partial rank correlation.

Figure 10. Part of the causality diagram of a measuring point in a power plant.

Figure 11. Experimental results of power plant fault measurement points prediction.

Figure 11. Continued.

Table

BESM	Based on simultaneous equations model
CB	CI test based methods and Bayesian methods
CS-NOD	Constraint-based causal Discovery from Nonstationary/heterogeneous Data
HSIC	Hilbert-Schmidt-Independence-Criterion
KCI	Kernel-based-Conditional-Independence
KPC	Kernel-PC
L1MB	L1-regularized-Markov-Blanket
LSMI	Least-Squares-Mutual-Information
MMHC	Max-Min-Hill-Climbing
PCB/PCS	Partial-Correlation-Based/Partial-Correlation-Statistic
PRCB/PRCS	Partial-Rank-Correlation-Based/Partial-Rank-Correlation-Statistics
RCS	Rank-Correlation-Based

Table

Multiple Linear Causal Discovery Algorithm	Multivariate Nonlinear Causal Discovery Algorithm
L1MB Algorithm	HSIC conditional independence test
TC Algorithm	MAP (Maximum A Posteriori)
Two-Phase Algorithm	KPC Algorithm
PCB/PCS Algorithm	CS-NOD Algorithm
PRCB/PRCS Algorithm