From d6232fdf417f7fe8ca5442bb5b886829937dcf9d Mon Sep 17 00:00:00 2001 From: Ivan Shalashilin Date: Fri, 8 Sep 2023 14:12:59 +0100 Subject: [PATCH] formatting and spelling --- docs/examples/oceanmodelling.py | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/docs/examples/oceanmodelling.py b/docs/examples/oceanmodelling.py index 107e3f12b..a40b0ea18 100644 --- a/docs/examples/oceanmodelling.py +++ b/docs/examples/oceanmodelling.py @@ -48,7 +48,7 @@ # $$ # where $\mathbf{x} = (x^{(0)}$,$x^{(1)})^\text{T}$, with a vector basis in the standard Cartesian directions (dimensions will be indicated by superscripts). # -# We shall label the ground truth $D_0=\left\{ \left(\mathbf{x}_{0,i} , \mathbf{y}_{0,i} \right)\right\}_{i=1}^N$, where $\mathbf{y}_{,i$ is the 2-dimensional velocity vector at the $i$-th location, $\mathbf{x}_{0,i}$. The training dataset contains simulated measurements from ocean drifters $D_T=\left\{\left(\mathbf{x}_{T,i}, \mathbf{y}_{T,i} \right)\right\}_{i=1}^{N_T}$, $N_T = 20$ in this case (the subscripts indicate the ground truth and the simulated measurements respectively). +# We shall label the ground truth $D_0=\left\{ \left(\mathbf{x}_{0,i} , \mathbf{y}_{0,i} \right)\right\}_{i=1}^N$, where $\mathbf{y}_{0,i}$ is the 2-dimensional velocity vector at the $i$-th location, $\mathbf{x}_{0,i}$. The training dataset contains simulated measurements from ocean drifters $D_T=\left\{\left(\mathbf{x}_{T,i}, \mathbf{y}_{T,i} \right)\right\}_{i=1}^{N_T}$, $N_T = 20$ in this case (the subscripts indicate the ground truth and the simulated measurements respectively). # @@ -129,7 +129,7 @@ def prepare_data(df): # \end{array}\right), # $$ # -# where each $f^{(z)}\left(\mathbf{x}\right), z \in \{0,1\}$ is a scalar valued function. +# where each $f^{(z)}\left(\mathbf{x}\right), z \in \{0,1\}$ is a scalar-valued function. # # Now consider the scalar-valued function $g: \mathbb{R}^2 \times\{0,1\} \rightarrow \mathbb{R}$, such that # @@ -397,7 +397,7 @@ def plot_fields( # %% [markdown] # ## Helmholtz decomposition -# In 2 dimensions, a twice continuously differentiable and compactly supported vector field $\mathbf{F}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ can be expressed as the sum of the gradient of a scalar potential $\Phi: \mathbb{R}^2 \rightarrow \mathbb{R}$, called the potential function, and the vorticity operator of another scalar potential $\Psi: \mathbb{R}^2 \rightarrow \mathbb{R}$, called the stream function ([Berlinghieri et. al, (2023)](https://arxiv.org/pdf/2302.10364.pdf)) such that +# In 2 dimensions, a twice continuously differentiable and compactly supported vector field $\mathbf{F}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ can be expressed as the sum of the gradient of a scalar potential $\Phi: \mathbb{R}^2 \rightarrow \mathbb{R}$, called the potential function, and the vorticity operator of another scalar potential $\Psi: \mathbb{R}^2 \rightarrow \mathbb{R}$, called the stream function ([Berlinghieri et al. (2023)](https://arxiv.org/pdf/2302.10364.pdf)) such that # $$ # \mathbf{F}=\operatorname{grad} \Phi+\operatorname{rot} \Psi, # $$ @@ -464,7 +464,7 @@ def __call__( # %% [markdown] -# ### GPJax iImplementation +# ### GPJax implementation # We repeat the same steps as with the velocity GP model, replacing `VelocityKernel` with `HelmholtzKernel`. # %% @@ -500,7 +500,7 @@ def __call__( # \mathrm{NLPD}=-\sum_{i=1}^{2N} \log \left( p\left(\mathcal{Y}_i = Y_{0,i} \mid \mathbf{X}_{i}\right) \right), # $$ # -# where each $p\left(\mathcal{Y}_i \mid \mathbf{X}_i \right)$ is the marginal Gaussian distribution at each test location, and $Y_{i,0}$ is the $i$th component of the (massaged) test data that we reserved at the beginning of the notebook in $D_0$. A smaller value is better, since the deviation of the ground truth and the model are small in this case. +# where each $p\left(\mathcal{Y}_i \mid \mathbf{X}_i \right)$ is the marginal Gaussian distribution over $\mathcal{Y}_i$ at each test location, and $Y_{i,0}$ is the $i$-th component of the (massaged) test data that we reserved at the beginning of the notebook in $D_0$. A smaller value is better, since the deviation of the ground truth and the model are small in this case. # %%