LOESS Smoothing in Excel
by Jon Peltier
Monday, March 9th, 2009
Peltier Technical Services, Inc., Copyright © 2012.
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
In 1979 William Cleveland published the LOESS (or LOWESS) technique for smoothing data, and in 1988 he and Susan J. Devlin published a refined version of the technique (references are given at the end of this article). For each X value where a Y value is to be calculated, the LOESS technique performs a regression on points in a moving range around the X value, where the values in the moving range are weighted according to their distance from this X value.
The NIST Engineering Statistics Handbook has a good description of the LOESS technique, including a worked example. A commenter named Nick used the NIST chapter as a starting point for his implementation of a LOESS function for Excel, and he posted in in a comment on JunkCharts. Nick’s approach was to create a UDF in VBA. The UDF accepts as inputs the X and Y data ranges, the number of points to use in the moving regression, and the X value for which to calculate Y. Nick’s UDF used Dectionary objects to hold intermediate values, and it outputs the Y value for the input X value.
I’ve expanded on Nick’s starting point, and produced the function presented later in this article. I’ve discarded the Dictionary objects in favor of VB arrays. It accepts the input X and Y data and the output X values either as ranges or as vertical arrays, and it outputs the calculated LOESS Y values as a vertical array. This means it can be called from within other procedures using arrays, or as a UDF from a worksheet, as an array formula. The original data must be sorted by X in either ascending or descending order (Nick’s Dictionaries do not require sorted input data, and I have an idea to remove the requirement from my function).
My algorithm, Nick’s original algorithm I based mine on, the NIST algorithm Nick based his upon, and all others I checked in the wild use a linear regression of the weighted values, and the weighting factor for input data point i is a sigmoidal curve based on
W(i) = (1 - X(i)^3)^3
where X(i) is the normalized distance (along the X axis) between input data point i and the output X value at which the LOESS smoothed value is being computed. The normalization X is the distance/(maximum distance among points in the moving regression).
To use the function as a UDF, select the multicell output Y range, and enter this formula:
=loess(C2:C22,D2:D22,F2:F21,7)
where C2:C22 and D2:D22 are the input X and Y ranges, F2:F21 is the output X range, and 7 is the number of points in the moving regression (see screenshot below).
Enter this as an array formula by holding Ctrl and Shift while pressing Enter, and the selection fills with the calculated Y values. Note the curly braces around the formula in the formula bar, which indicates the formula is an array formula.
This chart shows the original NIST data points and the smoothed LOESS curve.
In Local regression, Wikipedia has a decent description of LOESS, with some pros and cons of this approach compared to other smoothing methods.
Example Uses of LOESS
This chart compares LOESS smoothing of website statistics with a simple 7-day moving average. The LOESS captures the major trends in the data, but is less severely affected by week to week fluctuations such as those occuring around Thanksgiving and over the year-end and New Year holidays.
Using LOESS to analyze the body mass indexes (BMI) of Playboy playmates gives more insights than linear regression over the whole data set or over portions of the data. See the discussion in Wired Relates Playboy Playmate BMI and Average BMI, 1954-2008 on the FlowingData blog.
The LOESS Function
Public Function LOESS(X As Variant, Y As Variant, xDomain As Variant, nPts As Long) As Double()
Dim i As Long
Dim iMin As Long
Dim iMax As Long
Dim iPoint As Long
Dim iMx As Long
Dim mx As Variant
Dim maxDist As Double
Dim SumWts As Double, SumWtX As Double, SumWtX2 As Double, SumWtY As Double, SumWtXY As Double
Dim Denom As Double, WLRSlope As Double, WLRIntercept As Double
Dim xNow As Double
Dim distance() As Double
Dim weight() As Double
Dim yLoess() As Double
If TypeName(X) = "Range" Then
X = X.Value
End If
If TypeName(Y) = "Range" Then
Y = Y.Value
End If
If TypeName(xDomain) = "Range" Then
xDomain = xDomain.Value
End If
ReDim yLoess(LBound(xDomain, 1) To UBound(xDomain, 1), 1 To 1)
For iPoint = LBound(xDomain, 1) To UBound(xDomain, 1)
iMin = LBound(X, 1)
iMax = UBound(X, 1)
xNow = xDomain(iPoint, 1)
ReDim distance(iMin To iMax)
ReDim weight(iMin To iMax)
For i = iMin To iMax
' populate x, y, distance
distance(i) = Abs(X(i, 1) - xNow)
Next
Do
' find the nPts points closest to xNow
If iMax + 1 - iMin <= nPts Then Exit Do
If distance(iMin) > distance(iMax) Then
' remove first point
iMin = iMin + 1
ElseIf distance(iMin) < distance(iMax) Then
' remove last point
iMax = iMax - 1
Else
' remove both points?
iMin = iMin + 1
iMax = iMax - 1
End If
Loop
' Find max distance
maxDist = -1
For i = iMin To iMax
If distance(i) > maxDist Then maxDist = distance(i)
Next
' calculate weights using scaled distances
For i = iMin To iMax
weight(i) = (1 - (distance(i) / maxDist) ^ 3) ^ 3
Next
' do the sums of squares
SumWts = 0
SumWtX = 0
SumWtX2 = 0
SumWtY = 0
SumWtXY = 0
For i = iMin To iMax
SumWts = SumWts + weight(i)
SumWtX = SumWtX + X(i, 1) * weight(i)
SumWtX2 = SumWtX2 + (X(i, 1) ^ 2) * weight(i)
SumWtY = SumWtY + Y(i, 1) * weight(i)
SumWtXY = SumWtXY + X(i, 1) * Y(i, 1) * weight(i)
Next
Denom = SumWts * SumWtX2 - SumWtX ^ 2
' calculate the regression coefficients, and finally the loess value
WLRSlope = (SumWts * SumWtXY - SumWtX * SumWtY) / Denom
WLRIntercept = (SumWtX2 * SumWtY - SumWtX * SumWtXY) / Denom
yLoess(iPoint, 1) = WLRSlope * xNow + WLRIntercept
Next
LOESS = yLoess
End Function
The LOESS Utility
The flexibility of this LOESS function will make it easy to encapsulate into an add-in that uses a dialog to facilitate user selection of data and parameters. A working version uses the following dialog:
Update 24 June 2009
A LOESS utility for Excel has finally been made ready for public consumption. It is described in LOESS Utility for Excel, where there is a link to download the utility. It’s still in preliminary form, but runs pretty much trouble free. Users are encouraged to comment on it to drive further development.
Update 8 October 2009
The LOESS utility for Excel has been updated, and the interface made more flexible. It is described in LOESS Utility – Awesome Update, where there is a link to download the new utility.
References
Cleveland, W.S. (1979), “Robust Locally Weighted Regression and Smoothing Scatterplots,” Journal of the American Statistical Association, Vol. 74, pp. 829-836.
Cleveland, W.S. and Devlin, S.J. (1988), “Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting,” Journal of the American Statistical Association, Vol. 83, pp. 596-610.
NIST Engineering Statistics Handbook, 4.1.4.4. LOESS (aka LOWESS)
NIST Engineering Statistics Handbook, Example of LOESS Computations
Wikipedia, Local regression
Related Posts:
- LOESS Utility for Excel
- LOESS Utility – Awesome Update
- Deming Regression Utility
- UDF to Calculate an Arbitrary Formula
- Deming Regression
- LOESS Utility – What the Buttons Mean
Posted: Monday, March 9th, 2009 under Data Techniques.
Comments: 45
Comments
Comment from Colin Banfield
Time: Monday, March 9, 2009, 1:02 pm
Jon, nice stuff. I see that you’re added a link to this page on Wikipedia :D I haven’t done a lot of digging on the subject, but I was wondering how one would determine the optimum values for alpha and N. In moving average forecasing, for example, one would calculate the interval (simple moving average) or the damping factor (exponential smoothing) based on minimizing the mean absolute percentage error (which Excel’s ATP assumes you’re already calculated in your head!).
Comment from Jon Peltier
Time: Monday, March 9, 2009, 1:20 pm
Colin -
I don’t know how you’d optimize your alpha. A couple sources I looked at said “around 0.33″ or “below 0.5″, but I think you’re left to decide based on how it looks. If N gets too large, the curve doesn’t even follow most of the points, but if it isn’t large enough, the curve is too wiggly.
I used N=7 for the NIST data, because that’s what the example used, though I looked at a lot of other values. I used probably alpha=0.33 for the playmate data, and I looked at a few other values, but it didn’t make much difference. The web stats showed some difference with changing alpha, but mostly in the width and depth of the year-end dip.
I think the idea is to get a good overview which isn’t too badly affected by an outlying point.
If anyone has any guidelines other than “it looks nice”, please share with us.
Pingback from LOESS smoothing « Learning R
Time: Monday, March 9, 2009, 3:15 pm
[...] Posted by learnr under R | Tags: ggplot2, R | Jon Peltier writes about the LOESS smoothing in Excel, and presents a utility to facilitate adding smoothers to the data. He goes on to show how to use [...]
Comment from TV
Time: Tuesday, March 10, 2009, 7:32 am
Is changing this to LOESS function to a 2nd order difficult? It seems like a first order fit leads to discontinuities in the derivative…
Comment from Jon Peltier
Time: Tuesday, March 10, 2009, 11:05 am
TV -
You’d have to overhaul everything after this comment:
' do the sums of squares
I did a quick Google search, and found one useful bit of information in a pile of results pages:
http://www.eng-tips.com/viewthread.cfm?qid=55381
For y=ax^2+bx+c
P=Sum(x)
Q=Sum(x^2)
R=Sum(x^3)
S=Sum(x^4)
T=Sum(y)
U=Sum(xy)
V=Sum(x^2y)
W=N*Q*S+2*P*Q*R-Q^3-P^2*S-N*R^2
Then
a=(N*Q*V+P*R*T+P*Q*U-Q^2*T-P^2*V-N*R*U)/W
b=(N*S*U+P*Q*V+Q*R*T-Q^2*U-P*S*T-N*R*V)/W
c=(Q*S*T+Q*R*U+P*R*V-Q^2*V-P*S*U-R^2*T)/W
You’ll have to figure out what to do with the weights.
Comment from Will Dwinnell
Time: Saturday, March 14, 2009, 9:56 am
I like your article.
I came to the same conclusion you did about the Playboy data from Wired, and got results similar to yours, fitting a kernel regression in MATLAB.
Comment from Jon Peltier
Time: Saturday, March 14, 2009, 12:30 pm
Hi Will -
Thanks for the validation of my conclusions, especially since they involved a different tool. Two smart guys can’t be wrong, eh?
Pingback from dekay.org » links for 2009-03-15
Time: Sunday, March 15, 2009, 6:02 am
[...] LOESS Smoothing in Excel | PTS Blog (tags: loess data smoothing curve fit via:mento.info) [...]
Comment from Nick
Time: Wednesday, April 1, 2009, 4:27 pm
I stumbled upon that old comment of mine at JunkCharts, and saw your comment as well which led me here. First let me say: Wow, you’ve obviously got more VBA skills than I do, and although I haven’t tested it it looks great.
As for the question about using 2nd degree polynomials or higher instead of linear. You could replace the stuff after ‘do sum of squares’ with some trickery involving the LINEST function. (See Clyde38′s post at http://www.eng-tips.com/viewthread.cfm?qid=184726 ) but you’d lose the weighting.
Comment from Jon Peltier
Time: Wednesday, April 1, 2009, 4:30 pm
Nick -
I have to say that I wouldn’t have gotten too far without your coding of the original function.
As for the higher order fit, it seems most people that I noticed just use first order, and second, I would want somehow to keep the weighting. I thought maybe a little dimensional analysis might help, but I need at least an hour to get my brain around stuff like that.
Comment from Bang Seung Beom
Time: Monday, May 18, 2009, 9:04 pm
Hi.
Thanks for your code.
But I cannot find smoothing parameter (alpha) in your function?
Which variable is it?
Comment from Jon Peltier
Time: Monday, May 18, 2009, 9:08 pm
Alpha isn’t used directly in the calculations. The code uses nPts, the number of points to use for the moving regression. Alpha is the fraction of the whole data set used in the moving regression, or nPts/(number of values in X or Y). In the dialog-based version of this code, you can set either parameter, and the other is computed.
Comment from Andrew Bond
Time: Saturday, May 23, 2009, 3:02 pm
Jon
Thanks for a very interesting blog on Loess. I am in the process of writing a C++ implementation of Loess for use in processing information from ground investigations (I am a civil engineer by profession).
Having failed to port the available C implementations that I have found on the web, I have used your code as inspiration for mine.
I am now at the testing stage and wanted to compare results from my program with those posted in this blog. However, I can’t find the original (full) set of data. Can you provide a link to it here, please?
I would like to have the web statistics and Playmate data – the NIST data I already have. (I have found a subset of the Playmate data, containing just Playmates of the Year – not all Playmates).
Thanks in anticipation.
Andrew
Comment from Jon Peltier
Time: Saturday, May 23, 2009, 5:49 pm
Andrew -
Infoporn: Today’s Playmates Are More Like Anime Figures Than Real Humans in Wired Magazine has links to a PDF visualization and an Excel workbook of BMI data.
Comment from Andrew Bond
Time: Sunday, May 24, 2009, 5:51 am
Jon
Many thanks for the link – I missed it when I first visited the Wired website. What smoothing coefficient did you use for your loess fit? And is the website data as readily available as the Playmate data?
Again, thanks in anticipation.
Andrew
Comment from Jon Peltier
Time: Sunday, May 24, 2009, 6:53 am
Andrew -
I used 150 for the number of points in the analysis. Here is a comparison of three curves, for 100, 150, and 200 points in the moving regression:
The website data is from my own site’s history several months ago. I’m not sure I still have that particular data set.
Comment from Brian
Time: Friday, June 5, 2009, 4:06 pm
This is something I’ve been looking all over for. Thanks. I added a weighting capability, so that each case can be weighted with a third input range. I know next to nothing about VBA, but I think I did it right.
Public Function LOESSW(X As Variant, Y As Variant, PtWt As Variant, xDomain As Variant, nPts As Long) As Double()
Dim i As Long
Dim iMin As Long
Dim iMax As Long
Dim iPoint As Long
Dim iMx As Long
Dim mx As Variant
Dim maxDist As Double
Dim SumWts As Double, SumWtX As Double, SumWtX2 As Double, SumWtY As Double, SumWtXY As Double
Dim Denom As Double, WLRSlope As Double, WLRIntercept As Double
Dim xNow As Double
Dim distance() As Double
Dim Weight() As Double
Dim yLoess() As Double
Dim totalWeight As Double
If TypeName(X) = "Range" Then
X = X.Value
End If
If TypeName(Y) = "Range" Then
Y = Y.Value
End If
If TypeName(PtWt) = "Range" Then
PtWt = PtWt.Value
End If
If TypeName(xDomain) = "Range" Then
xDomain = xDomain.Value
End If
ReDim yLoess(LBound(xDomain, 1) To UBound(xDomain, 1), 1 To 1)
For iPoint = LBound(xDomain, 1) To UBound(xDomain, 1)
iMin = LBound(X, 1)
iMax = UBound(X, 1)
xNow = xDomain(iPoint, 1)
ReDim distance(iMin To iMax)
ReDim Weight(iMin To iMax)
For i = iMin To iMax
' populate x, y, distance
distance(i) = Abs(X(i, 1) - xNow)
Next
Do
' find the nPts points closest to xNow
If iMax + 1 - iMin distance(iMax) Then
' remove first point
iMin = iMin + 1
ElseIf distance(iMin) maxDist Then maxDist = distance(i)
totalWeight = totalWeight + PtWt(i, 1)
Next
' calculate weights using scaled distances
For i = iMin To iMax
Weight(i) = (PtWt(i, 1) / totalWeight) * (1 - (distance(i) / maxDist) ^ 3) ^ 3
Next
' do the sums of squares
SumWts = 0
SumWtX = 0
SumWtX2 = 0
SumWtY = 0
SumWtXY = 0
For i = iMin To iMax
SumWts = SumWts + Weight(i)
SumWtX = SumWtX + X(i, 1) * Weight(i)
SumWtX2 = SumWtX2 + (X(i, 1) ^ 2) * Weight(i)
SumWtY = SumWtY + Y(i, 1) * Weight(i)
SumWtXY = SumWtXY + X(i, 1) * Y(i, 1) * Weight(i)
Next
Denom = SumWts * SumWtX2 - SumWtX ^ 2
' calculate the regression coefficients, and finally the loess value
WLRSlope = (SumWts * SumWtXY - SumWtX * SumWtY) / Denom
WLRIntercept = (SumWtX2 * SumWtY - SumWtX * SumWtXY) / Denom
yLoess(iPoint, 1) = WLRSlope * xNow + WLRIntercept
Next
LOESSW = yLoess
End Function
Comment from Jon Peltier
Time: Friday, June 5, 2009, 5:52 pm
Thanks Brian. Maybe one of the smart people who read this blog will check it out for us.
Comment from Tian
Time: Thursday, September 24, 2009, 12:38 pm
Hi Jon,
Thank you for the nice code. It helped me easily added a function to a C++ App. There is a small bug you might not notice. The assignment for xNow should be:
xNow = X(xDomain(iPoint, 1) + 1, 1)
instead of:
xNow = xDomain(iPoint, 1)
Thank you,
Comment from Jon Peltier
Time: Thursday, September 24, 2009, 12:58 pm
Tian -
The line of code works fine in VBA, which assigns the values in a range to a 1-based array. You have to adjust the syntax to use the code in C++.
Comment from Tian
Time: Thursday, September 24, 2009, 1:17 pm
Jon,
Please see the difference of the output:
X input Y input X output original output new output
0.55782 18.63654 0 -13.07212037 20.59304051
2.021727 103.4965 1 47.06914445 107.1603064
2.577325 150.3539 2 105.8974689 139.7673806
3.414029 190.5103 3 159.6924955 174.2630716
4.301408 208.7012 4 196.8626739 207.2333938
4.744839 213.7114 5 219.0115635 216.6616039
5.107378 228.4935 6 227.6417493 220.5444981
6.541166 233.5539 7 230.2534309 229.8606994
6.721618 234.5505 8 227.0376643 229.8347242
7.260058 223.8923 9 221.246659 229.4301269
8.133587 227.6834 10 202.9876659 226.6044626
9.122438 223.9198 11 187.8518139 220.3904231
11.92967 168.02 12 170.6711911 172.3479193
12.37977 164.9575 13 162.210816 163.8416617
13.27286 152.6111 14 160.7756941 161.8489846
14.27675 160.7874 15 159.7155843 160.3350921
15.3731 168.5557 16 161.0838629 160.1920102
15.64766 152.4266 17 194.5740586 161.0555463
18.56054 221.707 18 215.4190384 227.3399984
18.58664 222.6904 19 236.7904956 227.8985782
18.75728 243.1883
Comment from Jon Peltier
Time: Thursday, September 24, 2009, 6:23 pm
Your original output is the result of LOESS calculation using a 7 point moving regression, using X and Y input as inputs and X output as the X output.
Your new output is the result of LOESS calculation using a 7 point moving regression, using X and Y input as inputs and X input as the X output.
Comment from Tian
Time: Friday, September 25, 2009, 9:40 am
Now I understand what you mean. Thank you for clarifying.
Comment from Don
Time: Tuesday, October 6, 2009, 7:01 pm
In MathCAD 14, you create a 2-column matrix of the data and set the first column = X and the second column = Y. I named my matrix NIST. Then enter the following:
NIST(x):=interp(loess(X,Y,.5),X,Y,x)
where x was set as a range variable x:=0,0.1,..20
This creates a function NIST(x) that you can plot versus the range variable x
Here is the MathCAD explanation for the two built-in functions ‘interp’ & ‘loess’:
loess(vx, vy, span) Returns a vector which interp uses to find a set of second-order polynomials that best fit the neighborhood of x and y data values in vx and vy in the least-squares sense. The size of the neighborhood is controlled by span.
interp(vs, vx, vy, x) Returns the interpolated y-value corresponding to x using the output vector vs from loess.
Comment from Tom
Time: Friday, January 29, 2010, 6:16 pm
I guess your page explains LOESS more practical than all the rocket science pages I tried before I came here.
You have a big glitch in your moving average example. MA always get a lag of n/2 points. The reason is that the MA is the 50% trend value of one linear regression of n x,y points. Both smoothing filters are comparable in the X axis if the MA is calculated from the n/2 days in the future. You would use the MA with OFFSET and AVG functions then and not take the diagram’s moving average.
However, some example charts indicate lags in the LOESS .. am I wrong?
By the way, I always was and am a big fan of your Excel experiences.
Big timesavers for me.
Comment from Jon Peltier
Time: Saturday, January 30, 2010, 10:06 am
Tom -
A comparison of moving average to loess would make more sense if the calculated moving average was plotted in the middle of the data for the moving average, but conventionally this is not done. Loess never used this convention.
Where I have noted an apparent lag in loess is in cases where the variation in the raw data is not symmetric, for example, when an increase is followed by a decrease which is much more or less steep. Changing the number of points in the moving regression may reduce the apparent lag. The apparent lag may be pronounced near the ends of the data set, where the regression consists of substantially more points on one side of the X value for the loess calculation.
Comment from tom
Time: Saturday, January 30, 2010, 2:57 pm
>The apparent lag may be pronounced near the ends of the data set
That’s what I know from moving average fitting.
It was designed to fit a time series where mostly the newest fit is the interesting one.
From the VBA code I would say that LOESS fitting is really a choice for sparse data series. Time series make only sense if there are gaps in between. I used a similar technique already on time series in Excel like in your example.
If the distances are always the same between two X points, then the algorithm is just too general and consuming. My experience is that you always find a moving average that is as almost as good fitting as the moving regression window.
By the way: Why did you use Redim in a for-loop? There is another redim going over
all available data. One second Redim before the for would take more memory and save
time.
Comment from Jon Peltier
Time: Saturday, January 30, 2010, 4:37 pm
“I would say that LOESS fitting is really a choice for sparse data series.”
The literature indicates that loess is better for heavily populated data sets.
Do your moving averages weight the data prior to averaging?
The ReDim statements and the iMin and iMax statements can all be taken out of the loop. Perhaps the original programmer initially had them inside the loop before I saw the code, and then I never put them in front of the loop.
Comment from Neha
Time: Thursday, July 29, 2010, 9:46 am
Hi,
I have count data (around 20,000 cases) for different days of the weeks. There are outliers in my series. Since my data is sparse, I cannot remove these outliers completely. I am planning to assign the weights to these outliers in order to smooth the data series.
For this I thought of using either Loess or rloess. Please suggest is it correct to do so?
Comment from Jon Peltier
Time: Thursday, July 29, 2010, 1:38 pm
Neha -
All I can say is “Try it”.
LOESS assigns its own weights to data for each of its regression calculations, I don’t know what kind of weights you plan to assign.
Comment from Scott
Time: Friday, September 10, 2010, 1:36 am
Does anyone have the C++ implementation of this? I have the Add-In working great after searching a while. Not having success with the C++ I have found on the net and I know someone out there has ported it over. Many thanks in advance!
Pingback from Creating More Effective Graphs [Book Review]
Time: Tuesday, February 1, 2011, 5:40 am
[...] using logarithmic scales and scatter plot smoothers like loess (Jon Peltier has a blog post on how to implement this in Excel). Robbins prefers dot plots to bars charts, and who can blame [...]
Comment from Bancam
Time: Monday, May 16, 2011, 3:11 am
Hi Jon,
I like this vanilla implementation – as a simple UDF. Could make both nPts and nAlpha optional with code to decide on what to use.
I am curious: what did mx and iMx represent in the original code?
Only thing I see is, it needs the Xvalues to be in ascending order. Some form of sorting would be handy when x,y data is not ordered.
Comment from Jon Peltier
Time: Monday, May 16, 2011, 7:52 am
iMx and mx are so secret, that I don’t even know what they do (or did).
I have some enhancements to the code in the works. Sorting wasn’t one of them, but it should be, so I’ll add it to the list. I’m also working on the way the utility handles N and alpha (which my OCD doesn’t like) and also making it accessible to other VBA procedures.
No schedule for the updates, you’ll just have to wait until I announce it in the blog.
Comment from Bancam
Time: Monday, May 16, 2011, 6:19 pm
Jon,
I did a couple of test where the x and y values were random numbers, and found that if there were large gaps in the x values, the function (with a sort) filled this OK.
I was concerned that sometimes, the ‘nPts’ closest data points to the xDomain value may be a long way off and to one side.. It would be better to determine the points surrounding the iPoint of interest within a specified distance along the x axis, not in terms of rank.. then having to either leave a gap if there were insufficient or expand the range to reach a minimum number of points.
I also discovered that often there are clusters of points. I know from experience that real data does this dues to vagaries of process and sampling times: in such clusters, the slope can be totally unrepresentative of the wider range of x values and when the iPoint is outside the cluster, you get extreme extrapolations.
I also read that in a particular commercial application, they determine the loess for each x value in the data and spline the curve together. That would avoid unrealistic interpolations.
Cheers
Comment from Iain
Time: Tuesday, May 31, 2011, 12:51 pm
Hi Jon,
I’m yet another person who is indebted to you for your tireless work with Excel. Thank you!
I am interested in the weighted version of LOESS that Brian posted in the comments, as it would be very useful to assign my own weights to each data point. If you could incorporate that into your major overhaul that would be brilliant =)
One other thing I would be interested in is being able to look at the confidence limits for the fit. In the R implementation, you can plot any confidence limit you choose (e.g. 5-95%), as the LOESS algorithm calculates and the standard errors. See about 2/3 of the way down this post for example: http://tinyurl.com/3b6a3lo
That would be another feature that would be very useful to have in your Excel version.
(sorry for being full of suggestions!)
Comment from Mauricio Prado
Time: Friday, June 10, 2011, 10:13 am
Hi Jon
This was a great find !
I need help though !
I implemented your code and I am trying to use if for a certain problem I have. I have a set of data Y(i), X(i) and I am trying to find a smothed trend of Y as a funciton of X. I played a little bit with your routine and I am getting a few problems. Basically what is happening is that for some values of nPts or sometimes for some points i, the Denominator (Denom) variable is becoming ZERO ! Do you happen to have any “easy” references on the math behind LOESS and how the Intercep and Slope are calculated. I would like to see what cases can yield a Denom of Zero and what can be done to go around this problem. If you want to take a look at the example I can send you the spreasheet.
thanks
mauricio
Comment from harry
Time: Wednesday, June 15, 2011, 11:03 am
if I have two predictors (x1, x2) and one output y, how can I use the LOESS?
This what I run into:
I ran regular multi regression, got a fitting function
y_hat = a0 + a1*x1 + a2*x2
But the R square is only 67%, can I use LOESS to improve the fit of the model?
Comment from Jon Peltier
Time: Wednesday, June 15, 2011, 12:38 pm
Harry -
I have not attempted anything more intricate than a first order, one factor weighted regression. You could probably Google for something; someone several months ago found a weighted second order algorithm.
I don’t think the point is really to reduce your R², since there’s already so much variability in the data. The point is to find a smoothed curve that passed reasonably well through the data.
Pingback from We Want Your Ideas – The Results | Chandoo.org – Learn Microsoft Excel Online
Time: Friday, July 22, 2011, 3:01 am
[...] http://peltiertech.com/WordPress/loess-smoothing-in-excel/ [...]
Comment from Bill McNair
Time: Friday, July 22, 2011, 4:04 pm
i can’t believe i missed this blog article until now. well done. thank you.
Comment from Chad S.
Time: Monday, December 12, 2011, 2:59 pm
Does a version of the above LOESS function (VBA code) exist for HORIZONTAL array inputs?
I have a model with X, Y, and xDomain in rows rather than columns, and use intermediary TRANSPOSE functions as a workaround. However, it would be nice to have both a Horizontal and Vertical version of the LOESS function (similar to how VLOOKUP has the HLOOKUP counterpart). I tried modifying the code to work for Horizontal inputs, but apparently I don’t understand the code well enough to make it work.
Any advice?
Thanks, by the way, for this excellent info.
Comment from Jon Peltier
Time: Monday, December 12, 2011, 3:40 pm
Well, it’s generally best practice to have data series in columns, since this is in accord with a database output’s layout of fields in columns and records in rows. So my advice is to transpose your whole workbook. Of course, I don’t know what you’re doing and why it’s arranged that way, I’m just generalizing.
You might adjust the input ranges like this:
If TypeName(X) = "Range" Then
If X.Columns.Count > X.Rows.Count Then
X = WorksheetFunction.Transpose(X.Value)
Else
X = X.Value
End If
End IfI have no plans to rewrite my add-in’s code to handle horizontal ranges.
Comment from Chad S.
Time: Monday, December 12, 2011, 4:59 pm
Jon -
Many thanks for your quick reply!
By the way, it’s a legacy financial model, which uses a SAS-summarized dataset of aggregated monthly transaction records for a segment of accounts. Rows are metrics like deposits, withdrawals, account attrition, etc. ‘Month’ is in columns. So the dataset, model, and graphs show how the metrics have moved/will move over time.
Comment from Jon Peltier
Time: Monday, December 12, 2011, 9:10 pm
Oh, “legacy”… That can get tough.
I’m about to drop support for Microsoft’s legacy product, Excel 2003, even though it’s my favorite. It’s just too much effort to support two of everything, and users of 2001/2010 now far outnumber users of 2003.
Write a comment
I welcome comments from my readers. If you have an opinion on this post, if you have a question or if there is anything to add, I want to hear from you. Whether you agree or disagree, please join the discussion.
If you want to include an image in your comment, post it on your own site or on one of the many free image sharing sites, and include a link in your comment. I'll download your image and insert the necessary html to display the image inline.
Read the PTS Blog Comment Policy.