use \sim for tilde

Author: Naeemkh

paper/paper.md

@@ -241,11 +241,11 @@ Original covariate balance:
 
 The time complexity of the standard Gaussian Process (GP) model is $O(n^3)$, while for the nearest neighbor GP (nnGP) model, it is $O(n * m ^ 3)$, where $m$ is the number of neighbors. An in-depth discussion on achieving these complexities is outside the scope of this paper. Readers interested in further details can refer to @Ren_2021_bayesian. This section focuses on comparing the wall clock time of standard GP and nnGP models in calculating the Conditional Exposure Response Function (CERF) at a specific exposure level, $w$. We set the hyper-parameters to values at $\alpha = \beta = \gamma/\sigma = 1$. \autoref{fig:performance} shows the comparison of standard GP model with nnGP utilizing 50 nearest neighbors. Due to the differing parallelization architectures of the standard GP and nnGP in our package, we conducted this benchmark on a single core. The sample size was varied from 3,000 to 10,000, a range where nnGP begins to demonstrate notable efficiency over the standard GP. We repeat the process 20 times with different seed values. We plotted wall clock time against sample size for both methods. To enhance the visualization of the increasing rate of wall clock time, we applied a log transformation to both axes. For this specific set of analyses the estimated slope of 3.09 (ideally 3) for standard GP aligns with its $O(n^3)$ time complexity. According to the results, a sample size of 10,000 data samples is not large enough to establish a meaningful relationship for the time complexity of the nnGP model effectively.
 
-![Representation of Wall Clock Time (s) vs. Data Samples for Standard GP and nnGP Models. All computations are conducted with $w=1$ and $\alpha = \beta = \gamma/\sigma = 1$. The process is repeated 20 times using various seed values to ensure robustness. A jitter effect is applied to enhance the visibility of data points. Both axes are displayed on log10 scales. The solid lines represent the linear regression modeled as $lm(log10(WC) \textasciitilde log10(n))$. \label{fig:performance}](figures/gp_vs_nngp.png ){ width=80% }
+![Representation of Wall Clock Time (s) vs. Data Samples for Standard GP and nnGP Models. All computations are conducted with $w=1$ and $\alpha = \beta = \gamma/\sigma = 1$. The process is repeated 20 times using various seed values to ensure robustness. A jitter effect is applied to enhance the visibility of data points. Both axes are displayed on log10 scales. The solid lines represent the linear regression modeled as $lm(log10(WC) \sim log10(n))$. \label{fig:performance}](figures/gp_vs_nngp.png ){ width=80% }
 
 \autoref{fig:performance_nn} compares the performance of the nnGP model across three nearest neighbor categories: 50, 100, and 200, using a data sample sequence ranging from 5,000 to 100,000 with intervals of 5,000. For each category, different sets of runs demonstrate a linear relationship, consistent with an $O(n)$ time complexity, assuming that $m^3$ remains constant for varying sample sizes within each category. 
 
-![Representation of Wall Clock Time (s) vs. Data Samples of the nnGP model across different nearest neighbor categories (50, 100, 200) over a range of data sample sizes from 5,000 to 100,000 in 5,000 increments. . All computations are conducted with $w=1$ and $\alpha = \beta = \gamma/\sigma = 1$. Both axes are displayed on log10 scales. The solid lines represent the linear regression modeled as $lm(log10(WC) \textasciitilde log10(n))$. \label{fig:performance_nn}](figures/nngp_nnsize.png ){ width=80% }
+![Representation of Wall Clock Time (s) vs. Data Samples of the nnGP model across different nearest neighbor categories (50, 100, 200) over a range of data sample sizes from 5,000 to 100,000 in 5,000 increments. . All computations are conducted with $w=1$ and $\alpha = \beta = \gamma/\sigma = 1$. Both axes are displayed on log10 scales. The solid lines represent the linear regression modeled as $lm(log10(WC) \sim log10(n))$. \label{fig:performance_nn}](figures/nngp_nnsize.png ){ width=80% }
 
 
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