Bayesian Gaussian Process Regression (GPR)

Gaussian Process Regression (GPR)

Gaussian process regression (GPR) is a machine learning method based on non-parametric regression method that can be used to fit arbitrary scalar and vectorial quantities. GPR provides a probabilistic model that can be used to make predictions and estimate the uncertainty of those predictions. A Gaussian process is a generalization of the Gaussian probability distribution to functions, where any finite set of function values has a joint Gaussian distribution. The mean function and covariance function of the Gaussian process describe the prior distribution of the function, and the observations are used to update the prior to the posterior distribution of the function. In GPR, the output variable is assumed to be a function of the input variables, and the function is modeled as a sample from a Gaussian process. The goal is to predict the value of the output variable at a new input point, given the observed data. The predicted value is given by the posterior mean of the Gaussian process, and the uncertainty of the prediction is given by the posterior variance. GPR is particularly useful when the data is noisy or when the function being modeled is complex and nonlinear. The key advantages of GPR over other regression techniques are its flexibility and its ability to provide a probabilistic framework for uncertainty quantification. GPR can be used for both regression and classification problems, and it can handle both scalar and vector-valued outputs. Moreover, GPR can be easily extended to handle non-stationary and non-Gaussian data. In practice, GPR is often implemented using the kernlab or gpflow packages in R or Python, respectively. These packages provide functions for specifying the kernel function, which is used to model the covariance between the input variables, and for estimating the hyperparameters of the kernel function using maximum likelihood or Bayesian methods.

Overfitting

Overfitting occurs when a model is too complex and fits the training data too closely, resulting in poor generalization to new data. Like some other machine learning techniques, GPR is prone to overfitting if the model is too complex relative to the amount of data available. Specifically, if the number of hyperparameters of the Gaussian process model is large, or if the covariance function is too flexible, the model may fit the noise in the data rather than the underlying signal. This can result in poor generalization performance, where the model performs well on the training data but poorly on new, unseen data. To mitigate the risk of overfitting in GPR, it is important to carefully select the kernel function and the hyperparameters of the model based on the available data. Cross-validation can be used to estimate the generalization error of the model and to select the optimal values of the hyperparameters. Regularization techniques, such as adding a prior distribution on the hyperparameters or using Bayesian model selection, can also be used to prevent overfitting. Another way to prevent overfitting in GPR is to use a simpler covariance function that captures the key features of the data, rather than trying to fit the noise in the data. Overall, while GPR is a powerful and flexible regression technique, it requires careful tuning of the hyperparameters and selection of the kernel function to prevent overfitting and achieve good generalization performance.

Healthcare

In recent years, the field of healthcare had an increase in the use of machine learning techniques to improve patient care, optimize treatment plans, and enhance medical decision-making. Along these lines, GPR can find diverse applications in healthcare. GPR could offer several benefits in healthcare, for instance, it provides uncertainty quantification, allowing healthcare professionals to assess the reliability of predictions and make informed decisions. Moreover, GPR demonstrates flexibility and adaptability, its non-parametric nature enhances versatility in predictive modeling. Lastly, GPR showcases data efficiency, accurately predicting outcomes even with limited data points. This feature is particularly valuable in healthcare where data collection can be challenging and costly, making GPR an optimal choice for applications with limited data availability.

Despite its numerous benefits, GPR use in healthcare comes with certain challenges and limitations that should be considered. Computational complexity poses a significant challenge, particularly with large datasets, necessitating efficient algorithms and computational resources to handle the complexity. Hyperparameter tuning is another consideration, involving the selection of optimal values for parameters such as the kernel function and noise level. This task can be challenging and may require expert knowledge or extensive experimentation. Furthermore, as GPR models complex relationships, the interpretability of the learned models can become intricate. Understanding the underlying factors contributing to predictions becomes more challenging in highly nonlinear models. These challenges highlight the need for careful consideration and expertise when applying GPR in healthcare settings. GPRs ability to model complex relationships, estimate uncertainties, and provide interpretable predictions makes it an invaluable asset for predictive modeling in healthcare, with a postential to enhance disease progression modeling, personalize treatment plans, detect diseases early, and improve medical imaging analysis. While challenges exist, ongoing research and advancements in computational techniques are addressing these limitations, making GPR an increasingly valuable tool in healthcare. As the field continues to evolve, GPR is poised to revolutionize healthcare by enabling more accurate predictions, better decision-making, and improved patient outcomes.

Code

# Load necessary packages
library(kernlab)
library(GPfit)
library(ggplot2)

# Generate simulated data
set.seed(123)
x <- seq(0, 10, length = 50)
y <- sin(x) + rnorm(50, 0, 0.2)
df <- data.frame(x = x, y = y)

# Fit Gaussian process regression model
gpr_model <- gausspr(y ~ x, data = df)
y_pred <- predict(gpr_model, x)

# Visualize results
ggplot(df, aes(x = x, y = y)) +
  geom_point() +
  geom_line(aes(y = y_pred), color = "red") +
  labs(title = "Gaussian Process Regression", x = "x", y = "y")

This R code performs Gaussian process regression (GPR) on simulated data and visualizes the results. Let’s break down each part of the code step-by-step:

Load Necessary Packages:

library(kernlab)
library(GPfit)
library(ggplot2)

This part loads the required R packages: kernlab for kernel-based machine learning, GPfit for Gaussian process modeling, and ggplot2 for data visualization.

Generate Simulated Data:

set.seed(123)
x <- seq(0, 10, length = 50)
y <- sin(x) + rnorm(50, 0, 0.2)
df <- data.frame(x = x, y = y)

Simulated data is generated for the predictor variable x and the response variable y. The x values are generated as a sequence from 0 to 10 with 50 points. The y values are generated by taking the sine of each x value and adding random noise from a normal distribution with mean 0 and standard deviation 0.2. The data is then combined into a data frame df.

Fit Gaussian Process Regression Model:

gpr_model <- gausspr(y ~ x, data = df)

A Gaussian process regression model is fitted using the gausspr function from the GPfit package. The model specification is y ~ x, indicating that we want to model y as a function of x using Gaussian process regression.

Predict Values of y and Visualize Results:

y_pred <- predict(gpr_model, x)

ggplot(df, aes(x = x, y = y)) +
  geom_point() +
  geom_line(aes(y = y_pred), color = "red") +
  labs(title = "Gaussian Process Regression", x = "x", y = "y")

This part predicts the values of the response variable y_pred for the predictor variable x using the fitted Gaussian process regression model. The predict function is used to make the predictions based on the model gpr_model.

The results are then visualized using ggplot2. A scatter plot of the original data points (x and y) is created with blue points (geom_point()). Overlaid on the scatter plot is a red line representing the predictions of the response variable (y_pred) from the Gaussian process regression model (geom_line(aes(y = y_pred), color = "red")).

Bayesian

Gaussian process regression (GPR) can also be implemented in a Bayesian context using Stan. In Bayesian GPR, we assume a prior distribution for the unknown function and then update our beliefs about the function based on the observed data. The prior distribution is typically specified as a Gaussian process with a mean function and covariance function that depend on hyperparameters. The likelihood function for the observed data is also assumed to be Gaussian with a mean function equal to the prior mean function and a covariance function equal to the sum of the prior covariance function and a noise term. The hyperparameters of the prior and likelihood functions are estimated from the data using Markov chain Monte Carlo (MCMC) methods.

Here is an example of R code for fitting a Bayesian GPR model using Stan. Let’s break down each part of the code step-by-step:

Generate Simulated Data:

set.seed(123)
x <- seq(0, 10, length = 50)
y <- sin(x) + rnorm(50, 0, 0.2)
df <- data.frame(x = x, y = y)

Specify Stan Model Code:

stan_model_code <- " ... "

The Stan model code is specified as a character string. The model defines the data, parameters, and the statistical model for Bayesian GPR. It uses a Gaussian process kernel to model the relationship between the predictor variable x and the response variable y. The parameters mu, sigma_f, sigma_n, and eta represent the mean function, the covariance function for the underlying Gaussian process, the noise standard deviation, and the latent function values, respectively.

Compile Stan Model:

gpr_stan_model <- stan_model(model_code = stan_model_code)

The Stan model is compiled using the stan_model function from the rstan package. This step converts the Stan model code into a C++ program that will be used for Bayesian inference.

Prepare Data for Stan Model:

stan_data <- list(x=df$x,
                  x2=df$x,
                  y=df$y,
                  N=length(df$x),
                  N2=length(df$x))

The data is prepared as a list stan_data with the number of rows N, the predictor variable x, and the response variable y. This data will be used as input to the Stan model during sampling.

Fit Bayesian GPR Model using Stan:

gpr_fit <- sampling(gpr_stan_model, data = stan_data)

The Bayesian GPR model is fitted using the sampling function from rstan. This step performs Markov chain Monte Carlo (MCMC) sampling to estimate the posterior distribution of the model parameters.

Extract Posterior Samples of f for Prediction:

f_samples <- extract(gpr_fit, "f")$f
sigma_samples <- extract(gpr_fit, "sigma")$sigma


df %>%
  mutate(Ef=colMeans(f_samples),
         sigma=mean(sigma_samples)) %>%  
  ggplot(aes(x=x,y=y))+
  geom_point()+
  labs(x="Time (ms)", y="Acceleration (g)")+
  geom_line(aes(y=Ef), color='red')+
  geom_line(aes(y=Ef-2*sigma), color='red',linetype="dashed")+
  geom_line(aes(y=Ef+2*sigma), color='red',linetype="dashed")

The extract function is used to extract the posterior samples of the latent function f from the fitted GPR model. These samples will be used to predict new values of f for new values of x.

The code above demonstrates how to perform Bayesian GPR on simulated data, which is useful for modeling non-linear relationships between variables and making predictions with uncertainty estimates. Bayesian GPR provides a flexible framework for dealing with complex and noisy data, making it a powerful tool for various data analysis tasks.

library(rstan)
library(ggplot2)

# Generate simulated data
set.seed(123)
x <- seq(0, 10, length = 50)
y <- sin(x) + rnorm(50, 0, 0.2)
df <- data.frame(x = x, y = y)

# Specify Stan model code
stan_model_code <- "
functions {
  vector gp_pred_rng(array[] real x2,
                     vector y1,
                     array[] real x1,
                     real sigma_f,
                     real lengthscale_f,
                     real sigma,
                     real jitter) {
    int N1 = rows(y1);
    int N2 = size(x2);
    vector[N2] f2;
    {
      matrix[N1, N1] L_K;
      vector[N1] K_div_y1;
      matrix[N1, N2] k_x1_x2;
      matrix[N1, N2] v_pred;
      vector[N2] f2_mu;
      matrix[N2, N2] cov_f2;
      matrix[N1, N1] K;
      K = gp_exp_quad_cov(x1, sigma_f, lengthscale_f);
      for (n in 1:N1)
        K[n, n] = K[n,n] + square(sigma);
      L_K = cholesky_decompose(K);
      K_div_y1 = mdivide_left_tri_low(L_K, y1);
      K_div_y1 = mdivide_right_tri_low(K_div_y1', L_K)';
      k_x1_x2 = gp_exp_quad_cov(x1, x2, sigma_f, lengthscale_f);
      f2_mu = (k_x1_x2' * K_div_y1);
      v_pred = mdivide_left_tri_low(L_K, k_x1_x2);
      cov_f2 = gp_exp_quad_cov(x2, sigma_f, lengthscale_f) - v_pred' * v_pred;

      f2 = multi_normal_rng(f2_mu, add_diag(cov_f2, rep_vector(jitter, N2)));
    }
    return f2;
  }
}
data {
  int<lower=1> N;      // number of observations
  vector[N] x;         // univariate covariate
  vector[N] y;         // target variable
  int<lower=1> N2;     // number of test points
  vector[N2] x2;       // univariate test points
}
transformed data {
  // Normalize data
  real xmean = mean(x);
  real ymean = mean(y);
  real xsd = sd(x);
  real ysd = sd(y);
  array[N] real xn = to_array_1d((x - xmean)/xsd);
  array[N2] real x2n = to_array_1d((x2 - xmean)/xsd);
  vector[N] yn = (y - ymean)/ysd;
  real sigma_intercept = 1;
  vector[N] zeros = rep_vector(0, N);
}
parameters {
  real<lower=0> lengthscale_f; // lengthscale of f
  real<lower=0> sigma_f;       // scale of f
  real<lower=0> sigman;         // noise sigma
}
model {
  // covariances and Cholesky decompositions
  matrix[N, N] K_f = gp_exp_quad_cov(xn, sigma_f, lengthscale_f)+
                     sigma_intercept^2;
  matrix[N, N] L_f = cholesky_decompose(add_diag(K_f, sigman^2));
  // priors
  lengthscale_f ~ normal(0, 1);
  sigma_f ~ normal(0, 1);
  sigman ~ normal(0, 1);
  // model
  yn ~ multi_normal_cholesky(zeros, L_f);
}
generated quantities {
  // function scaled back to the original scale
  vector[N2] f = gp_pred_rng(x2n, yn, xn, sigma_f, lengthscale_f, sigman, 1e-9)*ysd + ymean;
  real sigma = sigman*ysd;
}
"

# Compile Stan model
gpr_stan_model <- stan_model(model_code = stan_model_code)

# Prepare data for Stan model
stan_data <- list(x=df$x,
                  x2=df$x,
                  y=df$y,
                  N=length(df$x),
                  N2=length(df$x))

# Fit Bayesian GPR model using Stan
gpr_fit <- sampling(gpr_stan_model, data = stan_data)


f_samples <- extract(gpr_fit, "f")$f
sigma_samples <- extract(gpr_fit, "sigma")$sigma


df %>%
  mutate(Ef=colMeans(f_samples),
         sigma=mean(sigma_samples)) %>%  
  ggplot(aes(x=x,y=y))+
  geom_point()+
  labs(x="Time (ms)", y="Acceleration (g)")+
  geom_line(aes(y=Ef), color='red')+
  geom_line(aes(y=Ef-2*sigma), color='red',linetype="dashed")+
  geom_line(aes(y=Ef+2*sigma), color='red',linetype="dashed")

References

Duvenaud, D. K., Nickisch, H., & Rasmussen, C. E. (2013). Gaussian processes for machine learning: tutorial. In S. Sra, S. Nowozin, & S. J. Wright (Eds.), Optimization for Machine Learning (pp. 133-181). MIT Press.
Nguyen, T. D., & Nguyen, T. T. (2018). Multi-task Gaussian process models for biomedical applications. arXiv preprint arXiv:1806.03836.
Alaa, A. M., & van der Schaar, M. (2018). Prognostication and risk factors for cystic fibrosis via automated machine learning and Gaussian process regression. Scientific Reports, 8(1), 1-12.
Nguyen, T. T., Nguyen, H. T., Nguyen, T. L., & Chetty, G. (2017). Gaussian process regression for predicting 30-day readmission of heart failure patients. Journal of Biomedical Informatics, 71, 199-209.
Kazemi, S., & Soltanian-Zadeh, H. (2013). A new Gaussian process regression-based method for segmentation of brain tissues from MRI. Medical Image Analysis, 17(3), 225-234.
Gaussian process demonstration with Stan