I am trying to make a python script that will output a force based on a measured angle. The inputs are time, the curve and the angle, but I am having trouble using interpolation to fit the force to the curve. I looked at scipy.interpolate, but I'm not sure it will help me because the points aren't evenly spaced.
numpy.interp does not require your points to be evenly distributed. I'm not certain if you mean by "The inputs are time, the curve and the angle" that you have three independent variables, if so you will have to adapt it quite a bit... But for one-variable problems, interp is the way to go.
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I am doing some interpolation given 5000 points (t,x,y,z) where t is time. Some example points I have would be:
Sample data
I am using discrete velocity, change(Position)/change(time) for each consecutive point. I want to find a good way to interpolate the velocity at a given time. I know that I can use linear interpolation, but is there a better way to interpolate? I am using Python for this home project.
I am using the Scipy CubicSpline interpolation based on a certain number of points as shown in the diagram below:
My problem is, the second derivative of the Cubic Splive function looks a little bit edgy:
In order to smooth the second curve I need a higher degree of spline interpolation. Is there a Scipy build in function (similar to CubicSpline) or an easy way to do that? (A b-spline function want work)
make_interp_spline should be able to construct BSpline objects of higher degrees (FITPACK only goes up to k=5, which is hardcoded fairly deep down).
I have the energy spectrum of a certain number of particles N(E) v/s E.
However, I want to plot the differential energy spectrum i.e. dN/dE v/s E.I DO NOT intend to calculate the derivative here [ as the traditional way of representing a differential energy spectrum might suggest ] What I essentially need is the number of particles in the histogram to be divided by the bin-width.
Is there any way to do this automatically in matplotlib or something similar? Or do I actually need to do this manually, wherein I need to write some code to first put the particles in different bins and then divide by the bin-width and then redraw the histogram.
matplotlib is a graphical library. it can plot data and edit figures.
What you need to do there is apply a numerical method to differentiate your data. It shouldnt be difficult.
You could just apply the definition of the derivative, having as DeltaT the shortest measurement you got of E
Once you got the data you can just use matplotlib to plot it.
if you post the data here i would glady give you an example of how to do it.
or you can just check https://en.wikipedia.org/wiki/Numerical_differentiation
I have data points in x,y,z format. They form a point cloud of a closed manifold. How can I interpolate them using R-Project or Python? (Like polynomial splines)
It depends on what the points originally represented. Just having an array of points is generally not enough to derive the original manifold from. You need to know which points go together.
The most common low-level boundary representation ("brep") is a bunch of triangles. This is e.g. what OpenGL and Directx get as input. I've written a Python software that can convert triangular meshes in STL format to e.g. a PDF image. Maybe you can adapt that to for your purpose. Interpolating a triangle is usually not necessary, but rather trivail to do. Create three new points each halfway between two original point. These three points form an inner triangle, and the rest of the surface forms three triangles. So with this you have transformed one triangle into four triangles.
If the points are control points for spline surface patches (like NURBS, or Bézier surfaces), you have to know which points together form a patch. Since these are parametric surfaces, once you know the control points, all the points on the surface can be determined. Below is the function for a Bézier surface. The parameters u and v are the the parametric coordinates of the surface. They run from 0 to 1 along two adjecent edges of the patch. The control points are k_ij.
The B functions are weight functions for each control point;
Suppose you want to approximate a Bézier surface by a grid of 10x10 points. To do that you have to evaluate the function p for u and v running from 0 to 1 in 10 steps (generating the steps is easily done with numpy.linspace).
For each (u,v) pair, p returns a 3D point.
If you want to visualise these points, you could use mplot3d from matplotlib.
By "compact manifold" do you mean a lower dimensional function like a trajectory or a surface that is embedded in 3d? You have several alternatives for the surface-problem in R depending on how "parametric" or "non-parametric" you want to be. Regression splines of various sorts could be applied within the framework of estimating mean f(x,y) and if these values were "tightly" spaced you may get a relatively accurate and simple summary estimate. There are several non-parametric methods such as found in packages 'locfit', 'akima' and 'mgcv'. (I'm not really sure how I would go about statistically estimating a 1-d manifold in 3-space.)
Edit: But if I did want to see a 3D distribution and get an idea of whether is was a parametric curve or trajectory, I would reach for package:rgl and just plot it in a rotatable 3D frame.
If you are instead trying to form the convex hull (for which the word interpolate is probably the wrong choice), then I know there are 2-d solutions and suspect that searching would find 3-d solutions as well. Constructing the right search strategy will depend on specifics whose absence the 2 comments so far reflects. I'm speculating that attempting to model lower and higher order statistics like the 1st and 99th percentile as a function of (x,y) could be attempted if you wanted to use a regression effort to create boundaries. There is a quantile regression package, 'rq' by Roger Koenker that is well supported.
I'm performing a simulation of a simple queue using SimPy. One of the questions about the system is what is the distribution of the waiting times by a visitor. What I do is draw a normalized histogram of the sample I get during the simulation process.
This distribution is not purely continuous, we have a non-zero probability of the waiting time being exactly zero, hence the peak near the left end. I want it to be somehow obvious from the picture, what is the actual probability of hitting 0 exactly. Right now the height of the peak does not visualize that properly, the height is even higher than one (the reason is that many points are hitting a small segment near zero).
So the question is the general visualization technique of such distributions that are mixtures of a continuous and a discrete one.
(based on the discussion in the comments to OP).
For a distribution of some variable, call it t, being a mixture of a discrete and and continuous components, I'd write the pdf a sum of a set of delta-peaks and a continuous part,
p(t) = \sum_{a} p_a \delta(t-t_a) + f(t)
where a enumerates the discrete values t_a and p_a are probabilities of t_a, and f(t) is the pdf for the continuous part of the distribution, so that f(t)dt is the probability for t to belong to [t,t+dt).
Notice that the whole thing is normalized, \int p(t) =1 where the integral is over the approprite range of t.
Now, for visualizing this, I'd separate the discrete components, and plot them as discrete values (either as narrow bins or as points with droplines etc). Then for the rest, I'd use the histogram where you know the correct normalization from the equation above: the area under the histogram should sum up to 1-\sum_a p_a.
I'm not claiming this being the way, it's just what I'd do.