Fixed math rendering in README.md

README.md

@@ -3,21 +3,21 @@
 # Wasserstein distance code for Matlab
 This is a compact Matlab code to compute the 1- and 2-Wasserstein distances for one dimensional probability distributions.
 
-Refer to [Wikipedia](https://en.wikipedia.org/wiki/Wasserstein_metric) for a general definition of the Wasserstein distance ![w_p](https://render.githubusercontent.com/render/math?math=w_p). This implementation is based on the fact that for given distributions _u_ and _v_ the 1-Wasserstein distance can be written as
+Refer to [Wikipedia](https://en.wikipedia.org/wiki/Wasserstein_metric) for a general definition of the Wasserstein distance $w_p$. This implementation is based on the fact that for given distributions $u$ and $v$ the 1-Wasserstein distance can be written as
 
-![w_1(u,v) = \int_{-\infty}^\infty \left|U_\text{cdf}(x)-V_\text{cdf}(x)\right|\,dx](https://render.githubusercontent.com/render/math?math=w_1(u%2Cv)%20%3D%20%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20%5Cleft%7CU_%5Ctext%7Bcdf%7D(x)-V_%5Ctext%7Bcdf%7D(x)%5Cright%7C%5C%2Cdx)
+$$ w_1(u,v) = \int_{-\infty}^\infty \left|U_\text{cdf}(x)-V_\text{cdf}(x)\right| dx $$
 
 and the 2-Wasserstein distance as
 
-![w_2(u,v)^2 = \int_{0}^1 \left(U_\text{icdf}(p)-V_\text{icdf}(p)\right)^2\,dp.](https://render.githubusercontent.com/render/math?math=w_2(u%2Cv)%5E2%20%3D%20%5Cint_%7B0%7D%5E1%20%5Cleft(U_%5Ctext%7Bicdf%7D(p)-V_%5Ctext%7Bicdf%7D(p)%5Cright)%5E2%5C%2Cdp.)
+$$ w_2(u,v)^2 = \int_{0}^1 \left(U_\text{icdf}(p)-V_\text{icdf}(p)\right)^2 dp.$$
 
-Here ![U_\text{cdf}](https://render.githubusercontent.com/render/math?math=U_%5Ctext%7Bcdf%7D) and ![V_\text{cdf}](https://render.githubusercontent.com/render/math?math=V_%5Ctext%7Bcdf%7D) refer to the cumulative density functions of _u_ and _v_ and ![U_\text{icdf}](https://render.githubusercontent.com/render/math?math=U_%5Ctext%7Bicdf%7D) and ![V_\text{icdf}](https://render.githubusercontent.com/render/math?math=V_%5Ctext%7Bicdf%7D) to the corresponding pseudo inverse cumulative density functions defined by
+Here $U_\text{cdf}$ and $V_\text{cdf}$ refer to the cumulative density functions of $u$ and $v$ and $U_\text{icdf}$ and $V_\text{icdf}$ to the corresponding pseudo inverse cumulative density functions defined by
 
-![U_\text{icdf}(p) = \inf\{x: \, U_\text{cdf}(x) > p\}.](https://render.githubusercontent.com/render/math?math=U_%5Ctext%7Bicdf%7D(p)%20%3D%20%5Cinf%5C%7Bx%3A%20%5C%2C%20U_%5Ctext%7Bcdf%7D(x)%20%3E%20p%5C%7D.)
+$$ U_\text{icdf}(p) = \inf \\{ x:  U_\text{cdf}(x) > p \\}. $$
 
-The code assumes (unlike for instance the [bioconductor function](https://www.bioconductor.org/packages/devel/bioc/vignettes/waddR/inst/doc/wasserstein_metric.html)) that _u_ and _v_ are discrete and uniform probability distribution. In this case there are samples ![u_1, u_2, \dots, u_N](https://render.githubusercontent.com/render/math?math=u_1%2C%20u_2%2C%20%5Cdots%2C%20u_N) such that any _u_ distributed random variable ![X_u](https://render.githubusercontent.com/render/math?math=X_u) satisfies ![P(X_u = u_k)=1/N](https://render.githubusercontent.com/render/math?math={P(X_u=u_k)=1/N}) for all _k_. These samples are the inputs of the function, let's assume without loss of generality that they are increasingly sorted then the cumulative distribution function and its pseudo inverse are given by the step functions
+The code assumes (unlike for instance the [bioconductor function](https://www.bioconductor.org/packages/devel/bioc/vignettes/waddR/inst/doc/wasserstein_metric.html)) that $u$ and $v$ are discrete and uniform probability distribution. In this case there are samples $u_1, u_2, \dots, u_N$ such that any $u$ distributed random variable $X_u$ satisfies $P(X_u = u_k)=1/N$ for all $k$. These samples are the inputs of the function, let's assume without loss of generality that they are increasingly sorted then the cumulative distribution function and its pseudo inverse are given by the step functions
 
-![U_\text{cdf}(x) = \frac{\max\{i: \, u_i \leq x\}}{N}, \quad U_\text{icdf}(p)=u_{\left \lfloor{p/N} \right \rfloor + 1}\,.](https://render.githubusercontent.com/render/math?math=U_%5Ctext%7Bcdf%7D(x)%20%3D%20%5Cfrac%7B%5Cmax%5C%7Bi%3A%20%5C%2C%20u_i%20%5Cleq%20x%5C%7D%7D%7BN%7D%2C%20%5Cquad%20U_%5Ctext%7Bicdf%7D(p)%3Du_%7B%5Cleft%20%5Clfloor%7Bp%2FN%7D%20%5Cright%20%5Crfloor%20%2B%201%7D%5C%2C.)
+$$ U_\text{cdf}(x) = \frac{\max\\{i: u_i \leq x\\}}{N}, \qquad U_\text{icdf}(p)=u_{\left \lfloor{p/N} \right \rfloor + 1}. $$
 
 The code has been tested in Matlab R2017a and checked against the [scipy Wasserstein distance](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wasserstein_distance.html) (in case of the 1-Wasserstein distance) and the [earth mover distance code on Matlab Central File Exchange](https://www.mathworks.com/matlabcentral/fileexchange/22962-the-earth-mover-s-distance). The latter code allows for computations of the general p-Wasserstein distance but is computationally more complex than the present code for the described situation.