Constraints on drifts¶
In [1]:
import gstlearn as gl
import gstlearn.plot as gp
import gstlearn.document as gdoc
import numpy as np
import matplotlib.pyplot as plt
gdoc.setNoScroll()
This tutorial has two essential taks:
- cross-check the calculation of the Log-Likelihood (by hand or with gstlearn)
- design the definition pattern for the linkage of the drift functions in the multivariable framework
Parameters¶
In [2]:
# Data
np.random.seed(123)
ndat = 100
ndim = 2
# Model
rangev = 0.2
sill = 1.
nugget = 0.1
In [3]:
# Z : vecteur des données
# Covmat : matrice de covariance
# drift : matrice de drift
# A et c permettent d'encoder les contraintes sur le vecteur des coefficients beta :
# sous la forme A * beta = c
def estimCoeff(Z,Covmat,drift,A=None,c=None):
if A is not None and c is not None:
if A.shape[0]!= len(c) or A.shape[1]!=drift.shape[1]:
return np.nan
invcovmat = np.linalg.inv(Covmat)
invu = np.linalg.inv(X.T@invcovmat@X)
estimatedCoeffs = invu@X.T@invcovmat@Z
if A is None or c is None :
return estimatedCoeffs
temp = invu@A.T@np.linalg.inv(A@invu@A.T)
return estimatedCoeffs - temp@A@estimatedCoeffs+temp@c
def computeLogLikelihood(Z,Covmat,drift,coeffs=None,A=None,c=None):
if coeffs is None:
coeffs = estimCoeff(Z,Covmat,A,c)
Zc = Z - coeffs@drift.T
cholcovmat = np.linalg.cholesky(Covmat)
Zcstd = np.linalg.solve(cholcovmat,Zc)
quad = Zcstd.T@Zcstd
logdet = 2. * np.sum(np.log(np.diag(cholcovmat)))
return -0.5 * (quad + logdet + len(Z) * np.log(2.* np.pi))
Mono variate case¶
Model¶
In [4]:
model = gl.Model.createFromParam(gl.ECov.MATERN,param=1,range=rangev,sill=sill)
model.addCovFromParam(gl.ECov.NUGGET,sill=nugget)
model
Out[4]:
Model characteristics ===================== Space dimension = 2 Number of variable(s) = 1 Number of basic structure(s) = 2 Number of drift function(s) = 0 Number of drift equation(s) = 0 Covariance Part --------------- Matern (Third Parameter = 1) - Sill = 1.000 - Range = 0.200 - Theo. Range = 0.058 Nugget Effect - Sill = 0.100 Total Sill = 1.100 Known Mean(s) 0.000
Data¶
In [5]:
dat = gl.Db.createFillRandom(ndat, ndim, 0)
dat["drift"] = dat["x-1"]
gl.simtub(None,dat,model)
dat
Out[5]:
Data Base Characteristics ========================= Data Base Summary ----------------- File is organized as a set of isolated points Space dimension = 2 Number of Columns = 5 Total number of samples = 100 Variables --------- Column = 0 - Name = rank - Locator = NA Column = 1 - Name = x-1 - Locator = x1 Column = 2 - Name = x-2 - Locator = x2 Column = 3 - Name = drift - Locator = NA Column = 4 - Name = Simu - Locator = z1
In [6]:
truecoeffs = [0.5]
dat["Simu"] = truecoeffs[0] + dat["Simu"]
#dat.setLocator("drift",gl.ELoc.F)
dat
Out[6]:
Data Base Characteristics ========================= Data Base Summary ----------------- File is organized as a set of isolated points Space dimension = 2 Number of Columns = 5 Total number of samples = 100 Variables --------- Column = 0 - Name = rank - Locator = NA Column = 1 - Name = x-1 - Locator = x1 Column = 2 - Name = x-2 - Locator = x2 Column = 3 - Name = drift - Locator = NA Column = 4 - Name = Simu - Locator = z1
In [7]:
model.setDriftIRF(0,0)
model
Out[7]:
Model characteristics ===================== Space dimension = 2 Number of variable(s) = 1 Number of basic structure(s) = 2 Number of drift function(s) = 1 Number of drift equation(s) = 1 Covariance Part --------------- Matern (Third Parameter = 1) - Sill = 1.000 - Range = 0.200 - Theo. Range = 0.058 Nugget Effect - Sill = 0.100 Total Sill = 1.100 Drift Part ---------- Universality_Condition
In [8]:
X = model.evalDriftMatrix(dat).toTL()
Covmat = model.evalCovMatrixSymmetric(dat).toTL()
In [9]:
A = np.array([1]).reshape(1,1)
c = [0.3]
estimCoeff(dat["Simu"],Covmat,X,A,c)
Out[9]:
array([0.3])
In [10]:
estimCoeff(dat["Simu"],Covmat,X)
Out[10]:
array([0.72774768])
This lack of constraint can be emulated in gstlearn
In [11]:
likelihood = model.computeLogLikelihood(dat, True)
print(f"Compute manual loglikelihood = " + str(np.round(computeLogLikelihood(dat["Simu"],Covmat,X),6)))
Likelihood calculation:
- Number of active samples = 100
- Number of variables = 1
- Length of Information Vector = 100
- Number of drift conditions = 1
Optimal Drift coefficients =
0.728
Log-Determinant = -33.879144
Quadratic term = 90.846364
Log-likelihood = -120.377463
Compute manual loglikelihood = -120.377463
Multivariate¶
In [12]:
s1=0.4
s2=2
r = 0.8
sills = np.array([s1**2,r*s1*s2,r*s1*s2,s2**2]).reshape(2,2)
model = gl.Model.createFromParam(gl.ECov.MATERN,param=1,range=rangev,sills=sills.reshape(-1))
In [13]:
ndat=200
dat = gl.Db.createFillRandom(ndat, ndim, 0,2)
dat["drift"] = dat["x-1"]
gl.simtub(None,dat,model)
dat
Out[13]:
Data Base Characteristics ========================= Data Base Summary ----------------- File is organized as a set of isolated points Space dimension = 2 Number of Columns = 8 Total number of samples = 200 Variables --------- Column = 0 - Name = rank - Locator = NA Column = 1 - Name = x-1 - Locator = x1 Column = 2 - Name = x-2 - Locator = x2 Column = 3 - Name = f-1 - Locator = f1 Column = 4 - Name = f-2 - Locator = f2 Column = 5 - Name = drift - Locator = NA Column = 6 - Name = Simu.1 - Locator = z1 Column = 7 - Name = Simu.2 - Locator = z2
In [14]:
ax = plt.scatter(dat["Simu.1"],dat["Simu.2"])
In [15]:
truecoeffs1 = [0.5,3]
truecoeffs2 = [1.5,-2]
model.setDriftIRF(0,1)
dat["Simu.1"] = truecoeffs1[0] + truecoeffs1[1] * dat["drift"] + dat["Simu.1"]
dat["Simu.2"] = truecoeffs2[0] + truecoeffs2[1] * dat["drift"] + dat["Simu.2"]
dat.setLocator("drift",gl.ELoc.F)
No constraint¶
In [16]:
Covmat = model.evalCovMatrixSymmetric(dat).toTL()
X = model.evalDriftMatrix(dat).toTL()
In [17]:
Z = dat["Simu*"]
Z=Z.T.reshape(-1)
In [18]:
coeffs = estimCoeff(Z,Covmat,X)
fig,ax = plt.subplots(1,2)
ax[0].scatter(dat["x-1"],dat["Simu.1"])
ax[0].plot([0,1],[coeffs[0],coeffs[0]+coeffs[1]])
ax[1].scatter(dat["x-1"],dat["Simu.2"])
ax[1].plot([0,1],[coeffs[2],coeffs[2]+coeffs[3]])
Out[18]:
[<matplotlib.lines.Line2D at 0x7f5efc28edd0>]
In [19]:
print("No Constraint = ",coeffs)
No Constraint = [ 0.59138302 3.04949823 1.96840598 -2.14045187]
This option can be emulated in gstlearn
In [20]:
likelihood = model.computeLogLikelihood(dat, True)
Likelihood calculation:
- Number of active samples = 200
- Number of variables = 2
- Length of Information Vector = 400
- Number of drift conditions = 4
Optimal Drift coefficients =
0.591 3.049 1.968 -2.140
Log-Determinant = -676.517039
Quadratic term = 484.076013
Log-likelihood = -271.354901
Means of both variables are imposed¶
In [21]:
A = np.array([[1,0,0,0],[0,0,1,0]])
c = [0.5,1.5]
coeffs=estimCoeff(Z,Covmat,X,A,c)
In [22]:
fig,ax = plt.subplots(1,2)
ax[0].scatter(dat["x-1"],dat["Simu.1"])
ax[0].plot([0,1],[coeffs[0],coeffs[0]+coeffs[1]])
ax[1].scatter(dat["x-1"],dat["Simu.2"])
ax[1].plot([0,1],[coeffs[2],coeffs[2]+coeffs[3]])
Out[22]:
[<matplotlib.lines.Line2D at 0x7f5eb1c16bd0>]
In [23]:
print("a0=0.5 and b0=1.5) = ",coeffs)
a0=0.5 and b0=1.5) = [ 0.5 3.17779706 1.5 -1.48282494]
Same coefficients for mean and drift coefficients¶
In [24]:
A = np.array([[1,0,-1,0],[0,1,0,-1]])
c = [0,0]
coeffs = estimCoeff(Z,Covmat,X,A,c)
In [25]:
fig,ax = plt.subplots(1,2)
ax[0].scatter(dat["x-1"],dat["Simu.1"])
ax[0].plot([0,1],[coeffs[0],coeffs[0]+coeffs[1]])
ax[1].scatter(dat["x-1"],dat["Simu.2"])
ax[1].plot([0,1],[coeffs[2],coeffs[2]+coeffs[3]])
Out[25]:
[<matplotlib.lines.Line2D at 0x7f5eb1b055d0>]
In [26]:
print("a0=b0 and a1=b1",coeffs)
a0=b0 and a1=b1 [0.3618792 3.91448991 0.3618792 3.91448991]
This can be emulated with the current flagLinked option
In [27]:
model.setFlagLinked(True)
likelihood = model.computeLogLikelihood(dat, True)
Likelihood calculation:
- Number of active samples = 200
- Number of variables = 2
- Length of Information Vector = 400
- Number of drift conditions = 2
Optimal Drift coefficients =
0.362 3.914
Log-Determinant = -676.517039
Quadratic term = 528.711994
Log-likelihood = -293.672891
Means are equal¶
In [28]:
A = np.array([[1,0,-1,0]])
c = [0]
coeffs = estimCoeff(Z,Covmat,X,A,c)
In [29]:
fig,ax = plt.subplots(1,2)
ax[0].scatter(dat["x-1"],dat["Simu.1"])
ax[0].plot([0,1],[coeffs[0],coeffs[0]+coeffs[1]])
ax[1].scatter(dat["x-1"],dat["Simu.2"])
ax[1].plot([0,1],[coeffs[2],coeffs[2]+coeffs[3]])
Out[29]:
[<matplotlib.lines.Line2D at 0x7f5eb1b6b550>]
In [30]:
print("a0 = b0",coeffs)
a0 = b0 [0.3618792 3.37171422 0.3618792 0.11506005]
Means are linked, coefficient of drift on the first variable is imposed¶
In [31]:
A = np.array([[1,0,-1,0],[0,1,0,0]])
c = [0,1]
coeffs = estimCoeff(Z,Covmat,X,A,c)
In [32]:
fig,ax = plt.subplots(1,2)
ax[0].scatter(dat["x-1"],dat["Simu.1"])
ax[0].plot([0,1],[coeffs[0],coeffs[0]+coeffs[1]])
ax[1].scatter(dat["x-1"],dat["Simu.2"])
ax[1].plot([0,1],[coeffs[2],coeffs[2]+coeffs[3]])
Out[32]:
[<matplotlib.lines.Line2D at 0x7f5eb1b1c590>]
In [33]:
print("a0=b0 and a1=1 ",coeffs)
a0=b0 and a1=1 [ 1.24506848 1. 1.24506848 -5.65188878]