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(test-functions:cheng2d)=
# Two-dimensional Function from Cheng and Sandu (2010)

```{code-cell} ipython3
import numpy as np
import matplotlib.pyplot as plt
import uqtestfuns as uqtf
```

The two-dimensional test function from Cheng and Sandu (2002) (or
`Cheng2D` for short) is used in a metamodeling exercise via polynomial
chaos expansion in {cite}`Cheng2010`.

```{code-cell} ipython3
:tags: [remove-input]

from mpl_toolkits.axes_grid1 import make_axes_locatable

my_fun = uqtf.Cheng2D()

# --- Create 2D data
xx_1d = np.linspace(0.0, 1.0, 1000)[:, np.newaxis]
mesh_2d = np.meshgrid(xx_1d, xx_1d)
xx_2d = np.array(mesh_2d).T.reshape(-1, 2)
yy_2d = my_fun(xx_2d)

# --- Create two-dimensional plots
fig = plt.figure(figsize=(10, 5))

# Surface
axs_1 = plt.subplot(121, projection='3d')
axs_1.plot_surface(
    mesh_2d[0],
    mesh_2d[1],
    yy_2d.reshape(1000,1000).T,
    linewidth=0,
    cmap="plasma",
    antialiased=False,
    alpha=0.5
)
axs_1.set_xlabel("$x_1$", fontsize=14)
axs_1.set_ylabel("$x_2$", fontsize=14)
axs_1.set_zlabel("$\mathcal{M}(x_1, x_2)$", fontsize=14)
axs_1.set_title("Surface plot of Cheng2D", fontsize=14)

# Contour
axs_2 = plt.subplot(122)
cf = axs_2.contourf(
    mesh_2d[0], mesh_2d[1], yy_2d.reshape(1000, 1000).T, cmap="plasma", levels=10,
)
axs_2.set_xlabel("$x_1$", fontsize=14)
axs_2.set_ylabel("$x_2$", fontsize=14)
axs_2.set_title("Contour plot of Cheng2D", fontsize=14)
divider = make_axes_locatable(axs_2)
cax = divider.append_axes('right', size='5%', pad=0.05)
fig.colorbar(cf, cax=cax, orientation='vertical')
axs_2.axis('scaled')

fig.tight_layout(pad=4.0)
plt.gcf().set_dpi(75);
```


## Test function instance

To create a default instance of the test function:

```{code-cell} ipython3
my_testfun = uqtf.Cheng2D()
```

Check if it has been correctly instantiated:

```{code-cell} ipython3
print(my_testfun)
```

## Description

The test function is defined as follows[^location]:

$$
\mathcal{M}(\boldsymbol{x}) = \cos{(x_1 + x_2)} \exp{(x_1 x_2)},
$$
where $\boldsymbol{x} = \{ x_1, x_2 \}$
is the two-dimensional vector of input variables further defined below.

## Probabilistic input

The input consists of two uniformly distributed random variables as shown
below.

```{code-cell} ipython3
:tags: [hide-input]

print(my_testfun.prob_input)
```

## Reference results

This section provides several reference results of typical UQ analyses involving
the test function.

### Sample histogram

Shown below is the histogram of the output based on $100'000$ random points:

```{code-cell} ipython3
:tags: [hide-input]

xx_test = my_testfun.prob_input.get_sample(100000)
yy_test = my_testfun(xx_test)

plt.hist(yy_test, bins="auto", color="#8da0cb");
plt.grid();
plt.ylabel("Counts [-]");
plt.xlabel("$\mathcal{M}(\mathbf{X})$");
plt.gcf().set_dpi(150);
```

## References

```{bibliography}
:style: unsrtalpha
:filter: docname in docnames
```

[^location]: See Eq. (6), Section 4, p. 4 {cite}`Cheng2010`.