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Cubic Spline (cspline) Object

Main.ObjectCspline History

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February 10, 2020, at 03:32 AM by 73.77.66.3 -
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February 10, 2020, at 03:31 AM by 73.77.66.3 -
Changed lines 31-37 from:
!!!! [[https://gekko.readthedocs.io/en/latest/|GEKKO Python]] Example

Sample the function {`3\sin(x)-(x-3)`} at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the minimum of that function.

(:toggle hide gekko button show="Example GEKKO (Python) Code":)
(:div id=gekko:)
to:
!!!! [[https://apmonitor.com/wiki/index.php/Main/PythonApp|APM Python]] Example

Use the following x and y data to construct a cubic spline.

  x_data        ,  y_data
  -1.0000000e+00 ,  3.8461538e-02
  -8.0000000e-01 ,  5.8823529e-02
  -5.0000000e-01 ,  1.3793103e-01
  -2.5000000e-01 ,  3.9024390e-01
  0.0000000e+00 ,  1.0000000e+00
  1.0000000e-01 ,  8.0000000e-01
  2.0000000e-01 ,  5.0000000e-01
  5.0000000e-01 ,  1.3793103e-01 

Find the maximum of the interpolated function.

(:toggle hide apm button show="Example APMonitor (Python) Source":)
(:div id=apm:)
Deleted line 50:
from gekko import gekko
Changed lines 53-107 from:

"""
minimize y
s.t.    y = f(x)

using cubic spline with random sampling of data
"""

# Function to generate data for cspline
def f(x):
    return 3*np.sin(x) - (x-3)

# Create model
c = gekko()

# Cubic spline
x = c.Var(value=15)
y = c.Var()
x_data = np.random.rand(50)*10+10
y_data = f(x_data)
c.cspline(x,y,x_data,y_data,True)
c.Obj(y)

# Options
c.options.IMODE = 3
c.options.CSV_READ = 0
c.options.SOLVER = 3
c.solve()

# Generate continuous trend for plot
z = np.linspace(10,20,100)

# Check if solved successfully
if c.options.SOLVESTATUS == 1:
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b',label='data')
    plt.scatter(x.value,y.value,200,'k','x',label='minimum')
    plt.legend(loc='best')
else:
    print ('Failed to converge!')
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b')
    plt.legend(loc='best')
plt.show()
(:sourceend:)
(:divend:)

%width=550px%Attach:cspline_gekko.png

!!!! [[https://apmonitor.com/wiki/index.php/Main/PythonApp|APM Python]] Example

Use the following x and y data to construct a cubic spline.
to:
from APMonitor.apm import *

s = 'https://byu.apmonitor.com'
a = 'cspline'

model = '''
Objects
  c = cspline
End Objects

File c.csv
Deleted lines 72-100:

Find the maximum of the interpolated function.

(:toggle hide apm button show="Example APMonitor (Python) Source":)
(:div id=apm:)

(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
from APMonitor.apm import *

s = 'https://byu.apmonitor.com'
a = 'cspline'

model = '''
Objects
  c = cspline
End Objects

File c.csv
  x_data        ,  y_data
  -1.0000000e+00 ,  3.8461538e-02
  -8.0000000e-01 ,  5.8823529e-02
  -5.0000000e-01 ,  1.3793103e-01
  -2.5000000e-01 ,  3.9024390e-01
  0.0000000e+00 ,  1.0000000e+00
  1.0000000e-01 ,  8.0000000e-01
  2.0000000e-01 ,  5.0000000e-01
  5.0000000e-01 ,  1.3793103e-01 
Added lines 115-176:

!!!! [[https://gekko.readthedocs.io/en/latest/|GEKKO Python]] Example

Sample the function {3sin(x)-(x-3)} at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the minimum of that function.

(:toggle hide gekko button show="Example GEKKO (Python) Code":)
(:div id=gekko:)

(:source lang=python:)
from gekko import gekko
import numpy as np
import matplotlib.pyplot as plt

"""
minimize y
s.t.    y = f(x)

using cubic spline with random sampling of data
"""

# Function to generate data for cspline
def f(x):
    return 3*np.sin(x) - (x-3)

# Create model
c = gekko()

# Cubic spline
x = c.Var(value=15)
y = c.Var()
x_data = np.random.rand(50)*10+10
y_data = f(x_data)
c.cspline(x,y,x_data,y_data,True)
c.Obj(y)

# Options
c.options.IMODE = 3
c.options.CSV_READ = 0
c.options.SOLVER = 3
c.solve()

# Generate continuous trend for plot
z = np.linspace(10,20,100)

# Check if solved successfully
if c.options.SOLVESTATUS == 1:
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b',label='data')
    plt.scatter(x.value,y.value,200,'k','x',label='minimum')
    plt.legend(loc='best')
else:
    print ('Failed to converge!')
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b')
    plt.legend(loc='best')
plt.show()
(:sourceend:)
(:divend:)

%width=550px%Attach:cspline_gekko.png
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July 31, 2019, at 11:21 PM by 108.56.160.222 -
Added line 178:
See also [[Main/ObjectBspline|B-Spline Object]] for 2D surface function approximations from data
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October 16, 2018, at 01:34 PM by 67.60.8.226 -
Changed line 109 from:
(:toggle hide apm button show="Example APMonitor Model":)
to:
(:toggle hide apm button show="Example APMonitor (Python) Source":)
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October 16, 2018, at 01:32 PM by 67.60.8.226 -
Changed line 33 from:
Sample the function {3sin(x)-(x-3)} at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the maximum of that function.
to:
Sample the function {3sin(x)-(x-3)} at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the minimum of that function.
Restore
October 16, 2018, at 01:32 PM by 67.60.8.226 -
Changed lines 31-32 from:
!!!! GEKKO Example
to:
!!!! [[https://gekko.readthedocs.io/en/latest/|GEKKO Python]] Example

Sample the function {3sin(x)-(x-3)} at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the maximum of that function.
Deleted lines 37-38:
Sample the function {3sin(x)-(x-3)} at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the maximum of that function.
Changed lines 93-107 from:
!!!! APM Python Example
to:
!!!! [[https://apmonitor.com/wiki/index.php/Main/PythonApp|APM Python]] Example

Use the following x and y data to construct a cubic spline.

  x_data        ,  y_data
  -1.0000000e+00 ,  3.8461538e-02
  -8.0000000e-01 ,  5.8823529e-02
  -5.0000000e-01 ,  1.3793103e-01
  -2.5000000e-01 ,  3.9024390e-01
  0.0000000e+00 ,  1.0000000e+00
  1.0000000e-01 ,  8.0000000e-01
  2.0000000e-01 ,  5.0000000e-01
  5.0000000e-01 ,  1.3793103e-01 

Find the maximum of the interpolated function.
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October 16, 2018, at 01:29 PM by 67.60.8.226 -
Deleted lines 32-33:
%width=550px%Attach:cspline_gekko.png
Added lines 36-37:
Sample the function {3sin(x)-(x-3)} at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the maximum of that function.
Changed lines 91-93 from:
!!!! APMonitor Example
to:
%width=550px%Attach:cspline_gekko.png

!!!! APM Python Example
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October 16, 2018, at 01:25 PM by 67.60.8.226 -
Added lines 31-32:
!!!! GEKKO Example
Added line 91:
!!!! APMonitor Example
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October 16, 2018, at 01:24 PM by 67.60.8.226 -
Added lines 31-35:
%width=550px%Attach:cspline_gekko.png

(:toggle hide gekko button show="Example GEKKO (Python) Code":)
(:div id=gekko:)
Added line 37:
from gekko import gekko
Added lines 40-95:

"""
minimize y
s.t.    y = f(x)

using cubic spline with random sampling of data
"""

# Function to generate data for cspline
def f(x):
    return 3*np.sin(x) - (x-3)

# Create model
c = gekko()

# Cubic spline
x = c.Var(value=15)
y = c.Var()
x_data = np.random.rand(50)*10+10
y_data = f(x_data)
c.cspline(x,y,x_data,y_data,True)
c.Obj(y)

# Options
c.options.IMODE = 3
c.options.CSV_READ = 0
c.options.SOLVER = 3
c.solve()

# Generate continuous trend for plot
z = np.linspace(10,20,100)

# Check if solved successfully
if c.options.SOLVESTATUS == 1:
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b',label='data')
    plt.scatter(x.value,y.value,200,'k','x',label='minimum')
    plt.legend(loc='best')
else:
    print ('Failed to converge!')
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b')
    plt.legend(loc='best')
plt.show()
(:sourceend:)
(:divend:)


(:toggle hide apm button show="Example APMonitor Model":)
(:div id=apm:)

(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
Deleted lines 155-211:
(:sourceend:)

(:toggle hide gekko button show="Example GEKKO (Python) Code":)
(:div id=gekko:)

%width=550px%Attach:cspline_gekko.png

(:source lang=python:)
from gekko import gekko
import numpy as np
import matplotlib.pyplot as plt

"""
minimize y
s.t.    y = f(x)

using cubic spline with random sampling of data
"""

# Function to generate data for cspline
def f(x):
    return 3*np.sin(x) - (x-3)

# Create model
c = gekko()

# Cubic spline
x = c.Var(value=15)
y = c.Var()
x_data = np.random.rand(50)*10+10
y_data = f(x_data)
c.cspline(x,y,x_data,y_data,True)
c.Obj(y)

# Options
c.options.IMODE = 3
c.options.CSV_READ = 0
c.options.SOLVER = 3
c.solve()

# Generate continuous trend for plot
z = np.linspace(10,20,100)

# Check if solved successfully
if c.options.SOLVESTATUS == 1:
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b',label='data')
    plt.scatter(x.value,y.value,200,'k','x',label='minimum')
    plt.legend(loc='best')
else:
    print ('Failed to converge!')
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b')
    plt.legend(loc='best')
plt.show()
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February 17, 2018, at 05:55 AM by 184.254.42.171 -
Added lines 98-100:

%width=550px%Attach:cspline_gekko.png
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February 17, 2018, at 05:54 AM by 184.254.42.171 -
Added lines 95-150:

(:toggle hide gekko button show="Example GEKKO (Python) Code":)
(:div id=gekko:)
(:source lang=python:)
from gekko import gekko
import numpy as np
import matplotlib.pyplot as plt

"""
minimize y
s.t.    y = f(x)

using cubic spline with random sampling of data
"""

# Function to generate data for cspline
def f(x):
    return 3*np.sin(x) - (x-3)

# Create model
c = gekko()

# Cubic spline
x = c.Var(value=15)
y = c.Var()
x_data = np.random.rand(50)*10+10
y_data = f(x_data)
c.cspline(x,y,x_data,y_data,True)
c.Obj(y)

# Options
c.options.IMODE = 3
c.options.CSV_READ = 0
c.options.SOLVER = 3
c.solve()

# Generate continuous trend for plot
z = np.linspace(10,20,100)

# Check if solved successfully
if c.options.SOLVESTATUS == 1:
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b',label='data')
    plt.scatter(x.value,y.value,200,'k','x',label='minimum')
    plt.legend(loc='best')
else:
    print ('Failed to converge!')
    plt.figure()
    plt.plot(z,f(z),'r-',label='original')
    plt.scatter(x_data,y_data,5,'b')
    plt.legend(loc='best')
plt.show()
(:sourceend:)
(:divend:)
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February 14, 2018, at 04:29 PM by 174.148.12.56 -
Added lines 14-17:

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/s1jSLpDXvzs" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>
(:htmlend:)
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February 14, 2018, at 12:26 AM by 174.148.43.131 -
Changed line 23 from:
%width=400px%Attach:cspline.png
to:
%width=500px%Attach:cspline.png
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February 14, 2018, at 12:26 AM by 174.148.43.131 -
Changed lines 21-22 from:
The function is evaluated at the points x_data = [-1.0 -0.8 -0.5 -0.25 0.0 0.1 0.2 0.5].
to:
The function is evaluated at the points x_data = [-1.0 -0.8 -0.5 -0.25 0.0 0.1 0.2 0.5]. Evaluating at additional points [[Attach:cspline_plot.zip|shows the cubic spline interpolation function]]. The maximum of the original function is at ''x''=0 with a result ''y''=1. Because the cubic spline has only 8 points, there is some approximation error and the optimal solution of the cubic spline is slightly to the left of the true solution.
Changed line 25 from:
A cubic spline intersects the points to create the function approximations in the range of x between -1.0 and 0.5. There is extrapolation error outside of this range, as expected. Bounds on ''x'' should be added or additional cubic spline sample points should be added to avoid problems with optimizer performance in the extrapolation region.
to:
The cubic spline intersects the points to create the function approximations in the range of x between -1.0 and 0.5. There is extrapolation error outside of this range, as expected. Bounds on ''x'' should be added or additional cubic spline sample points should be added to avoid problems with optimizer performance in the extrapolation region.
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February 14, 2018, at 12:15 AM by 174.148.43.131 -
Added lines 17-22:
Find the maximum of a function defined by 8 points that approximate the true function.

{$y(x) = \frac{1}{1+25 x^2}$}

The function is evaluated at the points x_data = [-1.0 -0.8 -0.5 -0.25 0.0 0.1 0.2 0.5].
Added lines 24-25:

A cubic spline intersects the points to create the function approximations in the range of x between -1.0 and 0.5. There is extrapolation error outside of this range, as expected. Bounds on ''x'' should be added or additional cubic spline sample points should be added to avoid problems with optimizer performance in the extrapolation region.
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February 14, 2018, at 12:08 AM by 174.148.43.131 -
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%width=400px%Attach:cspline.png
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February 13, 2018, at 09:48 PM by 173.117.238.26 -
Changed lines 17-26 from:
 Objects
to:
(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
from APMonitor.apm import *

s = 'https://byu.apmonitor.com'
a = 'cspline'

model = '''
Objects
Changed lines 28-30 from:
 End Objects

 File c.csv
to:
End Objects

File c.csv
Changed lines 40-42 from:
 End File

 Connections
to:
End File

Connections
Changed lines 45-47 from:
 End Connections

Variables
to:
End Connections

Parameters
End Parameters
 
Variables
Changed lines 53-55 from:
 End Variables

 Equations
to:
End Variables

Equations
Changed lines 57-80 from:
 End Equations
to:
End Equations
'''

# write file
fid = open('model.apm','w')
fid.write(model)
fid.close()

# clear prior, load new model
apm(s,a,'clear all')
apm_load(s,a,'model.apm')

# set steady state optimiation and solve
apm_option(s,a,'apm.imode',3)
output = apm(s,a,'solve')
print(output)

# retrieve solution
z = apm_sol(s,a)

# print solution
print('x: ' + str(z['x']))
print('y: ' + str(z['y']))
(:sourceend:)
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February 13, 2018, at 06:57 PM by 10.37.41.255 -
Added lines 1-45:
(:title Cubic Spline (cspline) Object:)
(:keywords Cubic spline, Object, APMonitor, Option, Configure, Default, Description:)
(:description One dimensional cubic spline for nonlinear function approximation with multiple interpolating functions that have continuous first and second derivatives:)

%width=50px%Attach:apm.png [[Main/Objects|APMonitor Objects]]

 Type: Object
 Data: Two data vectors that define 1D function points
 Inputs: Name of first data column (e.g. x)
 Outputs: Name of second data column (e.g. y)
 Description: Cubic spline for nonlinear function approximation

A cubic spline is a nonlinear function constructed of multiple third-order polynomials. These polynomials pass through a set of control points and have continuous first and second derivatives everywhere. The second derivative is set to zero at the left and right endpoints, to provide a boundary condition to complete the system of equations. There is poor extrapolation when function retrievals are requested outside of the data points. The input should be constrained or else additional data points added to avoid extrapolation.

'''Example Usage'''

 Objects
  c = cspline
 End Objects

 File c.csv
  x_data        ,  y_data
  -1.0000000e+00 ,  3.8461538e-02
  -8.0000000e-01 ,  5.8823529e-02
  -5.0000000e-01 ,  1.3793103e-01
  -2.5000000e-01 ,  3.9024390e-01
  0.0000000e+00 ,  1.0000000e+00
  1.0000000e-01 ,  8.0000000e-01
  2.0000000e-01 ,  5.0000000e-01
  5.0000000e-01 ,  1.3793103e-01 
 End File

 Connections
  x = c.x_data
  y = c.y_data
 End Connections

 Variables
  x = -0.5  >= -1 <= 0.5
  y
 End Variables

 Equations
  maximize y
 End Equations
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