Linear Program Polynomial Interpolation Excel
Linear Interpolation Given a set of x vs y data, the linear interpolation routine, Interpolate, will find an appropriate value of y i given a value of x i. The routine will determine if x i is in between any of the x data values. If it is, then linear interpolation is performed the find the value of y i. If it is before the first x value, then the y i.
Recently, I received the following question from a reader:“I have an Excel question – Is there a way to interpolate a value from a table? I have an X and Y that are not on the table, but have correlated data so want to calculate the interpolated value”.Interpolation estimates data points within an existing data set.
As a simple example, if it took 15 minutes to walk 1 mile on Monday and 1 hour to walk 4 miles on Tuesday, we could reasonably estimate it would take 30 minutes to walk 2 miles. This is not to be confused with extrapolation, which estimates values outside of the data set. To estimate that it would take 2 hours to walk 8 miles would be extrapolation as the estimate is outside of the known values.Excel is a great tool for this type of analysis, as ultimately it is just a big visual calculator.In terms of answering my reader’s question, there are a number of scenarios which would lead to different solutions.
At first, I thought I could just use simple mathematics. This would work if the results were perfectly linear (i.e., the values move perfectly perfect sync with each other). But what if they are not perfectly correlated?I then thought about Excel’s FORECAST function. Based on its name, the FORECAST function seems like an odd choice.
It would appear to be a function specifically for extrapolation, however it is one of the best options for linear interpolation in Excel. FORECAST uses all the values in the dataset to estimate the result, therefore is excellent for linear relationships, even if they are not perfectly correlated.Then another thought, what if the X and Y relationship is not linear at all? How could we interpolate a value when the data is exponential?In this post we’ll look at these three scenarios:.Interpolation using simple mathematicsUsing simple mathematics works well when there are just two pairs of numbers or where the relationship between X & Y is perfectly linear. Here is a basic example. The formula in Cell E3 is: =B2+(E2-A2).(B3-B2)/(A3-A2)That might look a bit complicated to some, so I’ll just give a quick overview of this formula. =B2+(E2-A2).
(B3-B2)/(A3-A2)The last section (highlighted in red above) calculates how much the Y value moves whenever the X value moves by 1. In our example, Y moves by 1.67 for every 1 of X.
=B2+ (E2-A2).(B3-B2)/(A3-A2)The second section (in red above) calculates how far our interpolated X is away from the first X, then multiplies it by the value calculated above. Based on our example, it is 17.5 (Cell E2) minus 10 (Cell A2), the result of which is then multiplied by 1.67 (which equals 12.5). = B2+(E2-A2).(B3-B2)/(A3-A2)Finally, the first section of the formula (in red above); we add the first Y value. In our example, this provides the final result of 77.5 (65 + 12.5). For anybody who remembers high school mathematics the formula is as follows:Even if you don’t remember that from school, the good news is that Excel has given us a simpler option, the FORECAST function.
Interpolation using the FORECAST functionThe FORECAST function is great when we want to extrapolate values, to forecast the value of a future point. The FORECAST function in Cell E3 is interpolating the Y value based on an X value of 17.5. =FORECAST(E2,B2:B11,A2:A11)In these circumstances, FORECAST estimates a value based on all the available data, not just the start and end points. The result of the FORECAST function in Cell E3 is an estimate of 77.3 (rounded to 1 decimal place), which is more accurate than the simple linear interpolation applied above. Interpolation when the data is not linearBut, here’s the bigger question, what if the data is not linear at all? Here is our new data:Look at the chart below, which represents the data above.
If we took a simple linear approach, that would give us a value of 77.5, which as you can see, is quite a long way from the curve. Using the FORECAST function as above, it would provide a result of 70.8, which also a long way from the curve.In this circumstance, our approach needs to change. A reasonable option is to find the result above and below the X value, then apply straight-line interpolation between those two points.Using our example of 17.5, we would need to retrieve the results of 16 and 18 (both of which we have values for), then calculate the linear interpolation. To do this, we will use the INDEX, MATCH and FORECAST functions combined. This may sound tricky, but don’t worry, we’ll walk through it slowly.Firstly, we use the MATCH function to retrieve the position of the value lower than 17.5. =MATCH(E2,A2:A11,1)This formula is saying find the value in Cell E2 from the range of Cells A2-A11.
Bilinear Interpolation Excel
The 1 at the end of the formula tells the MATCH function that we wish to use an approximate match (i.e., the closest value below the lookup value). 16 is the closest value below 17.5. As 16 is the 5th item in the Cells A2-A11, it will return a value of 5.One key point to note, this method does require the range of known X’s to be listed in ascending order.Having identified in the previous stage that the 5th position contains the value below, we can use the INDEX function to identify the Cell reference for this value. INDEX(A2:A11,MATCH(E2,A2:A11,1))This would return cell reference A6.To find the value above, we can use the same function, but add 1 to the MATCH function. INDEX(A2:A11,MATCH(E2,A2:A11,1) +1)The formula above would return cell reference A7.Now things start to get a bit tricky.We can combine these functions together with a colon (: ) in the middle to for the two X values. INDEX(A2:A11,MATCH(E2,A2:A11,1)): INDEX(A2:A11,MATCH(E2,A2:A11,1)+1)The first INDEX function will return the cell reference of A6 (the result of the red highlighted section). The second INDEX function will return the cell reference of A7 (the result of the blue highlighted section).
These are separated by a colon (: ), to create a range – A6:A7We can do the same to create a range for the two Y values. The only difference is the INDEX function will look at Cells B2-B11. INDEX(B2:B11,MATCH(E2,A2:A11,1)):INDEX(B2:B11,MATCH(E2,A2:A11,1)+1)By using Cells B2-B11 in the INDEX function, it will create a range B6:B7.Now we have our two ranges the X values A6:A7 and the Y values B6:B7.Please note: it is also possible to use the OFFSET function if you prefer (but it is a, so may cause slower calculations).Finally, let’s combine all of this together within the FORECAST function. =FORECAST(E2,INDEX(B2:B11,MATCH(E2,A2:A11,1)):INDEX(B2:B11,MATCH(E2,A2:A11,1)+1),INDEX(A2:A11,MATCH(E2,A2:A11,1)):INDEX(A2:A11,MATCH(E2,A2:A11,1)+1))That’s a pretty big formula, right. Hopefully I’ve managed to explain it so that it’s not too scary.The result of the Non-Linear FORECAST function is 67.6 (to 1 decimal place, shown in Cell F5 below). Have a look at the chart again you will see it is a much more accurate estimation based on the data available.A word of warningUltimately, this is still a linear interpolation calculation but based on the two values either side of the X value.
The distance between the values above and below will have a direct impact on how accurate the interpolation is. Other uses for the FORECAST functionThe FORECAST function can also be used to extrapolate values to estimate future values. When the data follows a seasonal trend, we must use some Excel trickery –.
ConclusionInitially, what seemed like a simple question has lead us to three potential solutions for three different scenarios. You need to know your data to select the method which provides the most accurate results.In the process, we have covered the FORECAST function and seen that it is useful for interpolation as well as extrapolation. Also, we have seen that INDEX and MATCH can be used to create ranges for use within other formulas. Jos says:I used this guidance (forecast+index+match) to get the interpolation for my non-lineair data. But i have a problem that it’s not accurate enough.
I want to know the y value with x= 2000, between the points 500,0.02 and 12500,0.01.I suppose it is because of the big distance between the x values and the very small difference in the y values. The calculation gives me y=0.01875, while thats not on the line, it should be around 0.016. It is a very small difference, but for my purpose essential. Is there any way to get a more accurate answer? This website uses cookies to improve your experience while you navigate through the website.
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Polynomial Interpolation Excel
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FreeThree Dimensional Interpolateand Extrapolate Functions; Two Dimensional Interpolate, Extrapolate, CubicSpline, Polynomial Curve Fitting, Line Intersections, Minima, Maxima, FirstDerivative, Second Derivative, and other functions.Posted with the kind permission of:Free,XlXtrFun.xll is a collection of functions which extends thecapabilities of Microsoft Excel; developed primarily to facilitate,interpolation of 2-dimensional and 3-dimensional data, and simplify 2-variablecurve fitting. XlXtrFun has been used for years by engineering and research anddevelopment personnel on every continent who need to interpolate, extrapolate,and curve fit data rapidly, reliably, and with a virtually non-existent learningcurve.If you work with real-life data and want to interpolate, extrapolate, orcurve fit, then you will find these functions very useful.' There is a great satisfaction in building good tools forother people to use.'
Freeman Dyson (b. 1923); British-born U.S. Physicist, author.From a user in Minano, Italy: 'Your functions have become very popular in thelast two or three years, and it would be tough to abandon them.' Using Spline, Interpolate,Intersect, dYdX, and ddYdX functions.Example of the use of Spline,Interp,and Interpolate functions.KEY BENEFITS.Fast, reliable interpolated and extrapolated values in two and threedimensions.Three dimensional interpolation and extrapolation using either a set of(x, y, z) points, or matrix of evenly spaced z values.If the default method of interpolation and extrapolation is unsuitable, itcan be controlled using function arguments which can be cell references. Theinterpolation and extrapolation method can therefore be easily changed andresults can be viewed instantly.Polynomial curve fit using the least squares method (up to about order 49)without building elaborate matrices. Order of fit is a function argument whichcan also be a cell reference.
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