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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 <a href="https
Step1:  Now let's implement the  simulate some data from this model
Step2:  One thing to note here is that we've used named parameters in a dictionary, instead of an array of parameters as in some of the other examples.
This would be awkward (but not impossible) to fit using scipy, so instead we'll use optax for optimization
Step3:  Using our optimized model, over-plot the conditional predictions
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Python Code:
try:
    import tinygp
except ImportError:
    %pip install -q tinygp
try:
    import optax
except ImportError:
    %pip install -q optax
import tinygp
import jax
import jax.numpy as jnp
class SpectralMixture(tinygp.kernels.Kernel):
    def __init__(self, weight, scale, freq):
        self.weight = jnp.atleast_1d(weight)
        self.scale = jnp.atleast_1d(scale)
        self.freq = jnp.atleast_1d(freq)
    def evaluate(self, X1, X2):
        tau = jnp.atleast_1d(jnp.abs(X1 - X2))[..., None]
        return jnp.sum(
            self.weight
            * jnp.prod(
                jnp.exp(-2 * jnp.pi**2 * tau**2 / self.scale**2) * jnp.cos(2 * jnp.pi * self.freq * tau),
                axis=-1,
            )
        )
Explanation: <a href="https://colab.research.google.com/github/probml/probml-notebooks/blob/main/notebooks/gp_spectral_mixture.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a>
Spectral mixture kernel in 1d for GP
https://tinygp.readthedocs.io/en/latest/tutorials/kernels.html#example-spectral-mixture-kernel
In this section, we will implement the "spectral mixture kernel" proposed by Gordon Wilson & Adams (2013).
End of explanation
import numpy as np
import matplotlib.pyplot as plt
def build_gp(theta):
    kernel = SpectralMixture(
        jnp.exp(theta["log_weight"]),
        jnp.exp(theta["log_scale"]),
        jnp.exp(theta["log_freq"]),
    )
    return tinygp.GaussianProcess(kernel, t, diag=jnp.exp(theta["log_diag"]), mean=theta["mean"])
params = {
    "log_weight": np.log([1.0, 1.0]),
    "log_scale": np.log([10.0, 20.0]),
    "log_freq": np.log([1.0, 1.0 / 3.0]),
    "log_diag": np.log(0.1),
    "mean": 0.0,
}
random = np.random.default_rng(546)
t = np.sort(random.uniform(0, 10, 50))
true_gp = build_gp(params)
y = true_gp.sample(jax.random.PRNGKey(123))
plt.plot(t, y, ".k")
plt.ylim(-4.5, 4.5)
plt.title("simulated data")
plt.xlabel("x")
_ = plt.ylabel("y")
Explanation: Now let's implement the  simulate some data from this model:
End of explanation
import optax
@jax.jit
@jax.value_and_grad
def loss(theta):
    return -build_gp(theta).condition(y)
opt = optax.sgd(learning_rate=3e-4)
opt_state = opt.init(params)
for i in range(1000):
    loss_val, grads = loss(params)
    updates, opt_state = opt.update(grads, opt_state)
    params = optax.apply_updates(params, updates)
opt_gp = build_gp(params)
tau = np.linspace(0, 5, 500)
plt.plot(tau, true_gp.kernel(tau[:1], tau)[0], "--k", label="true kernel")
plt.plot(tau, opt_gp.kernel(tau[:1], tau)[0], label="inferred kernel")
plt.legend()
plt.xlabel(r"$\tau$")
plt.ylabel(r"$k(\tau)$")
_ = plt.xlim(tau.min(), tau.max())
plt.savefig("gp-spectral-mixture-learned-kernel.pdf")
Explanation: One thing to note here is that we've used named parameters in a dictionary, instead of an array of parameters as in some of the other examples.
This would be awkward (but not impossible) to fit using scipy, so instead we'll use optax for optimization:
End of explanation
x = np.linspace(-2, 12, 500)
plt.plot(t, y, ".k", label="data")
mu, var = opt_gp.predict(y, x, return_var=True)
plt.fill_between(
    x,
    mu + np.sqrt(var),
    mu - np.sqrt(var),
    color="C0",
    alpha=0.5,
    label="prediction",
)
plt.plot(x, mu, color="C0", lw=2)
plt.xlim(x.min(), x.max())
plt.ylim(-4.5, 4.5)
plt.legend(loc=2)
plt.xlabel("x")
_ = plt.ylabel("y")
plt.savefig("gp-spectral-mixture-pred.pdf")
Explanation: Using our optimized model, over-plot the conditional predictions:
End of explanation
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