---
title: "A Guide to the GauPro R package"
author: "Collin Erickson"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{A Guide to the GauPro R package}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, echo=FALSE}
knitr::opts_chunk$set(fig.width=7, fig.height=4) 

set.seed(0)
```

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## Overview

This package allows you to fit a Gaussian process regression model to a dataset. 
A Gaussian process (GP) is a commonly used model in computer simulation.
It assumes that the distribution of any set of points is multivariate
normal.
A major benefit of GP models is that they provide uncertainty estimates along
with their predictions.


## Installation

You can install like any other package through CRAN.

```
install.packages('GauPro')
```

The most up-to-date version can be downloaded from 
my Github account.

```
# install.packages("devtools")
devtools::install_github("CollinErickson/GauPro")
```


## Example in 1-Dimension

This simple shows how to fit the Gaussian process regression model to data.
The function `gpkm` creates a Gaussian process kernel model fit to the
given data.

```{r, libraryGauPro}
library(GauPro)
```


```{r, fitsine}
n <- 12
x <- seq(0, 1, length.out = n)
y <- sin(6*x^.8) + rnorm(n,0,1e-1)
gp <- gpkm(x, y)
```

Plotting the model helps us understand how accurate the model is and how much
uncertainty it has in its predictions.
The green and red lines are the 95% intervals for the mean and for samples,
respectively.

```{r, plotsine}
gp$plot1D()
```

## Factor data: fitting the `diamonds` dataset

The model fit using `gpkm` can also be used with data/formula input and
can properly handle factor data.

In this example, the `diamonds` data set is fit by specifying the formula
and passing a data frame with the appropriate columns.


```{r fit_dm}
library(ggplot2)
diamonds_subset <- diamonds[sample(1:nrow(diamonds), 60), ]
dm <- gpkm(price ~ carat + cut + color + clarity + depth,
           diamonds_subset)
```

Calling `summary` on the model gives details about the model, including
diagnostics about the model fit and the relative importance of the features.

```{r summary_dm}
summary(dm)
```

We can also plot the model to get a visual idea of how each input affects the
output.

```{r plot_dm}
plot(dm)
```

### Constructing a kernel

In this case, the kernel was chosen automatically by looking at which dimensions
were continuous and which were discrete, and then using a Matern 5/2 on the 
continuous dimensions (1,5), and separate ordered factor kernels on the other
dimensions since those columns in the data frame are all ordinal factors.
We can construct our own kernel using products and sums of any kernels,
making sure that the continuous kernels ignore the factor dimensions.

Suppose we want to construct a kernel for this example that uses the power
exponential kernel for the two continuous dimensions, ordered kernels on 
`cut` and `color`, and a Gower kernel on `clarity`.
First we construct the power exponential kernel that ignores the 3 factor
dimensions.
Then we construct 

```{r diamond_construct_kernel}
cts_kernel <- k_IgnoreIndsKernel(k=k_PowerExp(D=2), ignoreinds = c(2,3,4))
factor_kernel2 <- k_OrderedFactorKernel(D=5, xindex=2, nlevels=nlevels(diamonds_subset[[2]]))
factor_kernel3 <- k_OrderedFactorKernel(D=5, xindex=3, nlevels=nlevels(diamonds_subset[[3]]))
factor_kernel4 <- k_GowerFactorKernel(D=5, xindex=4, nlevels=nlevels(diamonds_subset[[4]]))

# Multiply them
diamond_kernel <- cts_kernel * factor_kernel2 * factor_kernel3 * factor_kernel4
```

Now we can pass this kernel into `gpkm` and it will use it.

```{r diamond_construct_kernel_fit}
dm <- gpkm(price ~ carat + cut + color + clarity + depth,
           diamonds_subset, kernel=diamond_kernel)
dm$plotkernel()
```







## Using kernels

A key modeling decision for Gaussian process models is the choice of kernel.
The kernel determines the covariance and the behavior of the model.
The default kernel is the Matern 5/2 kernel (`Matern52`), and is a good
choice for most cases.
The Gaussian, or squared exponential, kernel (`Gaussian`) is a common choice
but often leads to a bad fit since it assumes the process the data comes
from is infinitely differentiable.
Other common choices that are available include the `Exponential`,
Matern 3/2 (`Matern32`),
Power Exponential (`PowerExp`),
`Cubic`,
Rational Quadratic (`RatQuad`),
and Triangle (`Triangle`).

These kernels only work on numeric data.
For factor data, the kernel will default to a
Latent Factor Kernel (`LatentFactorKernel`) for character and unordered factors,
or an Ordered Factor Kernel (`OrderedFactorKernel`) for ordered factors.
As long as the input is given in as a data frame and the columns have the
proper types, then the default kernel will properly handle it by applying
the numeric kernel to the numeric inputs and
the factor kernel to the factor and character inputs.

Kernels are stored as R6 objects.
They can all be created using the R6 object generator (e.g., `Matern52$new()`),
or using the `k_<kernel name>` shortcut function (e.g., `k_Matern52()`).
The latter is easier to use (and recommended) since R will show the function
arguments and autocomplete.

The following table shows details on all the kernels available.

| Kernel | Function | Continuous/<br />discrete | Equation | Notes |
| --- | --- | --- | --- | --- |
| Gaussian | `k_Gaussian` | cts |  | Often causes issues since it assumes infinite differentiability. Experts don't recommend using it. |
| Matern 3/2 | `k_Matern32` | cts | | Assumes one time differentiability. This is often too low of an assumption. |
| Matern 5/2 | `k_Matern52` | cts | | Assumes two time differentiability. Generally the best. |
| Exponential | `k_Exponential` | cts | |  Equivalent to Matern 1/2. Assumes no differentiability. |
| Triangle | `k_Triangle` | cts | | |
| Power exponential | `k_PowerExp` | cts | | |
| Periodic | `k_Periodic` | cts | $k(x, y) = \sigma^2 * \exp(-\sum(\alpha_i*sin(p * (x_i-y_i))^2))$ | The only kernel that takes advantage of periodic data. But there is often incoherence far apart, so you will likely want to multiply by one of the standard kernels. |
| Cubic | `k_Cubic` | cts | | |
| Rational quadratic | `k_RatQuad` | cts | | |
| Latent factor kernel | `k_LatentFactorKernel` | factor | | This embeds each discrete value into a low dimensional space and calculates the distances in that space. This works well when there are many discrete values. |
| Ordered factor kernel | `k_OrderedFactorKernel` | factor | | This maintains the order of the discrete values. E.g., if there are 3 levels, it will ensure that 1 and 2 have a higher correlation than 1 and 3. This is similar to embedding into a latent space with 1 dimension and requiring the values to be kept in numerical order. |
| Factor kernel | `k_FactorKernel` | factor | | This fits a parameter for every pair of possible values. E.g., if there are 4 discrete values, it will fit 6 (4 choose 2) values. This doesn't scale well. When there are many discrete values, use any of the other factor kernels. |
| Gower factor kernel | `k_GowerFactorKernel` | factor | $k(x,y) = \begin{cases} 1,  & \text{if } x=y \\ p, & \text{if } x \neq y  \end{cases}$ | This is a very simple factor kernel. For the relevant dimension, the correlation will either be 1 if the value are the same, or $p$ if they are different. |
| Ignore indices | `k_IgnoreIndsKernel` | N/A | | Use this to create a kernel that ignores certain dimensions. Useful when you want to fit different kernel types to different dimensions or when there is a mix of continuous and discrete dimensions. |

Factor kernels: note that these all only work on a single dimension. If there
are multiple factor dimensions in your input, then they each will be given a
different factor kernel.

## Combining kernels

Kernels can be combined by multiplying or adding them directly.

The following example uses the product of a periodic and a Matern 5/2 kernel
to fit periodic data.

```{r combine seed, include=F}
set.seed(99)
```


```{r combine_periodic}
x <- 1:20
y <- sin(x) + .1*x^1.3
combo_kernel <- k_Periodic(D=1) * k_Matern52(D=1)
gp <- gpkm(x, y, kernel=combo_kernel, nug.min=1e-6)
gp$plot()
```

For an example of a more complex kernel being constructed,
see the diamonds section above.







## Intro to GPs

*(This section used to be the main vignette on CRAN for this package.)*


This R package provides R code for fitting Gaussian process models to data.
The code is created using the `R6` class structure, which is why `$` is used to access object methods.

A Gaussian process fits a model to a dataset, which gives a function that gives a prediction for the mean at any point along with a variance of this prediction.

Suppose we have the data below

```{r oldvignettedata}
x <- seq(0,1,l=10)
y <- abs(sin(2*pi*x))^.8
ggplot(aes(x,y), data=cbind(x,y)) +
  geom_point()
```

A linear model (LM) will fit a straight line through the data and clearly does not describe the underlying function producing the data.

```{r oldvignettedata_plot}
ggplot(aes(x,y), data=cbind(x,y)) +
    geom_point() +
    stat_smooth(method='lm')
```

A Gaussian process is a type of model that assumes that the distribution of points
follows a multivariate distribution.

In GauPro, we can fit a GP model with Gaussian correlation function using
the function `gpkm`.

```{r oldvignettedata_gpkm}
library(GauPro)
gp <- gpkm(x, y, kernel=k_Gaussian(D=1), parallel=FALSE)
```

Now we can plot the predictions given by the model.
Shown below, this model looks much better than a linear model.

```{r oldvignettedata_plot1D}
gp$plot1D()
```

A very useful property of GP's is that they give a predicted error.
The red lines give an approximate 95% confidence interval for the value at
each point (measure value, not the mean).
The width of the prediction interval is largest between points
and goes to zero near data points, which is what we would hope for.

GP models give distributions for the predictions.
Realizations from these distributions give an idea of
what the true function may look like.
Calling `$cool1Dplot()` on the 1-D gp object shows 20 realizations.
The realizations are most different away from the design points.

```{r oldvignettedata_cool1Dplot}
if (requireNamespace("MASS", quietly = TRUE)) {
  gp$cool1Dplot()
}
```


### Using kernels

The kernel, or covariance function, has a large effect on
the Gaussian process being estimated.
Many different kernels are available in the `gpkm()` function which
creates the GP object.

The example below shows what the Matern 5/2 kernel gives.

```{r oldvignettedata_maternplot}
kern <- k_Matern52(D=1)
gpk <- gpkm(matrix(x, ncol=1), y, kernel=kern, parallel=FALSE)
if (requireNamespace("MASS", quietly = TRUE)) {
  plot(gpk)
}
```

The exponential kernel is shown below. You can see that it
has a huge effect on the model fit.
The exponential kernel assumes the correlation between points
dies off very quickly, so there is much more uncertainty
and variation in the predictions and sample paths.


```{r oldvignettedata_exponentialplot}
kern.exp <- k_Exponential(D=1)
gpk.exp <- gpkm(matrix(x, ncol=1), y, kernel=kern.exp, parallel=FALSE)
if (requireNamespace("MASS", quietly = TRUE)) {
  plot(gpk.exp)
}
```


### Trends

Along with the kernel the trend can also be set.
The trend determines what the mean of a point is without
any information from the other points.
I call it a trend instead of mean because I refer to the
posterior mean as the mean, whereas the trend is the mean
of the normal distribution.
Currently the three options are to have a mean 0,
a constant mean (default and recommended), or  a linear model.

With the exponential kernel above we see some
regression to the mean. Between points the prediction
reverts towards the mean of `r gpk$trend$m`.
Also far away from any data the prediction will near this value.

Below when we use a mean of 0 we do not see
this same reversion.


```{r oldvignettedata_trendplot}
kern.exp <- k_Exponential(D=1)
trend.0 <- trend_0$new()
gpk.exp <- gpkm(matrix(x, ncol=1), y, kernel=kern.exp, trend=trend.0, parallel=FALSE)
if (requireNamespace("MASS", quietly = TRUE)) {
  plot(gpk.exp)
}
```
