## Spline Interpolation Alternatives

A recent post at Jon Peltier’s Blog looks at an “on-sheet” method of performing linear interpolation on a set of tabular data, and the following comments include a number of alternative ways of carrying out the same process, and also some alternatives using cubic splines.  This prompted me to have a look at the various ways of performing this calculation, including VBA based User Defined Functions (UDFs) that have been presented here previously.

One reason for the interest in the large number of alternatives is that Microsoft (in their wisdom) do not provide any built-in functions to perform this task, even for linear interpolation.  Interpolation of scattered data using least squares fitting has a large number of useful functions (Forecast, Trend, Linest, Slope and Intercept for instance), but if we want to interpolate between specified points we are on our own. The steps in the process for linear interpolation are (assuming we want to find the Y value for a specified X, given a table of X and Y values, with X monotonically either increasing or decreasing):

• Find   the nearest X values above and below the given value, and the associated Y values.
• Find the slope of the straight line between these two points (Y2 – Y1)/(X2 – X1)
• The interpolated Y value is then: Y = Y1 + (X-X1)  * Slope

I have added examples of two different examples of this procedure to the CSpline2 spreadsheet, which also includes VBA cubic spline functions, with full open source code.   The screenshot below shows the output:

The position of the highest X value less than the interpolation value (60 in the example) is found using the Match function: =MATCH(A17,B6:B13).  That value is then used with either the Index or Offset function to produce a 2 x 2 table of the X and Y values around the interpolation value:

• =INDEX(B\$6:B\$13,\$A21) or
• =OFFSET(B6,A21-1,0,2,2)

Note that the Offset function must be entered as an array function; see Using Array Formulas for details.

Finally this table is used to calculate, the interpolated Y value, either using an arithmetic formula (essentially the process used in Jon Peltier’s article), or using the Trend function.  The Trend function can also be combined with the offset function, to avoid the need to create a 2 x 2 table on the spreadsheet for each interpolation:

• =TREND(OFFSET(\$C\$6,A17-1,0,2,1),OFFSET(\$B\$6,A17-1,0,2,1),A17)

This allows the interpolation to be carried out on a single row, which is convenient if you have a column of values to interpolate, rather than a single value.

Four alternative “single cell” interpolation formulas are shown in the screen shot below (all taken from a comment by Ihem to Jon Peltier’s article at : http://peltiertech.com/WordPress/excel-interpolation-formulas/#more-3322)

UDFs carrying out the same process (other than the Lagrange Polynomial) can be found at:

Cubic Splines: CSpline2
More cubic splines, including functions for weighted and constrained fitting: AL-Spline-Matrix03.zip or AL-Spline-Matrix07.zip

The results of the UDFs, compared with the “on-sheet” formulas are shown in the screenshot below:

It can be seen that the InterpA UDF and the Percentile formula give exactly the same results, as do the CardSplineA UDF and the Catmull-Rom formula.  The results for all 7 methods are plotted below:

There is very little difference between any of the interpolation functions, and for this case simple straight line interpolation would be quite satisfactory.  It should be noted that the cubic polynomial least squares fit is not so good however, and even for smoothly varying data such as in this case it would not be appropriate.

With less well behaved data the differences between the alternative methods become more apparent, as can be seen in the screen shot below:

In this case the fitted cubic polynomial is clearly inappropriate. The most appropriate of the spline approximations would depend on the nature of the data, and the reason for the spike.

As for whether to use a UDF or an “on-sheet” method, this is really a matter of tatse.  My preference is for using a UDF, as being more convenient, easier to audit, and less likely to introduce error, but in any case checking the results by plotting togther with the original data is essential.

This entry was posted in Excel, Maths, Newton, UDFs, VBA and tagged , , , , , . Bookmark the permalink.

### 7 Responses to Spline Interpolation Alternatives

1. Arnold Fiction says:

Exceptional post!! I have enjoyed much from this bio. I hope everybody should follow this package and thus they will benefit too. Would you share me another important post?

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2. Lori Miller says:

Doug – As a regular visitor, you might have guessed the source of some of the later comments at Jon’s site (i sometimes go by initials, the first letter is an L). Thanks for bringing these suggestions to light, it’s good to see close agreement with the UDFs.

To extend the nonlinear methods to end intervals you can select B6:C7 and drag the fill handle up to row 5, select B12:C13 and drag down to row 14 then just use the same formulas to fill remaining values and these should also match the chart line.

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3. dougaj4 says:

Hi Lori – I didn’t guess, but I should have.

Thanks for the hint on extending the range of the data. I have updated the spreadsheet and I will amend the images in the article tomorrow.

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4. DaleW says:

Although I didn’t reference Newton Excel Bach in my comments on Jon’s blog, your excellent open source UDFs for cubic splines are my favorite way of implementing cubic splines within Excel.

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• Lori Miller says:

DaleW – I see you were following up further questions on the linked post at Jon Peltier’s Blog. I added a reply to your comments but it spilled over to a second page where it’s not easy to find. I agree that Doug’s UDFs would be an easier option for the data in question too.

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5. excelandaccess says:

It is a great post with really nice examples. Very cool of you to have the examples downloadable.

I too love UDFs.

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