In LiaDD/Fuzzy-Cognitive-Maps-FCMs: Fuzzy Cognitive Maps (FCMs) Inference
The fcm package
The fcm package includes the fcm.infer function which returns a dataframe of the concepts' values of each state after the inference procedure.
Fuzzy Cognitive Maps (FCMs)
Fuzzy Cognitive Map is a combination of fuzzy logic and cognitive mapping, and it is a way to represent knowledge of systems which are characterized of uncertainty and complex processes. FCMs were introduced by Kosko and since then they have gradually emerged as a powerful paradigm for knowledge representation. FCMs are ideal causal cognition tools for modeling and simulating dynamic systems.
FCMs are fuzzy signed directed graphs with feedback. Each interconnection between two concepts $C_{i}$ and $C_{j}$ has a weight, a directed edge $w_{ij}$, which indicates the strength of the causal relationships between concepts $C_{i}$ and $C_{j}$. The value of the weight $w_{ij}$ indicates how strongly the concept $C_{i}$ influences concept $C_{j}$.The time varying concept function $C_{i}(t)$ measures the non negative occurrence of some fuzzy event. There are three possible types of causal relationship between concept $C_{i}$ and concept $C_{j}$, according to the sign of the weight $w_{ij}$:
- $w_{ij}>0$ indicates a positive causality between concept $C_{i}$ and concept $C_{j}$. This means that an increase/decrease in the value of concept $C_{i}$ leads to the increase/decrease of the value of concept $C_{j}$. (positive causality)
- $w_{ij}<0$ indicates a negative (inverse) causality between concept $C_{i}$ and concept $C_{j}$. This means that an increase in the value of concept $C_{i}$ leads to a decrease of the value of concept $C_{j}$ and a decrease of the value of concept $C_{i}$ leads to an increase of the value of concept $C_{j}$. ( negative causality)
- $w_{ij}=0$ indicates no relationship between concept $C_{i}$ and $C_{j}$. (zero causality)
Inference Rules
Every concept in the FCM graph has a value $A_{i}$ that expresses the quantity of its corresponding physical value and it is derived by the transformation of the fuzzy values assigned by the experts to numerical values. The value $A_{i}$ of each concept $C_{i}$ is calculated during every at each the simulation step, computing the influence of other concepts to the specific concept by selecting one of the following equtations (inference rules):
| Inference Rule | Equation |
|----------------------------|:--------------------------------------------------------------------------------:|
| Kosko's inference:* | $A_{i}(k+1)=f\left(\sum_{j=1,j\neq1}^Nw_{ji}\times A_{j}(k)\right)$ |
| modified Kosko's inference:| $A_{i}(k+1)=f\left(A_{i}(k)+\sum_{j=1,j\neq1}^Nw_{ji}\times A_{j}(k)\right)$ |
| Rescale inference: | $A_{i}(k+1)=f\left((2\times A_{i}(k)-1)+\sum_{j=1,j\neq1}^Nw_{ji}\times (2\times A_{j}(k)-1)\right)$ |
| | |
where $A_{i}(k+1)$ is the value of concept $C_{i}$ at simulation step $k+1$, $A_{j}(k)$ is the value of concept $C_{j}$ at the simulationstep $k$, $w_{ij}$ is the weight of the interconnection between concept $C_{j}$ and concept $C_{i}$, k is the interaction index an every simulation step and $f(·)$ is the threshold (activation) function:
| Threshold function | Equation |
|------------------------|:--------------------------------------------------------------------------------------:|
| Bivalent: | [f(x)= \left{\begin{array}{ll} 1 & x > 0 \0 & x\leq 0 \\end{array} \right.] |
| Trivalent: | [f(x)= \left{\begin{array}{ll} 1 & x > 0 \0 & x=0 \-1 & x<0 \\end{array} \right.]|
| Sigmoid: | $f(x)=\frac{1}{1+e^{-\lambda\times x}}$ |
| Hyperbolic tangent: | $f(x)=\tanh(\lambda\times x)$ |
| | |
where \lambda is a real positive number ($\lambda>0$) which determines the steepness of the continuous function $f$ and $x$ is the value $A_{i}(k)$ on the equilibrium point. The sigmoid threshold function ensures that the calculated value of each concept will belong to the interval [0,1]. When the values of concepts can be negative and their values belong to the interval [-1,1], the following function is used: $f(x)=tanh(x)$. The FCM model of the system takes the initial values of concepts and weights based on experts knolegde and experience for the real system and it is free to interact. At each step, the value $A_{i}$ of a concept is influenced by the values of concepts connected to it and it is updated according to the inference rule.
How to obtain the FCM inference using R?
Step 1: Load the fcm package
library(fcm)
Step 2: Create (or import) the weight matrix and the activation vector
The weight matrix stores the weights assigned to the pairs of concepts which are usually normalized to the interval [0, 1] or [−1, +1]. The dimention of the weight matrix is m x m, where m denotes the number of the columns (nodes).
The activation vector contains the initial concept values which each concept is turned on or activated by making its vector element 1 or 0 or in [0,1]. The dimention of the activation matrix is 1 x m.
NOTE: Both of these input arguments should be tranformed as dataframes.
act.vec <- data.frame(1, 1, 1, 0, 0, 0) # Create the activation vector
colnames(act.vec) <- c("C1", "C2", "C3", "C4", "C5", "C6") # Change the column names
C1 = c(0.0, 0.0, 0.6, 0.9, 0.0, 0.0)
C2 = c(0.1, 0.0, 0.0, 0.0, 0.0, 0.0)
C3 = c(0.0, 0.7, 0.0, 0.0, 0.9, 0.0)
C4 = c(0.0, 0.0, 0.0, 0.0, 0.0, 0.9)
C5 = c(0.0, 0.0, 0.0, 0.0, 0.0, -0.9)
C6 = c(-0.3, 0.0, 0.0, 0.0, 0.0, 0.0)
w.mat <- matrix(c(C1, C2, C3, C4, C5, C6), nrow =6, ncol=6, byrow=TRUE) # Create the weight matrix
w.mat <- as.data.frame(w.mat) # Transform w.mat as a dataframe
colnames(w.mat) <- c("C1", "C2", "C3", "C4", "C5", "C6")
w.mat # View the weight matrix
NOTE: It is possible to import a file (for instance a .csv file), typing the file path. w.mat <- read.csv("C://type_the_file_path//name_of_file.csv")
For more information about reading .csv files, type in R console: ?read.csv.
Step 3: Use the fcm.infer function
There are seven input arguments in the fcm.infer function: the activation vector, the weight matrix, the number of iterations, the inference rule, the transformation function, lambda ($\lambda$) parameter and epsilon ($\epsilon$) parameter.
- the number of iteration is a possitive number (iter > 0). Default value:
iter = 20
- six different inference rules are available (
'k' Kosko, 'mk' modified Kosko, 'r' Rescale,'kc' Kosko-clamped, 'mkc' modified Kosko-clamped or 'rc' Rescale-clamped). Default value: infer = 'k'
- four transformation functions (
b Bivalent, tr Trivalent, s Sigmoid or t Hyperbolic tangent). Default value: transform = 's'
- the lambda parameter is a positive number, higher than zero and lower-equal to ten. Default value:
lambda = 1
- the possible e values are:
0.01, 0.001, 0.0001, 0.00001 or 0.000001. Default value: e = 0.001
The fcm.infer function prints a message to inform the user if the convergence has been reached or not.
-
If the convergence has been reached then the converged state (fixed point) is printed:
"The concepts' values are converged in ith state".
-
If the convergence has not been reached a message informs the user that more iterations required: "More iterations are required to reach the convergence".
Example 1: Estimate the Inference of FCM (using the default values of the function)
output1 <- fcm.infer(act.vec, w.mat)
Example 2: Estimate the FCM Inference (changing the default values)
The FCM Inference (fcm.infer) function is using the rescale 'r' inference rule, the sigmoid 's' tranformation function, the lambda parameter equals to 3 lambda = 3 and the e parameter e = 0.0001.
output2 <- fcm.infer(act.vec, w.mat, 35, "r", "s", lambda = 2, e = 0.0001)
output2$values # View the concepts' values for each iteration
Step 4: Visualize the concepts' values for each iteration
For the visualization of the concepts' values for each iteration after the inference process, it is crusial to load the libraries: reshape and ggplot2.
# load the libraries
library (reshape2)
library (ggplot2)
Visualization of Example 1:
iterations <- as.numeric(rownames(output1$values)) # create a numeric vector named "iterations"
df <- data.frame(iterations, output1$values) # add "iterations" in the "output1$values" dataframe
df2 <- melt(df, id="iterations") # transform the dataframe df into long formats
ggplot(data=df2, # Visualize the concepts' values
aes(x=iterations, y=value, group=variable, colour=variable)) +
theme_bw() + geom_line(size=0.7) + geom_point(size = 3)
Visualization of Example 2:
iterations <- as.numeric(rownames(output2$values))
df <- data.frame(iterations, output2$values)
df2 <- melt(df, id="iterations")
ggplot(data=df2,
aes(x=iterations, y=value, group=variable, colour=variable)) +
theme_bw() + geom_line(size=0.7) + geom_point(size = 3)
References
- Kosko B., "Fuzzy cognitive maps", International Journal of Man-Machine Studies 24, 65-75, 1986.
- Groumpos, P.P, Stylios, C.D.; "Modelling supervisory control systems using fuzzy cognitive maps", Chaos, Solitons & Fractals, Volume 11, Issues 1–3, p.p. 329–336, 2000.
- Papageorgiou E.I, "Fuzzy Cognitive Maps for Applied Sciences and Engineering From Fundamentals to Extensions and Learning Algorithms", Intelligent Systems Reference Library, Volume 54, 2014.
- Papageorgiou E.I., Stylios C.D., Groumpos P.P. , "Unsupervised learning techniques for finetuning fuzzy cognitive map causal links.", Int. J. Human Comput. Stud. Vol. 64, pp. 727–743, 2006.
LiaDD/Fuzzy-Cognitive-Maps-FCMs documentation built on May 5, 2019, 3:48 p.m.
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
The fcm package
The fcm package includes the fcm.infer function which returns a dataframe of the concepts' values of each state after the inference procedure.
Fuzzy Cognitive Maps (FCMs)
Fuzzy Cognitive Map is a combination of fuzzy logic and cognitive mapping, and it is a way to represent knowledge of systems which are characterized of uncertainty and complex processes. FCMs were introduced by Kosko and since then they have gradually emerged as a powerful paradigm for knowledge representation. FCMs are ideal causal cognition tools for modeling and simulating dynamic systems.
FCMs are fuzzy signed directed graphs with feedback. Each interconnection between two concepts $C_{i}$ and $C_{j}$ has a weight, a directed edge $w_{ij}$, which indicates the strength of the causal relationships between concepts $C_{i}$ and $C_{j}$. The value of the weight $w_{ij}$ indicates how strongly the concept $C_{i}$ influences concept $C_{j}$.The time varying concept function $C_{i}(t)$ measures the non negative occurrence of some fuzzy event. There are three possible types of causal relationship between concept $C_{i}$ and concept $C_{j}$, according to the sign of the weight $w_{ij}$:
- $w_{ij}>0$ indicates a positive causality between concept $C_{i}$ and concept $C_{j}$. This means that an increase/decrease in the value of concept $C_{i}$ leads to the increase/decrease of the value of concept $C_{j}$. (positive causality)
- $w_{ij}<0$ indicates a negative (inverse) causality between concept $C_{i}$ and concept $C_{j}$. This means that an increase in the value of concept $C_{i}$ leads to a decrease of the value of concept $C_{j}$ and a decrease of the value of concept $C_{i}$ leads to an increase of the value of concept $C_{j}$. ( negative causality)
- $w_{ij}=0$ indicates no relationship between concept $C_{i}$ and $C_{j}$. (zero causality)
Inference Rules
Every concept in the FCM graph has a value $A_{i}$ that expresses the quantity of its corresponding physical value and it is derived by the transformation of the fuzzy values assigned by the experts to numerical values. The value $A_{i}$ of each concept $C_{i}$ is calculated during every at each the simulation step, computing the influence of other concepts to the specific concept by selecting one of the following equtations (inference rules):
| Inference Rule | Equation | |----------------------------|:--------------------------------------------------------------------------------:| | Kosko's inference:* | $A_{i}(k+1)=f\left(\sum_{j=1,j\neq1}^Nw_{ji}\times A_{j}(k)\right)$ | | modified Kosko's inference:| $A_{i}(k+1)=f\left(A_{i}(k)+\sum_{j=1,j\neq1}^Nw_{ji}\times A_{j}(k)\right)$ | | Rescale inference: | $A_{i}(k+1)=f\left((2\times A_{i}(k)-1)+\sum_{j=1,j\neq1}^Nw_{ji}\times (2\times A_{j}(k)-1)\right)$ | | | |
where $A_{i}(k+1)$ is the value of concept $C_{i}$ at simulation step $k+1$, $A_{j}(k)$ is the value of concept $C_{j}$ at the simulationstep $k$, $w_{ij}$ is the weight of the interconnection between concept $C_{j}$ and concept $C_{i}$, k is the interaction index an every simulation step and $f(·)$ is the threshold (activation) function:
| Threshold function | Equation | |------------------------|:--------------------------------------------------------------------------------------:| | Bivalent: | [f(x)= \left{\begin{array}{ll} 1 & x > 0 \0 & x\leq 0 \\end{array} \right.] | | Trivalent: | [f(x)= \left{\begin{array}{ll} 1 & x > 0 \0 & x=0 \-1 & x<0 \\end{array} \right.]| | Sigmoid: | $f(x)=\frac{1}{1+e^{-\lambda\times x}}$ | | Hyperbolic tangent: | $f(x)=\tanh(\lambda\times x)$ | | | |
where \lambda is a real positive number ($\lambda>0$) which determines the steepness of the continuous function $f$ and $x$ is the value $A_{i}(k)$ on the equilibrium point. The sigmoid threshold function ensures that the calculated value of each concept will belong to the interval [0,1]. When the values of concepts can be negative and their values belong to the interval [-1,1], the following function is used: $f(x)=tanh(x)$. The FCM model of the system takes the initial values of concepts and weights based on experts knolegde and experience for the real system and it is free to interact. At each step, the value $A_{i}$ of a concept is influenced by the values of concepts connected to it and it is updated according to the inference rule.
How to obtain the FCM inference using R?
Step 1: Load the fcm package
library(fcm)
Step 2: Create (or import) the weight matrix and the activation vector
The weight matrix stores the weights assigned to the pairs of concepts which are usually normalized to the interval [0, 1] or [−1, +1]. The dimention of the weight matrix is m x m, where m denotes the number of the columns (nodes).
The activation vector contains the initial concept values which each concept is turned on or activated by making its vector element 1 or 0 or in [0,1]. The dimention of the activation matrix is 1 x m.
NOTE: Both of these input arguments should be tranformed as dataframes.
act.vec <- data.frame(1, 1, 1, 0, 0, 0) # Create the activation vector colnames(act.vec) <- c("C1", "C2", "C3", "C4", "C5", "C6") # Change the column names C1 = c(0.0, 0.0, 0.6, 0.9, 0.0, 0.0) C2 = c(0.1, 0.0, 0.0, 0.0, 0.0, 0.0) C3 = c(0.0, 0.7, 0.0, 0.0, 0.9, 0.0) C4 = c(0.0, 0.0, 0.0, 0.0, 0.0, 0.9) C5 = c(0.0, 0.0, 0.0, 0.0, 0.0, -0.9) C6 = c(-0.3, 0.0, 0.0, 0.0, 0.0, 0.0) w.mat <- matrix(c(C1, C2, C3, C4, C5, C6), nrow =6, ncol=6, byrow=TRUE) # Create the weight matrix w.mat <- as.data.frame(w.mat) # Transform w.mat as a dataframe colnames(w.mat) <- c("C1", "C2", "C3", "C4", "C5", "C6") w.mat # View the weight matrix
NOTE: It is possible to import a file (for instance a .csv file), typing the file path.
w.mat <- read.csv("C://type_the_file_path//name_of_file.csv")
For more information about reading .csv files, type in R console: ?read.csv.
Step 3: Use the fcm.infer function
There are seven input arguments in the fcm.infer function: the activation vector, the weight matrix, the number of iterations, the inference rule, the transformation function, lambda ($\lambda$) parameter and epsilon ($\epsilon$) parameter.
- the number of iteration is a possitive number (iter > 0). Default value:
iter = 20 - six different inference rules are available (
'k'Kosko,'mk'modified Kosko,'r'Rescale,'kc'Kosko-clamped,'mkc'modified Kosko-clamped or'rc'Rescale-clamped). Default value:infer = 'k' - four transformation functions (
bBivalent,trTrivalent,sSigmoid ortHyperbolic tangent). Default value:transform = 's' - the lambda parameter is a positive number, higher than zero and lower-equal to ten. Default value:
lambda = 1 - the possible e values are:
0.01, 0.001, 0.0001, 0.00001 or 0.000001. Default value:e = 0.001
The fcm.infer function prints a message to inform the user if the convergence has been reached or not.
-
If the convergence has been reached then the converged state (fixed point) is printed:
"The concepts' values are converged in ith state". -
If the convergence has not been reached a message informs the user that more iterations required:
"More iterations are required to reach the convergence".
Example 1: Estimate the Inference of FCM (using the default values of the function)
output1 <- fcm.infer(act.vec, w.mat)
Example 2: Estimate the FCM Inference (changing the default values)
The FCM Inference (fcm.infer) function is using the rescale 'r' inference rule, the sigmoid 's' tranformation function, the lambda parameter equals to 3 lambda = 3 and the e parameter e = 0.0001.
output2 <- fcm.infer(act.vec, w.mat, 35, "r", "s", lambda = 2, e = 0.0001) output2$values # View the concepts' values for each iteration
Step 4: Visualize the concepts' values for each iteration
For the visualization of the concepts' values for each iteration after the inference process, it is crusial to load the libraries: reshape and ggplot2.
# load the libraries library (reshape2) library (ggplot2)
Visualization of Example 1:
iterations <- as.numeric(rownames(output1$values)) # create a numeric vector named "iterations" df <- data.frame(iterations, output1$values) # add "iterations" in the "output1$values" dataframe df2 <- melt(df, id="iterations") # transform the dataframe df into long formats ggplot(data=df2, # Visualize the concepts' values aes(x=iterations, y=value, group=variable, colour=variable)) + theme_bw() + geom_line(size=0.7) + geom_point(size = 3)
Visualization of Example 2:
iterations <- as.numeric(rownames(output2$values)) df <- data.frame(iterations, output2$values) df2 <- melt(df, id="iterations") ggplot(data=df2, aes(x=iterations, y=value, group=variable, colour=variable)) + theme_bw() + geom_line(size=0.7) + geom_point(size = 3)
References
- Kosko B., "Fuzzy cognitive maps", International Journal of Man-Machine Studies 24, 65-75, 1986.
- Groumpos, P.P, Stylios, C.D.; "Modelling supervisory control systems using fuzzy cognitive maps", Chaos, Solitons & Fractals, Volume 11, Issues 1–3, p.p. 329–336, 2000.
- Papageorgiou E.I, "Fuzzy Cognitive Maps for Applied Sciences and Engineering From Fundamentals to Extensions and Learning Algorithms", Intelligent Systems Reference Library, Volume 54, 2014.
- Papageorgiou E.I., Stylios C.D., Groumpos P.P. , "Unsupervised learning techniques for finetuning fuzzy cognitive map causal links.", Int. J. Human Comput. Stud. Vol. 64, pp. 727–743, 2006.
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