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subpsd.py
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subpsd.py
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#!/usr/bin/env python
# coding: utf-8
# # Computing Optimal Submatrix of an Invalid Correlation Matrix
#
#
# ### Goal: Given invalid correlation matrix C,find the optimal submatrix that has the largest similarity to the corresponding principal submatrix.
#
# ### Key Mathematical Operations:
#
# * $C \in R^{n \times n}$ : fake correlation matrix
# * [$J(C)$](https://www.researchgate.net/publication/280050954_The_most_simple_methodology_to_create_a_valid_correlation_matrix_for_risk_management_and_option_pricing_purposes): function computing the optimal valid correlation matrix representing $C$
# * $\hat{C} = J(C),\in R^{n×n}$
# * $similarity ∶= 1 - \frac{2}{n(n-1)} \times \sum_{i<j}|\frac{C_{ij}-\hat{C}_{ij}}{C_{ij}}|$
# * $change(i) := \sum_{j}|C_{ij}-\hat{C}_{ij}|$
# * $i^* = max_{_{i=1,2,3,...,n}}change(i)$
# * $j^* = i^*$
# * $C_{sub\setminus i^*,j^*} := \{ C_{ij} \}_{i\neq i^*,j\neq j^*},\in R^{ n-1 \times n-1}$
# * $\hat{C}_{sub\setminus i^*,j^*} := J(C_{sub\setminus i^*,j^*}), \in R^{ n-1 \times n-1}$
#
# ### Algorithm
# $C_{optimal}=C$ <br />
# $\hat{C}_{optimal}=\hat{C}$ <br />
# $similarity_{optimal}=similarity$ <br />
# $for \quad m = n-1,n-2,...,5,4,3:$ <br />
# $\quad \quad C=C_{sub\setminus i^*,j^*}$ <br />
# $\quad \quad \hat{C}=\hat{C}_{sub\setminus i^*,j^*}$ <br />
# $\quad \quad if \quad similarity > similarity_{optimal}:$ <br />
# $\quad \quad \quad \quad C_{optimal}=C$ <br />
# $\quad \quad \quad \quad \hat{C}_{optimal}=\hat{C}$ <br />
# $\quad \quad \quad \quad similarity_{optimal}=similarity$ <br />
# $\quad \quad \quad \quad if \quad similarity_{optimal} = 1:$ <br />
# $\quad \quad \quad \quad \quad \quad stop$
# ## Python Scripting
# In[1]:
import numpy as np
import numpy.linalg as LA
from statsmodels.stats.moment_helpers import cov2corr
# ## $J(C): find\_valid\_corr$
# In[2]:
def find_valid_corr(C,search_size=1000,phi_size=100000,max_attempt=500,step_unit=1e-17,disp=False):
'''
Compute the optimal valid correlation matrix representing matrix C
Parameters:
-----------
C: numpy array, symmetric square matrix whose diagonal values equal to 1 with any other element from -1 to 1
search_size: int, number of valid correlation matrices for C in the search space
phi_size: int, number of standard normal draws
max_attempt: int, maximum attempts to construct a positive definite seed matrix by slightly changing eigenvalues
step_unit: float, at each attempt, increase non-positive eigenvalues by one step_unit
disp: boolean, if true print information on seed matrix, eigenvalues, and upper matrix after Cholesky decomposition
Return:
-------
optimal_C_hat: numpy array, optimal valid correlation matrix representing C
'''
if C.shape[0] != C.shape[1]:
raise ValueError('Matrix is not square matrix')
if np.sqrt(np.sum((C.T - C)**2)) > (C.shape[0]**2)*1e-5:
raise ValueError('Matrix is not symmetric')
val_,vec = LA.eigh(C)
if (val_>=-1e-08).all(): # positive is defined as >= -1e-8 due to numerical precision
print('C is already a PSD matrix')
return C
chol_done = False
attempt = 0
while (not chol_done) and attempt <= max_attempt:
try:
#print('{}th attempt'.format(attempt))
val = val_.copy()
val[val<0] = attempt*step_unit # set negative eigenvalues into 0
# 0 here represented by 1e-16, this is to make psd matrix a little more positive
# to allow Cholesky Decomposition to work
B_star = vec*(np.sqrt(val)) # take square root for every eigen values
B = B_star/np.linalg.norm(B_star,axis=1,keepdims=True) # normalize row vectors
seed_C = np.matmul(B,B.T)
U = LA.cholesky(seed_C) # return upper matrix
chol_done = True
except:
if attempt == max_attempt:
raise RuntimeError('Failed to find Seed Matrix')
attempt += 1
if disp:
print('seed correlation matrix:')
print(seed_C)
print('eigen values of seed matrix:')
print(LA.eigh(seed_C)[0])
print('chol_done:',chol_done,'at {}th attempt'.format(attempt))
print('Upper Matrix:')
print(U)
optimal_C_hat = None
optimal_error = np.inf
for _ in range(search_size):
phi = np.random.randn(phi_size,C.shape[0])
R = phi.dot(U)
C_hat = cov2corr(np.cov(R,rowvar=False))
error = np.sum((C_hat - C)**2)
if error < optimal_error and (LA.eigh(C_hat)[0]>=-1e-8).all(): # positive is defined as >= -1e-5 due to computing reason
optimal_C_hat = C_hat
optimal_error = error
else:
continue
print('element-wise error:',optimal_error)
print('distance error:',error_measure(C,optimal_C_hat,'d'))
print('similarity:',error_measure(C,optimal_C_hat,'s'))
return optimal_C_hat
#return optimal_error
# ## $ Similarity: error\_measure$
# In[3]:
def error_measure(C,C_hat,method='d'):
'''
Measure the difference/similarity between Matrix C and its simulated counterparty C_hat
Parameters:
-----------
C: numpy array, symmetric square matrix whose diagonal values equal to 1 with any other element from -1 to 1
C_hat: numpy array, symmetric square matrix whose diagonal values equal to 1 with any other element from -1 to 1
method: str, distance/d to compute the square root of sum of square of element-wise difference
similarity/s to compute the 1 - sum of element-wise change of the upper matrix devided by number of elements
Return:
-------
float, distance or similarity
'''
if C.shape != C_hat.shape:
raise ValueError('Simulated Matrix does not have the same dimension the original Matrix')
if C.shape[0] != C.shape[1]:
raise ValueError('Matrices are not square matrix')
if np.sqrt(np.sum((C.T - C)**2)) > (C.shape[0]**2)*1e-6:
raise ValueError('Matrix C is not symmetric')
if np.sqrt(np.sum((C_hat.T - C_hat)**2)) > (C_hat.shape[0]**2)*1e-6:
raise ValueError('Matrix C_hat is not symmetric')
if method in ('distance','d'):
return np.sqrt(np.sum((C - C_hat)**2))
elif method in ('similarity','s'):
n = C.shape[0]
return 1-np.sum(np.abs(np.triu((C - C_hat),k=1)/C))/(n*(n-1)/2)
else:
raise NameError('method can only be distance(d)/similarity(s)')
# ## $ Algorithm: find\_optimal\_subcorr$
# In[4]:
def find_optimal_subcorr(C,stopping_dim=3):
'''
Compute the optimal valid submatrix that has the largest similarity to the corresponding principal submatrix of a given matrix
Parameters:
-----------
C: numpy array, symmetric square matrix whose diagonal values equal to 1 with any other element from -1 to 1
stopping_dim: int, the lowest dimension of interest to search on
Return:
-------
opt_C: numpy array, the principal submatrix
opt_C_hat: numpy array, optimal valid submatrix that has the largest similarity to the corresponding principal submatrix
'''
cur_C = C.copy()
print('Dimension: '+str(C.shape[0]))
cur_C_hat = find_valid_corr(cur_C)
opt_C = cur_C
opt_C_hat = cur_C_hat
opt_sim = error_measure(cur_C,cur_C_hat,'s')
for m in range(C.shape[0]-1,stopping_dim-1,-1):
print('Dimension: '+str(m))
change_vec = np.sum(np.abs(cur_C - cur_C_hat),axis=1)
ind_max = change_vec.argmax()
remain_ind = [i for i in range(cur_C.shape[0]) if i != ind_max]
cur_C = cur_C[np.ix_(remain_ind,remain_ind)]
cur_C_hat = find_valid_corr(cur_C)
sim = error_measure(cur_C,cur_C_hat,'s')
if sim > opt_sim:
opt_C = cur_C
opt_C_hat = cur_C_hat
opt_sim = sim
if opt_sim > 1-1e-10:
break
print('optimal similarity {} at dimension {}'.format(opt_sim,opt_C.shape[0]))
return opt_C, opt_C_hat
# In[ ]: