`dgpsi`

provides a flexible toolbox for Gaussian process
(GP), deep Gaussian process (DGP) and linked (D)GP emulations. In this
guide, we show how to use the package to emulate a step function with a
three-layered DGP structure. There are other examples showing the
functionality of the package in `Articles`

on the package website, and how to customize a DGP structure. A
comprehensive reference of available functions is documented in `Reference`

section of the package website.

## Set up the step function

`dgpsi`

provides a function `init_py()`

that
helps us set up, initialize, re-install, or uninstall the underlying
Python environment. We could run `init_py()`

every time after
`dgpsi`

is loaded to manually initiate the Python
environment. Alternatively, we could activate the Python environment by
simply executing a function from `dgpsi`

. For example, the
Python environment will be automatically loaded after we run
`set_seed()`

from the package to specify a seed for the
reproducibility:

`set_seed(9999)`

We then define the step function:

and generate ten training data points:

## Training

We now build and train a DGP emulator with three layers:

`m <- dgp(X, Y, depth = 3)`

```
## Auto-generating a 3-layered DGP structure ... done
## Initializing the DGP emulator ... done
## Training the DGP emulator:
## Iteration 500: Layer 3: 100%|██████████| 500/500 [00:03<00:00, 148.10it/s]
## Imputing ... done
```

The progress bar displayed shows how long it takes to finish the
training. We are able to switch off the progress bar and the trace
information by setting `verb = FALSE`

. Note that if we want
to train the DGP emulator `m`

for additional iterations, we
can simply do `m <- continue(m)`

instead of rebuilding the
DGP emulator.

The trained DGP emulator can be viewed using the
`summary()`

function:

`summary(m)`

```
## +-----------+----------+------------------+-----------------+---------------+-------------------+------------+-------------------+
## | Layer No. | Node No. | Type | Length-scale(s) | Variance | Nugget | Input Dims | Global Connection |
## +-----------+----------+------------------+-----------------+---------------+-------------------+------------+-------------------+
## | Layer 1 | Node 1 | GP (Squared-Exp) | [0.385] | 1.000 (fixed) | 1.000e-06 (fixed) | [1] | No |
## | Layer 2 | Node 1 | GP (Squared-Exp) | [1.183] | 1.000 (fixed) | 1.000e-06 (fixed) | [1] | [1] |
## | Layer 3 | Node 1 | GP (Squared-Exp) | [2.745] | 0.480 | 1.000e-06 (fixed) | [1] | [1] |
## +-----------+----------+------------------+-----------------+---------------+-------------------+------------+-------------------+
## 1. 'Input Dims' presents the indices of GP nodes in the feeding layer whose outputs feed into the GP node referred by 'Layer No.' and 'Node No.'.
## 2. 'Global Connection' indicates the dimensions (i.e., column indices) of the global input data that are used as additional input dimensions to the GP node referred by 'Layer No.' and 'Node No.'.
```

The summary table gives key information about the trained DGP
emulator. Note that in the auto-generated emulator we have nugget terms
fixed to `1e-6`

for all GP nodes because we are emulating a
deterministic step function (i.e., we would like our emulator to
interpolate training data points). The prior scales (i.e., variances)
for GP nodes in the first and second layers are fixed to
`1.0`

while that for the GP node in the final layer is
estimated due to its attachment to the output. For further information
on how to change the default settings to construct and train a DGP
emulator, see `?dgp`

.

At this point, you could use `write()`

to save the
emulator `m`

to a local file and then load it using
`read()`

when you would like to make predictions from the
emulator, e.g., on another laptop that also has the package
installed.

## Validation

After we have the emulator, we can validate it by drawing the validation plots. There are two types of validation plots provided by the package. The first one is the Leave-One-Out (LOO) cross validation plot:

`plot(m)`

```
## Initializing ... done
## Post-processing LOO results ... done
## Plotting ... done
```

The second validation plot is the Out-Of-Sample (OOS) validation plot that requires an out-of-sample testing data set. Here we generate an OOS data set that contains 10 testing data points

for the validation:

`plot(m,oos_x,oos_y)`

```
## Initializing ... done
## Post-processing OOS results ... done
## Plotting ... done
```

## Prediction

Once the validation is done, we can make predictions from the DGP emulator. We generate 200 data points from the step function over \([0,1]\)

for predictions:

`m <- predict(m, x = test_x)`

The `predict()`

function returns an updated DGP emulator
`m`

that contains a slot named `results`

that
gives the posterior predictive means and variances at testing positions.
We can extract these information and plot the emulation results to check
the predictive performance of our constructed DGP emulator:

```
mu <- m$results$mean # extract the predictive means
sd <- sqrt(m$results$var) # extract the predictive variance and compute the predictive standard deviations
# compute predictive bounds which are two predictive standard deviations above and below the predictive means
up <- mu + 2*sd
lo <- mu - 2*sd
plot(test_x, mu, type = 'l', lty = 2, lwd = 1.5, col = 'black', xlab = 'x', cex.axis = 1.3, cex.lab = 1.3, ylab = 'y', ylim = c(-1.5,1.5)) # predictive means
polygon(c(test_x, rev(test_x)), c(up,rev(lo)), col = 'grey80', border = F) # credible interval
lines(test_x, test_y, type = 'l', col = "#D55E00", lwd = 2) # Underlying truth
lines(test_x, mu, type = 'l', lty = 2, lwd = 1.5, col = 'black')
lines(X, Y, type = 'p', pch = 16, cex = 1, col = "#0072B2") # Training data points
```