Polynomials as functions in R - r

Sorry in advance if the question is stupid/was answered somewhere else/... I could not find any nice solution.
Based on the idea of power series I have
A) a vector of real coefficients of lengths n which comes from an other loop and which can be rather long, but lets assume it is simple, for instance,
a<-1:10
and
B) a real center, e.g.
c<-3
I would like to define the polynomial (in my example)
a[1]+a[2]*(x-3)+ a[3]*(x-3)^2+ .... + a[10]*(x-3)^9
as a function. Unfortunately
1) the function as.polynomial(a) only allows center 0 (as far as I understand) so I cannot use it and
2) the list of coefficients can be long, too long to do it by hand
3) I might later ever need a multivariable version.
I would prefer to use a loop to define this "finite power series" but I do not know how loops and sums of functions can be realized in a clean fashion (and I did not find it either).
Something like (very naive)
t<-function(x) 0
for(i in 1:length(a))
{t<-function(x) {t(x) + a[i]*(x-c)^(i-1}}
Thanks so much for your help.

i think this works
my_polynomial = function(x) {
sum(sapply(seq_along(a), function(ii) a[ii] * (x - c) ^ (ii - 1L)))
}

Just for the future reference. To change the center using the package polynom use change.origin
For example:
change.origin(as.polynomial(a),3)

Related

More efficient way to compute the rowNorms in R?

I wrote a program using an unsupervised K-means algorithm to try and compress images. It now works but in comparison to Python it's incredibly slow! Specifically it's finding the rowNorms thats slow. The array X is 350000+ elements.
This is the particular function:
find_closest_centroids <- function(X, centroids) {
m <- nrow(X)
c <- integer(m)
for(i in 1:m){
distances = rowNorms(sweep(centroids,2,X[i,]))
c[i] = which.min(distances)
}
return(c)
}
In Python I am able to do it like this:
def find_closest_centroids(X, centroids):
m = len(X)
c = np.zeros(m)
for i in range(m):
distances = np.linalg.norm(X[i] - centroids, axis=1)
c[i] = np.argmin(distances)
return c
Any recommendations?
Thanks.
As dvd280 has noted in his comment, R tends to do worse than many other languages in terms of performance. If are content with the performance of your code in Python, but need the function available in R, you might want to look into the reticulate package which provides an interface to python like the Rcpp package mentioned by dvd280 does for C++.
If you still want to implement this natively in R, be mindful of the data structures you use. For rowwise operations, data frames are a poor choice as they are lists of columns. I'm not sure about the data structures in your code, but rowNorms() seems to be a matrix method. You might get more mileage out of a list of rows structure.
If you feel like getting into dplyr, you could find this vignette on row-wise operations helpful. Make sure you have the latest version of the package, as the vignette is based on dplyr 1.0.
The data.table package tends to yield the best performance for large data sets in R, but I'm not familiar with it, so I can't give you any further directions on that.

R: Creating a function with an arbitrarily long expression

Although my original question is more general, in order to keep things more comprehensive, I'm formulating below just its partial case, - I expect that a solution/ answer for it will serve as an answer for the more general question.
Question:
to integrate a function f(x)=(...(((x^x)^x)^x)...^x)^x (... powered x n times) on the interval (0,1) ?
Thanks a lot for any ideas!
P.S.: please, do not try to solve the problem mathematically or to simplify an expression (e.g., to approximate the result with a Taylor expansion, whatever), since it's not the main topic (however, I've tried to choose such an example, which should not have any simple transformations)
P.S.2: Original question (which does not require an answer here, since it's expected that an answer for posted question is valid for original one):
if it's possible in R to create a function with an arbitrarily long expression (avoiding "manual" defining). For example, it's easy to set up manually a given function for n=5:
f<-function(x) {
((((x^x)^x)^x)^x)^x
}
But what if n=1'000, or 1'000'000 ?
It seems that simple looping is not appropriate here...
Copied from Rhelp: You should look at:
# ?funprog Should have worked but didn't. Try instead ...
?Reduce
There are several examples of repeated applications of a functional argument. Also composition of list of functions.
One instance:
Funcall <- function(f, ...) f(...) # sort of like `do.call`
Iterate <- function(f, n = 1)
function(x) Reduce(Funcall, rep.int(list(f), n), x, right = TRUE)
Iterate(function(x) x^1.1, 30)(1.01)
#[1] 1.189612

R optim same function for fn and gr

I would like to use optim() to optimize a cost function (fn argument), and I will be providing a gradient (gr argument). I can write separate functions for fn and gr. However, they have a lot of code in common and I don't want the optimizer to waste time repeating those calculations. So is it possible to provide one function that computes both the cost and the gradient? If so, what would be the calling syntax to optim()?
As an example, suppose the function I want to minimize is
cost <- function(x) {
x*exp(x)
}
Obviously, this is not the function I'm trying to minimize. That's too complicated to list here, but the example serves to illustrate the issue. Now, the gradient would be
grad <- function(x) {
(x+1)*exp(x)
}
So as you can see, the two functions, if called separately, would repeat some of the work (in this case, the exponential function). However, since optim() takes two separate arguments (fn and gr), it appears there is no way to avoid this inefficiency, unless there is a way to define a function like
costAndGrad <- function(x) {
ex <- exp(x)
list(cost=x*ex, grad=(x+1)*ex)
}
and then pass that function to optim(), which would need to know how to extract the cost and gradient.
Hope that explains the problem. Like I said my function is much more complicated, but the idea is the same: there is considerable code that goes into both calculations (cost and gradient), which I don't want to repeat unnecessarily.
By the way, I am an R novice, so there might be something simple that I'm missing!
Thanks very much
The nlm function does optimization and it expects the gradient information to be returned as an attribute to the value returned as the original function value. That is similar to what you show above. See the examples in the help for nlm.

Sliding FFT in R

Is there a function or package in R for calculating the Sliding FFT of a sample? By this I mean that given the output of fft(x[n:m]), calculate fft(x[1+(n:m)]) efficiently.
Ideally I'd find both an online version (where I don't have access to the full time series at the beginning, or it's too big to fit in memory, and I'm not going to try to save the whole running FFT in memory either) and a batch version (where I give it the whole sample x and tell it the running window width w, resulting in a complex matrix of dimension c(w,length(x)/w)).
An example of such an algorithm is presented here (but I've never tried implementing it in any language yet):
http://cnx.org/content/m12029/latest/
If no such thingy exists already in R, that doesn't look too hard to implement I guess.
As usually happens when I post something here, I kept working on it and came up with a solution:
fft.up <- function(x1, xn, prev) {
b <- length(prev)
vec <- exp(2i*pi*seq.int(0,b-1)/b)
(prev - x1 + xn) * vec
}
# Test it out
x <- runif(6)
all.equal(fft.up(x[1], x[6], fft(x[1:5])), fft(x[2:6]))
# [1] TRUE
Still interested to know if some library offers this, because then it might offer other handy things too. =) But for now my problem's solved.

How to vectorize equations?

I'm trying to implement the Softmax regression algorithm to solve the K-classifier problem after watching Professor Andrew Ng's lectures on GLM. I thought I understood everything he was saying until it finally came to writing the code to implement the cost function for Softmax regression, which is as follows:
The problem I am having is trying to figure out a way to vectorize this. Again I thought I understood how to go about vectorizing equations like this since I was able to do it for linear and logistic regression, but after looking at that formula I am stuck.
While I would love to figure out a vectorized solution for this (I realize there is a similar question posted already: Vectorized Implementation of Softmax Regression), what I am more interested in is whether any of you can tell me a way (your way) to methodically convert equations like this into vectorized forms. For example, for those of you who are experts or seasoned veterans in ML, when you read of new algorithms in the literature for the first time, and see them written in similar notation to the equation above, how do you go about converting them to vectorized forms?
I realize I might be coming off as being like the student who is asking Mozart, "How do you play the piano so well?" But my question is simply motivated from a desire to become better at this material, and assuming that not everyone was born knowing how to vectorize equations, and so someone out there must have devised their own system, and if so, please share! Many thanks in advance!
Cheers
This one looks pretty hard to vectorize since you are doing exponentials inside of your summations. I assume you are raising e to arbitrary powers. What you can vectorize is the second term of the expression \sum \sum theta ^2 just make sure to use .* operator in matlab enter link description here to computer \theta ^2
Same goes for the inner terms of the ratio of the that goes into the logarithm. \theta ' x^(i) is vectorizable expression.
You might also benefit from a memoization or dynamic programming technique and try to reuse the results of computations of e^\theta' x^(i).
Generally in my experience the way to vectorize is first to get non-vectorized implementation working. Then try to vectorize the most obvious parts of your computation. At every step tweak your function very little and always check if you get the same result as non-vectorized computation. Also, having multiple test cases is very helpful.
The help files that come with Octave have this entry:
19.1 Basic Vectorization
To a very good first approximation, the goal in vectorization is to
write code that avoids loops and uses whole-array operations. As a
trivial example, consider
for i = 1:n
for j = 1:m
c(i,j) = a(i,j) + b(i,j);
endfor
endfor
compared to the much simpler
c = a + b;
This isn't merely easier to write; it is also internally much easier to
optimize. Octave delegates this operation to an underlying
implementation which, among other optimizations, may use special vector
hardware instructions or could conceivably even perform the additions in
parallel. In general, if the code is vectorized, the underlying
implementation has more freedom about the assumptions it can make in
order to achieve faster execution.
This is especially important for loops with "cheap" bodies. Often it
suffices to vectorize just the innermost loop to get acceptable
performance. A general rule of thumb is that the "order" of the
vectorized body should be greater or equal to the "order" of the
enclosing loop.
As a less trivial example, instead of
for i = 1:n-1
a(i) = b(i+1) - b(i);
endfor
write
a = b(2:n) - b(1:n-1);
This shows an important general concept about using arrays for
indexing instead of looping over an index variable.  Index Expressions.
Also use boolean indexing generously. If a condition
needs to be tested, this condition can also be written as a boolean
index. For instance, instead of
for i = 1:n
if (a(i) > 5)
a(i) -= 20
endif
endfor
write
a(a>5) -= 20;
which exploits the fact that 'a > 5' produces a boolean index.
Use elementwise vector operators whenever possible to avoid looping
(operators like '.*' and '.^').  Arithmetic Ops. For simple
inline functions, the 'vectorize' function can do this automatically.
-- Built-in Function: vectorize (FUN)
Create a vectorized version of the inline function FUN by replacing
all occurrences of '', '/', etc., with '.', './', etc.
This may be useful, for example, when using inline functions with
numerical integration or optimization where a vector-valued
function is expected.
fcn = vectorize (inline ("x^2 - 1"))
=> fcn = f(x) = x.^2 - 1
quadv (fcn, 0, 3)
=> 6
See also:  inline,  formula,
 argnames.
Also exploit broadcasting in these elementwise operators both to
avoid looping and unnecessary intermediate memory allocations.
 Broadcasting.
Use built-in and library functions if possible. Built-in and
compiled functions are very fast. Even with an m-file library function,
chances are good that it is already optimized, or will be optimized more
in a future release.
For instance, even better than
a = b(2:n) - b(1:n-1);
is
a = diff (b);
Most Octave functions are written with vector and array arguments in
mind. If you find yourself writing a loop with a very simple operation,
chances are that such a function already exists. The following
functions occur frequently in vectorized code:
Index manipulation
* find
* sub2ind
* ind2sub
* sort
* unique
* lookup
* ifelse / merge
Repetition
* repmat
* repelems
Vectorized arithmetic
* sum
* prod
* cumsum
* cumprod
* sumsq
* diff
* dot
* cummax
* cummin
Shape of higher dimensional arrays
* reshape
* resize
* permute
* squeeze
* deal
Also look at these pages from a Stanford ML wiki for some more guidance with examples.
http://ufldl.stanford.edu/wiki/index.php/Vectorization
http://ufldl.stanford.edu/wiki/index.php/Logistic_Regression_Vectorization_Example
http://ufldl.stanford.edu/wiki/index.php/Neural_Network_Vectorization

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