Mean-Variance Optimization with risky asset and cash¶

Purpose:¶

  • Reinforcement learning is applied to the problem of optimal allocation
  • The least squares policy iteration (LSPI) algorithm is applied to a Monte Carlo simulation of a stock's price movements, constructing a basis over the state-action space using B-spline basis functions at each time period.
  • The optimal Q-function is approximated with a dynamic programming approach, and this is shown to approach the exact solution
In [1]:
import numpy as np
import pandas as pd

from scipy.interpolate import interp1d
from scipy.stats import norm

from bspline import Bspline, splinelab

import matplotlib.pyplot as plt
%matplotlib inline

Parameters for MC simulation of stock prices¶

In [2]:
S0 = 100 # Initial stock price
mu = 0.03 # Drift
sigma = 0.20 # Volatility
r = 0.01 # Risk-free rate
M = 1 # Maturity

T = 30 # Number of time steps
delta_t = M / T # Time interval

N_MC = 2000 # Number of paths

gamma = np.exp(- r * delta_t) # Discount factor
lmbda = 10.0 # Risk aversion
alpha = 1.0 # Learning rate

Create dataset¶

In [3]:
# Stock price
S = pd.DataFrame([], index=range(1, N_MC+1), columns=range(T+1))
S.loc[:, 0] = S0

# Standard normal random numbers
RN = pd.DataFrame(np.random.randn(N_MC, T), index=range(1, N_MC+1), columns=range(1, T+1))

for t in range(1, T+1):
    S.loc[:, t] = S.loc[:, t-1] * np.exp((mu - 1/2 * sigma**2) * delta_t + sigma * np.sqrt(delta_t) * RN.loc[:, t])
    
    # Or, using Euler for alpha-stable distributions:
    #S.loc[:, t] = S.loc[:, t-1]*(1 + mu*delta_t + sigma *S.loc[:,t-1]**(alpha-1)*np.sqrt(delta_t) * RN.loc[:,t])

# Plot 10 paths
step_size = N_MC // 10
idx_plot = np.arange(step_size, N_MC, step_size)
plt.plot(S.T.iloc[:, idx_plot])
plt.xlabel('Time Steps')
plt.title('Stock Price Sample Paths')
plt.show()
No description has been provided for this image

$$C_t = -R_t= -r_t + \lambda (r_t-\mu)^2 = -(1-u_t)r_f - u_t \phi_t + \lambda u_t^2 Var(\phi_t | S_t)$$

In [4]:
def negative_reward(mu, var, a, rf, lmbda):
    C = -(1-a)*rf - a*mu + lmbda*(a**2)*var
    return C

Define spline basis functions¶

In [5]:
X = S # set the wealth (but not the total wealth) of the portfolio to be the stock price
p = 4 # order of spline (as-is; 3 = cubic, 4: B-spline)
ncolloc = 16
a_min = -1
a_max = 1
In [6]:
def get_basis_functions(X_min, X_max, a_min, a_max, ncolloc, p=3):
    tau_x = np.linspace(X_min, X_max, ncolloc)  # These are the sites to which we
    tau_a = np.linspace(a_min, a_max, ncolloc)  # would like to interpolate

    # k is a knot vector that adds endpoints repeats as appropriate for a spline of order p
    # To get meaninful results, one should have ncolloc >= p+1
    k_x = splinelab.aptknt(tau_x, p)
    k_a = splinelab.aptknt(tau_a, p)
                             
    # Spline basis of order p on knots k
    basis_x = Bspline(k_x, p)
    basis_a = Bspline(k_a, p)
    return basis_x, basis_a

Make data matrices with feature values¶

"Features" here are the values of basis functions at data points The outputs are 3D arrays of dimensions num_tSteps x num_MC x num_basis

In [7]:
num_t_steps = T + 1
num_basis =  ncolloc**2 

data_mat_t = np.zeros((num_t_steps, N_MC, num_basis ))
In [8]:
x = X.values[:, 0]
np.shape(x)
Out[8]:
(2000,)

Compute the optimal Q-function with the DP approach¶

Coefficients for expansions of the optimal Q-function $Q_t^\star\left(X_t,a_t^\star\right)$ are solved by

$$W_t=\mathbf S_t^{-1}\mathbf M_t$$

where $\mathbf S_t$ and $\mathbf M_t$ are matrix and vector respectively with elements given by

$$S_{nm}^{\left(t\right)}=\sum_{k=1}^{N_{MC}}{\Phi_n\left(X_t^k,a_t^k\right)\Phi_m\left(X_t^k,a_t^k\right)}\quad\quad M_n^{\left(t\right)}=\sum_{k=1}^{N_{MC}}{\Phi_n\left(X_t^k,a_t^k\right)\left(C\left(X_t^k,a_t^k,X_{t+1}^k\right)+\gamma\min_{a_{t+1}\in\mathcal{A}}Q_{t+1}^\star\left(X_{t+1}^k,a_{t+1}^k\right)\right)}$$

Define function function_S and function_M to compute the value of matrix $\mathbf S_t$ and vector $\mathbf M_t$.

In [9]:
def function_S_vec(t, data_mat_t):
    # Compute the matrix S_{nm} 
    X_mat = data_mat_t[t, :, :]
    num_basis_funcs = X_mat.shape[1]    
    S_mat = np.dot(X_mat.T, X_mat)
    return S_mat

def function_M_vec(t, Q, R, data_mat_t, gamma=0.1):
    X_mat = data_mat_t[t,:,:]
    tmp = R + gamma * np.min(Q[:, t+1])  # note that the second argument in Q is t+1
    M = np.dot(X_mat.T, tmp)
    return M

Least Squares Policy Iteration¶

Call function_S and function_M for $t=T-1,...,0$ together with basis function $\Phi_n\left(X_t,a_t\right)$ to compute optimal action Q-function $Q_t^\star\left(X_t,a_t^\star\right)=\sum_n^N{\omega_{nt}\Phi_n\left(X_t,a_t^\star\right)}$ backward recursively with terminal condition $Q_T^\star\left(X_T,a_T=0\right)=0$.

Initialize data structures¶

In [10]:
mu = np.zeros(T)
var = np.zeros(T)

# optimal action
a_opt = np.zeros((N_MC, T))
a_star = pd.DataFrame([], index=range(1, N_MC+1), columns=range(T+1))
a_star.iloc[:, -1] = 0

# optimal Q-function with optimal action
max_Q_star = np.zeros((N_MC, T))

a_mean = []

initialize actions in feasible region [0, a_max]¶

In [11]:
grid_size_x = 20
grid_size_a = 500 

# set up a small grid, sufficiently nested inside the support of the basis functions
a_min_prime = a_min + 0.1
a_max_prime = a_max - 0.1

a_grid = np.linspace(a_min_prime, a_max_prime, grid_size_a)

Q_star = np.zeros((N_MC, T))
tau = 1e-6
a = a_min_prime + (a_max_prime - a_min_prime) * np.random.rand(N_MC)

There are grid_size_x $\times$ grid_size_a inner grid points and 256 basis functions. Let $(k,l)$ denote the indices of the smaller grid $\Omega^h$. Let $(i,j)$ denote the indices of the knot points of the basis functions. If you sum over the last index, then you can check the interpolation of ones $$f(x_k,a_l)=\sum_{ij} \Phi(x_k,a_l)f_{ij}$$ check if $f_{ij}=1$ everywhere $$f(x,a)=\sum_{ij} \Phi(x,a)1 =1, \forall x,a ?$$

The backward loop¶

Make sure to rerun cells under the heading 'Initialize data structures' to reset the initial conditions before running this cell

In [15]:
for t in np.arange(T - 2, 0, -1):
    error = tau
    q_prev = np.zeros(N_MC)  
    ret = (S.loc[:, t+1] - S.loc[:, t]) / S.loc[:, t]
    mu[t] = np.mean(ret)
    var[t] = np.var(ret)
    count = 0
    max_iter = 100
    x = X.values[:, t]
    basis_x, basis_a = get_basis_functions(np.min(x)-10, np.max(x)+10, a_min, a_max, ncolloc, p)
    x_grid = np.linspace(np.min(x), np.max(x), grid_size_x)
    Phi_mat = np.array([[np.kron(basis_x(x_grid[i]), basis_a(a_grid[j])).reshape(num_basis, 1) 
                         for i in range(grid_size_x)] for j in range(grid_size_a)])[:, :, :, 0]   
        
    while (np.abs(error)>=tau) and (count<max_iter):
       
        R = negative_reward(mu[t], var[t], a, r, lmbda)
        data_mat_t[t, :, :] = np.array([np.kron(basis_x(x[i]), basis_a(a[i])).reshape(num_basis, 1) for i in range(N_MC)])[:, :, 0]
        
        # Check partition of unity
        h = np.dot(data_mat_t[t],np.ones(np.shape(data_mat_t[t])[1]))
        if np.sum(h) != N_MC:
            print("error: loss of partition of unity")
        S_t = function_S_vec(t, data_mat_t) 
        M_t = function_M_vec(t, Q_star, R, data_mat_t, gamma)
        W_t = np.dot(np.linalg.pinv(S_t), M_t)
       
        # Compute Q_t matrix over small grid
        # Phi_mat is the matrix for interpolating over the smaller x * a grid
        Q_t = np.dot(Phi_mat, W_t) # gridded Q_t
        
        print("residual error: ||r||=||Sw-M||")
        print(np.linalg.norm(np.dot(S_t, W_t) - M_t))
        
        # Find the optimal action on the small grid
        a_idx =  np.argmin(Q_t, axis=0)
        # Need to interpolate over X 
        a_star_ = np.zeros(grid_size_x, dtype='float64')
        
        for j in range(grid_size_x):
            a_star_[j] = a_grid[a_idx[j]]
    
        f = interp1d(x_grid, a_star_, kind='cubic')
        a_prime = f(x)
        
        # Only update the actions along the paths where the neg. reward is lowered
        R_prime = negative_reward(mu[t], var[t], a_prime, r, lmbda)
        idx = np.where(R_prime>R)
        a_prime[idx] = a[idx]
        a = a_prime
        a_mean.append(np.mean(a))
        Q_star[:,t] = np.dot(data_mat_t[t, :, :], W_t)
        error = np.linalg.norm(Q_star[:, t] - q_prev)
        
        q_prev = np.copy(Q_star[:, t])
        
        print('count, Q error, E[R], E[a]')
        print(count, error, np.mean(R), np.mean(a))
        count += 1 
       
    a_opt[:, t] = a  

residual error: ||r||=||Sw-M||
4.054087113470703e-09
count, Q error, E[R], E[a]
0 0.39985090486548763 -0.0064244171795789255 -0.38409046036259487
residual error: ||r||=||Sw-M||
0.0016271308689962385
count, Q error, E[R], E[a]
1 0.37221550157859185 -0.011958626753660277 -0.38419634873003156
residual error: ||r||=||Sw-M||
0.001724869490438883
count, Q error, E[R], E[a]
2 0.00043257994172365814 -0.01195880276161595 -0.38427276043711833
residual error: ||r||=||Sw-M||
0.000686165596517034
count, Q error, E[R], E[a]
3 0.0001865506479181699 -0.01195880336297234 -0.3842866140073686
residual error: ||r||=||Sw-M||
0.0008138967279655943
count, Q error, E[R], E[a]
4 0.00013478255458155967 -0.011958803475966265 -0.3842884727274255
residual error: ||r||=||Sw-M||
0.0002586494714753655
count, Q error, E[R], E[a]
5 8.904912691955464e-05 -0.011958808239279702 -0.3842902778013504
residual error: ||r||=||Sw-M||
0.0003211161745436431
count, Q error, E[R], E[a]
6 5.2007692918097614e-05 -0.011958812829682598 -0.3842902778013504
residual error: ||r||=||Sw-M||
0.0003211161745436431
count, Q error, E[R], E[a]
7 0.0 -0.011958812829682598 -0.3842902778013504
residual error: ||r||=||Sw-M||
0.00027128554073471613
count, Q error, E[R], E[a]
0 1.0676404373631583 -0.011916159898305204 -0.3842902778013504
residual error: ||r||=||Sw-M||
0.00027128554073471613
count, Q error, E[R], E[a]
1 0.0 -0.011916159898305204 -0.3842902778013504
residual error: ||r||=||Sw-M||
0.0007025668992294378
count, Q error, E[R], E[a]
0 1.6176934963654577 -0.01230632377503389 -0.3844119193149309
residual error: ||r||=||Sw-M||
0.0005547494509953625
count, Q error, E[R], E[a]
1 0.00017524360139859626 -0.012306428252995539 -0.3849095214114395
residual error: ||r||=||Sw-M||
0.00155285605615567
count, Q error, E[R], E[a]
2 0.00018879427019954137 -0.012306639378564228 -0.3864356946940989
residual error: ||r||=||Sw-M||
0.00020539038044391667
count, Q error, E[R], E[a]
3 0.0001606682089911478 -0.012307102268700614 -0.38887530740857573
residual error: ||r||=||Sw-M||
0.0004537046871139951
count, Q error, E[R], E[a]
4 0.00011228791433104998 -0.0123078488306605 -0.39520041327054223
residual error: ||r||=||Sw-M||
0.00044625589432227276
count, Q error, E[R], E[a]
5 0.00020143711967503207 -0.012310092846248438 -0.3992959300573166
residual error: ||r||=||Sw-M||
0.0019624055942185953
count, Q error, E[R], E[a]
6 0.00023049900056702053 -0.012311220736391197 -0.40487669877792587
residual error: ||r||=||Sw-M||
0.0029952180024731437
count, Q error, E[R], E[a]
7 0.00038826282546714814 -0.012313124537409617 -0.40850130458039174
residual error: ||r||=||Sw-M||
0.014057388595526904
count, Q error, E[R], E[a]
8 0.0012771849980811334 -0.012314277674339614 -0.40847601037204434
residual error: ||r||=||Sw-M||
0.006315677889401548
count, Q error, E[R], E[a]
9 0.001787467374283835 -0.012314281721361968 -0.40854303841758816
residual error: ||r||=||Sw-M||
0.006093746262542528
count, Q error, E[R], E[a]
10 0.0007953369074083744 -0.01231428418519095 -0.40851967244152976
residual error: ||r||=||Sw-M||
0.01011269027519587
count, Q error, E[R], E[a]
11 0.0013011926003446955 -0.012314284909715464 -0.4085193275705825
residual error: ||r||=||Sw-M||
0.017373271227261535
count, Q error, E[R], E[a]
12 0.0021179803481542942 -0.012314285338888158 -0.4085245651777541
residual error: ||r||=||Sw-M||
0.005665189974524261
count, Q error, E[R], E[a]
13 0.0016466993356203858 -0.012314288278177392 -0.40850660005829276
residual error: ||r||=||Sw-M||
0.02584871741383835
count, Q error, E[R], E[a]
14 0.0019124671983266903 -0.012314288443668211 -0.40855932472917983
residual error: ||r||=||Sw-M||
0.06037606841164746
count, Q error, E[R], E[a]
15 0.004896516655173087 -0.012314292344854517 -0.40855196016034506
residual error: ||r||=||Sw-M||
0.022029695010324285
count, Q error, E[R], E[a]
16 0.004728409952199698 -0.012314292395182747 -0.4085401761541372
residual error: ||r||=||Sw-M||
0.026740018808734542
count, Q error, E[R], E[a]
17 0.0019866728001371584 -0.012314292544746686 -0.40852319706110024
residual error: ||r||=||Sw-M||
0.03396784940965752
count, Q error, E[R], E[a]
18 0.005792711458096325 -0.012314292697867902 -0.40851801531010534
residual error: ||r||=||Sw-M||
0.045767036163330437
count, Q error, E[R], E[a]
19 0.006467662339748758 -0.012314292780124683 -0.40851344253372984
residual error: ||r||=||Sw-M||
0.05353591090209379
count, Q error, E[R], E[a]
20 0.003088913745122548 -0.012314292850513173 -0.40852114348434077
residual error: ||r||=||Sw-M||
0.04549717820963136
count, Q error, E[R], E[a]
21 0.0054682092119877415 -0.01231429340077594 -0.408515018860958
residual error: ||r||=||Sw-M||
0.028230738504452898
count, Q error, E[R], E[a]
22 0.002875659545249235 -0.012314293470713398 -0.4084991192508953
residual error: ||r||=||Sw-M||
0.07980415139479773
count, Q error, E[R], E[a]
23 0.004343820252339387 -0.012314293627549017 -0.40849329981178584
residual error: ||r||=||Sw-M||
0.01637270367152446
count, Q error, E[R], E[a]
24 0.006934930536702326 -0.012314293793512931 -0.4084830835578016
residual error: ||r||=||Sw-M||
0.014020911345883342
count, Q error, E[R], E[a]
25 0.0028116529590529126 -0.012314293932699802 -0.408467975437622
residual error: ||r||=||Sw-M||
0.014350964502201873
count, Q error, E[R], E[a]
26 0.0027081188782055723 -0.012314294143381086 -0.408467975437622
residual error: ||r||=||Sw-M||
0.014350964502201873
count, Q error, E[R], E[a]
27 0.0 -0.012314294143381086 -0.408467975437622
residual error: ||r||=||Sw-M||
0.04137337052242329
count, Q error, E[R], E[a]
0 2.110856708828953 -0.010920540063843895 -0.32588619482592424
residual error: ||r||=||Sw-M||
0.0018886760171382383
count, Q error, E[R], E[a]
1 0.007138423297795844 -0.011094121561365597 -0.3033125791716161
residual error: ||r||=||Sw-M||
0.005016043283167808
count, Q error, E[R], E[a]
2 0.0015315796492860175 -0.011117952295955589 -0.2952415849547424
residual error: ||r||=||Sw-M||
0.0008171925223870515
count, Q error, E[R], E[a]
3 0.0005971467608432888 -0.011120079115376413 -0.29273862715499155
residual error: ||r||=||Sw-M||
0.0015655245373831282
count, Q error, E[R], E[a]
4 0.00020335010500003805 -0.01112089381497256 -0.2871651311318194
residual error: ||r||=||Sw-M||
0.0016425913657666545
count, Q error, E[R], E[a]
5 0.00034182704110586297 -0.011122230341553123 -0.28643510365632635
residual error: ||r||=||Sw-M||
0.0033003872425509113
count, Q error, E[R], E[a]
6 0.0004462797312687441 -0.011122436671738116 -0.2865562086276061
residual error: ||r||=||Sw-M||
0.0016597885300755758
count, Q error, E[R], E[a]
7 0.00018271718540083747 -0.011122462216666169 -0.28658229095627313
residual error: ||r||=||Sw-M||
0.00309043210387628
count, Q error, E[R], E[a]
8 0.0003947251839746551 -0.011122462888618134 -0.2865468558169001
residual error: ||r||=||Sw-M||
0.0026894214289633536
count, Q error, E[R], E[a]
9 0.0004954948463204276 -0.011122463750144298 -0.28683957251161163
residual error: ||r||=||Sw-M||
0.0008084371308017781
count, Q error, E[R], E[a]
10 0.0002804744009066606 -0.011122475742155052 -0.286839654652495
residual error: ||r||=||Sw-M||
0.00042481640682551154
count, Q error, E[R], E[a]
11 0.00014511257384556937 -0.011122475742178279 -0.286839654724371
residual error: ||r||=||Sw-M||
0.0007752514054847174
count, Q error, E[R], E[a]
12 6.592137271626193e-05 -0.011122475742178301 -0.286839654724371
residual error: ||r||=||Sw-M||
0.0007752514054847174
count, Q error, E[R], E[a]
13 0.0 -0.011122475742178301 -0.286839654724371
residual error: ||r||=||Sw-M||
0.012500669967107659
count, Q error, E[R], E[a]
0 2.6364113124727755 -0.01160043397890189 -0.3176041537917467
residual error: ||r||=||Sw-M||
0.00011969517893616828
count, Q error, E[R], E[a]
1 0.0022104270270279633 -0.011635731455672364 -0.34857633258341336
residual error: ||r||=||Sw-M||
0.00038411629447343706
count, Q error, E[R], E[a]
2 0.0005489446664248584 -0.011643397267046787 -0.34695404471521385
residual error: ||r||=||Sw-M||
0.00241657055407753
count, Q error, E[R], E[a]
3 0.0002403719768981456 -0.011647415140162348 -0.3460385952487054
residual error: ||r||=||Sw-M||
0.0010311910495972565
count, Q error, E[R], E[a]
4 0.0002511047463750817 -0.011648416362091692 -0.3455013701038399

residual error: ||r||=||Sw-M||
3.3233991785938706e-05
count, Q error, E[R], E[a]
5 0.00013285191576742122 -0.011648649493199397 -0.34512906301794893
residual error: ||r||=||Sw-M||
2.3042502172395993e-05
count, Q error, E[R], E[a]
6 2.370332169790823e-05 -0.011648690938317607 -0.3451208137791208
residual error: ||r||=||Sw-M||
3.7159421208453256e-05
count, Q error, E[R], E[a]
7 7.356768655097124e-06 -0.011648695985750232 -0.34511570478606085
residual error: ||r||=||Sw-M||
6.23682055136617e-05
count, Q error, E[R], E[a]
8 6.444158676366542e-06 -0.011648696203050833 -0.34511570478606085
residual error: ||r||=||Sw-M||
6.23682055136617e-05
count, Q error, E[R], E[a]
9 0.0 -0.011648696203050833 -0.34511570478606085
residual error: ||r||=||Sw-M||
7.945324456164468e-06
count, Q error, E[R], E[a]
0 3.1434532954596883 -0.01130923374135247 -0.3236553212599598
residual error: ||r||=||Sw-M||
0.03494810469257471
count, Q error, E[R], E[a]
1 0.0038938493505896324 -0.011323607472175987 -0.3190316732917447
residual error: ||r||=||Sw-M||
0.0001076869268684607
count, Q error, E[R], E[a]
2 0.0034769114218665877 -0.011324796051584513 -0.3141085013097217
residual error: ||r||=||Sw-M||
0.00017854296689090928
count, Q error, E[R], E[a]
3 8.070842295549497e-05 -0.011325769456841633 -0.31226691347094
residual error: ||r||=||Sw-M||
0.00042205955792132965
count, Q error, E[R], E[a]
4 6.209622393360427e-05 -0.01132606831625536 -0.3117422865183364
residual error: ||r||=||Sw-M||
0.0026072060628796775
count, Q error, E[R], E[a]
5 0.0002117955857574514 -0.01132610599203855 -0.3117475034067371
residual error: ||r||=||Sw-M||
0.006176820946999051
count, Q error, E[R], E[a]
6 0.0005157647744755061 -0.011326123893668334 -0.31137402332669295
residual error: ||r||=||Sw-M||
0.0020081593061197847
count, Q error, E[R], E[a]
7 0.00047975854589560866 -0.011326155177985587 -0.31119883983612406
residual error: ||r||=||Sw-M||
0.00026675905827568044
count, Q error, E[R], E[a]
8 0.0002689415719543737 -0.011326158470850608 -0.3110621227549436
residual error: ||r||=||Sw-M||
0.006960236408651707
count, Q error, E[R], E[a]
9 0.0008021320891831546 -0.011326163627922626 -0.3107534944478697
residual error: ||r||=||Sw-M||
0.002592112484814513
count, Q error, E[R], E[a]
10 0.0006443361413595912 -0.011326166198515189 -0.3104531825993862
residual error: ||r||=||Sw-M||
0.0022531885639844166
count, Q error, E[R], E[a]
11 0.0003216550796335158 -0.011326172203144363 -0.31040932879660144
residual error: ||r||=||Sw-M||
0.013577489530741822
count, Q error, E[R], E[a]
12 0.0010265397844019784 -0.011326175105141797 -0.31041130107428383
residual error: ||r||=||Sw-M||
0.004260522059460088
count, Q error, E[R], E[a]
13 0.0011675869148625338 -0.011326175112129572 -0.31044049948004476
residual error: ||r||=||Sw-M||
0.008477307895586372
count, Q error, E[R], E[a]
14 0.0012040315306013928 -0.011326175627938669 -0.3102742556685864
residual error: ||r||=||Sw-M||
0.006231854655201606
count, Q error, E[R], E[a]
15 0.00064152372890282 -0.011326177668404509 -0.3102414660252206
residual error: ||r||=||Sw-M||
0.0036045274361425233
count, Q error, E[R], E[a]
16 0.001005048692882299 -0.011326178949515113 -0.3102356326493202
residual error: ||r||=||Sw-M||
0.007633305684363094
count, Q error, E[R], E[a]
17 0.0009632598250835508 -0.011326179081812536 -0.31023794336490595
residual error: ||r||=||Sw-M||
0.010103576441308834
count, Q error, E[R], E[a]
18 0.001277789143441901 -0.011326179702323528 -0.3102330075001624
residual error: ||r||=||Sw-M||
0.00905198150890191
count, Q error, E[R], E[a]
19 0.000668510489625639 -0.011326180202227672 -0.31023398612984115
residual error: ||r||=||Sw-M||
0.00704675845828353
count, Q error, E[R], E[a]
20 0.00034692422398491436 -0.011326180209486524 -0.3102468857172446
residual error: ||r||=||Sw-M||
0.00626109822139568
count, Q error, E[R], E[a]
21 0.0011668178451977854 -0.011326180705520468 -0.31024672016494753
residual error: ||r||=||Sw-M||
0.007126305673587684
count, Q error, E[R], E[a]
22 0.0010199857688053622 -0.011326180707433523 -0.31011222492013707
residual error: ||r||=||Sw-M||
0.007250379938845547
count, Q error, E[R], E[a]
23 0.0005331469293560706 -0.011326182388183063 -0.310128785767674
residual error: ||r||=||Sw-M||
0.0021204412238816123
count, Q error, E[R], E[a]
24 0.0009088401418443116 -0.011326182531825626 -0.3101248573591248
residual error: ||r||=||Sw-M||
0.0036400452999419157
count, Q error, E[R], E[a]
25 0.0003354824833001308 -0.011326182587511825 -0.3101143875048205
residual error: ||r||=||Sw-M||
0.013816608249026816
count, Q error, E[R], E[a]
26 0.0008914557488586641 -0.011326182921870898 -0.3101143875048205
residual error: ||r||=||Sw-M||
0.013816608249026816
count, Q error, E[R], E[a]
27 0.0 -0.011326182921870898 -0.3101143875048205
residual error: ||r||=||Sw-M||
0.008571589947873496
count, Q error, E[R], E[a]
0 3.656017908209628 -0.01143256432866967 -0.3159065962318168
residual error: ||r||=||Sw-M||
0.0004981026779307583
count, Q error, E[R], E[a]
1 0.0009611912412529334 -0.011433242825430092 -0.3165251265148125
residual error: ||r||=||Sw-M||
0.0008027007430590854
count, Q error, E[R], E[a]
2 0.0001199779249108913 -0.01143341631084816 -0.3175023928546829
residual error: ||r||=||Sw-M||
0.0008185191109345113
count, Q error, E[R], E[a]
3 0.00013449569744799638 -0.011433551630297316 -0.3181206680092856
residual error: ||r||=||Sw-M||
0.002584737560702182
count, Q error, E[R], E[a]
4 0.00030570930626623057 -0.01143366831224289 -0.31901838881867284
residual error: ||r||=||Sw-M||
0.0005149345686898807
count, Q error, E[R], E[a]
5 0.00021362083065112517 -0.011433821748526377 -0.319430969630489
residual error: ||r||=||Sw-M||
0.00580206954382098
count, Q error, E[R], E[a]
6 0.0004947985518974892 -0.011433924075243579 -0.319639028000915
residual error: ||r||=||Sw-M||
0.013943976168678377
count, Q error, E[R], E[a]
7 0.0007352541988094087 -0.011433965387210325 -0.31980528537408565
residual error: ||r||=||Sw-M||
0.009889203775843434
count, Q error, E[R], E[a]
8 0.002056558964586769 -0.0114340032816953 -0.3198092435286362
residual error: ||r||=||Sw-M||
0.002539224287454606
count, Q error, E[R], E[a]
9 0.0009283123264788613 -0.011434016011995831 -0.31990590400013924
residual error: ||r||=||Sw-M||
0.004601449261575329
count, Q error, E[R], E[a]
10 0.0004338028696202745 -0.011434024592548716 -0.31993902372826505
residual error: ||r||=||Sw-M||
0.00014181335878621064
count, Q error, E[R], E[a]
11 0.0003936769487080567 -0.011434031724066424 -0.3199486464187745
residual error: ||r||=||Sw-M||
0.00018602525312990057
count, Q error, E[R], E[a]
12 2.439988446611457e-05 -0.011434031961918749 -0.3199521684109831
residual error: ||r||=||Sw-M||
0.00013790007272329447
count, Q error, E[R], E[a]
13 2.7555715876608714e-05 -0.011434033317133416 -0.3199521684109831
residual error: ||r||=||Sw-M||
0.00013790007272329447
count, Q error, E[R], E[a]
14 0.0 -0.011434033317133416 -0.3199521684109831
residual error: ||r||=||Sw-M||
7.29021515789785e-05
count, Q error, E[R], E[a]
0 4.177305358613513 -0.01167263772331904 -0.3693888870138245
residual error: ||r||=||Sw-M||
0.06996670803241194
count, Q error, E[R], E[a]
1 0.0065746095950712105 -0.011678678849499588 -0.36854596093737757
residual error: ||r||=||Sw-M||
0.07276557287701355
count, Q error, E[R], E[a]
2 0.012988137389281625 -0.011678951735954162 -0.3639801831096093
residual error: ||r||=||Sw-M||
0.00047653764014162647
count, Q error, E[R], E[a]
3 0.006663322624263075 -0.011680246023908939 -0.34944086441136557
residual error: ||r||=||Sw-M||
0.001703530951192091
count, Q error, E[R], E[a]
4 0.00023119023628701717 -0.011682486915089078 -0.3506310669542579
residual error: ||r||=||Sw-M||
0.0002537064832687594
count, Q error, E[R], E[a]
5 0.00021085790575956046 -0.01168270007357884 -0.3518510014433594
residual error: ||r||=||Sw-M||
0.00031097374021430275
count, Q error, E[R], E[a]
6 0.00011241736954956031 -0.011683162062946685 -0.34884986195364515
residual error: ||r||=||Sw-M||
0.0005006978164044641
count, Q error, E[R], E[a]
7 0.0001327568005441745 -0.011684424992796554 -0.3489648230905989
residual error: ||r||=||Sw-M||
0.00017479352733493632
count, Q error, E[R], E[a]
8 6.769664931639641e-05 -0.011684429465875163 -0.3487520487281006

residual error: ||r||=||Sw-M||
0.0017270164667982143
count, Q error, E[R], E[a]
9 0.00016886591472191267 -0.011684435369819156 -0.34859467764802066
residual error: ||r||=||Sw-M||
8.43828086600416e-05
count, Q error, E[R], E[a]
10 0.00017053490707780086 -0.011684438674136032 -0.34859492970056927
residual error: ||r||=||Sw-M||
0.00015526364382177182
count, Q error, E[R], E[a]
11 1.8927195217583547e-05 -0.01168443870902257 -0.34859492970056927
residual error: ||r||=||Sw-M||
0.00015526364382177182
count, Q error, E[R], E[a]
12 0.0 -0.01168443870902257 -0.34859492970056927
residual error: ||r||=||Sw-M||
9.311849903561253e-05
count, Q error, E[R], E[a]
0 4.702174397606315 -0.011755081159362156 -0.35529058940908365
residual error: ||r||=||Sw-M||
0.006182641511297971
count, Q error, E[R], E[a]
1 0.0006994371079297109 -0.011756715531787055 -0.3553006297154633
residual error: ||r||=||Sw-M||
0.018427415285904157
count, Q error, E[R], E[a]
2 0.0029567335952147126 -0.011756717715441905 -0.35596478975048657
residual error: ||r||=||Sw-M||
0.0017912456316930634
count, Q error, E[R], E[a]
3 0.0023300365700779794 -0.011756750841259225 -0.35613165700511573
residual error: ||r||=||Sw-M||
0.01436078634638188
count, Q error, E[R], E[a]
4 0.001426270053751095 -0.011756766354370014 -0.3565022715940918
residual error: ||r||=||Sw-M||
0.07254891417600833
count, Q error, E[R], E[a]
5 0.006074686869692684 -0.011756811778511057 -0.3584466876027124
residual error: ||r||=||Sw-M||
0.003979059261204244
count, Q error, E[R], E[a]
6 0.006900228189100384 -0.011756954832748827 -0.358853793145684
residual error: ||r||=||Sw-M||
0.003998052856905164
count, Q error, E[R], E[a]
7 0.0008040783783193896 -0.011756975902514193 -0.35934301019630766
residual error: ||r||=||Sw-M||
0.007491752668971494
count, Q error, E[R], E[a]
8 0.0008821704910991438 -0.011757023188330492 -0.3594900482068226
residual error: ||r||=||Sw-M||
0.012009796543006554
count, Q error, E[R], E[a]
9 0.001653152000672536 -0.011757044070572242 -0.35979925581069894
residual error: ||r||=||Sw-M||
0.01569965692167054
count, Q error, E[R], E[a]
10 0.0012199132368332175 -0.011757061846325457 -0.3601393938417714
residual error: ||r||=||Sw-M||
0.011026440361721592
count, Q error, E[R], E[a]
11 0.0022747579485904385 -0.011757077962544965 -0.36091493954921744
residual error: ||r||=||Sw-M||
0.010401948869867606
count, Q error, E[R], E[a]
12 0.0016447127384918657 -0.011757118894514005 -0.36102226128292686
residual error: ||r||=||Sw-M||
0.004439498470836814
count, Q error, E[R], E[a]
13 0.0011078218621690767 -0.011757123766781449 -0.3610527786704918
residual error: ||r||=||Sw-M||
0.012273142083219139
count, Q error, E[R], E[a]
14 0.0012497088383069158 -0.011757127652927699 -0.36113388901577836
residual error: ||r||=||Sw-M||
0.006174474834492622
count, Q error, E[R], E[a]
15 0.001219704084017374 -0.011757131400527396 -0.36112124578084537
residual error: ||r||=||Sw-M||
0.013605976922234657
count, Q error, E[R], E[a]
16 0.0016161808198321699 -0.011757131712570958 -0.3611238313674708
residual error: ||r||=||Sw-M||
0.008996130535154216
count, Q error, E[R], E[a]
17 0.0009032160332693233 -0.011757131809044896 -0.36111712224478676
residual error: ||r||=||Sw-M||
0.022633209035050634
count, Q error, E[R], E[a]
18 0.0020890857575939603 -0.011757133425247682 -0.3612470458956683
residual error: ||r||=||Sw-M||
0.008524974333694348
count, Q error, E[R], E[a]
19 0.0021889898417381524 -0.011757135238874639 -0.3613908739258377
residual error: ||r||=||Sw-M||
0.011692057431697978
count, Q error, E[R], E[a]
20 0.001141786394029798 -0.01175713662260167 -0.3613432717927435
residual error: ||r||=||Sw-M||
0.005021264514681122
count, Q error, E[R], E[a]
21 0.0014351032855447305 -0.011757137527910771 -0.3613241726990352
residual error: ||r||=||Sw-M||
0.004298415804618989
count, Q error, E[R], E[a]
22 0.0008064591860486415 -0.011757138888997001 -0.3613214900077965
residual error: ||r||=||Sw-M||
0.0046578855223077045
count, Q error, E[R], E[a]
23 0.0009147681151211455 -0.011757139038404855 -0.36132007276843314
residual error: ||r||=||Sw-M||
0.005235441231148892
count, Q error, E[R], E[a]
24 0.0007079860236831431 -0.011757139185958507 -0.3613868244716589
residual error: ||r||=||Sw-M||
0.003846007189600943
count, Q error, E[R], E[a]
25 0.0008389202417329938 -0.011757139834623885 -0.3613868244716589
residual error: ||r||=||Sw-M||
0.003846007189600943
count, Q error, E[R], E[a]
26 0.0 -0.011757139834623885 -0.3613868244716589
residual error: ||r||=||Sw-M||
0.002732162880075242
count, Q error, E[R], E[a]
0 5.231693541605084 -0.011853705788733508 -0.3657181609161548
residual error: ||r||=||Sw-M||
0.0005384198724469536
count, Q error, E[R], E[a]
1 0.0003751214152394276 -0.011854409063323308 -0.37095280807403014
residual error: ||r||=||Sw-M||
0.0036560026989214634
count, Q error, E[R], E[a]
2 0.0003815171860283179 -0.011855230123367212 -0.37300743947068943
residual error: ||r||=||Sw-M||
0.0017806784075619522
count, Q error, E[R], E[a]
3 0.00037792487909408694 -0.011855437005113512 -0.37284314419573833
residual error: ||r||=||Sw-M||
0.011956365346462817
count, Q error, E[R], E[a]
4 0.0010143768530827073 -0.01185545056208296 -0.372955866775952
residual error: ||r||=||Sw-M||
0.004895681804138091
count, Q error, E[R], E[a]
5 0.0009096166521201543 -0.011855456315685802 -0.3729911065257266
residual error: ||r||=||Sw-M||
0.005276619586082832
count, Q error, E[R], E[a]
6 0.0006474989107398343 -0.01185545794009061 -0.3729885214389475
residual error: ||r||=||Sw-M||
0.010518672612399471
count, Q error, E[R], E[a]
7 0.0016226305741083966 -0.011855457977815932 -0.3729851882229091
residual error: ||r||=||Sw-M||
0.012719908812171324
count, Q error, E[R], E[a]
8 0.002061891512890371 -0.011855457995241533 -0.37299802731826864
residual error: ||r||=||Sw-M||
0.00898038501001162
count, Q error, E[R], E[a]
9 0.0018848346758334445 -0.011855458142495839 -0.3729940419331
residual error: ||r||=||Sw-M||
0.005876063107028963
count, Q error, E[R], E[a]
10 0.0010509729001507465 -0.011855458179111196 -0.37300929633962004
residual error: ||r||=||Sw-M||
0.003290061170307557
count, Q error, E[R], E[a]
11 0.000531628924317995 -0.011855458383406157 -0.3730161747358877
residual error: ||r||=||Sw-M||
0.0037987791512560133
count, Q error, E[R], E[a]
12 0.00023416520238795228 -0.011855458727549478 -0.3730231124904266
residual error: ||r||=||Sw-M||
0.00924175694087593
count, Q error, E[R], E[a]
13 0.0015217377288540249 -0.011855458745015658 -0.3730222700534365
residual error: ||r||=||Sw-M||
0.03015432605957345
count, Q error, E[R], E[a]
14 0.0034015214580488703 -0.011855458750115 -0.37302990872590114
residual error: ||r||=||Sw-M||
0.03264374910767372
count, Q error, E[R], E[a]
15 0.005646475294888366 -0.01185545901550498 -0.37302168810638875
residual error: ||r||=||Sw-M||
0.01774660516789422
count, Q error, E[R], E[a]
16 0.0045881744392186575 -0.0118554591040464 -0.3730211792380993
residual error: ||r||=||Sw-M||
0.009096287758428433
count, Q error, E[R], E[a]
17 0.0024146614633104766 -0.011855459112953704 -0.373021416532555
residual error: ||r||=||Sw-M||
0.0495701344638631
count, Q error, E[R], E[a]
18 0.004941976902034578 -0.01185545912679341 -0.3730144363023511
residual error: ||r||=||Sw-M||
0.006124263828678848
count, Q error, E[R], E[a]
19 0.004095018689105477 -0.01185545915593394 -0.3730148000709646
residual error: ||r||=||Sw-M||
0.03548151604593327
count, Q error, E[R], E[a]
20 0.002923501207101733 -0.011855459159053386 -0.3730133891692208
residual error: ||r||=||Sw-M||
0.01433744526481409
count, Q error, E[R], E[a]
21 0.002760382473786654 -0.011855459177390752 -0.3730135539233081
residual error: ||r||=||Sw-M||
0.013048888398087495
count, Q error, E[R], E[a]
22 0.0011861305230570405 -0.011855459189237045 -0.3730135539233081
residual error: ||r||=||Sw-M||
0.013048888398087495
count, Q error, E[R], E[a]
23 0.0 -0.011855459189237045 -0.3730135539233081
residual error: ||r||=||Sw-M||
0.00844131620061978
count, Q error, E[R], E[a]
0 5.744978915197822 -0.011436149360312165 -0.3257030653936248
residual error: ||r||=||Sw-M||
0.004339280360476673
count, Q error, E[R], E[a]
1 0.0010204815869926452 -0.01145456428347002 -0.33614328921788

residual error: ||r||=||Sw-M||
0.00261553494522653
count, Q error, E[R], E[a]
2 0.0005574445722658425 -0.011460194718865762 -0.33145925488242844
residual error: ||r||=||Sw-M||
0.0003907640554730167
count, Q error, E[R], E[a]
3 0.0003156654652824479 -0.011461427818005007 -0.32883962776358566
residual error: ||r||=||Sw-M||
0.001673669599611927
count, Q error, E[R], E[a]
4 0.00023497087533933596 -0.011461686556107301 -0.3294739306934683
residual error: ||r||=||Sw-M||
0.0019278533605981798
count, Q error, E[R], E[a]
5 0.00029862606873591325 -0.011461748733628603 -0.32946495321252917
residual error: ||r||=||Sw-M||
0.0007457293156077845
count, Q error, E[R], E[a]
6 0.00022385381796159306 -0.011461751265319979 -0.32946507270684716
residual error: ||r||=||Sw-M||
0.0017730768898527442
count, Q error, E[R], E[a]
7 0.0002063753369778312 -0.011461751269530827 -0.32950306018366826
residual error: ||r||=||Sw-M||
0.0019737433820017475
count, Q error, E[R], E[a]
8 0.00033240649951809143 -0.011461758617686349 -0.32949289433155277
residual error: ||r||=||Sw-M||
0.0011958513022710222
count, Q error, E[R], E[a]
9 0.00032455122384984476 -0.01146176003435841 -0.3294926373636408
residual error: ||r||=||Sw-M||
0.002101880405357663
count, Q error, E[R], E[a]
10 0.0003153210215854653 -0.011461760036105864 -0.3296898036982343
residual error: ||r||=||Sw-M||
0.004836626874175011
count, Q error, E[R], E[a]
11 0.00031686954881898283 -0.011461764690819769 -0.3296759942096957
residual error: ||r||=||Sw-M||
0.0003245608779039603
count, Q error, E[R], E[a]
12 0.000412028717117526 -0.011461765990268062 -0.32967599420394134
residual error: ||r||=||Sw-M||
0.0002481356451698769
count, Q error, E[R], E[a]
13 2.1534520522135537e-05 -0.011461765990272073 -0.3296759213133938
residual error: ||r||=||Sw-M||
0.00034102642735262624
count, Q error, E[R], E[a]
14 6.189005308375678e-05 -0.01146176604094575 -0.3296759213133938
residual error: ||r||=||Sw-M||
0.00034102642735262624
count, Q error, E[R], E[a]
15 0.0 -0.01146176604094575 -0.3296759213133938
residual error: ||r||=||Sw-M||
0.0002583865991187686
count, Q error, E[R], E[a]
0 6.268179832190101 -0.011723282695841202 -0.35869095787502936
residual error: ||r||=||Sw-M||
0.10062359023490408
count, Q error, E[R], E[a]
1 0.00971884826605818 -0.011734588930541915 -0.35873109598886943
residual error: ||r||=||Sw-M||
0.030236893211362026
count, Q error, E[R], E[a]
2 0.008235027402380727 -0.011734604819548413 -0.3587504013533714
residual error: ||r||=||Sw-M||
0.045393366107426836
count, Q error, E[R], E[a]
3 0.005486891387853868 -0.011734607158371496 -0.3588456641951236
residual error: ||r||=||Sw-M||
0.00946480110367436
count, Q error, E[R], E[a]
4 0.0035433738681524846 -0.011734610806369951 -0.3588449102778942
residual error: ||r||=||Sw-M||
0.049050925301396875
count, Q error, E[R], E[a]
5 0.004005258406867472 -0.011734610820244958 -0.35884418378712574
residual error: ||r||=||Sw-M||
0.03207965868390211
count, Q error, E[R], E[a]
6 0.00540861902450508 -0.011734610821398352 -0.35884415655670787
residual error: ||r||=||Sw-M||
0.024802870168843326
count, Q error, E[R], E[a]
7 0.0062137070519772625 -0.011734610821477401 -0.35884415655670787
residual error: ||r||=||Sw-M||
0.024802870168843326
count, Q error, E[R], E[a]
8 0.0 -0.011734610821477401 -0.35884415655670787
residual error: ||r||=||Sw-M||
0.009512230229776517
count, Q error, E[R], E[a]
0 6.7880187628769555 -0.011542988548300602 -0.35552449088652377
residual error: ||r||=||Sw-M||
0.0024589068009239485
count, Q error, E[R], E[a]
1 0.0010367267301801574 -0.011543414655598386 -0.34819843148169477
residual error: ||r||=||Sw-M||
0.0007548771016478453
count, Q error, E[R], E[a]
2 0.0001832162436412769 -0.011543974508728167 -0.3420326540455534
residual error: ||r||=||Sw-M||
0.0005942720838059269
count, Q error, E[R], E[a]
3 0.0001647865361120151 -0.01154529653599742 -0.3433843981735562
residual error: ||r||=||Sw-M||
0.0014768790765460299
count, Q error, E[R], E[a]
4 0.0003786520043248514 -0.01154550516983426 -0.34452545994283773
residual error: ||r||=||Sw-M||
0.0781483416944471
count, Q error, E[R], E[a]
5 0.006378387306390465 -0.011545563854260697 -0.3445830601813364
residual error: ||r||=||Sw-M||
0.01972640677319635
count, Q error, E[R], E[a]
6 0.006256141895577669 -0.011545565920931172 -0.34458388302125936
residual error: ||r||=||Sw-M||
0.05164181780623279
count, Q error, E[R], E[a]
7 0.0044971765757196224 -0.011545565960760569 -0.3445843909772823
residual error: ||r||=||Sw-M||
0.022278583877154916
count, Q error, E[R], E[a]
8 0.005459356701012489 -0.01154556596183835 -0.34459074504889975
residual error: ||r||=||Sw-M||
0.034121071678265245
count, Q error, E[R], E[a]
9 0.0021695650621878925 -0.011545565979331568 -0.34459124622270504
residual error: ||r||=||Sw-M||
0.028290017886441768
count, Q error, E[R], E[a]
10 0.004986085185537514 -0.011545565980850662 -0.3445916066441388
residual error: ||r||=||Sw-M||
0.0462034981009337
count, Q error, E[R], E[a]
11 0.006041983400020814 -0.011545565981720534 -0.3445919428826802
residual error: ||r||=||Sw-M||
0.012806306648127479
count, Q error, E[R], E[a]
12 0.004298479775887919 -0.011545566039004876 -0.34459557814495684
residual error: ||r||=||Sw-M||
0.0075489336725295935
count, Q error, E[R], E[a]
13 0.0009113308557221076 -0.01154556605649311 -0.34459753786366465
residual error: ||r||=||Sw-M||
0.0046327621795304
count, Q error, E[R], E[a]
14 0.001346102584581817 -0.0115455660651808 -0.34459697392864846
residual error: ||r||=||Sw-M||
0.00930795169256093
count, Q error, E[R], E[a]
15 0.001218573888483707 -0.011545566065595893 -0.34459619578671313
residual error: ||r||=||Sw-M||
0.02248592296645486
count, Q error, E[R], E[a]
16 0.002779027645470393 -0.0115455660722164 -0.34460080480082833
residual error: ||r||=||Sw-M||
0.03666705480605153
count, Q error, E[R], E[a]
17 0.00457642428621013 -0.011545566083993071 -0.3446017778589509
residual error: ||r||=||Sw-M||
0.013941907029432465
count, Q error, E[R], E[a]
18 0.00402638538365622 -0.011545566127615743 -0.34460207331496645
residual error: ||r||=||Sw-M||
0.0018844736886442796
count, Q error, E[R], E[a]
19 0.0011869282393436196 -0.011545566127787414 -0.3446026354786679
residual error: ||r||=||Sw-M||
0.027717590774017514
count, Q error, E[R], E[a]
20 0.0023990493698181426 -0.011545566128801482 -0.3446040276131341
residual error: ||r||=||Sw-M||
0.00951719496142848
count, Q error, E[R], E[a]
21 0.003089961388422323 -0.011545566133382309 -0.3446075181162747
residual error: ||r||=||Sw-M||
0.023093728203736334
count, Q error, E[R], E[a]
22 0.002759226435974493 -0.011545566140765597 -0.3446075181162747
residual error: ||r||=||Sw-M||
0.023093728203736334
count, Q error, E[R], E[a]
23 0.0 -0.011545566140765597 -0.3446075181162747
residual error: ||r||=||Sw-M||
0.004957706113039446
count, Q error, E[R], E[a]
0 7.282274927866643 -0.011022114195769699 -0.3247290997112699
residual error: ||r||=||Sw-M||
0.0009359339238765834
count, Q error, E[R], E[a]
1 0.000984722704680024 -0.011051867184073527 -0.3167450716216655
residual error: ||r||=||Sw-M||
0.0035303446201190303
count, Q error, E[R], E[a]
2 0.0004577937408860099 -0.011061019590722634 -0.3104139991799436
residual error: ||r||=||Sw-M||
0.016367247068266814
count, Q error, E[R], E[a]
3 0.0013550239808286836 -0.011066529947799358 -0.30360307234695083
residual error: ||r||=||Sw-M||
0.0193143418590383
count, Q error, E[R], E[a]
4 0.0009574996200347892 -0.01107153648255431 -0.2976376489262781
residual error: ||r||=||Sw-M||
0.005404760220257216
count, Q error, E[R], E[a]
5 0.002135963614156971 -0.011074966558783943 -0.2884315557764876
residual error: ||r||=||Sw-M||
0.02134541265180257
count, Q error, E[R], E[a]
6 0.0019568821769060008 -0.011078392306436197 -0.28553308423368273
residual error: ||r||=||Sw-M||
0.0013347690148573267
count, Q error, E[R], E[a]
7 0.0017917660023262167 -0.011078920827218742 -0.2839892049666678
residual error: ||r||=||Sw-M||
0.05011463965241042
count, Q error, E[R], E[a]
8 0.004552317501108881 -0.01107921766896968 -0.2835393672074244
residual error: ||r||=||Sw-M||
0.010786380063581213
count, Q error, E[R], E[a]
9 0.003634286128186087 -0.011079301075271778 -0.28401743150095016

residual error: ||r||=||Sw-M||
0.001381674336522486
count, Q error, E[R], E[a]
10 0.0011768985427297444 -0.011079403667371902 -0.2831428632823293
residual error: ||r||=||Sw-M||
0.002006336930411817
count, Q error, E[R], E[a]
11 0.0003027298417763846 -0.011079505530138032 -0.2823866797716928
residual error: ||r||=||Sw-M||
0.006795153079861341
count, Q error, E[R], E[a]
12 0.0007050273840174087 -0.01107952052031704 -0.28103672791763334
residual error: ||r||=||Sw-M||
0.003910031011944382
count, Q error, E[R], E[a]
13 0.0006481441904449657 -0.01107960541529135 -0.28000583568371906
residual error: ||r||=||Sw-M||
0.12286103908920414
count, Q error, E[R], E[a]
14 0.012778685480832648 -0.011079772712891227 -0.2800031776511989
residual error: ||r||=||Sw-M||
0.02464335163578047
count, Q error, E[R], E[a]
15 0.01500204787763564 -0.011079772804964015 -0.27994033788555567
residual error: ||r||=||Sw-M||
0.01835007809712463
count, Q error, E[R], E[a]
16 0.0036624355363651254 -0.011079775587825297 -0.27994033788555567
residual error: ||r||=||Sw-M||
0.01835007809712463
count, Q error, E[R], E[a]
17 0.0 -0.011079775587825297 -0.27994033788555567
residual error: ||r||=||Sw-M||
0.03105545120310265
count, Q error, E[R], E[a]
0 7.787578371632503 -0.011240135859771584 -0.3019090111630838
residual error: ||r||=||Sw-M||
6.193285648853983e-05
count, Q error, E[R], E[a]
1 0.004764962219611094 -0.011247397256866477 -0.3183157216040439
residual error: ||r||=||Sw-M||
0.012164173538958163
count, Q error, E[R], E[a]
2 0.001615233902122273 -0.011255154471757211 -0.319860705621185
residual error: ||r||=||Sw-M||
0.022079965703313352
count, Q error, E[R], E[a]
3 0.0016388016086353225 -0.011255474394881755 -0.31983800949408986
residual error: ||r||=||Sw-M||
0.032545316379094975
count, Q error, E[R], E[a]
4 0.00676793350699448 -0.011255511818016314 -0.31687830350125185
residual error: ||r||=||Sw-M||
0.003936582105414812
count, Q error, E[R], E[a]
5 0.003712387200216026 -0.011256095930413587 -0.31624155741496734
residual error: ||r||=||Sw-M||
0.05118553263000062
count, Q error, E[R], E[a]
6 0.005924607020686042 -0.011256156366358681 -0.31611185205521475
residual error: ||r||=||Sw-M||
0.01843353464570155
count, Q error, E[R], E[a]
7 0.004994228189113239 -0.011256159895230603 -0.3158954818760284
residual error: ||r||=||Sw-M||
0.05215894570290667
count, Q error, E[R], E[a]
8 0.007844619893797583 -0.011256166502819482 -0.3158954717469139
residual error: ||r||=||Sw-M||
0.009582736804318188
count, Q error, E[R], E[a]
9 0.007205324624781927 -0.011256166503175059 -0.31588501104986
residual error: ||r||=||Sw-M||
0.09191295361152506
count, Q error, E[R], E[a]
10 0.009332706132659145 -0.011256167445652998 -0.3158849936306659
residual error: ||r||=||Sw-M||
0.011861104001111181
count, Q error, E[R], E[a]
11 0.0097523115035347 -0.011256167445770394 -0.3158849936306659
residual error: ||r||=||Sw-M||
0.011861104001111181
count, Q error, E[R], E[a]
12 0.0 -0.011256167445770394 -0.3158849936306659
residual error: ||r||=||Sw-M||
0.01107945461552675
count, Q error, E[R], E[a]
0 8.308424059980203 -0.011648677295403144 -0.3371989660304638
residual error: ||r||=||Sw-M||
0.002260821760270106
count, Q error, E[R], E[a]
1 0.0014874111062833457 -0.01166613314636176 -0.35843339317492284
residual error: ||r||=||Sw-M||
0.0002337796172079636
count, Q error, E[R], E[a]
2 0.0005437796148675827 -0.01167388990076642 -0.35999024014771974
residual error: ||r||=||Sw-M||
0.0034151873742206715
count, Q error, E[R], E[a]
3 0.00043464798039834305 -0.011673987813306176 -0.3615291016009157
residual error: ||r||=||Sw-M||
0.010313201315862226
count, Q error, E[R], E[a]
4 0.001134678936766863 -0.011674209379739123 -0.36123884878038054
residual error: ||r||=||Sw-M||
0.007654569292979012
count, Q error, E[R], E[a]
5 0.0007313966823649586 -0.01167422508722269 -0.36120893880297755
residual error: ||r||=||Sw-M||
0.007632622984626769
count, Q error, E[R], E[a]
6 0.0015740832443468884 -0.011674237180505934 -0.3612217823600635
residual error: ||r||=||Sw-M||
0.01579079005727014
count, Q error, E[R], E[a]
7 0.0017721226013356329 -0.011674238713134905 -0.3605540867343979
residual error: ||r||=||Sw-M||
0.015218478557386735
count, Q error, E[R], E[a]
8 0.000600882856276566 -0.011674245724426087 -0.36055430570678176
residual error: ||r||=||Sw-M||
0.013582656017884148
count, Q error, E[R], E[a]
9 0.0005963750385111536 -0.011674245741501589 -0.36054114669874554
residual error: ||r||=||Sw-M||
0.005155395991767151
count, Q error, E[R], E[a]
10 0.0017006765849908023 -0.01167424581771531 -0.36052030355142256
residual error: ||r||=||Sw-M||
0.008547334278923499
count, Q error, E[R], E[a]
11 0.0011668102766316567 -0.011674249113264241 -0.36016822796015774
residual error: ||r||=||Sw-M||
0.011232844522148126
count, Q error, E[R], E[a]
12 0.0017399139638799566 -0.01167425313706908 -0.36016822796015774
residual error: ||r||=||Sw-M||
0.011232844522148126
count, Q error, E[R], E[a]
13 0.0 -0.01167425313706908 -0.36016822796015774
residual error: ||r||=||Sw-M||
0.00881436497061759
count, Q error, E[R], E[a]
0 8.81750983270341 -0.01137835848894095 -0.344339873433444
residual error: ||r||=||Sw-M||
0.0031793292683202044
count, Q error, E[R], E[a]
1 0.0008111826801460261 -0.011384580881591427 -0.34030826623955346
residual error: ||r||=||Sw-M||
0.001018441720011474
count, Q error, E[R], E[a]
2 0.00030801966006539143 -0.011385647778781129 -0.3350451703084665
residual error: ||r||=||Sw-M||
0.00032768620487923477
count, Q error, E[R], E[a]
3 0.00017128952969979057 -0.01138745419271504 -0.3326120165105857
residual error: ||r||=||Sw-M||
0.000745017637374385
count, Q error, E[R], E[a]
4 0.0001031691405785234 -0.0113878647817308 -0.3329911676487734
residual error: ||r||=||Sw-M||
0.00014382885228623085
count, Q error, E[R], E[a]
5 5.8868075391082455e-05 -0.011387926312370074 -0.33333967723271574
residual error: ||r||=||Sw-M||
0.00015942965912185618
count, Q error, E[R], E[a]
6 2.3585801977643537e-05 -0.01138795141496112 -0.33331940226022005
residual error: ||r||=||Sw-M||
0.0001412325745683
count, Q error, E[R], E[a]
7 2.902364985237458e-05 -0.011387951590142244 -0.33322543574057895
residual error: ||r||=||Sw-M||
0.00032551728728061585
count, Q error, E[R], E[a]
8 4.174694963846626e-05 -0.011387952596486986 -0.33312356016917205
residual error: ||r||=||Sw-M||
0.0001505385583451534
count, Q error, E[R], E[a]
9 3.93745823990431e-05 -0.011387954594532342 -0.33312092791498926
residual error: ||r||=||Sw-M||
0.00012816946726594325
count, Q error, E[R], E[a]
10 1.9375977009488208e-05 -0.011387954611257651 -0.33312092814175814
residual error: ||r||=||Sw-M||
0.00021710424212569322
count, Q error, E[R], E[a]
11 2.9351543954557938e-05 -0.011387954611257902 -0.3331209294609884
residual error: ||r||=||Sw-M||
0.000236240297046829
count, Q error, E[R], E[a]
12 1.701038124387447e-05 -0.011387954611261747 -0.3331209294609884
residual error: ||r||=||Sw-M||
0.000236240297046829
count, Q error, E[R], E[a]
13 0.0 -0.011387954611261747 -0.3331209294609884
residual error: ||r||=||Sw-M||
8.249894580468851e-05
count, Q error, E[R], E[a]
0 9.315769896855892 -0.011183156122735233 -0.3228629844330949
residual error: ||r||=||Sw-M||
0.025394342521748055
count, Q error, E[R], E[a]
1 0.0027048106675452103 -0.011193479063074956 -0.3227793243113467
residual error: ||r||=||Sw-M||
0.06871549856962189
count, Q error, E[R], E[a]
2 0.006514167253524003 -0.01119355140233322 -0.3221847012719995
residual error: ||r||=||Sw-M||
0.004470619309825517
count, Q error, E[R], E[a]
3 0.007122976162119142 -0.01119404062187649 -0.32116201144926537
residual error: ||r||=||Sw-M||
0.008161503544499633
count, Q error, E[R], E[a]
4 0.001503717720595522 -0.011194836817150118 -0.3110201513917109
residual error: ||r||=||Sw-M||
0.002653657409772685
count, Q error, E[R], E[a]
5 0.0014407682802173266 -0.011201566603874547 -0.31096863914535117
residual error: ||r||=||Sw-M||
0.005082839700205039
count, Q error, E[R], E[a]
6 0.0004574053146408596 -0.011201600859898264 -0.309910508305397
residual error: ||r||=||Sw-M||
0.0007336438489074241
count, Q error, E[R], E[a]
7 0.0005400734889049969 -0.011202174513309526 -0.3082238588393766
residual error: ||r||=||Sw-M||
0.0006527816870721333
count, Q error, E[R], E[a]
8 0.00015295204061633195 -0.011202727085504452 -0.30678020316772564

residual error: ||r||=||Sw-M||
0.010988386438905526
count, Q error, E[R], E[a]
9 0.0011671233889318344 -0.011203058046953738 -0.3046492200090265
residual error: ||r||=||Sw-M||
0.007227526419270682
count, Q error, E[R], E[a]
10 0.001735050277674186 -0.011203595377091488 -0.2929720274557239
residual error: ||r||=||Sw-M||
0.024516796914529498
count, Q error, E[R], E[a]
11 0.001978381401777481 -0.011206537002986256 -0.2929648130268663
residual error: ||r||=||Sw-M||
0.015601743998120515
count, Q error, E[R], E[a]
12 0.0017770979798986493 -0.011206538068954419 -0.29289185998705625
residual error: ||r||=||Sw-M||
0.040684829647864115
count, Q error, E[R], E[a]
13 0.0034889700618370087 -0.011206539809694764 -0.29290528245340625
residual error: ||r||=||Sw-M||
0.09237278894002196
count, Q error, E[R], E[a]
14 0.013731953031973635 -0.01120655527562437 -0.29289833984273367
residual error: ||r||=||Sw-M||
0.016236605771745946
count, Q error, E[R], E[a]
15 0.009084380921888426 -0.011206555408135525 -0.29285539130067856
residual error: ||r||=||Sw-M||
0.02691775946428913
count, Q error, E[R], E[a]
16 0.003042800355346812 -0.011206556460022625 -0.29284881820829567
residual error: ||r||=||Sw-M||
0.06116014363288406
count, Q error, E[R], E[a]
17 0.008466268870383266 -0.011206556592348657 -0.2927274963364882
residual error: ||r||=||Sw-M||
0.02382030243474209
count, Q error, E[R], E[a]
18 0.008089680721579753 -0.011206564648024083 -0.2927589291878413
residual error: ||r||=||Sw-M||
0.002610772675164732
count, Q error, E[R], E[a]
19 0.0026248060540883986 -0.0112065698662029 -0.29259688275758244
residual error: ||r||=||Sw-M||
0.007798189399356774
count, Q error, E[R], E[a]
20 0.0014907572872316132 -0.011206573806795354 -0.29269829492676924
residual error: ||r||=||Sw-M||
0.027786021278655253
count, Q error, E[R], E[a]
21 0.0027337032130477154 -0.011206577504971298 -0.2927047412690043
residual error: ||r||=||Sw-M||
0.033185613525725875
count, Q error, E[R], E[a]
22 0.005691350692334063 -0.01120657780348517 -0.29273827061805435
residual error: ||r||=||Sw-M||
0.02461708391472046
count, Q error, E[R], E[a]
23 0.001177038803165029 -0.011206578998340724 -0.2926175589943982
residual error: ||r||=||Sw-M||
0.022934722967546103
count, Q error, E[R], E[a]
24 0.0052262755421768016 -0.011206579414691577 -0.2926473780042782
residual error: ||r||=||Sw-M||
0.021489246366181214
count, Q error, E[R], E[a]
25 0.002648217787645424 -0.01120657986481546 -0.2926419790529935
residual error: ||r||=||Sw-M||
0.018906428140110763
count, Q error, E[R], E[a]
26 0.001355156879854316 -0.011206580804244904 -0.2926190157929943
residual error: ||r||=||Sw-M||
0.032826205473770576
count, Q error, E[R], E[a]
27 0.004881728653178445 -0.011206582111731521 -0.29266574649337335
residual error: ||r||=||Sw-M||
0.014530587330218697
count, Q error, E[R], E[a]
28 0.0043205833333576915 -0.011206583836561283 -0.2927605087073914
residual error: ||r||=||Sw-M||
0.02515944626485698
count, Q error, E[R], E[a]
29 0.0023607432697693885 -0.011206586867639496 -0.29278114716879894
residual error: ||r||=||Sw-M||
0.01594993503685893
count, Q error, E[R], E[a]
30 0.004240292813502712 -0.011206587106320765 -0.29291326908698884
residual error: ||r||=||Sw-M||
0.04119824309266852
count, Q error, E[R], E[a]
31 0.004389279618057567 -0.011206590579817098 -0.2929058390911524
residual error: ||r||=||Sw-M||
0.017270087811584258
count, Q error, E[R], E[a]
32 0.004918878642237601 -0.011206590642876428 -0.29289188663311905
residual error: ||r||=||Sw-M||
0.023184674829475033
count, Q error, E[R], E[a]
33 0.0034234645991401648 -0.011206591268701174 -0.29288674471393494
residual error: ||r||=||Sw-M||
0.0175043105508068
count, Q error, E[R], E[a]
34 0.0028191484834663816 -0.011206591327023967 -0.2928921358782294
residual error: ||r||=||Sw-M||
0.024550779657227382
count, Q error, E[R], E[a]
35 0.0034409948397239438 -0.011206591640202901 -0.29278239572654163
residual error: ||r||=||Sw-M||
0.02849373697530736
count, Q error, E[R], E[a]
36 0.004713574400545503 -0.011206593192356774 -0.2927636715693451
residual error: ||r||=||Sw-M||
0.011774348736302601
count, Q error, E[R], E[a]
37 0.0030984969560371705 -0.011206593340342845 -0.2927716379788669
residual error: ||r||=||Sw-M||
0.04485256559480961
count, Q error, E[R], E[a]
38 0.0042322500004101805 -0.011206593632716224 -0.29274973715195907
residual error: ||r||=||Sw-M||
0.029640586969040254
count, Q error, E[R], E[a]
39 0.002126119020988307 -0.011206594095606539 -0.2927391179125193
residual error: ||r||=||Sw-M||
0.024288459504918373
count, Q error, E[R], E[a]
40 0.0016526154431711158 -0.011206594394073517 -0.292741866123253
residual error: ||r||=||Sw-M||
0.02588855758230612
count, Q error, E[R], E[a]
41 0.001702902524311374 -0.011206594495936546 -0.2927469245525387
residual error: ||r||=||Sw-M||
0.022594550670978774
count, Q error, E[R], E[a]
42 0.003438962226017555 -0.011206594601190159 -0.2927429400412853
residual error: ||r||=||Sw-M||
0.03430550851243453
count, Q error, E[R], E[a]
43 0.006079613655229907 -0.011206594619277898 -0.29274155276819397
residual error: ||r||=||Sw-M||
0.04303889992532922
count, Q error, E[R], E[a]
44 0.007881172944094584 -0.011206594633501439 -0.2927434765433891
residual error: ||r||=||Sw-M||
0.0937814536127156
count, Q error, E[R], E[a]
45 0.01066606899231417 -0.011206594646018685 -0.29275315894606624
residual error: ||r||=||Sw-M||
0.031505163093099496
count, Q error, E[R], E[a]
46 0.010272468514546242 -0.01120659479850695 -0.29275298650752846
residual error: ||r||=||Sw-M||
0.10060574049030288
count, Q error, E[R], E[a]
47 0.011087420333624283 -0.011206594798540838 -0.2927501780598955
residual error: ||r||=||Sw-M||
0.026627907178418835
count, Q error, E[R], E[a]
48 0.009399276801610379 -0.01120659482085598 -0.292777383161771
residual error: ||r||=||Sw-M||
0.022656883814082224
count, Q error, E[R], E[a]
49 0.0027061778886346005 -0.011206595436664217 -0.29277190534564135
residual error: ||r||=||Sw-M||
0.03991304614424011
count, Q error, E[R], E[a]
50 0.006422283999974683 -0.011206595693158043 -0.29275552414171097
residual error: ||r||=||Sw-M||
0.03064434282032846
count, Q error, E[R], E[a]
51 0.005217985277743961 -0.011206595908531278 -0.2927544741242797
residual error: ||r||=||Sw-M||
0.04339919482236615
count, Q error, E[R], E[a]
52 0.006062319595133859 -0.01120659602481971 -0.2927696146708672
residual error: ||r||=||Sw-M||
0.06570719451704458
count, Q error, E[R], E[a]
53 0.00912906390534542 -0.011206596369123786 -0.29275361380783044
residual error: ||r||=||Sw-M||
0.08396245289692401
count, Q error, E[R], E[a]
54 0.00631398544694559 -0.011206596636023755 -0.2927532278247091
residual error: ||r||=||Sw-M||
0.04469173513722917
count, Q error, E[R], E[a]
55 0.011548744747403056 -0.011206596646167636 -0.2927532278247091
residual error: ||r||=||Sw-M||
0.04469173513722917
count, Q error, E[R], E[a]
56 0.0 -0.011206596646167636 -0.2927532278247091
residual error: ||r||=||Sw-M||
0.020475818411690508
count, Q error, E[R], E[a]
0 9.84162875880131 -0.01155151619412568 -0.30449741205310843
residual error: ||r||=||Sw-M||
0.004274171709172197
count, Q error, E[R], E[a]
1 0.001632830865796195 -0.01156118111653317 -0.31428827733213316
residual error: ||r||=||Sw-M||
0.004111017293631496
count, Q error, E[R], E[a]
2 0.0007116340523097212 -0.011567890527400494 -0.33004349335737765
residual error: ||r||=||Sw-M||
0.003593807857415982
count, Q error, E[R], E[a]
3 0.0005205570463081578 -0.011573911495189148 -0.32810899107793257
residual error: ||r||=||Sw-M||
0.001020668116987622
count, Q error, E[R], E[a]
4 0.000379006200507362 -0.01157408428276224 -0.3285432036479084
residual error: ||r||=||Sw-M||
0.0014171684931226383
count, Q error, E[R], E[a]
5 0.00023794288036338939 -0.011574146242236837 -0.329257138792227
residual error: ||r||=||Sw-M||
0.0012076736843565433
count, Q error, E[R], E[a]
6 0.00026351608742791716 -0.011574257394141899 -0.333030504250783
residual error: ||r||=||Sw-M||
0.0015108738034742735
count, Q error, E[R], E[a]
7 0.0002774556795826873 -0.011574800881326442 -0.3327876685829181
residual error: ||r||=||Sw-M||
0.0024887014135367524
count, Q error, E[R], E[a]
8 0.00024050634231324264 -0.011574828242519646 -0.33283587137748855
residual error: ||r||=||Sw-M||
0.0011332905692691807
count, Q error, E[R], E[a]
9 0.0004254039159905557 -0.011574830462165972 -0.3335205701132792

residual error: ||r||=||Sw-M||
0.0024515408158344385
count, Q error, E[R], E[a]
10 0.0004278633108513567 -0.011574853041324502 -0.33343431394612494
residual error: ||r||=||Sw-M||
0.0023921696001492003
count, Q error, E[R], E[a]
11 0.0003259623092063793 -0.011574853670854594 -0.3334289896431571
residual error: ||r||=||Sw-M||
0.00667394802935473
count, Q error, E[R], E[a]
12 0.0007435484657782206 -0.011574853686081923 -0.3334281746785153
residual error: ||r||=||Sw-M||
0.007066002765180291
count, Q error, E[R], E[a]
13 0.0006530633482579014 -0.01157485368902071 -0.33340432481984544
residual error: ||r||=||Sw-M||
0.005713235712635963
count, Q error, E[R], E[a]
14 0.0010387744633322013 -0.011574853760295685 -0.3334033729691346
residual error: ||r||=||Sw-M||
0.002607176368327692
count, Q error, E[R], E[a]
15 0.0006038403008173356 -0.011574853764351311 -0.33340549551966403
residual error: ||r||=||Sw-M||
0.0029643559589351376
count, Q error, E[R], E[a]
16 0.0005316236484042865 -0.01157485389545484 -0.3333910881273906
residual error: ||r||=||Sw-M||
0.009697369231445076
count, Q error, E[R], E[a]
17 0.0011173016456077513 -0.011574854155932283 -0.3334140677187178
residual error: ||r||=||Sw-M||
0.007599877225588321
count, Q error, E[R], E[a]
18 0.0014857284543326006 -0.01157485429881687 -0.3334140677187178
residual error: ||r||=||Sw-M||
0.007599877225588321
count, Q error, E[R], E[a]
19 0.0 -0.01157485429881687 -0.3334140677187178
residual error: ||r||=||Sw-M||
0.0014929992513654001
count, Q error, E[R], E[a]
0 10.344585504212196 -0.011272104895527477 -0.3281997481721605
residual error: ||r||=||Sw-M||
0.00226421484161871
count, Q error, E[R], E[a]
1 0.00048461714559193376 -0.011273954827625136 -0.32232582883776534
residual error: ||r||=||Sw-M||
0.14523076301532947
count, Q error, E[R], E[a]
2 0.013375433231068344 -0.011275322791672544 -0.32220061269636646
residual error: ||r||=||Sw-M||
0.3617447906114805
count, Q error, E[R], E[a]
3 0.04678851643243306 -0.011275341803692121 -0.32179275902412985
residual error: ||r||=||Sw-M||
0.04598573891515357
count, Q error, E[R], E[a]
4 0.02933256138268234 -0.011275389952985607 -0.31882311697693594
residual error: ||r||=||Sw-M||
0.001879886553311296
count, Q error, E[R], E[a]
5 0.004089793409854638 -0.011275572838546162 -0.31626192351615645
residual error: ||r||=||Sw-M||
0.006394385551168456
count, Q error, E[R], E[a]
6 0.0009061869866754231 -0.011275793879827743 -0.3161462100687818
residual error: ||r||=||Sw-M||
0.010956600617722786
count, Q error, E[R], E[a]
7 0.0015087951512573667 -0.011275795897223514 -0.3157936596385463
residual error: ||r||=||Sw-M||
0.0144932988256401
count, Q error, E[R], E[a]
8 0.0014324257055732708 -0.011275807333573704 -0.31580949173128453
residual error: ||r||=||Sw-M||
0.007274474166049826
count, Q error, E[R], E[a]
9 0.002174676035730522 -0.011275808491659692 -0.315808295884697
residual error: ||r||=||Sw-M||
0.00658502591046205
count, Q error, E[R], E[a]
10 0.0006715755714930755 -0.011275808815798209 -0.3158123391520413
residual error: ||r||=||Sw-M||
0.010672824414543623
count, Q error, E[R], E[a]
11 0.0016675289752829526 -0.011275808934388341 -0.3158123391520413
residual error: ||r||=||Sw-M||
0.010672824414543623
count, Q error, E[R], E[a]
12 0.0 -0.011275808934388341 -0.3158123391520413
residual error: ||r||=||Sw-M||
0.012068090245828913
count, Q error, E[R], E[a]
0 10.865449946903416 -0.011682774137587493 -0.3469281104155717
residual error: ||r||=||Sw-M||
0.0015803486484168821
count, Q error, E[R], E[a]
1 0.0007856967290949374 -0.011705727129351081 -0.3558043610801684
residual error: ||r||=||Sw-M||
0.004437869584561002
count, Q error, E[R], E[a]
2 0.0005812813011450531 -0.011710378184989434 -0.35904286954813625
residual error: ||r||=||Sw-M||
0.0024955647174569806
count, Q error, E[R], E[a]
3 0.00035147669738436194 -0.011711354449114286 -0.36303462325032515
residual error: ||r||=||Sw-M||
0.0006705043777305657
count, Q error, E[R], E[a]
4 0.00033209989575141085 -0.011711911819019196 -0.36251095441142744
residual error: ||r||=||Sw-M||
0.010960394917716002
count, Q error, E[R], E[a]
5 0.0011431649583016751 -0.011711974219668766 -0.3625922439241893
residual error: ||r||=||Sw-M||
0.005810690641970051
count, Q error, E[R], E[a]
6 0.0016935798726720518 -0.011711989881877646 -0.3625644408826571
residual error: ||r||=||Sw-M||
0.012521841722602614
count, Q error, E[R], E[a]
7 0.0011603940202758543 -0.011711991977809876 -0.3625701881952553
residual error: ||r||=||Sw-M||
0.015735891839529288
count, Q error, E[R], E[a]
8 0.0025509042595417528 -0.011711992138388375 -0.3625701881952553
residual error: ||r||=||Sw-M||
0.015735891839529288
count, Q error, E[R], E[a]
9 0.0 -0.011711992138388375 -0.3625701881952553
residual error: ||r||=||Sw-M||
0.0025973924466972886
count, Q error, E[R], E[a]
0 11.376901008031659 -0.01147797597674788 -0.33975966331374885
residual error: ||r||=||Sw-M||
0.0016719902927263032
count, Q error, E[R], E[a]
1 0.0004525087810175382 -0.011490562703692515 -0.33177872521555535
residual error: ||r||=||Sw-M||
0.0024929018845163717
count, Q error, E[R], E[a]
2 0.0003846459182722035 -0.011493109580431664 -0.3292306963985839
residual error: ||r||=||Sw-M||
0.0006915633576171713
count, Q error, E[R], E[a]
3 0.00035505251013600463 -0.011493529298611927 -0.3293496326672338
residual error: ||r||=||Sw-M||
0.004261866352141035
count, Q error, E[R], E[a]
4 0.0003673509428624925 -0.01149365675714261 -0.3293223058890348
residual error: ||r||=||Sw-M||
0.003651428347038552
count, Q error, E[R], E[a]
5 0.0006335152919784548 -0.011493692742214612 -0.32890284800363356
residual error: ||r||=||Sw-M||
0.014689659819448192
count, Q error, E[R], E[a]
6 0.0016253920410506714 -0.011493781617919516 -0.3288688347585678
residual error: ||r||=||Sw-M||
0.005905391984826962
count, Q error, E[R], E[a]
7 0.0012852735141880922 -0.011493783839597139 -0.3288620449286779
residual error: ||r||=||Sw-M||
0.009063989374600205
count, Q error, E[R], E[a]
8 0.00041969021971259606 -0.011493785299503649 -0.3288610058232104
residual error: ||r||=||Sw-M||
0.005930831419313065
count, Q error, E[R], E[a]
9 0.0012144487235031707 -0.011493785642112921 -0.3288424043041186
residual error: ||r||=||Sw-M||
0.004273218485420092
count, Q error, E[R], E[a]
10 0.00018616472063426963 -0.011493785756908844 -0.3288349724033117
residual error: ||r||=||Sw-M||
0.005394023605394595
count, Q error, E[R], E[a]
11 0.0002697391139099396 -0.011493785873428619 -0.32852742673067903
residual error: ||r||=||Sw-M||
0.018395181224455995
count, Q error, E[R], E[a]
12 0.002035248964672354 -0.011493788926922769 -0.3284867991841376
residual error: ||r||=||Sw-M||
0.0033071315881596734
count, Q error, E[R], E[a]
13 0.0019986807244085714 -0.011493789614443351 -0.3284850145086266
residual error: ||r||=||Sw-M||
0.008886457119116873
count, Q error, E[R], E[a]
14 0.0010602626631860822 -0.011493789872901537 -0.3284654683459579
residual error: ||r||=||Sw-M||
0.008604543470220457
count, Q error, E[R], E[a]
15 0.0008596911388189688 -0.011493790183720232 -0.3284654683459579
residual error: ||r||=||Sw-M||
0.008604543470220457
count, Q error, E[R], E[a]
16 0.0 -0.011493790183720232 -0.3284654683459579
residual error: ||r||=||Sw-M||
0.0017643528912718572
count, Q error, E[R], E[a]
0 11.880157173670607 -0.01128349043381829 -0.3201412401896729
residual error: ||r||=||Sw-M||
0.002945597458857236
count, Q error, E[R], E[a]
1 0.00038445434144940586 -0.011286857366951463 -0.32013677503288507
residual error: ||r||=||Sw-M||
0.003912248214958779
count, Q error, E[R], E[a]
2 0.0006797604645233148 -0.011286859532339119 -0.320103162817908
residual error: ||r||=||Sw-M||
0.0006538662798273026
count, Q error, E[R], E[a]
3 0.00044488049044475765 -0.011286861737626925 -0.3200736365829581
residual error: ||r||=||Sw-M||
8.420967457575923e-05
count, Q error, E[R], E[a]
4 0.00012276174894303938 -0.011286866964767345 -0.3199034280714519
residual error: ||r||=||Sw-M||
0.00024664365236480557
count, Q error, E[R], E[a]
5 2.0817962259828277e-05 -0.011286880727046426 -0.31945402822854724
residual error: ||r||=||Sw-M||
0.002543150530074294
count, Q error, E[R], E[a]
6 0.00021809457089182142 -0.011286924172826606 -0.31866413161271867

residual error: ||r||=||Sw-M||
0.0072423870053955385
count, Q error, E[R], E[a]
7 0.001100242493296044 -0.011287044811734552 -0.31732460806806023
residual error: ||r||=||Sw-M||
0.0016868608868680606
count, Q error, E[R], E[a]
8 0.0008754515332217769 -0.011287181085645222 -0.3135422395680505
residual error: ||r||=||Sw-M||
0.01142178203379157
count, Q error, E[R], E[a]
9 0.0014677100222630797 -0.011287782025755602 -0.3120814069660814
residual error: ||r||=||Sw-M||
0.009736772817420252
count, Q error, E[R], E[a]
10 0.0011488703859781012 -0.011288083872358709 -0.31052271572685336
residual error: ||r||=||Sw-M||
0.03020887611654973
count, Q error, E[R], E[a]
11 0.0037732432910729035 -0.011288321521750098 -0.30970334350049156
residual error: ||r||=||Sw-M||
0.020091624276342618
count, Q error, E[R], E[a]
12 0.001316214481424766 -0.011288392775722045 -0.30929394934312887
residual error: ||r||=||Sw-M||
0.0164815103261339
count, Q error, E[R], E[a]
13 0.002610711405433097 -0.011288471539966368 -0.3093119497369141

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
<ipython-input-15-80c5e371c109> in <module>()
     19 
     20         # Check partition of unity
---> 21         h = np.dot(data_mat_t[t],np.ones(np.shape(data_mat_t[t])[1]))
     22         if np.sum(h) != N_MC:
     23             print("error: loss of partition of unity")

<__array_function__ internals> in dot(*args, **kwargs)

KeyboardInterrupt:
In [ ]:
a_star_exact = []
a_star_approx = []
for t in range(1, T-1):
    a_star_exact.append((mu[t] - r) / (2 * lmbda * var[t]))
    a_star_approx.append(np.mean(a_opt[:, t]))
In [ ]:
plt.plot(a_star_approx)
plt.plot(a_star_exact, color='red')
plt.xlabel('time')
plt.ylabel('action');
In [ ]:
a_star_exact = (mu[t] - r) / (2 * lmbda * var[t])
In [ ]:
a_star_exact
In [ ]:
negative_reward(mu[t], var[t], a_star_exact, r, lmbda)
In [ ]:
a_ = np.arange(-1, 1, 0.01)
plt.plot(a_, negative_reward(mu[t], var[t], a_, r, lmbda))
plt.xlabel('S')
plt.ylabel('R');
In [ ]:
plt.plot(a_mean)
plt.plot(x_grid, a_star_)
plt.plot(x_grid, np.ones(len(x_grid)) * a_star_exact, color='black')
plt.ylim([-1, 1])

optimal action depends on lambda

In [ ]: