human
Solve the following python programming problem given the following code:
# 1-D example
from ipywidgets import interactive, interact, widgets
import matplotlib.pyplot as plt
import numpy as np
import scipy.spatial as SP

# defining Squared Exponential Kernel and plot it

def k(length_scale):
    x = np.arange(0., 5., 0.1)
    plt.figure(figsize=(10, 7))
    plt.ylim([0, 1.05])
    plt.xlabel('$x$', fontsize=16)
    plt.ylabel('$k(x,0)$', fontsize=16)
    plt.plot(x, np.exp(-.5 * x**2/length_scale**2), 'b-')
    plt.show()

controls = {r'length_scale': widgets.FloatSlider(
    min=0.01, max=5.0, step=0.1, value=1., continuous_update=False, description=r'$\ell$')}

from ipywidgets import interactive
import matplotlib.pyplot as plt
import numpy as np

def GP(length_scale, Test, Training, sigma):
    np.random.seed(100)

    """ This is code for simple GP regression. It assumes a zero mean GP Prior """

    # This is the true unknown function we are trying to approximate
    def f(x): return np.sin(0.9*x.flatten())

    # Define the kernel
    def kernel(a, b):
        sqdist = SP.distance.cdist(a, b, 'sqeuclidean')
        return np.exp(-.5 * sqdist/(length_scale**2))

    N = Training    # number of training points.
    n = Test        # number of test points.
    s = sigma       # noise variance.

    # Sample some input points and noisy versions of the function evaluated at
    # these points.
    X = np.random.uniform(-5, 5, size=(N, 1))
    y = f(X) + s*np.random.randn(N)

    K = kernel(X, X)
    L = np.linalg.cholesky(K + s*np.eye(N))

    # points we're going to make predictions at.
    Xtest = np.linspace(-5, 5, n)[:, None]

    # compute the mean at our test points.
    Lk = np.linalg.solve(L, kernel(X, Xtest))
    mu = np.dot(Lk.T, np.linalg.solve(L, y))

    # compute the variance at our test points.
    K_ = kernel(Xtest, Xtest)
    s2 = np.diag(K_) - np.sum(Lk**2, axis=0)
    s  = np.sqrt(s2)

    # PLOTS:
    plt.figure(figsize=(9, 7))
    plt.clf()
    plt.plot(X, y, 'r+', ms=18, label="Training points")
    plt.plot(Xtest, f(Xtest), 'b-', label="Function")
   ...
Given the above code, generate some markdown instructions for it.
assistant
#Here is Python program to compute the mean at our test points.

Linear model
The model described above is a special case of linear model: we fix a basis set and obtain
$$f(x) \approx \sum_{k=1}^M c_k \phi_k(x).$$
For $x \in \mathbb{R}^d$ what basis set to choose?
Why tensor-product polynomials are bad for large $d$ ? 
What are the alternatives to tensor-product basis?
Good approach: sparse polynomial bases
Instead of taking all possible $\mathbf{x}^{\mathbf{\alpha}}$, we take only a subset, such as:

Total degree: $\vert \mathbf{\alpha} \vert \leq T$
Hyperbolic cross scheme

For very smooth functions, such approximations work really well (and simple to use!).
Experiment design
Given a linear model, 
$$f(x) \approx \sum_{k=1}^M c_k \phi_k(x).$$
How to find coefficients?
Sampling
Answer: do sampling, 
and solve linear least squares
$$f(x_i) \approx \sum_{k=1}^M c_k \phi_k(x_i).$$
How to sample?
Sampling methods
$$f(x_i) \approx \sum_{k=1}^M c_k \phi_k(x_i).$$

Non-adaptive schemes: Monte-Carlo, Quasi-Monte Carlo, Latin Hypercube Sampling (LHS)
Adaptive: optimize for points $x_1, \ldots, x_N$.

There are many criteria.
D-optimality
If we select $N = M$ and select points such that 
$$\vert \det M \vert \rightarrow \max,$$
where 
$$M_{kj} = \phi_k(x_i)$$ is the design matrix.
Why it is good?
Linear algebra: maximum volume
Let $A \in \mathbb{R}^{n \times r}$, $n \gg r$.
Let $\widehat{A}$ be the submatrix of maximum volume.
Then, all coefficients in $A \widehat{A}^{-1}$ are less than $1$ in modulus.
As a simple consequence, 
we have
$$E_{D} \leq (r+1) E_{best},$$
where $E_D$ is the approximation from optimal design, and $E_{best}$ is the best possibble approximation error in the Chebyshev norm.
Problem setting

We have an unknown multivariate function $f(\mathbf{x})$ that we would like to approximate on some specified domain.<br> 
We have a dataset $\mathcal{D}$ of $n$ function observations, $\mathcal{D} = {(\mathbf{x}_i,y_i),i = 1,\ldots,n}$.

Given $\mathcal{D}$ we wish to make predictions for new inputs $\mathbf{x}_*$ within the domain.

To solve this problem we must make...