LinearNDInterpolator -- Qhull 精度误差：初始单纯形是平坦的

Question

I am attempting to use scipy.interpolate.LinearNDInterpolator() to interpolate data points in an 8-dimensional space and am getting an error I don't understand:

scipy.spatial.qhull.QhullError: QH6154 Qhull precision error: Initial simplex is flat (facet 1 is coplanar with the interior point)

followed by much more text which I will post below. Using what I can find online I cannot fine the error in my code. It all looks right to me (I only copied relevant portions):

from scipy import interpolate as inter
from numpy import array

npPoints = array(points)
npS = array(s)

inter.LinearNDInterpolator(npPoints, npS)

where points is an Nx8 nested list of floats and s is an Nx1 list of floats, both defined previously.

From what I see in the documentation it seems to me that I'm doing it right. Where is my error? Should I be using a different method?

Here is the full Qhull error :

Traceback (most recent call last):
  File "BellDataFit", line 83, in <module>
    inter.LinearNDInterpolator(npPoints, npS)
  File "interpnd.pyx", line 248, in scipy.interpolate.interpnd.LinearNDInterpolator.__init__
  File "qhull.pyx", line 1826, in scipy.spatial.qhull.Delaunay.__init__
  File "qhull.pyx", line 354, in scipy.spatial.qhull._Qhull.__init__
scipy.spatial.qhull.QhullError: QH6154 Qhull precision error: Initial simplex is flat (facet 1 is coplanar with the interior point)

While executing:  | qhull d Qbb Qx Qz Q12 Qt Qc
Options selected for Qhull 2015.2.r 2016/01/18:
  run-id 704299719  delaunay  Qbbound-last  Qxact-merge  Qz-infinity-point
  Q12-no-wide-dup  Qtriangulate  Qcoplanar-keep  _zero-centrum  Qinterior-keep
  Q3-no-merge-vertices-dim-high  Pgood  _max-width 5.6  Error-roundoff 3.5e-14
  _one-merge 6.7e-13  Visible-distance 2.1e-13  U-coplanar-distance 2.1e-13
  Width-outside 4.2e-13  _wide-facet 1.3e-12

precision problems (corrected unless 'Q0' or an error)
      2 flipped facets
     11 nearly singular or axis-parallel hyperplanes
     11 zero divisors during back substitute
 119436 zero divisors during gaussian elimination

The input to qhull appears to be less than 9 dimensional, or a
computation has overflowed.

Qhull could not construct a clearly convex simplex from points:
- p3(v9): -0.89 -0.89     0     0     0  -1.7  -3.1  -3.1   2.1
- p2(v8): -0.89 -0.89     0     0     0  -2.1  -3.1  -3.1   2.7
- p1(v7): -0.89 -0.89     0     0     0  -2.4  -3.1  -3.1   3.4
- p16(v6): -0.89 -0.89     0     0     0   2.8  -3.1  -3.1   4.3
- p2720(v5):     0     0 -0.89  0.89     0  -2.8  -3.1  -3.1   4.3
- p2448(v4):     0     0 -0.89 -0.89     0  -2.8  -3.1  -3.1   4.3
- p7055(v3):     0     0  0.89 -0.89     0  -2.8  -3.1  -3.1   4.3
- p272(v2): -0.89  0.89     0     0     0  -2.8  -3.1  -3.1   4.3
- p0(v1): -0.89 -0.89     0     0     0  -2.8  -3.1  -3.1   4.3
- p9503(v0):  0.89 -0.89     0     0     0  -2.8  -3.1  -3.1   4.3

The center point is coplanar with a facet, or a vertex is coplanar
with a neighboring facet.  The maximum round off error for
computing distances is 3.5e-14.  The center point, facets and distances
to the center point are as follows:

center point  -0.4444  -0.4444 -0.08889 -0.08889        0   -2.025   -3.142   -3.142    3.806

facet p2 p1 p16 p2720 p2448 p7055 p272 p0 p9503 distance=    0
facet p3 p1 p16 p2720 p2448 p7055 p272 p0 p9503 distance=    0
facet p3 p2 p16 p2720 p2448 p7055 p272 p0 p9503 distance=    0
facet p3 p2 p1 p2720 p2448 p7055 p272 p0 p9503 distance=    0
facet p3 p2 p1 p16 p2448 p7055 p272 p0 p9503 distance=    0
facet p3 p2 p1 p16 p2720 p7055 p272 p0 p9503 distance= -0.13
facet p3 p2 p1 p16 p2720 p2448 p272 p0 p9503 distance=    0
facet p3 p2 p1 p16 p2720 p2448 p7055 p0 p9503 distance=    0
facet p3 p2 p1 p16 p2720 p2448 p7055 p272 p9503 distance=    0
facet p3 p2 p1 p16 p2720 p2448 p7055 p272 p0 distance=    0

These points either have a maximum or minimum x-coordinate, or
they maximize the determinant for k coordinates.  Trial points
are first selected from points that maximize a coordinate.

Because of the high dimension, the min x-coordinate and max-coordinate
points are used if the determinant is non-zero.  Option 'Qs' will
do a better, though much slower, job.  Instead of 'Qs', you can change
the points by randomly rotating the input with 'QR0'.

The min and max coordinates for each dimension are:
  0:   -0.8889    0.8889  difference= 1.778
  1:   -0.8889    0.8889  difference= 1.778
  2:   -0.8889    0.8889  difference= 1.778
  3:   -0.8889    0.8889  difference= 1.778
  4:         0         0  difference=    0
  5:    -2.793     2.793  difference= 5.585
  6:    -3.142  -2.225e-308  difference= 3.142
  7:    -3.142  -2.225e-308  difference= 3.142
  8:  1.776e-15     5.585  difference= 5.585

If the input should be full dimensional, you have several options that
may determine an initial simplex:
  - use 'QJ'  to joggle the input and make it full dimensional
  - use 'QbB' to scale the points to the unit cube
  - use 'QR0' to randomly rotate the input for different maximum points
  - use 'Qs'  to search all points for the initial simplex
  - use 'En'  to specify a maximum roundoff error less than 3.5e-14.
  - trace execution with 'T3' to see the determinant for each point.

If the input is lower dimensional:
  - use 'QJ' to joggle the input and make it full dimensional
  - use 'Qbk:0Bk:0' to delete coordinate k from the input.  You should
    pick the coordinate with the least range.  The hull will have the
    correct topology.
  - determine the flat containing the points, rotate the points
    into a coordinate plane, and delete the other coordinates.
  - add one or more points to make the input full dimensional.

Answer 1

It seems to me that the problem arises because, as the error says, your data (or a given set of your data) are really less than 9-dimensional even though you're passing it a 9-column array. As you can see in the data it's printed out in the error message, the 5th, 7th, and 8th columns are constant. This means those columns are linearly dependent and that set of data only has 7 dimensions. As it's going through, it's trying to form a 9-dimensional simplex but can't form one that's convex .

This issue showed up for me when I was coding an airfoil database that runs Xfoil to get the airfoil coefficients at different angles of attack, Reynolds numbers, flap deflections, etc. Xfoil was being finnicky and crash and not return any result for a lot of points in the space. That meant I ended up with a database where all the points were at the same Reynolds number. This made the database not dependent upon Reynolds number, even though I said it was. My data had fewer dimensions than I though it did.

I would fix this by making sure all the points in your data are unique and each column actually adds dimension to the set.