I have the following system, that I am trying to solve in OpenMDAO, as a part of a larger system:
C*x = F1(x) + F2(x)
where F1 and F2 are some responses (6x1 vectors) computed for the given x (6x1 vector) in separate explicit components (F1, F2 being the outputs of the components, and x being the input in both components). The relationship between F and x cannot be described analytically. C does not depend on x, and is a 6x6 matrix the inverse of which happens to be singular.
What would be the best way to solve this system for x in OpenMDAO?
You have the right instincts here, that this is am implicit problem. You only need to rearrange things to form a residual --- i.e. you want the left hand side to be equal to 0. Once in that form, it becomes more clear how to handle this in OpenMDAO.
Option 1) restructure the components a little and use a BalanceComp
0 = F1(x)/Cx - (F2(x)/Cx-1)
You may not want to express it in that form, but lets assume for now that you were willing to modify you two components that output F1 and F2 into the above form.
Then you could add the BalanceComp to create the necessary residual equation. Like you pointed out, the residual owning component (the balance comp) by itself is singular. You need a group with all three components in it, and you need to use the Newton solver at that group level to create a non singular Jacobian for the solver to work with.
Option 2) Keep F1 and F2 the way they are and define a custom balance comp
What if you didn't want to rearrange those components to us the balance comp? Then you will need to write you own simple implicit component instead. You still need to move the C*x term over to the right hand side in order to form a proper residual though.
import openmdao.api as om
class CustomBalance(om.ImplicitComponent):
def setup(self):
self.add_input('F1')
self.add_input('F2')
self.add_input('C')
self.add_output('x')
self.declare_partials('*', '*', method='cs')
def apply_nonlinear(self, inputs, outputs, residuals):
residuals['x'] = inputs['F1'] + inputs['F2'] - inputs['C']*outputs['x']
prob = om.Problem()
ec1 = om.ExecComp('F1=2*x**2')
ec2 = om.ExecComp('F2=-x/2')
prob.model.add_subsystem(name='func1', subsys=ec1, promotes=['x', 'F1'])
prob.model.add_subsystem(name='func2', subsys=ec2, promotes=['x', 'F2'])
prob.model.add_subsystem(name='balance', subsys=CustomBalance(), promotes=['x', 'F1', 'F2', 'C'])
prob.model.linear_solver = om.DirectSolver(assemble_jac=True)
prob.model.nonlinear_solver = om.NewtonSolver(solve_subsystems=False, maxiter=100, iprint=2)
prob.setup()
prob.set_val('C', 2.55)
# initial guess
prob['x'] = 1.0
prob.run_model()
print(prob.get_val('balance.x'))
That gives you a model structure that looks like this:
And a convergence history like this:
NL: Newton 0 ; 1.88480768 1
NL: Newton 1 ; 2.4432133 1.29626663
NL: Newton 2 ; 0.413841411 0.219566916
NL: Newton 3 ; 0.0271563582 0.014408026
NL: Newton 4 ; 0.000154934262 8.22016293e-05
NL: Newton 5 ; 5.16021093e-09 2.73779175e-09
NL: Newton 6 ; 4.4408921e-16 2.35615131e-16
NL: Newton Converged
[1.525]
Related
I'm plotting this dataset and making a logarithmic fit, but, for some reason, the fit seems to be strongly wrong, at some point I got a good enough fit, but then I re ploted and there were that bad fit. At the very beginning there were a 0.0 0.0076 but I changed that to 0.001 0.0076 to avoid the asymptote.
I'm using (not exactly this one for the image above but now I'm testing with this one and there is that bad fit as well) this for the fit
f(x) = a*log(k*x + b)
fit = fit f(x) 'R_B/R_B.txt' via a, k, b
And the output is this
Also, sometimes it says 7 iterations were as is the case shown in the screenshot above, others only 1, and when it did the "correct" fit, it did like 35 iterations or something and got a = 32 if I remember correctly
Edit: here is again the good one, the plot I got is this one. And again, I re ploted and get that weird fit. It's curious that if there is the 0.0 0.0076 when the good fit it's about to be shown, gnuplot says "Undefined value during function evaluation", but that message is not shown when I'm getting the bad one.
Do you know why do I keep getting this inconsistence? Thanks for your help
As I already mentioned in comments the method of fitting antiderivatives is much better than fitting derivatives because the numerical calculus of derivatives is strongly scattered when the data is slightly scatered.
The principle of the method of fitting an integral equation (obtained from the original equation to be fitted) is explained in https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales . The application to the case of y=a.ln(c.x+b) is shown below.
Numerical calculus :
In order to get even better result (according to some specified criteria of fitting) one can use the above values of the parameters as initial values for iterarive method of nonlinear regression implemented in some convenient software.
NOTE : The integral equation used in the present case is :
NOTE : On the above figure one can compare the result with the method of fitting an integral equation to the result with the method of fitting with derivatives.
Acknowledgements : Alex Sveshnikov did a very good work in applying the method of regression with derivatives. This allows an interesting and enlightening comparison. If the goal is only to compute approximative values of parameters to be used in nonlinear regression software both methods are quite equivalent. Nevertheless the method with integral equation appears preferable in case of scattered data.
UPDATE (After Alex Sveshnikov updated his answer)
The figure below was drawn in using the Alex Sveshnikov's result with further iterative method of fitting.
The two curves are almost indistinguishable. This shows that (in the present case) the method of fitting the integral equation is almost sufficient without further treatment.
Of course this not always so satisfying. This is due to the low scatter of the data.
In ADDITION , answer to a question raised in comments by CosmeticMichu :
The problem here is that the fit algorithm starts with "wrong" approximations for parameters a, k, and b, so during the minimalization it finds a local minimum, not the global one. You can improve the result if you provide the algorithm with starting values, which are close to the optimal ones. For example, let's start with the following parameters:
gnuplot> a=47.5087
gnuplot> k=0.226
gnuplot> b=1.0016
gnuplot> f(x)=a*log(k*x+b)
gnuplot> fit f(x) 'R_B.txt' via a,k,b
....
....
....
After 40 iterations the fit converged.
final sum of squares of residuals : 16.2185
rel. change during last iteration : -7.6943e-06
degrees of freedom (FIT_NDF) : 18
rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 0.949225
variance of residuals (reduced chisquare) = WSSR/ndf : 0.901027
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = 35.0415 +/- 2.302 (6.57%)
k = 0.372381 +/- 0.0461 (12.38%)
b = 1.07012 +/- 0.02016 (1.884%)
correlation matrix of the fit parameters:
a k b
a 1.000
k -0.994 1.000
b 0.467 -0.531 1.000
The resulting plot is
Now the question is how you can find "good" initial approximations for your parameters? Well, you start with
If you differentiate this equation you get
or
The left-hand side of this equation is some constant 'C', so the expression in the right-hand side should be equal to this constant as well:
In other words, the reciprocal of the derivative of your data should be approximated by a linear function. So, from your data x[i], y[i] you can construct the reciprocal derivatives x[i], (x[i+1]-x[i])/(y[i+1]-y[i]) and the linear fit of these data:
The fit gives the following values:
C*k = 0.0236179
C*b = 0.106268
Now, we need to find the values for a, and C. Let's say, that we want the resulting graph to pass close to the starting and the ending point of our dataset. That means, that we want
a*log(k*x1 + b) = y1
a*log(k*xn + b) = yn
Thus,
a*log((C*k*x1 + C*b)/C) = a*log(C*k*x1 + C*b) - a*log(C) = y1
a*log((C*k*xn + C*b)/C) = a*log(C*k*xn + C*b) - a*log(C) = yn
By subtracting the equations we get the value for a:
a = (yn-y1)/log((C*k*xn + C*b)/(C*k*x1 + C*b)) = 47.51
Then,
log(k*x1+b) = y1/a
k*x1+b = exp(y1/a)
C*k*x1+C*b = C*exp(y1/a)
From this we can calculate C:
C = (C*k*x1+C*b)/exp(y1/a)
and finally find the k and b:
k=0.226
b=1.0016
These are the values used above for finding the better fit.
UPDATE
You can automate the process described above with the following script:
# Name of the file with the data
data='R_B.txt'
# The coordinates of the last data point
xn=NaN
yn=NaN
# The temporary coordinates of a data point used to calculate a derivative
x0=NaN
y0=NaN
linearFit(x)=Ck*x+Cb
fit linearFit(x) data using (xn=$1,dx=$1-x0,x0=$1,$1):(yn=$2,dy=$2-y0,y0=$2,dx/dy) via Ck, Cb
# The coordinates of the first data point
x1=NaN
y1=NaN
plot data using (x1=$1):(y1=$2) every ::0::0
a=(yn-y1)/log((Ck*xn+Cb)/(Ck*x1+Cb))
C=(Ck*x1+Cb)/exp(y1/a)
k=Ck/C
b=Cb/C
f(x)=a*log(k*x+b)
fit f(x) data via a,k,b
plot data, f(x)
pause -1
I wanted to get beta coefficients in my panel data random effects regression model in Stata. But then I noticed that the option "beta" is not allowed in the xtreg command.
It made me think if it is probably wrong to want standardised coefficients in a random effects model?
my model looks something like this -
xtreg y x##z, re
You can manually get standardized coefficients by 0-1 standardizing your variables before the command:
foreach v of varlist x y z {
qui sum `v'
replace `v' = (`v'-`r(mean)') / `r(sd)'
xtreg y x##z, re
#DCTLib, do you recall this discussion below? You suggested a recursive equation, which was the right approach.
Cudd_PrintMinterm, accessing the individual minterms in the sum of products
Now, I am considering multistate reliability, where we can have either not fail or fail to n-1 different states, with n >= 2. Tulip-dd implements MDDs as described in:
https://github.com/tulip-control/dd/blob/master/doc.md#multi-valued-decision-diagrams-mdd
https://github.com/tulip-control/dd/issues/71
https://github.com/tulip-control/dd/issues/66
In the diagrams in the drawings below, we have defined an MDD declared by:
aut.declare_variable(x=(0,3))
u = aut.add_expr(‘x=i’)
Each value/state of the multi-value variable (MSV) x, x=0, x=1, x=2, or x=3 leads to a specific BDD as shown in the diagrams at the bottom, taking a four-state variable x as example here. The notation is that state 0 represents the normal state and x can fail to different states 1, 2, and 3. The failure probabilities are assigned in table below. In the BDDs below, we (and tulip as well) use the binary coding with two bits x_1 and x_0 to represent each state/value of the MSV. The least significant bit (LSB), i.e., x_0, is always the ancestor. Each of the BDD diagrams below is a representation of a specific value, or state.
To quantify the BDD of a specific state, i.e., the top node, we must know probabilities of binary variables x_0 and x_1 taking different branches (then or else) in the BDD. These branch probabilities are not given directly but need to be calculated according to the BDD structure.
The key here is that the child node probabilities and the branch probabilities of the parent node must be known prior to the calculation of the parent node probability.
In the previous BDD quantification, we knew the probabilities of branches from node x_1 to leaf nodes when calculating node x_1 probability. We did not need to know how node x_1 was connected to node x_0.
Now, for this four-state variable x, we need to know how node x_1 is connected to node x_0, the binary variable representing the least significant bit, to determine the probabilities of branches from node x_1 to leaf nodes. The question is how to implement this?
Here’s one attempt that falls short:
import numpy as np
from omega.symbolic import temporal as trl
aut = trl.Automaton()
# Example of function that returns branch probabilities
def prntr(v, pvars):
assert sum(pvars)==1.0,"Probs must add to 1!"
if (v.high).var == 'x_1':
if (v.low) == aut.true:
return pvars[1] + pvars[3], pvars[1]
else:
return pvars[1] + pvars[3], pvars[3]
if (v.low).var == 'x_1':
if (v.low).negated:
return pvars[1] + pvars[3], pvars[0]
else:
return pvars[1] + pvars[3], pvars[2]
aut.declare_variables(x=(0, 3))
u=aut.add_expr('x = 3')
pvars = np.array([0.1, 0.2, 0.3, 0.4])
prntr(u,pvars)
The key here is that the child node probabilities and the branch probabilities of the parent node must be known prior to the calculation of the parent node probability.
Yes, exactly. In this case, a fully recursive bottom-up computation, like normally done with BDDs, will not work for the reason that you wrote.
However, the approach will start to work again when you treat the variables that together form a state to be a block. So in your recursive function for the probability calculation, whenever you encounter a variable for a block, you treat the node and the successor nodes for the same state component as a block and only recurse when you encounter a node not belonging to the block.
Note that this approach requires that the variables for the state appear continuously in the variable ordering. For the CUDD library, you can constrain the automatic variable reordering to guarantee this.
The following code is a modification of yours implementing this idea:
#!/usr/bin/env python3
import numpy as np
from omega.symbolic import temporal as trl
aut = trl.Automaton()
# Example of function that returns branch probabilities
# Does not yet use result caching and does not yet support assigning probabilities
# to more than one state variable set
def prntr(v, pvars):
assert abs(sum(pvars)-1.0)<=0.0001,"Probs must add to 1!"
if v == aut.true:
return 1.0
elif v == aut.false:
return 0.0
if v.var in ["x_0","x_1"]:
thisSum = 0.0
# Compute probabilities
for i,p in enumerate(pvars):
# Find successor
# Assuming that x_2 always comes after x_1
thisV = v
negate = thisV.negated
if thisV.var == 'x_0':
if i & 1:
thisV = thisV.high
else:
thisV = thisV.low
negate = negate ^ thisV.negated
if thisV.var == 'x_1':
if i & 2:
thisV = thisV.high
else:
thisV = thisV.low
if negate:
thisSum += p*prntr(~thisV, pvars)
else:
thisSum += p*prntr(thisV, pvars)
return thisSum
# TODO: What is the semantics for variables outside of the current block?
return 0.5*prntr(v.high, pvars) + 0.5*prntr(v.low, pvars)
pvars = np.array([0.1, 0.2, 0.3, 0.4])
aut.declare_variables(x=(0, 3))
u= aut.add_expr('x = 0')
print(prntr(u,pvars))
u2 = aut.add_expr('x = 3') | aut.add_expr('x = 2')
print(prntr(u2,pvars))
Writing a python script to calc Implied Normal Vol ; in line with Jekel article (Industry Standard).
https://jaeckel.000webhostapp.com/ImpliedNormalVolatility.pdf
They say they are using a Generalized Incomplete Gamma Function Inverse.
For a call:
F(x)=v/(K - F) -> find x that makes this true
Where F is Inverse Incomplete Gamma Function
And x = (K - F)/(T*sqrt(T) ; v is the value of a call
for that x, IV is =(K-F)/x*sqrt(T)
Example I am working with:
F=40
X=38
T=100/365
v=5.25
Vol= 20%
Using the equations I should be able to backout Vol of 20%
Scipy has upper and lower Incomplete Gamma Function Inverse in their special functions.
Lower: scipy.special.gammaincinv(a, y) : {a must be positive param}
Upper: scipy.special.gammainccinv(a, y) : {a must be positive param}
Implementation:
SIG= sympy.symbols('SIG')
F=40
T=100/365
K=38
def Objective(sig):
SIG=sig
return(special.gammaincinv(.5,((F-K)**2)/(2*T*SIG**2))+special.gammainccinv(.5,((F-K)**2)/(2*T*SIG**2))+5.25/(K-F))
x=optimize.brentq(Objective, -20.00,20.00, args=(), xtol=1.48e-8, rtol=1.48e-8, maxiter=1000, full_output=True)
IV=(K-F)/x*T**.5
Print(IV)
I know I am wrong, but Where am I going wrong / how do I fix it and use what I read in the article ?
Did you also post this on the Quantitative Finance Stack Exchange? You may get a better response there.
This is not my field, but it looks like your main problem is that brentq requires the passed Objective function to return values with opposite signs when passed the -20 and 20 arguments. However, this will not end up happening because according to the scipy docs, gammaincinv and gammainccinv always return a value between 0 and infinity.
I'm not sure how to fix this, unfortunately. Did you try implementing the analytic solution (rather than iterative root finding) in the second part of the paper?
I understand how to render (two dimensional) "Escape Time Group" fractals (Julia and Mandelbrot), but I can't seem to get a Mobius Transformation or a Newton Basin rendered.
I'm trying to render them using the same method (by recursively using the polynomial equation on each pixel 'n' times), but I have a feeling these fractals are rendered using totally different methods. Mobius 'Transformation' implies that an image must already exist, and then be transformed to produce the geometry, and the Newton Basin seems to plot each point, not just points that fall into a set.
How are these fractals graphed? Are they graphed using the same iterative methods as the Julia and Mandelbrot?
Equations I'm Using:
Julia: Zn+1 = Zn^2 + C
Where Z is a complex number representing a pixel, and C is a complex constant (Correct).
Mandelbrot: Cn+1 = Cn^2 + Z
Where Z is a complex number representing a pixel, and C is the complex number (0, 0), and is compounded each step (The reverse of the Julia, correct).
Newton Basin: Zn+1 = Zn - (Zn^x - a) / (Zn^y - a)
Where Z is a complex number representing a pixel, x and y are exponents of various degrees, and a is a complex constant (Incorrect - creating a centered, eight legged 'line star').
Mobius Transformation: Zn+1 = (aZn + b) / (cZn + d)
Where Z is a complex number representing a pixel, and a, b, c, and d are complex constants (Incorrect, everything falls into the set).
So how are the Newton Basin and Mobius Transformation plotted on the complex plane?
Update: Mobius Transformations are just that; transformations.
"Every Möbius transformation is
a composition of translations,
rotations, zooms (dilations) and
inversions."
To perform a Mobius Transformation, a shape, picture, smear, etc. must be present already in order to transform it.
Now how about those Newton Basins?
Update 2: My math was wrong for the Newton Basin. The denominator at the end of the equation is (supposed to be) the derivative of the original function. The function can be understood by studying 'NewtonRoot.m' from the MIT MatLab source-code. A search engine can find it quite easily. I'm still at a loss as to how to graph it on the complex plane, though...
Newton Basin:
f(x) = x - f(x) / f'(x)
In Mandelbrot and Julia sets you terminate the inner loop if it exceeds a certain threshold as a measurement how fast the orbit "reaches" infinity
if(|z| > 4) { stop }
For newton fractals it is the other way round: Since the newton method is usually converging towards a certain value we are interested how fast it reaches its limit, which can be done by checking when the difference of two consecutive values drops below a certain value (usually 10^-9 is a good value)
if(|z[n] - z[n-1]| < epsilon) { stop }