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Update README.md
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hdlugas authored Dec 2, 2024
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Expand Up @@ -122,7 +122,7 @@ functionality for computing the following similarity measures:
```math
S_{Cosine}(I,J)=\frac{I\circ J}{|I|_{2}\cdot |J|_{2}}
```
where multiplication in the numerator refers to the dot product $I\circ J=a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n}$ of $I$ and $J$ and multiplication in the denominator refers to multiplication of the $L^{2}$-norm of $I$ and $J$, $\vertI\vert_{2}=\sqrt{a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}}, \vert J\vert_{2}=\sqrt{b_{1}^{2}+b_{2}^{2}+...+b_{n}^{2}}$.
where multiplication in the numerator refers to the dot product $I\circ J=a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n}$ of $I$ and $J$ and multiplication in the denominator refers to multiplication of the $L^{2}$-norm of $I$ and $J$, $\vert I\vert_{2}=\sqrt{a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}}, \vert J\vert_{2}=\sqrt{b_{1}^{2}+b_{2}^{2}+...+b_{n}^{2}}$.

- Shannon Entropy Similarity Measure:

Expand All @@ -136,13 +136,21 @@ H_{Shannon}(I)=-\sum_{i=1}^{n}a_{i}\cdot ln(a_{i})

- Tsallis Entropy Similarity Measure:

$S_{Tsallis}(I,J,q)=1-\frac{2\times H_{Tsallis}(I/2+J/2,q)-H_{Tsallis}(I,q)-H_{Tsallis}(J,q)}{N_{Tsallis}(I,J,q)}$

$N_{Tsallis}(I,J,q):=\frac{\sum_{i=1}^{n}\left(2\left(\frac{a_{i}}{2}\right)^{q}+2\left(\frac{b_{i}}{2}\right)^{q}-a_{i}^{q}-b_{i}^{q}\right)}{1-q}$

$H_{Tsallis}(I,q)=\frac{\left(\sum_{i=1}^{n}a_{i}^{q}\right)-1}{1-q}$

$q\neq 1, \ q\textgreater 0$
```math
S_{Tsallis}(I,J,q)=1-\frac{2\times H_{Tsallis}(I/2+J/2,q)-H_{Tsallis}(I,q)-H_{Tsallis}(J,q)}{N_{Tsallis}(I,J,q)}
```

```math
N_{Tsallis}(I,J,q):=\frac{\sum_{i=1}^{n}\left(2\left(\frac{a_{i}}{2}\right)^{q}+2\left(\frac{b_{i}}{2}\right)^{q}-a_{i}^{q}-b_{i}^{q}\right)}{1-q}
```

```math
H_{Tsallis}(I,q)=\frac{\left(\sum_{i=1}^{n}a_{i}^{q}\right)-1}{1-q}
```

```math
q\neq 1, \ q\textgreater 0
```

- Rényi Entropy Similarity Measure:

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