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Home > English site > Articles > Least Squares

Fitting curves to your data using least squares

Introduction

If you're an engineer (like I used to be in a previous life), you have probably done your bit of experimenting. Usually, you then need a way to fit your measurement results with a curve. If you're a proper engineer, you also have some idea what type of equation should theoretically fit your data.

Perhaps you did some measurements with results like this:

Fitting data with an equation
Fitting data with an equation.

A well known way to fit data to an equation is by using the least squares method (LS). I won't repeat the theory behind the method here, just read up on the matter by clicking that link to Wikipedia.

Fitting simple linear equations

Excel provides us with a couple of tools to perform Least Squares calculations, but they are all centered around the simpler functions: simple Linear functions of the shape
y=a.x+b, y-a.exp(b.x), y=a.x^b and etcetera. With some tricks you can also perform LS on polynomes using Excel.

Regression tools in the Analysis Toolpak Add-in

Activate the Analysis Toolpak in your list of Add-ins (File button or Office button, Excel Options, Add-ins tab, click Go):

Add-ins list of Excel
The add-ins list of Excel with the Analysis toolpak activated

This adds the "Data Analysis" button to your ribbon, on the Data tab, Analysis group (this is also the location where you can find the Solver button mentioned later on):

Ribbon with Data Analysis button
Ribbon with Data Analysis button

Click that button to explore which regression tools are available.

Worksheet functions

There is a number of worksheet functions which you can also use to do regression analysis. To quickly access them, select an empty cell and click shift+F3 to open the function wizard. In the search box, enter "Regression" (without the quotes of course). Excel will list the relevant functions:

Function wizard showing Regression functions
Function wizard showing Regression functions

Pick one and click on the "Help on this function" link at the bottom of the function wizard to find out more about its use.

Fitting more complex functions

What if you want to fit a more complex function, like y=exp(a.x).sin(x) + b ? How can that be done using Excel?

I devised a way to do this which involves the following steps:

Explanation of the Example file

I created an example file you can put to use directly. Below you will find a link to the file and an explanation on how the file is put together.

Download

Download this file:

Non linear least squares example

How the file works

Data

The calculations and the data are concentrated on Sheet1 of the file. The most important area is the table starting in cell A1:

Data table in LS file
Data table in LS file

Column A holds your x-values and column B holds the y-values. The third column holds the formula that calculates the result of the fitted equation using the constants and the x-values. The sample file has this formula in column C:

=EXP(Const_a*xValues)*SIN(xValues)+Const_b

The fourth column of the table is used to calculate the sum of squares. Formula:

=(B2-C2)^2

As you probably noted already, I used a couple of range names. I explain those below.

Range names

To ease working with the file I created some range names. Instead of using the table references that Excel 2007, 2010 and 2013 offer, I included some dynamic range names that point to the data. This means the workbook also works in Excel 2003 and before.

Range name Refers To Description
Const_a =Sheet1!$G$2 Model constant
Const_b =Sheet1!$G$3 Model constant
Const_c =Sheet1!$G$4 Model constant
Const_d =Sheet1!$G$5 Model constant
Const_e =Sheet1!$G$6 Model constant
Const_f =Sheet1!$G$7 Model constant
Const_g =Sheet1!$G$8 Model constant
Const_h =Sheet1!$G$9 Model constant
Constants =Sheet1!$G$2:$G$9 constants of equation
xValues =OFFSET(Sheet1!$A$2,0,0,COUNT(Sheet1!$A$1:$A$65551),1) Column with x values
yDelta =OFFSET(xValues,0,3) Column with Squared differences
yhat =OFFSET(xValues,0,2) Column with model fit results
yValues =OFFSET(xValues,0,1) Column with y values

Constants of the equation

The const range names point to a second table in the file:

Constants table
Constants table

This table is where you enter your first initial guesses for the resulting constants and where the Solver add-in also returns the results. As you can see, below that table the residual Sum of Squares is shown. Formula:

=SUM(yDelta)

It is this cell G11 that we try to minimize using the Solver add-in.

Using Solver

First of all, you need to install the Solver add-in. Use the Add-ins dialog I showed at the top of this article and check the box next to "Solver Add-in". This adds the Solver button in the same location on the ribbon as the "Data Analysis" button I showed before.

After you have ensured the model formula is correctly entered in column C and the calculations work, click the Solver button. The dialog below is shown:

The Solver dialog
The Solver dialog

Make sure the "Set Objective" box points to the cell that contains the sum of squares. Select "Min" next to "To".

The "By Changing Variable cells" box must ONLY point to the cells that are used by your model, otherwise the degrees of freedom calculation (on the ANOVA sheet) will be wrong. Also ensure that any unused constant cells are empty by selecting them and hitting the del key.

Note that depending on your model type you may have to change the solver settings. A bit of experimenting may be needed for best results. You can save and load Solver settings using the appropriate button.

So be prudent and critical on whether or not you have actually reached a best fit, the Solver may come up with non-optimal results, depending on your model equation and solver settings.

If you're happy with the current Solver settings, click Solve. After some time the "Solver Results" dialog opens, giving you some options on how to continue. Note that it also enables you to ask for a couple of reports.

The example file shows the end result:

The end result
The end result

Analysis of Variance

On the ANOVA tab, you can find the ANalysis Of VAriance table, which looks like this:

The ANOVA table
The ANOVA table

The most important cell here is cell F2. If the value in that cell is less than 0.05, there is a 95% probability your model is correctly fitting the data. So less is more for this cell, you want it to stay below 0.05. The cell will turn red for values over 0.05.

Please check whether the value in cell B2 is exactly one less than the number of constants you used for the model. If not, go back to Sheet1 and empty the cells not used by your model. So if you used const_a and const_b, then the value of B2 (model degrees of freedom) should be 1.

Conclusion

As you've seen fitting complex functions to your data isn't very hard to do. A combination of some relatively simple formulas and the Solver Add-in comes to the rescue here.

Some advice as one engineer to another; Be critical please. Don't believe everything Excel tells you! Carefully analyse the results it returns, as Solver may get things wrong and not give you the best possible result!


Comments

Showing last 8 comments of 52 in total (Show All Comments):

 


Comment by: Jan Karel Pieterse (3/10/2015 1:52:05 PM)

Hi Kevin,

I'm not entirely sure, my first reponse was that r-squared only applies to linear fits. But I'm not even sure on that :-)

 


Comment by: DC (3/26/2015 1:23:58 PM)

Thank you for your informative resource.

I've just launched a free online tool to do this type of curve-fitting in a much simpler way. It's free to use, does not require any download and exports results to Excel. See http://www.mycurvefit.com.

I welcome any feedback you can provide.

 


Comment by: DC (3/26/2015 1:44:46 PM)

And here is how your example data appears on the site:

http://www.mycurvefit.com/index.html?action=openshare&id=08cfa3e5-37a3-4187-b4ac-97108639acb2

I hope to hear from you!

All the best,

Darren

 


Comment by: Jan Karel Pieterse (3/26/2015 2:04:32 PM)

Hi DC,

Interstingly, it seems your ANOVA returns slightly different results from mine?

 


Comment by: DC (3/26/2015 5:51:58 PM)

Thanks Jan,

The earlier link included only the first 9 rounded values (from the screen image). The following includes the whole data set pasted in from your Excel document (from your downloads page):

http://www.mycurvefit.com/index.html?action=openshare&id=e9975460-3d24-46b9-a6b4-18f98d1c17e7

Here the results are much closer.

The results displayed on the web-page show values to 7 sig figs. In this case the calculated coefficients are almost equal to Solver's. I would expect a slight difference in the results based on the different implementations of the minimisation process. In both cases the results are very good.

 


Comment by: Jan Karel Pieterse (3/26/2015 6:24:08 PM)

Hi DC,

I wasn't referring to the fitted constants but rather to the value of F. yours shows 954, I seem to have 383. Not sure which is correct :-)

 


Comment by: Duncan (3/27/2015 3:01:16 PM)

Hello,
Am a university student working on my final year project.

Using an empirical formula in the form of y=A*x^(B)*z^(C), I want to determine the constants A,B, and C based on experimental resuts of y,x, and z using least-square fit method but i have failed. Any help would be welcome.
Thank you

 


Comment by: Jan Karel Pieterse (3/30/2015 3:21:29 PM)

Hi Duncan,

It may very well be that Solver is having difficulties solving your problem. The only advice I can give you is to experiment with the settings of solver.
If you have an idea in what range the A, B and C values should be, it will also help setting thoes as initial values prior to running solver.

 


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