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I am by no means a math person, but I am really trying to figure out how create a graphable function from some data plots I measure from a chemical titration. I have been trying to learn R and I would like to know if anyone can explain to me or point me to a guide to create a mathmatic function of the titration graph below.
Thanks in advance.
What you are looking for is a Interpolation. I'm not a R programmer, but I'll try to answer anyway.
Some of the more common ways to achieve this function you want is by Polynomial Interpolation which usually gives back a Nth degree polynomial function, where N is the number of data points minus one (1 point gives a constant, 2 points make a line, 3 makes a*x^2 + b*x + c and so on).
Other common alternatives I've learn are used in Computer Graphics are Splines, B-spline, Bézier curve and Hermite interpolation. Those make the curve smoother and good looking (I've told they originated in the car industry so they are less true to the data points).
TL;DR: I've found evidence that there is a implementation of spline in R from the question Interpolation in R which may lead you to your solution.
Hope you get to know better your tool and do a great work.
When doing this kind of work in Computer Science we call it Numerical Methods (at least here in my university), I've done some class and homework in this area while attending to the Numerical Methods Course (it can be found at github) but it's nothing worth noting.
I would add a lot of links to Wikipedia but StackOverflow didn't allow it.
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I have a large p (~20K) and small n (~500) problem. The first thing I was thinking is dimension reduction. After trying PCA, robust PCA, ICA, removing highly correlated features, I was thinking to use Random Projection. However, there is no simple R implementation of Random Projection.
I have found a few random projection R packages, like
MCLUST: https://cran.r-project.org/web/packages/mclust/index.html
RPEnsemble: https://cran.r-project.org/web/packages/RPEnsemble/index.html
But, it seems that they don't support random projection directly for dimension reduction. I have limited knowledge about random projection, but I found two functions in sklearn support this:
Gaussian random projection and Sparse random projection:
http://scikit-learn.org/stable/modules/random_projection.html
And it has pretty simple function interface.
Is that possible to implement random projection easily in R? Or, taking advantage of existing tools to do dimension reduction with Random Projection in R?
I concur that the RPEnsemble package doesn't seem to expose the low-level methods that would allow you to use only that feature in any convenient form.
I did however come across this R source code which seems fairly straight-forward and reasonably documented: R source code for random projections. This is part of the clusterv package and you can download it there.
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So - edited because some of us thought that this question is off-topic.
I need to build spline (approximation) on 100 points in one of environments listed in tags. But I need it with exact number of intervals (maximum of 6 intervals - separate equations - in whole domain). Packages / libraries in R and Maxima which I know let me for building spline on this points but with 25-30 intervals (separate equations). Does anyone know how to build spline with set number of intervals without coding whole algorithm all over again?
What you're looking for might be described as "local regression" or "localized regression"; searching for those terms might turn up some hits.
I don't know if you can find exactly what you've described. But implementing it doesn't seem too complicated: (1) Split the domain into N intervals (say N=10). For each interval, (2) make a list of the data in the interval, (3) fit a low-order polynomial (e.g. cubic) to the data in the interval using least squares.
If that sounds interesting to you, I can go into details, or maybe you can work it out yourself.
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i'd like to know how to solve a definite integral in Mathematica.
I do know all variables except b, and need to solve for F(b)=0.
How can i solve it in Mathematica?
Here is my try:
NSolve[Integrate[1/(8*(1 - ff) (2 Pi)^0.5) E^(-0.5*((x - 1.1)/(1 - ff)/8)^2), {x, 0, 9999}] == -0.44531779637243296, ff]
These integrals can be trivially expressed in terms of an error function: Wiki, Mathworld. Hence what you need here is a library to (i) calculate error functions, (ii) numerically solve non-linear equations. Virtually any language has this, so pick anything you're familiar with. In Mathematica, look up Erf and NSolve.
I'd start by plugging it into Wolfram Alpha and see what it gives you.
Mathematica should be able to do it. I think of statistics first when R comes up; I don't know about its calculus capabilities. Excel is not the first choice.
If I were you, I'd be less worried about the software and more worried about the solution itself. A function of this form might be well known. Plot each one and visually check to see what the functions look like and how easy they might be to integrate.
Like this:
http://www.wolframalpha.com/input/?i=graph+exp%28-%28%28x%2B5%29%2F1.5%29%5E2%29
You should be wondering why it's three similar looking integrals. Those singularities in the plot tell you why.
If there's no closed form solutions, you'll have to go with a numerical one. You'll have to choose an algorithm (simple Euler or Runga Kutta or something else), interval sizes, etc. You'll want to know about singular points and how best to tackle them.
Choosing a package is just the start.
You might find http://r.789695.n4.nabble.com/calculus-using-R-td1676727.html helpful.
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I didn't find a function to calculate the orthogonal regression (TLS - Total Least Squares).
Is there a package with this kind of function?
Update: I mean calculate the distance of each point symmetrically and not asymmetrically as lm() does.
You might want to consider the Deming() function in package MethComp [function info]. The package also contains a detailed derivation of the theory behind Deming regression.
The following search of the R Archives also provide plenty of options:
Total Least Squares
Deming regression
Your multiple questions on CrossValidated, here and R-Help imply that you need to do a bit more work to describe exactly what you want to do, as the terms "Total least squares" and "orthogonal regression" carry some degree of ambiguity about the actual technique wanted.
Two answers:
gx.rma in the rgr package appears to do this.
Brian Ripley has given a succinct answer on this thread. Basically, you're looking for PCA, and he suggests princomp. I do, too.
I got the following solution from this url:
https://www.inkling.com/read/r-cookbook-paul-teetor-1st/chapter-13/recipe-13-5
r <- prcomp( ~ x + y )
slope <- r$rotation[2,1] / r$rotation[1,1]
intercept <- r$center[2] - slope*r$center[1]
Basically you performa PCA that will fit a line between x and y minimizing the orthogonal residuals. Then you can retrieve the intercept and slope for the first component.
For anyone coming across this question again, there exists a dedicated package 'onls' by now for that purpose. It is similar handled as the nls package (which implements ordinary least square algorithms)
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I needed an application for solving linear systems of equations (N up to 10), so I got different codes, and compile them, and they seem to work, but I get lots of problems with precision. I mean, the solvers are really very sensitive to small changes of the system.
So, could somebody recommend to me a reliable commandl ine application for this purpose? Or some useful open source code (and easy to compile)
Thanks
GNU Octave is essentially a free version of Matlab (the syntax is identical for basic operations), so you can try things out there and see how they compare to the answers that you're getting.
Having said that, if your answer is very sensitive to the input, it's possible that your problem is ill-conditioned - you can check this by computing the condition number of the matrix in Octave. It's hard to say what to do in that case without knowing more specifics on the problem.
Also, you don't mention which method you're currently using. Gaussian elimination (i.e. "what you learned in math class") is notoriously numerically unstable if you don't use pivoting (see the wikipedia entry for "Pivoting"); adding that might be enough to improve the quality of the results.
An approach is to use the numpy package in Python. You can create a 2d matrix A and a 1d vector b, then solve Ax=b for x using solve(A, x). It's part of the linalg subpackage of numpy.