I want to generate a function which would non-linearly interpolate data. Let's say the fraction x (varies between 0 and 1) represents the distance from the Point A towards Point B. So, if x=0, we are at Point A and if x=1, we are at Point B. Now the condition I have if that at x=0.1, we would need to have travelled 90% of the distance from Point A towards Point B. Notice, I would not be able to connect this with the actual values of the parameter data at Point A or B unlink is the case in exponential interpolation techniques [https://www.mrmath.com/misfit/algebra-stuff/linear-and-non-linear-interpolation/#:~:text=ExponentialThe%20second%20most%20popular,smooth%2C%20concave%20curve%20between%20points].
The following resources are too tough for me and I am looking for the simplest solutions:-
https://www.sciencedirect.com/science/article/pii/0022247X82901378
https://ieeexplore.ieee.org/document/1054589
As an example of linear interpolation, let's say that the value of the parameter at Point A is 'a' and at Point B is 'b'. The linear interpolation formula is then given as:-
a+(b-a)*y
where y=x for linear interpolations. I wish to develop a non-linear function for y=f(x). Is there an easy general way for this?
Related
I have a function given by a list of points, ex:
f = [0.03, 0.05, 0.02, 1.3, 1.0, 5.6, ..., 13.4, 12.45]
I need an algorithm (with linear complexity) to "cut" this function/list into K intervals/sublists so that each interval/sublist contains points that "lie near a line segment" (take a look at the image)
The number K may be decided either by the algorithm itself or be a parameter of the algorithm. (preferable is to be decided by the algorithm itself)
Is there such a known algorithm I could use ?
i am writing with smartphone so this is short. Basically a function is nearly linear if the difference between two consecutive values is approximately equal see http://psn.virtualnerd.com/viewtutorial/PreAlg_13_01_0006
As an algorithm for traversing an unsorted array Sliding Window is nice ( https://www.geeksforgeeks.org/window-sliding-technique/ ) and can be implemented by a single pass (1-pass solution)
Update because comment :
So with a sliding window you can implement the vagueness or fuzziness of the values you mentioned in the comment this is why nearly linear and approximately, i.e.
if(abs(abs(x[i]-x[i+1]) - abs(x[i+1]-x[i+2])) < 0.5)
{linearity_flag=1;}
else
{linearity_flag=0;}
where x[i]-x[i+1] and x[i+1]-x[i+2] are two consecutive differences of two consecutive values and 0.5 is a deliberately chosen threshold that fixes what you define as a straight line or linear function in an x-y graph (or what 'jittering' of the line you allow). So you have to use the difference of differences of consecutive values. Instead of 3 points you can also include more points with this approach (sliding window)
If you want a strict mathematical ansatz you could use other curve analysis techniques : https://openstax.org/books/calculus-volume-1/pages/4-5-derivatives-and-the-shape-of-a-graph (actually the difference of differences of consecutive values is a discrete realization of a 2nd derivative)
All examples that I can find on the Internet just visualize the result array of the function computeSpectrum, but I am tasked with something else.
I generate a music note and I need by analyzing the result array to be able to say what note is playing. I figured out that I need to set the second parameter of the function call 'FFTMode' to true and then it returns sound frequencies. I thought that really it should return only one non-zero value which I could use to determine what note I generated using Math.sin function, but it is not the case.
Can somebody suggest a way how I can accomplish the task? Using the soundMixer.computeSpectrum is a requirement because I am going to analyze more complex sounds later.
FFT will transform your signal window into set of Nyquist sine waves so unless 440Hz is one of them you will obtain more than just one nonzero value! For a single sine wave you would obtain 2 frequencies due to aliasing. Here an example:
As you can see for exact Nyquist frequency the FFT response is single peak but for nearby frequencies there are more peaks.
Due to shape of the signal you can obtain continuous spectrum with peaks instead of discrete values.
Frequency of i-th sample is f(i)=i*samplerate/N where i={0,1,2,3,4,...(N/2)-1} is sample index (first one is DC offset so not frequency for 0) and N is the count of samples passed to FFT.
So in case you want to detect some harmonics (multiples of single fundamental frequency) then set the samplerate and N so samplerate/N is that fundamental frequency or divider of it. That way you would obtain just one peak for harmonics sinwaves. Easing up the computations.
So, I have a vector that corresponds to a given feature (same dimensionality). Is there a package in Julia that would provide a mathematical function that fits these data points, in relation to the original feature? In other words, I have x and y (both vectors) and need to find a decent mapping between the two, even if it's a highly complex one. The output of this process should be a symbolic formula that connects x and y, e.g. (:x)^3 + log(:x) - 4.2454. It's fine if it's just a polynomial approximation.
I imagine this is a walk in the park if you employ Genetic Programming, but I'd rather opt for a simpler (and faster) approach, if it's available. Thanks
Turns out the Polynomials.jl package includes the function polyfit which does Lagrange interpolation. A usage example would go:
using Polynomials # install with Pkg.add("Polynomials")
x = [1,2,3] # demo x
y = [10,12,4] # demo y
polyfit(x,y)
The last line returns:
Poly(-2.0 + 17.0x - 5.0x^2)`
which evaluates to the correct values.
The polyfit function accepts a maximal degree for the output polynomial, but defaults to using the length of the input vectors x and y minus 1. This is the same degree as the polynomial from the Lagrange formula, and since polynomials of such degree agree on the inputs only if they are identical (this is a basic theorem) - it can be certain this is the same Lagrange polynomial and in fact the only one of such a degree to have this property.
Thanks to the developers of Polynomial.jl for leaving me just to google my way to an Answer.
Take a look to MARS regression. Multi adaptive regression splines.
I have 3 arrays of X, Y and Z. Each have 8 elements. Now for each possible combination of (X,Y,Z) I have a V value.
I am looking to find a formula e.g. V=f(X,Y,Z). Any idea about how that can be done?
Thank you in advance,
Astry
You have a function sampled on a (possibly nonuniform) 3D grid, and want to evaluate the function at any arbitrary point within the volume. One way to approach this (some say the best) is as a multivariate spline evaluation. https://en.wikipedia.org/wiki/Multivariate_interpolation
First, you need to find which rectangular parallelepiped contains the (x,y,z) query point, then you need to interpolate the value from the nearest points. The easiest thing is to use trilinear interpolation from the nearest 8 points. If you want a smoother surface, you can use quadratic interpolation from 27 points or cubic interpolation from 64 points.
For repeated queries of a tricubic spline, your life would be a bit easier by preprocessing the spline to generate Hermite patches/volumes, where your sample points not only have the function value, but also its derivatives (∂/∂x, ∂/∂y, ∂/∂z). That way you don't need messy code for the boundaries at evaluation time.
I was asked to create a leg follower robot (I already did it) and in the second part of this assignment I have to develop a Kalman filter in order to improve the following process of the robot. The robot gets from the person the distance where she is to the robot and also the angle (it is a relative angle, because the reference is the robot itself, not absolute x-y coordinates)
About this assignment I have a serious doubt. Everything I have read, every sample I have seen about kalman filter has been in one dimension (a car running distance or a rock falling from a building) and according to the task I would have to apply it in 2 dimensions. Is it possible to apply a kalman filter like this?
If it is possible to calculate kalman filter in 2 dimensions then I would understand that what is asked to do is to follow the legs in a linnearized way, despite a person walks weirdly (with random movements) --> About this I have the doubt of how to establish the function of the state matrix, could anyone please tell me how to do it or to tell me where I can find more information about this?
thanks.
Well you should read up on Kalman Filter. Basically what it does is estimate a state through its mean and variance separately. The state can be whatever you want. You can have local coordinates in your state but also global coordinates.
Note that the latter will certainly result in nonlinear system dynamics, in which case you could use the Extended Kalman Filter, or to be more correct the continuous-discrete Kalman Filter, where you treat the system dynamics in a continuous manner and the measurements in discrete time.
Example with global coordinates:
Assuming you have a small cubic mass which can drive forward with velocity v. You could simply model the dynamics in local coordinates only, where your state s would be s = [v], which is a linear model.
But, you could also incorporate the global coordinates x and y, assuming we are moving on a plane only. Then you would have s = [x, y, phi, v]'. We need phi to keep track of the current orientation since the cube can only move forward in respect to its orientation of course. Let's define phi as the angle between the cube's forward direction and the x-axis. Or in other words: With phi=0 the cube would move along the x-axis, with phi=90° it would move along the y-axis.
The nonlinear system dynamics with global coordinates can then be written as
s_dot = [x_dot, y_dot, phi_dot, v_dot]'
with
x_dot = cos(phi) * v
y_dot = sin(phi) * v
phi_dot = ...
v_dot = ... (Newton's Law)
In EKF (Extended Kalman Filter) Prediction step you would use the (discretized) equations above to predict the mean of the state in the first step of and the linearized (and discretized) equations for prediction of the Variance.
There are two things to keep in mind when you decide what your state vector s should look like:
You might be tempted to use my linear example s = [v] and then integrate the velocity outside of the Kalman Filter in order to obtain the global coordinate estimates. This would work, but you would lose the awesomeness of the Kalman Filter since you would only integrate the mean of the state, not its variance. In other words, you would have no idea what the current uncertainties for your global coordinates are.
The second step of the Kalman Filter, the measurement or correction update, requires that you can describe your sensor output as a function of your states. So you may have to add states to your representation just so that you can express your measurements correctly as z[k] = h(s[k], w[k]) where z are measurements and w is a noise vector with Gaussian distribution.