9.06 Additive noise map

We reproduce here the figure 9.6 of the book. Utilitary functions can be found next to this file. Here, we only define codpy-related functions.

Necessary Imports

import os
import sys

import matplotlib.pyplot as plt
import numpy as np

from codpy.kernel import Sampler

try:
    CURRENT_DIR = os.path.dirname(os.path.abspath(__file__))
except NameError:
    CURRENT_DIR = os.getcwd()
data_path = os.path.join(CURRENT_DIR, "data")
PARENT_DIR = os.path.abspath(os.path.join(CURRENT_DIR, ".."))
sys.path.insert(0, PARENT_DIR)

import utils.ch9.mapping as maps
from utils.ch9.data_utils import stats_df
from utils.ch9.market_data import retrieve_market_data
from utils.ch9.path_generation import generate_paths
from utils.ch9.plot_utils import display_historical_vs_generated_distribution

Parameter definition

def get_cdpres_param():
    return {
        "rescale_kernel": {"max": 2000, "seed": None},
        "rescale": True,
        "grid_projection": True,
        "reproductibility": False,
        "date_format": "%d/%m/%Y",
        "begin_date": "01/06/2020",
        "end_date": "01/06/2022",
        "today_date": "01/06/2022",
        "symbols": ["AAPL", "GOOGL", "AMZN"],
    }

Get the market data

params = retrieve_market_data()

Defining the map

The additive noise map is defined as:

\[\eta_Y(\varepsilon) = \varepsilon - f(Y), \qquad \varepsilon = \eta_Y^{-1}(\eta) = \eta + f(Y),\]

Where, \(\eta_Y\) is a white noise, that is an independent random variable, and \(f : \mathbb{R}^{D_Y} \to \mathbb{R}^{D_{\varepsilon}}\) is a smooth function modeling the dependence of \(\varepsilon\) on \(Y\). If unknown, \(f\) can be estimated from historical data using the denoising algorithm

params["map"] = maps.composition_map(
    [maps.additive_noise_map(), maps.log_map, maps.remove_time()]
)
params = maps.apply_map(params)

We define our sampler on the mapped data using codpy’s Sampler

You can define your own latent generator function, here we use a simple uniform distribution. But if not provided, a default one will be used by the Sampler class.

mapped_data = params["transform_h"].values
generator = lambda n: np.random.uniform(size=(n, mapped_data.shape[1]))
sampler = Sampler(mapped_data, latent_generator=generator)
params["sampler"] = sampler

We plot the original distribution vs the generated one

params = display_historical_vs_generated_distribution(params)
params["graphic"](params)
plt.show()

Reproductibility test

We regenerate the same path by generating from the latent representation We make sure we get the original data back.

params["reproductibility"] = True
params = generate_paths(params)
params["graphic"](params)
plt.show()

We now generate a new set of 10 paths

params["reproductibility"] = False
params["Nz"] = 100
params = generate_paths(params)
params["graphic"](params)
plt.show()

stats = stats_df(params["transform_h"], params["transform_g"]).T
print(stats)

                         0                 1                 2
Mean      -2.4e-06(0.0011)  -4.8e-06(0.0009)  1.3e-06(0.00052)
Variance       0.19(0.039)      -0.36(-0.19)       -0.1(-0.16)
Skewness    0.0003(0.0003)  0.00034(0.00033)  0.00025(0.00023)
Kurtosis         2.1(0.62)         3.4(0.72)        1.3(0.058)
KS test         0.11(0.05)        0.36(0.05)        0.58(0.05)