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(test-functions:mclain-s4)=
# McLain S4 Function

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

The McLain S4 function is a two-dimensional scalar-valued function.
The function was introduced in {cite}`McLain1974` as a test function for
procedures to construct contours from a given set of points.

```{note}
The McLain's test functions are a set of five two-dimensional functions 
that mathematically defines surfaces. The functions are:

- {ref}`S1 <test-functions:mclain-s1>`: A part of a sphere
- {ref}`S2 <test-functions:mclain-s2>`: A steep hill rising from a plain
- {ref}`S3 <test-functions:mclain-s3>`: A less steep hill
- {ref}`S4 <test-functions:mclain-s4>`: A long narrow hill (_this function_)
- {ref}`S5 <test-functions:mclain-s5>`: A plateau and plain separated by a steep cliff
```

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

from mpl_toolkits.axes_grid1 import make_axes_locatable

my_fun = uqtf.McLainS4()

# --- Create 2D data
xx_1d = np.linspace(1.0, 10.0, 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 McLain S4", 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"
)
axs_2.set_xlabel("$x_1$", fontsize=14)
axs_2.set_ylabel("$x_2$", fontsize=14)
axs_2.set_title("Contour plot of McLain S4", 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);
```

As shown in the plots above, the resulting surface consists of a long narrow
hill running diagonally from $(0.0, 10.0)$ to $(10.0, 0.0)$.
The maximum height is $1.0$ at $(5.5, 5.5)$.

## Test function instance

To create a default instance of the McLain S4 function:

```{code-cell} ipython3
my_testfun = uqtf.McLainS4()
```

Check if it has been correctly instantiated:

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

## Description

The McLain S4 function is defined as follows:

$$
\mathcal{M}(\boldsymbol{x}) = \exp{\left[ -1 \left( (x_1 + x_2 - 11)^2 + \frac{(x_1 - x_2)^2}{10} \right) \right]}
$$
where $\boldsymbol{x} = \{ x_1, x_2 \}$
is the two-dimensional vector of input variables further defined below.

## Probabilistic input

Based on {cite}`McLain1974`, the probabilistic input model
for the function consists of two independent 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, 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
```