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sofenn
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Self-Organizing Fuzzy Neural Network
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# sofenn: Self-Organizing Fuzzy Neural Network Welcome to sofenn (sounds like soften)! This package is a Keras/TensorFlow implementation of a Self-Organizing Fuzzy Neural Network. The **sofenn** framework consists of two models: <br /> (1) **Fuzzy Network** - Underlying fuzzy network whose parameters are optimized during training <br /> (2) **Self-Organizer** - Metamodel that optimizes the architecture of the Fuzzy Network ## FuzzyNetwork Underlying neural network model that contains parameters to be optimized during training ## FuzzySelfOrganizer Metamodel to optimize the architecture of underlying fuzzy network ## Installation You can install the package using pip: ```bash pip install sofenn ``` ## Usage [Demo notebooks](https://github.com/andrewre23/sofenn/tree/master/demos) available on Github. Importing the model and self-organizer: ```python from sofenn import FuzzyNetwork, FuzzySelfOrganizer # initialize model separate, and attach to self-organizer model = FuzzyNetwork(input_shape, **_init_params) sofnn = FuzzySelfOrganizer(model=model) sofnn.self_organize(x, y) # initialize sofnn directly sofnn = FuzzySelfOrganizer(input_shape, **_init_params) sofnn.self_organize(x, y) ``` ## Model Description The model is implemented per the description in: <br /> **'An on-line algorithm for creating self-organizing fuzzy neural networks\'** <br /> Leng, Prasad, McGinnity (2004)  *Fuzzy Neural Network Architecture* Credit: Leng, Prasad, McGinnity (2004) ## Layers ### Inputs Layer (0) **Input layer of network** - **input** : <a href="https://www.codecogs.com/eqnedit.php?latex=input" target="_blank"><img src="https://latex.codecogs.com/gif.latex?input" title="input" /></a> - shape: (*, features) ### Fuzzy Layer (1) **Radial (Ellipsoidal) Basis Function Layer** - Each neuron represents "if-part" or premise of a fuzzy rule - Individual Membership Functions (MF) are applied to each feature for each neuron - Output is product of Membership Functions - Each MF is a Gaussian function: <a href="https://www.codecogs.com/eqnedit.php?latex=\mu_{ij}&space;=&space;\exp([-\frac{(x_i&space;-&space;c_{ij})^2}{2\sigma^2_{ij}}])" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\mu_{ij}&space;=&space;\exp([-\frac{(x_i&space;-&space;c_{ij})^2}{2\sigma^2_{ij}}])" title="\mu_{ij} = \exp([-\frac{(x_i - c_{ij})^2}{2\sigma^2_{ij}}])" /></a> - for i features and j neurons: - <a href="https://www.codecogs.com/eqnedit.php?latex=\mu_{ij}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\mu_{ij}" title="\mu_{ij}" /></a> = ith MF of jth neuron - <a href="https://www.codecogs.com/eqnedit.php?latex=c_{ij}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?c_{ij}" title="c_{ij}" /></a> = center of ith MF of jth neuron - <a href="https://www.codecogs.com/eqnedit.php?latex=\sigma_{ij}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\sigma_{ij}" title="\sigma_{ij}" /></a> = width of ith MF of jth neuron - output for Fuzzy Layer is: <a href="https://www.codecogs.com/eqnedit.php?latex=\Phi_j&space;=&space;\exp(\sum_{i=1}^{r}&space;\frac{(x_i&space;-&space;c_{ij})^2}{2\sigma^2_{ij}})" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Phi_j&space;=&space;\exp(\sum_{i=1}^{r}&space;\frac{(x_i&space;-&space;c_{ij})^2}{2\sigma^2_{ij}})" title="\Phi_j = \exp(\sum_{i=1}^{r} \frac{(x_i - c_{ij})^2}{2\sigma^2_{ij}})" /></a> - **input** : <a href="https://www.codecogs.com/eqnedit.php?latex=x" target="_blank"><img src="https://latex.codecogs.com/gif.latex?x" title="x" /></a> - shape: (*, features) - **output** : <a href="https://www.codecogs.com/eqnedit.php?latex=\Phi" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Phi" title="\Phi" /></a> - shape: (*, neurons)  *Information flow of* **r** *features within neuron* **j** Credit: Leng, Prasad, McGinnity (2004) ### Normalize Layer (2) **Normalization Layer** - Output of each neuron is normalized by total output from the previous layer - Number of outputs equal to the previous layer (# of neurons) - Output for Normalize Layer is: <a href="https://www.codecogs.com/eqnedit.php?latex=\Psi_j&space;=&space;\frac{\Phi_j}{\sum_{k=1}^{u}&space;\Phi_k}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Psi_j&space;=&space;\frac{\Phi_j}{\sum_{k=1}^{u}&space;\Phi_k}" title="\Psi_j = \frac{\Phi_j}{\sum_{k=1}^{u} \Phi_k}" /></a> <a href="https://www.codecogs.com/eqnedit.php?latex=\Psi_j" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Psi_j" title="\Psi_j" /></a> = output of Fuzzy Layer neuron j - **input** : <a href="https://www.codecogs.com/eqnedit.php?latex=\Phi" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Phi" title="\Phi" /></a> - shape : (*, neurons) - **output** : <a href="https://www.codecogs.com/eqnedit.php?latex=\Psi" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Psi" title="\Psi" /></a> - shape : (*, neurons) ### Weighted Layer (3) **Weighting of ith MF of each feature** - Yields the "consequence" of the *j*th fuzzy rule of the fuzzy model - Each neuron has two inputs: - <a href="https://www.codecogs.com/eqnedit.php?latex=j" target="_blank"><img src="https://latex.codecogs.com/gif.latex?j" title="j" /></a> = output of previous related neuron - <a href="https://www.codecogs.com/eqnedit.php?latex=w_2j" target="_blank"><img src="https://latex.codecogs.com/gif.latex?w_2j" title="w_2j" /></a> = weighted bias - with: <a href="https://www.codecogs.com/eqnedit.php?latex=r" target="_blank"><img src="https://latex.codecogs.com/gif.latex?r" title="r" /></a> = number of original input features <a href="https://www.codecogs.com/eqnedit.php?latex=B&space;=&space;[1,&space;x_1,&space;x_2,&space;...&space;,&space;x_r]" target="_blank"><img src="https://latex.codecogs.com/gif.latex?B&space;=&space;[1,&space;x_1,&space;x_2,&space;...&space;,&space;x_r]" title="B = [1, x_1, x_2, ... , x_r]" /></a> <a href="https://www.codecogs.com/eqnedit.php?latex=A_j&space;=&space;[a_{j0},&space;a_{j1},&space;a_{j2},&space;...,&space;a_{jr}]" target="_blank"><img src="https://latex.codecogs.com/gif.latex?A_j&space;=&space;[a_{j0},&space;a_{j1},&space;a_{j2},&space;...,&space;a_{jr}]" title="A_j = [a_{j0}, a_{j1}, a_{j2}, ..., a_{jr}]" /></a> <a href="https://www.codecogs.com/eqnedit.php?latex=w_2j&space;=&space;A_j&space;*&space;B&space;=&space;a_{j0}&space;+&space;a_{j1x_1}&space;+&space;a_{j2x_2}&space;+&space;...&space;+&space;a_{jr}x_r" target="_blank"><img src="https://latex.codecogs.com/gif.latex?w_2j&space;=&space;A_j&space;*&space;B&space;=&space;a_{j0}&space;+&space;a_{j1x_1}&space;+&space;a_{j2x_2}&space;+&space;...&space;+&space;a_{jr}x_r" title="w_2j = A_j * B = a_{j0} + a_{j1x_1} + a_{j2x_2} + ... + a_{jr}x_r" /></a> <a href="https://www.codecogs.com/eqnedit.php?latex=\Psi_j" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Psi_j" title="\Psi_j" /></a> = output of jth neuron from normalize layer - output for weighted layer is: <a href="https://www.codecogs.com/eqnedit.php?latex=f_j&space;=&space;w_{2j}\Psi_j" target="_blank"><img src="https://latex.codecogs.com/gif.latex?f_j&space;=&space;w_{2j}\Psi_j" title="f_j = w_{2j}\Psi_j" /></a> - **inputs** : <a href="https://www.codecogs.com/eqnedit.php?latex=[x,&space;\Psi]" target="_blank"><img src="https://latex.codecogs.com/gif.latex?[x,&space;\Psi]" title="[x, \Psi]" /></a> - shape: \[(\*, 1+features), (*, neurons)] - **output** : <a href="https://www.codecogs.com/eqnedit.php?latex=f" target="_blank"><img src="https://latex.codecogs.com/gif.latex?f" title="f" /></a> - shape: (*, neurons) ### Output Layer (4) **Final Output** - Unweighted sum of each output of the previous layer (<a href="https://www.codecogs.com/eqnedit.php?latex=f" target="_blank"><img src="https://latex.codecogs.com/gif.latex?f" title="f" /></a>) - Provide activation function to layer - Function choice determines output shape (e.g., linear vs. softmax) - Output for fuzzy layer is: <a href="https://www.codecogs.com/eqnedit.php?latex=\sum_{k=1}^{u}&space;f(k)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\sum_{k=1}^{u}&space;f(k)" title="\sum_{k=1}^{u} f(k)" /></a> for *u* neurons - Provide activation function to layer (default: linear) - Activation function determines output dimensions ### Examples Regression output: - **input** : <a href="https://www.codecogs.com/eqnedit.php?latex=f" target="_blank"><img src="https://latex.codecogs.com/gif.latex?f" title="f" /></a> - shape: (*, neurons) - **output** : <a href="https://www.codecogs.com/eqnedit.php?latex=output" target="_blank"><img src="https://latex.codecogs.com/gif.latex?output" title="output" /></a> - shape: (*,) Softmax classification output: - **input** : <a href="https://www.codecogs.com/eqnedit.php?latex=output" target="_blank"><img src="https://latex.codecogs.com/gif.latex?output" title="output" /></a> - shape: (*, ) - **output** : <a href="https://www.codecogs.com/eqnedit.php?latex=softmax" target="_blank"><img src="https://latex.codecogs.com/gif.latex?softmax" title="softmax" /></a> - shape: (*, classes)
