Marcenko-Pastur Theorem - Generate Probability Density Function

## mpPDF.py

import numpy as np, pandas as pd

import matplotlib.pyplot as plt
# -Generate Probability Density Function as per Marcenko Pastur Theorem, for random process with zero mean, and specified variance
def mpPDF(var,q,pts):
    # Var - Variance of the underlying process. Different variance will generate different PDF
    #q = T/N T is #observations, N is #Independent Variables
    #pts - How granularly you want to plot eigen values from Min Eigen Value to Max Eigen Value
    eMax = var*(1+((1./q)**.5)) # Max Eigen Value
    eMin = var*(1-((1./q)**.5)) # Min Eigen Value
    eVal = np.linspace(eMin,eMax,pts) # Divide space b/w Min & Max Eigen into equally spaced intervals
    pdf = (q/(2*np.pi*eVal*var))*(((eMax-eVal)*(eVal-eMin))**.5) # PDF as per Theorem
    pdf = pd.Series(pdf,index=eVal) # Use Pandas to store result as series, index with Eigen value
    return pdf # Return the Probability Density Function

Comments

Post a Comment

Popular posts from this blog

1.2 Structured Data: Information Driven Bars

Image
This post will describe how to process & store High Frequency financial data to support your Machine Learning Algorithm, based on how much information is contained in the data The post is directly based on content from the book "Advances in Financial Machine Learning" from Marcos Lopez de Prado Physical meaning: This approach is inspired by Claude Shannon's Information Theory, that is the basis of most compression algorithms. Shannon argued that only departure from the norm is new information. A famous example, is that the sun has risen in the morning. Yes, the event has happened, but because the probability of this event was 1.0, this event does not represent information. In information driven bars, we attempt to use the same insight, to compress the financial data. Here, we know that financial data is inherently noisy, and put guardrails around what level of variance we expect for a particular behavior. It is only when the variance exceeds these guardrails, that we ...
Read more

2.2 Labeling: Triple barrier method

Image
 This post will describe how to label a financial dataset to support supervised learning The post is directly based on content from the book "Advances in Financial Machine Learning" from Marcos Lopez de Prado Physical meaning: Algorithm description: Python code: # Import libraries import numpy as np import pandas as pd import matplotlib.pyplot as plt import math # Import data df = pd.read_csv( r'C:\Users\josde\OneDrive\Denny\Deep-learning\Data-sets\Trade-data\ES_Trades.csv' ) df = df.iloc[: , 0 : 5 ] df[ 'Dollar' ] = df[ 'Price' ] * df[ 'Volume' ] print (df.columns) # Generate thresholds d = pd.DataFrame(pd.pivot_table(df , values = 'Dollar' , aggfunc = 'sum' , index = 'Date' )) DOLLAR_THRESHOLD = ( 1 / 50 ) * np.average(d[ 'Dollar' ]) # Generate bars def bar_gen (df , DOLLAR_THRESHOLD): collector , dollarbar_tmp = [] , [] dollar_cusum = 0 for i , (price , dollar) in enumerate ( zip (df[ 'Price...
Read more

Denoise a Covariance Matrix using Constant Residual Eigen Value Method

Image
 This post will describe how to denoise a Covariance matrix using Constant Residual Eigen Value Method Practical use: The Eigen Values are naturally stored in decreasing order. Once we find the range of Eigen Values that correspond to signal (using fit to Marcenko Pastur Distribution), we can then reset all other eigen values below this threshold to a constant value. This basically vaccums all the eigen values containing random noise into a single Eigen value. These transformed Eigen values are used with the original Eigen Vectors, to recreate a denoised covariance matrix, that will be used for further processing. Compute using Python: # File: Denoise-Constant-Residual-Eigen.py import numpy as np,pandas as pd import matplotlib.pyplot as plt #--------------------------------------------------- def mpPDF(var,q,pts): # Marcenko-Pastur pdf # q=T/N eMin,eMax=var*( 1 -( 1. /q)** .5 )** 2 ,var*( 1 +( 1. /q)** .5 )** 2 eVal=np.linspace(eMin,eMax,pts) pdf=q/( 2 *np.pi*var*eVal)*...
Read more