From: Chris Michalski <chris.michalski@gm...>  20090828 17:43:31

Recently, I had a need for a monotonic piecewise 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 piecewise 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 “dated” and the code very difficult to “read”. 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 – 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 – a Python pchip() equivalent ==> 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 subinterval # at a time (piecewise). 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 # being evaluated. The second call, pchip_eval(), takes the slopes computed by # 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 m = pchip_init(x, y) # Use these slopes (along with the Hermite basis function) to interpolate ynew = 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 = len(x) m = len(xvec) # Make sure "x" is in sorted order (brute force, but works...) xx = x.copy() xx.sort() total_matches = (xx == x).sum() if total_matches != 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 delta = x[1:]  x[:1] if (delta == 0.0).any(): print "*" * 50 print "x_is_okay()" print "x values weren't monotonic aborting" return False # Check for inrange xvec values (beyond upper edge) check = 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 = ", x[1] indices = P.compress(check, range(m)) print "outofrange xvec's = ", xvec[indices] print "outofrange xvec indices = ", indices return False # Second  check for inrange xvec values (beyond lower edge) check = xvec< x[0] if check.any(): print "*" * 50 print "x_is_okay()" print "Certain 'xvec' values are beyond the lower end of 'x'" print "x_min = ", x[0] indices = P.compress(check, range(m)) print "outofrange xvec's = ", xvec[indices] print "outofrange xvec indices = ", indices return False return True #========================================================= def pchip_eval(x, y, m, xvec): # Evaluate the piecewise cubic Hermite interpolant with monoticity preserved # # x = array containing the xdata # y = array containing the ydata # m = slopes at each (x,y) point [computed to preserve monotonicity] # xnew = 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" n = len(x) mm = 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 2dimensional vector xx = P.resize(x,(mm,n)).transpose() xxx = xx > xvec # Compute column by column differences z = xxx[:1,:]  xxx[1:,:] # Collapse over rows... k = z.argmax(axis=0) # Create the Hermite coefficients h = x[k+1]  x[k] t = (xvec  x[k]) / h[k] # Hermite basis functions h00 = (2 * t**3)  (3 * t**2) + 1 h10 = t**3  (2 * t**2) + t h01 = (2* t**3) + (3 * t**2) h11 = t**3  t**2 # Compute the interpolated value of "y" ynew = h00*y[k] + h10*h*m[k] + h01*y[k+1] + h11*h*m[k+1] return ynew #========================================================= def pchip_init(x,y): # Evaluate the piecewise cubic Hermite interpolant with monoticity preserved # # x = array containing the xdata # y = array containing the ydata # # x must be sorted low to high... (no repeats) # y can have repeated values # # x input conditioning is assumed but not checked n = len(x) # Compute the slopes of the secant lines between successive points delta = (y[1:]  y[:1]) / (x[1:]  x[:1]) # Initialize the tangents at every points as the average of the secants m = P.zeros(n, dtype='d') # At the endpoints  use onesided differences m[0] = delta[0] m[n1] = delta[1] # In the middle  use the average of the secants m[1:1] = (delta[:1] + delta[1:]) / 2.0 # Special case: intervals where y[k] == y[k+1] # Setting these slopes to zero guarantees the spline connecting # these points will be flat which preserves monotonicity indices_to_fix = P.compress((delta == 0.0), range(n)) # print "zero slope indices to fix = ", indices_to_fix for ii in indices_to_fix: m[ii] = 0.0 m[ii+1] = 0.0 alpha = m[:1]/delta beta = m[1:]/delta dist = alpha**2 + beta**2 tau = 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] = tau[k]alpha[k]delta[k] and m[k+1] = tau[k]beta[b]delta[k] # where tau = 3/sqrt(alpha**2 + beta**2). # Find the indices that need adjustment over = (dist > 9.0) indices_to_fix = P.compress(over, range(n)) # print "overshoot indices to fix... = ", indices_to_fix for ii in indices_to_fix: m[ii] = tau[ii] * alpha[ii] * delta[ii] m[ii+1] = tau[ii] * beta[ii] * delta[ii] return m #= ======================================================================= def CubicHermiteSpline(x, y, x_new): # Piecewise Cubic Hermite Interpolation using CatmullRom # method for computing the slopes. # # Note  this only works if deltax is uniform? # Find the two points which "bracket" "x_new" found_it = False for ii in range(len(x)1): if (x[ii] <= x_new) and (x[ii+1] > x_new): found_it = True break if not found_it: print print "requested x=<%f> outside X range[%f,%f]" % (x_new, x[0], x[1]) STOP_CubicHermiteSpline() # Starting and ending data points x0 = x[ii] x1 = x[ii+1] y0 = y[ii] y1 = y[ii+1] # Starting and ending tangents (using CatmullRom spline method) # Handle special cases (hit one of the endpoints...) if ii == 0: # Hit lower endpoint m0 = (y[1]  y[0]) m1 = (y[2]  y[0]) / 2.0 elif ii == (len(x)  2): # Hit upper endpoints m0 = (y[ii+1]  y[ii1]) / 2.0 m1 = (y[ii+1]  y[ii]) else: # Inside the field... m0 = (y[ii+1]  y[ii1])/ 2.0 m1 = (y[ii+2]  y[ii]) / 2.0 # Normalize to x_new to [0,1] interval h = (x1  x0) t = (x_new  x0) / h # Compute the four Hermite basis functions h00 = ( 2.0 * t**3)  (3.0 * t**2) + 1.0 h10 = ( 1.0 * t**3)  (2.0 * t**2) + t h01 = (2.0 * t**3) + (3.0 * t**2) h11 = ( 1.0 * t**3)  (1.0 * t**2) h = 1 y_new = (h00 * y0) + (h10 * h * m0) + (h01 * y1) + (h11 * h * m1) return y_new #============================================================== def main(): ############################################################ # Sine wave test ############################################################ # Create a example vector containing a sine wave. x = P.arange(30.0)/10. y = P.sin(x) # Interpolate the data above to the grid defined by "xvec" xvec = P.arange(250.)/100. # Initialize the interpolator slopes m = pchip_init(x,y) # Call the monotonic piecewise Hermite cubic interpolator yvec2 = pchip_eval(x, y, m, xvec) P.figure(1) P.plot(x,y, 'ro') P.title("pchip() Sin test code") # Plot the interpolated points P.plot(xvec, yvec2, 'b') ######################################################################### # Step function test... ######################################################################### P.figure(2) P.title("pchip() step function test") # Create a step function (will demonstrate monotonicity) x = P.arange(7.0)  3.0 y = P.array([1.0, 1,1,0,1,1,1]) # Interpolate using monotonic piecewise Hermite cubic spline xvec = P.arange(599.)/100.  3.0 # Create the pchip slopes slopes m = pchip_init(x,y) # Interpolate... yvec = pchip_eval(x, y, m, xvec) # Call the Scipy cubic spline interpolator from scipy.interpolate import interpolate function = interpolate.interp1d(x, y, kind='cubic') yvec2 = function(xvec) # Nonmontonic cubic Hermite spline interpolator using # CatmulRom method for computing slopes... yvec3 = [] for xx in xvec: yvec3.append(CubicHermiteSpline(x,y,xx)) yvec3 = P.array(yvec3) # Plot the results P.plot(x, y, 'ro') P.plot(xvec, yvec, 'b') P.plot(xvec, yvec2, 'k') P.plot(xvec, yvec3, 'g') P.xlabel("X") P.ylabel("Y") P.title("Comparing pypchip() vs. Scipy interp1d() vs. non monotonic CHS") legends = ["Data", "pypchip()", "interp1d","CHS"] P.legend(legends, loc="upper left") P.show() ################################################################### main() 