I smooth a series of data points using the algorithm described here: http://www.scipy.org/Cookbook/SignalSmooth .
How could I compare the smoothed signal with the input signal afterward? I'm hoping I could get a scalar describing how "close" the output is from the input. Is there any standard way to do this? Some term I could look for?
I have no idea what to even look for. Thanks!
I used normalized root mean squared deviation. That gives me a number between 0 and 1. The bigger the number, the further away the two data series are. 0 means perfect match between the signal and the smoothed signal.
Discrete correlation is a way to detect a known waveform in a noisy background. Just find the correlation between two signals. Discrete correlation is simply a vector dot product:
for n in range(N):
y[n] = sum( [x1[i]*x2[i+n] for i in range(N)] )
in pure Python, or:
y = xcorr(x1,x2);
in Matlab, or:
y = correlate(x1,x2)
in Python+Scipy.
Correlation is a very sensitive measure of similarity of two signals. It is maximized when the two signals are similar in frequency content and are in phase with each other.
Assuming you smoothed the signal to remove noise, the most natural figure of merit would be the SNR.
So something like:
mean((smoothed[n] - original[n])^2) / mean( (smoothed[n])^2 )
The above assume the average of the signal is ~0.
Related
I have implemented an algorithm that uses two other algorithms for calculating the shortest path in a graph: Dijkstra and Bellman-Ford. Based on the time complexity of the these algorithms, I can calculate the running time of my implementation, which is easy giving the code.
Now, I want to experimentally verify my calculation. Specifically, I want to plot the running time as a function of the size of the input (I am following the method described here). The problem is that I have two parameters - number of edges and number of vertices.
I have tried to fix one parameter and change the other, but this approach results in two plots - one for varying number of edges and the other for varying number of vertices.
This leads me to my question - how can I determine the order of growth based on two plots? In general, how can one experimentally determine the running time complexity of an algorithm that has more than one parameter?
It's very difficult in general.
The usual way you would experimentally gauge the running time in the single variable case is, insert a counter that increments when your data structure does a fundamental (putatively O(1)) operation, then take data for many different input sizes, and plot it on a log-log plot. That is, log T vs. log N. If the running time is of the form n^k you should see a straight line of slope k, or something approaching this. If the running time is like T(n) = n^{k log n} or something, then you should see a parabola. And if T is exponential in n you should still see exponential growth.
You can only hope to get information about the highest order term when you do this -- the low order terms get filtered out, in the sense of having less and less impact as n gets larger.
In the two variable case, you could try to do a similar approach -- essentially, take 3 dimensional data, do a log-log-log plot, and try to fit a plane to that.
However this will only really work if there's really only one leading term that dominates in most regimes.
Suppose my actual function is T(n, m) = n^4 + n^3 * m^3 + m^4.
When m = O(1), then T(n) = O(n^4).
When n = O(1), then T(n) = O(m^4).
When n = m, then T(n) = O(n^6).
In each of these regimes, "slices" along the plane of possible n,m values, a different one of the terms is the dominant term.
So there's no way to determine the function just from taking some points with fixed m, and some points with fixed n. If you did that, you wouldn't get the right answer for n = m -- you wouldn't be able to discover "middle" leading terms like that.
I would recommend that the best way to predict asymptotic growth when you have lots of variables / complicated data structures, is with a pencil and piece of paper, and do traditional algorithmic analysis. Or possibly, a hybrid approach. Try to break the question of efficiency into different parts -- if you can split the question up into a sum or product of a few different functions, maybe some of them you can determine in the abstract, and some you can estimate experimentally.
Luckily two input parameters is still easy to visualize in a 3D scatter plot (3rd dimension is the measured running time), and you can check if it looks like a plane (in log-log-log scale) or if it is curved. Naturally random variations in measurements plays a role here as well.
In Matlab I typically calculate a least-squares solution to two-variable function like this (just concatenates different powers and combinations of x and y horizontally, .* is an element-wise product):
x = log(parameter_x);
y = log(parameter_y);
% Find a least-squares fit
p = [x.^2, x.*y, y.^2, x, y, ones(length(x),1)] \ log(time)
Then this can be used to estimate running times for larger problem instances, ideally those would be confirmed experimentally to know that the fitted model works.
This approach works also for higher dimensions but gets tedious to generate, maybe there is a more general way to achieve that and this is just a work-around for my lack of knowledge.
I was going to write my own explanation but it wouldn't be any better than this.
I have a set of first 25 Zernike polynomials. Below are shown few in Cartesin co-ordinate system.
z2 = 2*x
z3 = 2*y
z4 = sqrt(3)*(2*x^2+2*y^2-1)
:
:
z24 = sqrt(14)*(15*(x^2+y^2)^2-20*(x^2+y^2)+6)*(x^2-y^2)
I am not using 1st since it is piston; so I have these 24 two-dim ANALYTICAL functions expressed in X-Y Cartesian co-ordinate system. All are defined over unit circle, as they are orthogonal over unit circle. The problem which I am describing here is relevant to other 2D surfaces also apart from Zernike Polynomials.
Suppose that origin (0,0) of the XY co-ordinate system and the centre of the unit circle are same.
Next, I take linear combination of these 24 polynomials to build a 2D wavefront shape. I use 24 random input coefficients in this combination.
w(x,y) = sum_over_i a_i*z_i (i=2,3,4,....24)
a_i = random coefficients
z_i = zernike polynomials
Upto this point, everything is analytical part which can be done on paper.
Now comes the discretization!
I know that when you want to re-construct a signal (1Dim/2Dim), your sampling frequency should be at least twice the maximum frequency present in the signal (Nyquist-Shanon principle).
Here signal is w(x,y) as mentioned above which is nothing but a simple 2Dim
function of x & y. I want to represent it on computer now. Obviously I can not take all infinite points from -1 to +1 along x axis and same for y axis.
I have to take finite no. of data points (which are called sample points or just samples) on this analytical 2Dim surface w(x,y)
I am measuring x & y in metres, and -1 <= x <= +1; -1 <= y <= +1.
e.g. If I divide my x-axis from -1 to 1, in 50 sample points then dx = 2/50= 0.04 metre. Same for y axis. Now my sampling frequency is 1/dx i.e. 25 samples per metre. Same for y axis.
But I took 50 samples arbitrarily; I could have taken 10 samples or 1000 samples. That is the crux of the matter here: how many samples points?How will I determine this number?
There is one theorem (Nyquist-Shanon theorem) mentioned above which says that if I want to re-construct w(x,y) faithfully, I must sample it on both axes so that my sampling frequency (i.e. no. of samples per metre) is at least twice the maximum frequency present in the w(x,y). This is nothing but finding power spectrum of w(x,y). Idea is that any function in space domain can be represented in spatial-frequency domain also, which is nothing but taking Fourier transform of the function! This tells us how many (spatial) frequencies are present in your function w(x,y) and what is the maximum frequency out of these many frequencies.
Now my question is first how to find out this maximum sampling frequency in my case. I can not use MATLAB fft2() or any other tool since it means already I have samples taken across the wavefront!! Obviously remaining option is find it analytically ! But that is time consuming and difficult since I have 24 polynomials & I will have to use then continuous Fourier transform i.e. I will have to go for pen and paper.
Any help will be appreciated.
Thanks
Key Assumptions
You want to use the "Nyquist-Shanon" theorem to determine sampling frequency
Obviously remaining option is find it analytically ! But that is time
consuming and difficult since I have 21 polynomials & I have to use
continuous Fourier transform i.e. done by analytically.
Given the assumption I have made (and noting that consideration of other mathematical techniques is out of scope for StackOverflow), you have no option but to calculate the continuous Fourier Transform.
However, I believe you haven't considered all the options for calculating the transform other than a laborious paper exercise e.g.
Numerical approximation of the continuous F.T. using code
Symbolic Integration e.g. Wolfram Alpha
Surely a numerical approximation of the Fourier Transform will be adequate for your solution?
I am assuming this is for coursework or research rather, so all you really care about as a physicist is a solution that is the quickest solution that is accurate within the scope of your problem.
So to conclude, IMHO, don't waste time searching for a more mathematically elegant solution or trick and just solve the problem with one of the above methods
Basically, I have a set of up to 100 co-ordinates, along with the desired tangents to the curve at the first and last point.
I have looked into various methods of curve-fitting, by which I mean an algorithm with takes the inputted data points and tangents, and outputs the equation of the cure, such as the gaussian method and interpolation, but I really struggled understanding them.
I am not asking for code (If you choose to give it, thats acceptable though :) ), I am simply looking for help into this algorithm. It will eventually be converted to Objective-C for an iPhone app, if that changes anything..
EDIT:
I know the order of all of the points. They are not too close together, so passing through all points is necessary - aka interpolation (unless anyone can suggest something else). And as far as I know, an algebraic curve is what I'm looking for. This is all being done on a 2D plane by the way
I'd recommend to consider cubic splines. There is some explanation and code to calculate them in plain C in Numerical Recipes book (chapter 3.3)
Most interpolation methods originally work with functions: given a set of x and y values, they compute a function which computes a y value for every x value, meeting the specified constraints. As a function can only ever compute a single y value for every x value, such an curve cannot loop back on itself.
To turn this into a real 2D setup, you want two functions which compute x resp. y values based on some parameter that is conventionally called t. So the first step is computing t values for your input data. You can usually get a good approximation by summing over euclidean distances: think about a polyline connecting all your points with straight segments. Then the parameter would be the distance along this line for every input pair.
So now you have two interpolation problem: one to compute x from t and the other y from t. You can formulate this as a spline interpolation, e.g. using cubic splines. That gives you a large system of linear equations which you can solve iteratively up to the desired precision.
The result of a spline interpolation will be a piecewise description of a suitable curve. If you wanted a single equation, then a lagrange interpolation would fit that bill, but the result might have odd twists and turns for many sets of input data.
I am using the code in this website http://blog.chrislowis.co.uk/2008/11/24/ruby-gsl-pearson.html to implement a Pearson Correlation given two time series data like so:
require 'gsl'
pearson_correlation = GSL::Stats::correlation(
GSL::Vector.alloc(first_metrics),GSL::Vector.alloc(second_metrics)
)
This returns a number such as -0.2352461593569471.
I'm currently using the highcharts library and am feeding it two sets of timeseries data. Given that I have a finite time series for both sets, can I do something with this number (-0.2352461593569471) to create a third time series showing the slope of this curve? If anyone can point me in the right direction I'd really appreciate it!
No, correlation doesn't tell you anything about the slope of the line of best fit. It just tells you approximately how much of the variability in one variable (or one time series, in this case) can be explained by the other. There is a reasonably good description here: http://www.graphpad.com/support/faqid/1141/.
How you deal with the data in your specific case is highly dependent on what you're trying to achieve. Are you trying to show that variable X causes variable Y? If so, you could start by dropping the time-series-ness, and just treat the data as paired values, and use linear regression. If you're trying to find a model of how X and Y vary together over time, you could look at multivariate linear regression (I'm not very familiar with this, though).
I am computing a similarity matrix based on Euclidean distance in MATLAB. My code is as follows:
for i=1:N % M,N is the size of the matrix x for whose elements I am computing similarity matrix
for j=1:N
D(i,j) = sqrt(sum(x(:,i)-x(:,j)).^2)); % D is the similarity matrix
end
end
Can any help with optimizing this = reducing the for loops as my matrix x is of dimension 256x30000.
Thanks a lot!
--Aditya
The function to do so in matlab is called pdist. Unfortunately it is painfully slow and doesnt take Matlabs vectorization abilities into account.
The following is code I wrote for a project. Let me know what kind of speed up you get.
Qx=repmat(dot(x,x,2),1,size(x,1));
D=sqrt(Qx+Qx'-2*x*x');
Note though that this will only work if your data points are in the rows and your dimensions the columns. So for example lets say I have 256 data points and 100000 dimensions then on my mac using x=rand(256,100000) and the above code produces a 256x256 matrix in about half a second.
There's probably a better way to do it, but the first thing I noticed was that you could cut the runtime in half by exploiting the symmetry D(i,j)==D(i,j)
You can also use the function norm(x(:,i)-x(:,j),2)
I think this is what you're looking for.
D=zeros(N);
jIndx=repmat(1:N,N,1);iIndx=jIndx'; %'# fix SO's syntax highlighting
D(:)=sqrt(sum((x(iIndx(:),:)-x(jIndx(:),:)).^2,2));
Here, I have assumed that the distance vector, x is initalized as an NxM array, where M is the number of dimensions of the system and N is the number of points. So if your ordering is different, you'll have to make changes accordingly.
To start with, you are computing twice as much as you need to here, because D will be symmetric. You don't need to calculate the (i,j) entry and the (j,i) entry separately. Change your inner loop to for j=1:i, and add in the body of that loop D(j,i)=D(i,j);
After that, there's really not much redundancy left in what that code does, so your only room left for improvement is to parallelize it: if you have the Parallel Computing Toolbox, convert your outer loop to a parfor and before you run it, say matlabpool(n), where n is the number of threads to use.