Content-Type: multipart/related; boundary=Apple-Mail-5-443763252; type="text/html" --Apple-Mail-5-443763252 Content-Type: text/html; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable
Recently, I had a need for a = monotonic piece-wise cubic Hermite interpolator.   Matlab provides = the function "pchip" (Piecewise Cubic Hermite Interpolator), but when I = Googled I didn't find any Python equivalent.   I tried "interp1d()" = from scipy.interpolate but this was a standard cubic spline using all of = the data - not a piece-wise cubic spline.

I had access to Matlab = documentation, so I spent a some time tracing through the code to figure = out how I might write a Python duplicate.  This was an massive = exercise in frustration and a potent reminder on why I love Python and = use Matlab only under duress.  I find typical Matlab code is = poorly documented (if at all) and that apparently includes the code = included in their official releases.  I also find Matlab syntax = =93dated=94 and the code very difficult to =93read=94.

Wikipedia to the = rescue. 

Not to be = deterred, I found a couple of very well written Wikipedia entries, which = explained in simple language how to compute the interpolant.  Hats = off to whoever wrote these entries =96 they are excellent.  The = result was a surprising small amount of code considering the Matlab code = was approaching 10 pages of incomprehensible code. Again - strong = evidence that things are just better in Python...

Offered for those who might = have the same need =96 a Python pchip() equivalent =3D=3D> pypchip(). =  Since I'm not sure how attachments work (or if they work at = all...), I copied the code I used below, followed by a PNG showing = "success":

#
#    pychip.py
#    Michalski
#    20090818
#    Piecewise = cubic Hermite = interpolation (monotonic...) in Python
#
#    References:
#
#      =   Wikipedia:  Monotone cubic = interpolation
#      =               Cubic Hermite spline
#
#    A cubic = Hermte spline is a = third degree spline with each polynomial of the spline
#    in Hermite form.  The Hermite form consists of two control points and two = control
#    tangents = for each polynomial.  Each interpolation is performed on one = sub-interval
#    at a time = (piece-wise).  A monotone cubic interpolation is a variant of = cubic
#    = interpolation that preserves monotonicity of the data to be interpolated = (in other
#    words, it = controls overshoot).  Monotonicity is preserved by linear = interpolation
#    but not by = cubic interpolation.
#
#    Use:
#    There are = two separate calls, the first call, pchip_init(), computes the slopes = that
#    the interpolator needs.  If = there are a large number of points to compute,
#    it is more efficient to compute the = slopes once, rather than for every point
#  =   pchip_init() along with X, Y, and a vector of desired "xnew"s and computes a = vector
#    of "ynew"s.  If only a = handful of points is needed, pchip() is a third function
#    which combines a call to pchip_init() = followed by pchip_eval().
#

import pylab as = P

def pchip(x, y, xnew):
    # Compute the slopes used by the piecewise = cubic Hermite interpolator

    # Use these slopes (along = with the Hermite basis = function) to interpolate
    ynew =3D pchip_eval(x, = y, xnew)

return = ynew

def x_is_okay(x,xvec):
    # Make sure "x" and "xvec" satisfy the conditions = for
    # running the pchip interpolator

    n =3D len(x)

    = # Make sure "x" is in sorted order (brute force, but = works...)
    xx =3D x.copy()
    total_matches =3D (xx = =3D=3D x).sum()
    if total_matches !=3D n:
        = print "*" * 50
      =   print "x_is_okay()"
      =   print "x values weren't in sorted order --- = aborting"
        return False

    = # Make sure 'x' doesn't have any repeated values
1:] - = x[:-1]
if (delta =3D=3D = 0.0).any():
print "*" * 50
        print = "x_is_okay()"
        print "x values = weren't monotonic--- aborting"
        return False
    # Check for in-range xvec values (beyond upper = edge)
    check =3D xvec > x[-1]
    if check.any():
        print "*" * 50
      =   print "x_is_okay()"
      =   print "Certain 'xvec' values are beyond the upper end of = 'x'"
        print "x_max =3D = ", x[-1]
        print "out-of-range = xvec's =3D ", xvec[indices]
        print = "out-of-range xvec indices =3D ", = indices
        return False

    = # Second - check for in-range xvec values (beyond lower edge)
0]
    if check.any():
        print "*" * 50
      =   print "x_is_okay()"
      =   print "Certain 'xvec' values are beyond the lower end of = 'x'"
        print "x_min =3D = ", x[0]
        print "out-of-range = xvec's =3D ", xvec[indices]
        print = "out-of-range xvec indices =3D ", = indices
        return False

    = return True

def pchip_eval(x, y, m, xvec):

    # Evaluate the piecewise = cubic Hermite interpolant with monoticity = preserved
    #
    #    x =3D array containing the = x-data
    #    y =3D array containing the = y-data
    #    m =3D slopes at each (x,y) = point [computed to preserve monotonicity]
    #    = xnew =3D new "x" value = where the interpolation is desired
    #
    #    x must be sorted low to = high... (no repeats)
    #    y can have repeated = values
    #
    # This works with either a scalar or = vector of "xvec"

    mm =3D = len(xvec)

    = ############################
    # Make sure there aren't = problems with the input data
    = ############################
    if not = x_is_okay(x, xvec):
        print = "pchip_eval2() - ill formed 'x' vector!!!!!!!!!!!!!"

        # Cause a = hard crash...
        = STOP_pchip_eval2

   # = Find the indices "k" such that x[k] < xvec < x[k+1]

    = # Create "copies" of "x" as rows in a mxn 2-dimensional = vector
    xx =3D = P.resize(x,(mm,n)).transpose()
    xxx =3D xx > = xvec

    = # Compute column by column differences
1,:] - = xxx[1:,:]

    # Collapse over = rows...
    k =3D z.argmax(axis=3D0)

    = # Create the Hermite coefficients
    h =3D x[k+1] - x[k]
    t =3D (xvec - = x[k]) / h[k]

    = # Hermite basis = functions
    h00 =3D (2 * t**3) - (3 * t**2) + 1
3  - (2 * = t**2) + t
2* = t**3) + (3 * t**2)
    h11 =3D      t**3  -      t**2

    = # Compute the interpolated value of "y"
1] + h11*h*m[k+1]

return = ynew

def pchip_init(x,y):

    = # Evaluate the piecewise cubic Hermite interpolant with monoticity preserved
    #
    #    x =3D array = containing the x-data
    #    y =3D array containing the = y-data
    #
    #    x must be sorted low to = high... (no repeats)
    #    y can have repeated = values
    #
    #    x input conditioning is = assumed but not checked


    = # Compute the slopes of the secant lines between successive points
1:] - = y[:-1]) / (x[1:] - x[:-1])

    = # Initialize the tangents at every points as the average of the = secants
'd')

    = # At the endpoints - use one-sided differences
0] =3D = delta[0]
1] =3D = delta[-1]

    # In the middle - use the = average of the secants
    m[1:-1] =3D = (delta[:-1] + delta[1:]) / 2.0

    = # Special case: intervals where y[k] =3D=3D y[k+1]

    # Setting these slopes to = zero guarantees the spline connecting
    # these points = will be flat which preserves monotonicity
    indices_to_fix = =3D P.compress((delta =3D=3D 0.0), = range(n))

#    print "zero slope indices to fix =3D = ", indices_to_fix

for ii in indices_to_fix:
0.0
        m[ii+1] =3D 0.0

1]/delta
    beta  =3D m[1:]/delta
    dist  =3D = alpha**2 + beta**2
    tau   =3D 3.0 / P.sqrt(dist)

    # To prevent overshoot or = undershoot, restrict = the position vector
    # (alpha, beta) to a circle of radius = 3.  If (alpha**2 + beta**2)>9,
    # then set m[k] =3D = tau[k]alpha[k]delta[k] and m[k+1] =3D tau[k]beta[b]delta[k]
    = # where tau =3D 3/sqrt(alpha**2 + beta**2).
    # Find the indices that need = adjustment
    over =3D (dist > 9.0)
    indices_to_fix =3D = P.compress(over, range(n))

#    print "overshoot indices to fix... =3D = ", indices_to_fix

for ii in indices_to_fix:
        m[ii+1] =3D tau[ii] * beta[ii]  * = delta[ii]

return m

def CubicHermiteSpline(x, y, = x_new):

    = # Piecewise Cubic Hermite Interpolation using Catmull-Rom
    = # method for computing the slopes.
    #
    # Note - this only works = if delta-x is uniform?

    = # Find the two points which "bracket" "x_new"
False

for ii in range(len(x)-1):
        if (x[ii] <=3D x_new) and (x[ii+1] > = x_new):
            found_it = =3D True
break

if not found_it:
print
        print "requested = x=3D<%f> outside X range[%f,%f]" % = (x_new, x[0], x[-1])

    # Starting and ending data = points
    x0 =3D x[ii]
1]

    y1 =3D y[ii+1]

    = # Starting and ending tangents (using Catmull-Rom spline method)

    # Handle special cases = (hit one of the endpoints...)
    if ii =3D=3D 0:

      =   # Hit lower endpoint
        m0 =3D = (y[1] - y[0])
        m1 =3D (y[2] - y[0]) / 2.0

elif ii =3D=3D = (len(x) - 2):

        # Hit upper = endpoints
1] - y[ii-1]) / = 2.0
        = m1 =3D (y[ii+1] - y[ii])

    else:

      =   # Inside the field...
        m0 =3D = (y[ii+1] - y[ii-1])/ 2.0
        m1 =3D (y[ii+2] - y[ii])  / 2.0

    = # Normalize to x_new to [0,1] interval
    t =3D (x_new - x0) / = h

    # Compute the four Hermite basis = functions
    h00 =3D ( 2.0 * t**3) - = (3.0 * t**2) + 1.0
1.0 * = t**3) - (2.0 * t**2) + = t
2.0 * = t**3) + (3.0 * t**2)
    h11 =3D ( 1.0 * t**3) - = (1.0 * t**2)

1

    y_new =3D (h00 * y0) + = (h10 * h * m0) + (h01 * y1) + (h11 * h * m1)

    return y_new

def main():

    = ############################################################
<= div style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; font: normal normal normal 11px/normal Monaco; color: = rgb(192, 35, 30); ">    = # Sine wave test
    = ############################################################
<= div style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; font: normal normal normal 11px/normal Monaco; = min-height: 15px; ">
    # Create a example vector = containing a sine wave.
    x =3D P.arange(30.0)/10.
    y =3D P.sin(x)

    # Interpolate the data = above to the grid defined by "xvec"
    xvec =3D = P.arange(250.)/100.

    = # Initialize the interpolator slopes
    m =3D = pchip_init(x,y)

    = # Call the monotonic piece-wise Hermite cubic interpolator
    yvec2 =3D = pchip_eval(x, y, m, xvec)

1)
    P.plot(x,y, 'ro')
    = P.title("pchip() Sin test code")

    = # Plot the interpolated points
    P.plot(xvec, = yvec2, 'b')

    = ###################################################################= ######
    # Step function test...
    = ###################################################################= ######

2)
    P.title("pchip() step function test")

    = # Create a step function (will demonstrate = monotonicity)
    x =3D P.arange(7.0) - 3.0
    y =3D P.array([-1.0, -1,-1,0,1,1,1])

    = # Interpolate using monotonic piecewise Hermite cubic = spline
    xvec =3D P.arange(599.)/100. - 3.0

    = # Create the pchip slopes slopes
    m    = =3D pchip_init(x,y)

    = # Interpolate...
    yvec =3D pchip_eval(x, = y, m, xvec)

    = # Call the Scipy cubic spline interpolator
    from scipy.interpolate import interpolate
    function =3D = interpolate.interp1d(x, y, kind=3D'cubic')
    yvec2 =3D = function(xvec)

    = # Non-montonic = cubic Hermite spline = interpolator = using
    # Catmul-Rom method for computing slopes...
    for xx in = xvec:
        = yvec3.append(CubicHermiteSpline(x,y,xx))
    yvec3 =3D = P.array(yvec3)

    = # Plot the results
    P.plot(x,    = y,     'ro')
    P.plot(xvec, = yvec,  'b')
'k')
    P.plot(xvec, yvec3, 'g')
    P.xlabel("X")
    P.ylabel("Y")
    P.title("Comparing pypchip() vs. Scipy interp1d() vs. non-monotonic CHS")
    legends =3D ["Data", "pypchip()", = "interp1d","CHS"]
    P.legend(legends, = loc=3D"upper left")

    P.show()








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