Interpolation of data

Question

Consider a set of data points

data = {{1.76842, -4.05181}, {1.78353, -4.08368}, {1.7926, -4.10526}, \
{1.83347, -4.19819}, {1.83984, -4.21376}, {1.87722, -4.31579}, \
{1.8896, -4.35321}, {1.91579, -4.44367}, {1.93133, -4.50411}, \
{1.93709, -4.52632}, {1.95047, -4.57586}, {1.97969, -4.64556}, \
{2.0356, -4.73684}, {2.04833, -4.75802}, {2.06316, -4.78248}, \
{2.11756, -4.86965}, {2.16706, -4.94737}, {2.18808, -4.97944}, \
{2.21053, -5.01398}, {2.25946, -5.08799}, {2.30551, -5.15789}, \
{2.33156, -5.19552}, {2.35789, -5.23314}, {2.40539, -5.30058}, \
{2.45345, -5.36842}, {2.47936, -5.40543}, {2.50526, -5.43997}, \
{2.55506, -5.50781}, {2.60946, -5.57895}, {2.6322, -5.60814}, \
{2.65263, -5.63466}, {2.70962, -5.70806}, {2.77582, -5.78947}, \
{2.78877, -5.80551}, {2.8, -5.81898}, {2.8685, -5.90214}, {2.92031, \
-5.96135}, {2.94737, -5.9926}, {2.95435, -6.}, {3.03285, -6.0884}, \
{3.09474, -6.15707}, {3.11604, -6.1801}, {3.14424, -6.21053}, \
{3.20008, -6.27056}, {3.24211, -6.31538}, {3.28413, -6.36102}, \
{3.34169, -6.42105}, {3.36933, -6.44984}, {3.38947, -6.47122}, \
{3.45452, -6.53865}, {3.51554, -6.60115}, {3.53684, -6.62294}, \
{3.54037, -6.62654}, {3.5449, -6.63158}, {3.61053, -6.6986}, \
{3.62607, -6.71464}, {3.64794, -6.73684}, {3.68421, -6.77385}, \
{3.71213, -6.80222}, {3.75113, -6.84211}, {3.79762, -6.89062}, \
{3.83158, -6.92516}, {3.88353, -6.97841}, {3.95477, -7.05263}, \
{3.96866, -7.06733}, {3.97895, -7.07854}, {4.0509, -7.15543}, \
{4.05278, -7.15769}, {4.12632, -7.23931}, {4.14733, -7.26316}, \
{4.21626, -7.34519}, {4.27368, -7.4192}, {4.29268, -7.44655}, \
{4.31168, -7.47368}, {4.36291, -7.55674}, {4.41472, -7.67516}, \
{4.4176, -7.68421}, {4.41817, -7.68832}, {4.42105, -7.69696}, \
{4.43372, -7.78947}, {4.42926, -7.88302}, {4.42681, -7.89474}, \
{4.42623, -7.90214}, {4.42105, -7.92933}, {4.40321, -8.}};

I want to interpolate it, but when performing interpolation, Interpolation[Data], it writes that some points corresponding to 1st coordinate are duplicated:

Interpolation::inddp: The point 4.42105` in dimension 1 is duplicated.

This is, however, not the case, and I can't find out a reason for this.

Can you please help me?

Answer 1

You can find the duplicated point with Position:

Position[data, 4.42105`]
(*  {{79, 1}, {84, 1}}  *)

However, Interpolation is less sensitive than Position in determining distinct points:

data2 = {{1., 2.}, {1 + $MachineEpsilon, 2.}, {2., 2.}, {3., 2.}};
Interpolation[data2]

Interpolation::inddp: The point 1.0000000000000002` in dimension 1 is duplicated.

In this case, the simple call to Position fails, because all points are treated as distinct:

Position[data2, 1.0000000000000002`]
(*  {{2, 1}}  *)

Interpolation is more sensitive than Equal, so the following will identify the offending points and possibly others.

Position[data2, x_Real /; x == 1.0000000000000002`]
(*  {{1, 1}, {2, 1}}  *)

---

Update: More possible debugging

It's bit embarrassing not to do what you exhort others to do when things seem awry, namely, MAG or Make-A-Graph; perhaps in this case, the advanced variant MAGS or Make-A-Graph-Stupid should be used.

A data plot suggests that if the data were stored as they were collected, then the dependent and independent variables are switched:

ListPlot[data]

Reversing them makes a plot that looks like a function:

ListPlot[Reverse /@ data]

And the following returns an InterpolatingFunction without errors:

Interpolation[Reverse /@ data]