`dgpsi`

provides a flexible toolbox for Gaussian process
(GP), deep Gaussian process (DGP) and linked (D)GP emulations. In this
guide, we shows 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`

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

.

`dgpsi`

in R provides an initialization function
`init_py()`

that helps us set up and initialize the required
Python environment automatically. We just need to run
`init_py()`

every time after the package is loaded to
initiate the underlying Python environment.

```
library(dgpsi)
init_py()
```

```
## Connecting to Python ... done
## Importing required Python modules ... done
## The Python environment for 'dgpsi' is successfully loaded.
```

We first define the step function:

```
<- function(x) {
f if (x < 0.5) return(-1)
if (x >= 0.5) return(1)
}
```

Then we generate ten training data points:

```
<- seq(0, 1, length = 10)
X <- sapply(X, f) Y
```

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

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

```
## Auto-generating a 3-layered DGP structure ... done
## Initializing the DGP emulator ... done
## Training the DGP emulator:
## Iteration 500: Layer 3: 100%|██████████| 500/500 [00:04<00:00, 123.37it/s]
```

The progress bar displayed shows how long it takes to finish the
training. We are able to change some default settings. E.g., we can set
`verb = 2`

so a summary table of the initialized DGP emulator
will be printed out before we proceed to training. This is particularly
useful if a customized structure is provided to `struc`

and
we want to check it before the training is implemented. 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) | [0.285] | 1.000 (fixed) | 1.000e-06 (fixed) | [1] | No |
## | Layer 2 | Node 1 | GP (Squared-Exp) | [1.773] | 1.000 (fixed) | 1.000e-06 (fixed) | [1] | [1] |
## | Layer 3 | Node 1 | GP (Squared-Exp) | [2.339] | 0.475 | 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
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()`

next time when you would like to make predictions
from the emulator.

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

```
<- sample(seq(0, 1, length = 200),10)
oos_x <- sapply(oos_x, f) oos_y
```

for the validation:

`plot(m,oos_x,oos_y)`

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

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

```
<- seq(0, 1, length = 200)
test_x <- sapply(test_x, f) test_y
```

for predictions:

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

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:

```
<- m$results$mean # extract the predictive means
mu <- sqrt(m$results$var) # extract the predictive variance and compute the predictive standard deviations
sd # compute predictive bounds which are two predictive standard deviations above and below the predictive means
<- mu + 2*sd
up <- mu - 2*sd
lo
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) # predictive bounds
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
```