advanced-algorithmic-trading

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# Use a linear model (y ~ beta_0 + beta_1*x + epsilon) to
# generate a column ’y’ of responses based on ’x’
eps_mean = 0.0
df["y"] = beta_0 + beta_1*df["x"] + np.random.RandomState(42).normal(
eps_mean, eps_sigma_sq, N
)
return df
if __name__ == "__main__":
# These are our "true" parameters
beta_0 = 1.0 # Intercept
beta_1 = 2.0 # Slope
# Simulate 100 data points, with a variance of 0.5
N = 100
eps_sigma_sq = 0.5
# Simulate the "linear" data using the above parameters
df = simulate_linear_data(N, beta_0, beta_1, eps_sigma_sq)
# Plot the data, and a frequentist linear regression fit
# using the seaborn package
sns.lmplot(x="x", y="y", data=df, size=10)
plt.xlim(0.0, 1.0)
The output is given in Figure 5.1:
We’ve simulated 100 datapoints, with an intercept β 0 = 1 and a slope of β 1 = 2. The epsilon
values are normally distributed with a mean of zero and variance σ 2 = 1 2
. The data has been
plotted using the sns.lmplot method. In addition, the method uses a frequentist MLE approach
to fit a linear regression line to the data.
Now that we have carried out the simulation we want to fit a Bayesian linear regression to
the data. This is where the glm module comes in. It uses a model specification syntax that is
similar to how the R statistical language specifies models. To achieve this we make implicit use
of the Patsy library.
In the following snippet we are going to import PyMC3, utilise the with context manager, as
described in the previous chapter on MCMC and then specify the model using the glm module.
We are then going to find the maximum a posteriori (MAP) estimate for the MCMC sampler
to begin sampling from. Finally, we are going to use the No-U-Turn Sampler[53] to carry out the
actual inference and then plot the trace of the model, discarding the first 500 samples as "burn
in":
def glm_mcmc_inference(df, iterations=5000):
"""