Update app.py

app.py

@@ -53,13 +53,30 @@ oly_y = np.array([   4.47083333,    4.46472926,    5.22208333,    4.15467867,
 
 st.title("Heteroscedastic Gaussian Processes")
 
-st.markdown(r"We are learning the noise v/s inputs relationship with a neural net.")
-
-data = st.selectbox("Data", ["Motorcycle", "Olympic", 'GPflow'])
+st.markdown(r"""
+    Gaussian processes generally assume Homoskedastic noise such as:
+    $$
+    y_{i}=f\left(\mathbf{x}_{i}\right)+\epsilon_{i}, \quad \epsilon_{i} \stackrel{\text { i.i.d. }}{\sim} \mathcal{N}\left(0, \sigma_{\epsilon}^{2}\right), \quad 1 \leq i \leq n
+    $$
+    We can also assume separate distribution of noise over each data point:       
+    $$
+    y_{i}=f\left(\mathbf{x}_{i}\right)+\epsilon_{i}, \quad \epsilon_{i} {\sim} \mathcal{N}\left(0, \sigma_{\epsilon_i}^{2}\right), \quad 1 \leq i \leq n
+    $$ 
+    However, this may not be straightforward to extend for inference or conditioning. A simple idea can be to learn a non-linear neural network function to model the relationship between inputs and noise:
+    $$
+    \sigma_{\epsilon_i} = f(\mathbf{x}_i)
+    $$
+    This demo is an attempt to see how this idea works on several synthetic and real datasets with Heteroskedasticity.
+             """)
+
+data = st.selectbox("Data", ["Motorcycle", "Olympic", "Linear", 'GPflow'])
 if data == "Motorcycle":
     data_x, data_y = mcycle_x, mcycle_y
 elif data == "Olympic":
     data_x, data_y = oly_x, oly_y
+elif data == "Linear":
+    data_x = np.linspace(0,1,99)
+    data_y = 3 * data_x + 2 + (np.random.randn_like(data_x) * data_x)
 elif data == 'GPflow':
     N = 1001