What is Multidimensional: Definition and 46 Discussions

Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset. MDS is used to translate "information about the pairwise 'distances' among a set of



n


{\textstyle n}
objects or individuals" into a configuration of



n


{\textstyle n}
points mapped into an abstract Cartesian space.More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction.
Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N=1, 2, and 3, the resulting points can be visualized on a scatter plot.Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the father of functional data analysis.

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  1. D

    Multidimensional time simulation

    If there are computer simulations of four-dimensional space are there any possibilities to digitally simulate a space -time with time having more than one dimension? Please, leave some related links, if possible.
  2. SaschaSIGI

    I Understanding Hessian for multidimensional function

    Hello everybody, I have a question regarding this visualization of a multidimensional function. Given f(u, v) = e^{−cu} sin(u) sin(v). Im confused why the maximas/minimas have half positive Trace and half negative Trace. I thought because its maxima it only has to be negative. 3D vis 2D...
  3. Math Amateur

    MHB Exploring Proposition 6.1.2 from D&K's Multidimensional Real Analysis II (Integration)

    I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ... I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of...
  4. Purplepixie

    MHB Multidimensional Newton Raphson

    Hello, I am having difficulty in translating the univariate Newton's approximation {Xn = Xn-1 - [ f(Xn-1) / f'(Xn-1)]} into the multidimensional case. My multidimensional equation system is y = F.x where y and x are (nx1) column vectors and the coefficients matrix F is (nxn), so that (nx1) =...
  5. E

    How to Use a Multidimensional Array to SED?

    I have a .dat file which contains an ##m \times n## (specifically, a ##9 \times 2##) array and I have a file which has this kind of format, variable_x xx variable_y yy where xx and yy are numbers (I'll call this file the input_file). This file serves as an input to an external program, which I...
  6. N

    Numpy multidimensional array assigment unknown logic error

    I'm confused as to why this bubble sort I'm implementing is setting _all_ items to one of the items (I have no idea which one as array is too big) the data type is of ... [[1128 1026 1192 1023]] [[ 771 195 858 196]] [[ 953 1799 955 1738]]] when I have an array of int, this same algorithm...
  7. Telemachus

    Running a multidimensional problem on a single core

    Hi. I have written a Fortran program to solve a pde, which has six dimensions. One of these dimensions is time, 3 are for space, and the other are related to velocities. The program works fine when I run it with ##38^3## points in space (one for each space dimension), and 80 other dimensions...
  8. Math Amateur

    MHB Multidimensional Real Analysis - Duistermaat and Kolk, Lemma 1.1.7 ....

    I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Lemma 1,1,7 (ii) ... Duistermaat and Kolk"s Lemma 1.1.7 reads as follows: In the above Lemma part (ii)...
  9. M

    Library that supports multidimensional Arrays

    I'm am looking for a C++ library that supports high-dimensional arrays ( 3=< ) and element-wise calculations. Is there such a thing?
  10. J

    2D Integrating With Quadratic Arg. of Delta Function

    Homework Statement I have a 2D integral that contains a delta function: ##\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp{-((x_2-x_1)^2)+(a x_2^2+b x_1^2-c x_2+d x_1+e))}\delta(p x_1^2-q x_2^2) dx_1 dx_2##, where ##x_1## and ##x_2## are variables, and a,b,c,d,e,p and q are some real...
  11. M

    I What will be the 4th axis of a 3d curl?

    Suppose we have do a curl of two 2-d vectors... we get the 3rd axis about which it is rotating. But when we do the curl of two 3-d vectors.. we get a answer like x-y plane is rotating wrt z axis, y-z plane rotating wrt to x-axis and similarly x-z plane rotating wrt to y axis. My question is...
  12. Ben Wilson

    MATLAB Replace NaN with next element in Multidimensional Array

    I have a a multidimensional array (49x49x49) which has NaN values scattered randomly. For each NaN, I want to replace the value with the previous non NaN value. e.g. if i had a 3 x 3 matrix 4 7 9 2 7 NaN 1 2 NaN should become 4 7 9 2 7 7 1 2 2 I have 3 dimensions [i,j,k], but...
  13. G

    Gravitational Waves & Multidimensional Spacetime: Experiments & Detection

    So I saw that claims are being made that LIGO may have detected gravitational waves. http://www.nature.com/news/has-giant-ligo-experiment-seen-gravitational-waves-1.18449 My question is, if the universe were in fact multidimensional as string theory predicts, would gravitational waves propagate...
  14. S

    Convolution and space-time Fourier transform

    Hi, I have a general function u(x,y,z,t). Then, (1) what would be the space-time Fourier transform of G⊗(∂nu/∂tn) and (2) would the relation G⊗(∂nu/∂tn) = ∂n(G⊗u)/∂tn hold true? Here, note that the symbol ⊗ represents convolution and G is a function of (x,y,z) only (i.e. it does not depend on...
  15. S

    How to fit plane onto sampling data?

    For example I have the variables x, y and a probability distribution p(x,y). I want to approximate p(x,y) as a linear function, a plane in this case, at least somewhere in the domain. However I only have samples from the distribution. In case of big amount of data the it is easy to collect them...
  16. H

    Multidimensional first order linear PDE

    Hello, I have the following PDE : v_y \frac{\partial f}{\partial y} + \Omega_x(y)\left(v_z\frac{\partial f}{\partial v_y} - v_y\frac{\partial f}{\partial v_z}\right) + \Omega_z(y)\left(v_y\frac{\partial f}{\partial v_x} - v_x\frac{\partial f}{\partial v_y}\right) = 0 (which is the steady...
  17. SalfordPhysics

    Comp Sci Efficient Fortran Method for Trajectory Plotting with 4d Vector-First Order ODE

    Homework Statement I have a code that successfully plots the trajectory of a ball moving under gravity and air resistance, but my method is rather long-winded and I want to use a 4d vector-first order ODE instead - but I don't know how to do it. I've tried writing some simple skeletons but...
  18. Hanyu Ye

    How to compute multidimensional inverse Fourier transform

    Hello, everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field: \tilde{p}=-\frac{i}{{{k}^{2}}}\mathbf{F}\centerdot \mathbf{k} where i is the imaginary unit...
  19. D

    Multidimensional fitting of two sets of data

    Hello, my problem is the following: A lasers gives out a bunch of data points which are reflected off a metal surface and recorded by a camera attached to the side of the laser. The image the camera receives is however distorted. In order to calibrate the camera I need to find a function...
  20. R

    What is the importance of Phase in case of Multidimensional signals?

    I am beginner in image processing and want to do filtering in Frequency domain. I can understand that the frequency spectrum in case of 1D waves. It denotes what frequencies are present in a wave. If we draw the phase spectrum of cos(2πft) , we get an impulse signal at −f and +f, and it is...
  21. M

    Multidimensional chain rule (tough)

    hey pf! suppose i have a function ##f( x , y)##. i make a change of variables such that ##z(x,y)## in such a way that now ##f( z , y)##. how do i find $$\frac{\partial f}{\partial y}$$ $$\frac{\partial f}{\partial x}$$ $$\frac{\partial^2 f}{\partial y^2}$$ $$\frac{\partial^2 f}{\partial x}$$...
  22. M

    Latest research in multidimensional space.

    Their are three proven dimensions ( + time ) and we can only see two right? Google insists their could be more or even less. What was the latest research/expirements on multiple dimensions in the quantum world. What were the results? -(a question coming from someone understanding a 1/4 of...
  23. dexterdev

    How do we represent multidimensional data vectors on a 2D plot.

    Hi all, I would like to know if there exists any method to represent multidimensional vectors on a 2D plot so using extracting any unique features of those vectors. Can eigen values be used for such purposes like dimensional reduction. If so how ? I would like to know more about these...
  24. J

    C/C++ [C/C++] Using a double pointer to create a multidimensional array

    Tomorrow I'm taking a programing assessment for a potential job. I was wondering whether, if I ever have to create a 2-D array for one of the challenge problems, I should use the method here: http://stackoverflow.com/questions/13974001/two-dimensional-array-implementation-using-double-pointer...
  25. C

    Sampling with multidimensional transformations

    I am not sure if I have the title right, but here is my problem: I have a ray which 'should be' shot vertically from a point p, but depending on the situation it can: 1) either be shot in any direction in the hemisphere above p 2) shot with an angle of no more than σ off the vertical 3) shot...
  26. B

    Algorithm for multidimensional constrained root finding

    Hi all, I'm looking for an algorithm for multidimensional constrained root finding, implemented in Fortran. It's intended for finding a steady-state solution for a dynamic model. I have n state variables and n coupled differential equations (n~=60), and I need to find the value for the state...
  27. G

    Computer simulation of geologic processes using multidimensional functions

    Hello all, this is my first topic here at PF, though I have been using this site as a homework aid for quite a while. Just to clarify, this problem is NOT a homework problem. I have been attempting to create a MOD for the 3D lego-like indy game called minecraft. Minecraft features a...
  28. F

    Multidimensional Gaussian integral with constraints

    Homework Statement The larger context is that I'm looking at the scenario of fitting a polynomial to points with Gaussian errors using chi squared minimization. The point of this is to describe the likelihood of measuring a given parameter set from the fit. I'm taking N equally spaced x values...
  29. C

    Multidimensional machine learning

    Hi, I am working on a project that involves classifying many 'situations' where there are some number of objects and the objects can be defined by the makeup and weighting of their parameters. There are probably 200-2000 parameters per object, 1 to 100 objects and hundreds of situations...
  30. S

    Maximum final component of a simple symmetric multidimensional random walk?

    I am developing a simple probabilistic model of my own mistakes in (1) solving math problems and (2) implementing algorithms on a computer. I have reduced the problem to one which seems simple enough, but which I have been unable to solve, due to my mathematical inexperience. I figured I would...
  31. M

    Multidimensional Fourier transform oddity

    Hi all, I'm trying to compute the Fourier transform of a slightly odd function, a pair of monomials in k cobbled together with heaviside theta functions: f(k)=\theta(1-k) k^{n-2}+\theta(k-1) k^{-2} where n is some integer >2. A complicating factor is that k is really the modulus of a vector...
  32. W

    Numerical multidimensional integration over function of six variables

    So, I'm writing a program in matlab. I have a function of six variables, say f(x1,x2,x3,x4,x5,x6). I want to integrate over x4, x5, and x6 numerically. f is defined over a 10 sided 6-cube of points. I also want to integrate over the whole cube. So I want, F(x_1, x_2, x_3) = \int\int\int...
  33. O

    Multidimensional cross product vector

    Does anyone knows how to compute cross product vector of more than 3 dimensions? It seems all the linear algebra textbooks only discuss 3D cross product vector. What are the formulas?
  34. Z

    A quesiton about multidimensional Fourier transform

    my question is the following let be the Fourier transform \int_{-\infty}^{\infty}d^{4}p \frac{exp( ip*k)}{p^{2}+a^{2}} here p^{2}= p_{0}^{2}+p_{1}^{2}+p_{2}^{2}+p_{3}^{2} is the modulus of vector 'p' , here * means scalar product for the scalar product i can use the definition...
  35. V

    Fortran Memory storage of multidimensional arrays in fortran 90

    Hello, I think i heard somewhere that a multidimensional array in fortran 90 may be stored non-contiguously in memory. Is this true? Even if it the size was known at compile time? If your answer is it depends on a compiler, what about ifort and gfortran?
  36. M

    Swapping numbers in a multidimensional array

    i'm trying to swap the 43 and the 435, but instead 435 is printed where 43 should be. #include <iostream> using namespace std; void print(const int matrix[][2]); void swap(int matrix[][2]); int main(){ int matrix[2][2] = {{14, 435}, {43, 65}}; print(matrix); cout << endl...
  37. A

    Pointers and multidimensional arrays in C

    /* Demonstrates passing a pointer to a multidimensional */ /* array to a function. */ #include <stdio.h> void printarray_1(int (*ptr)[4]); void printarray_2(int (*ptr)[4], int n); main() { int multi[3][4] = { { 1, 2, 3, 4 }, { 5, 6, 7, 8 }...
  38. A

    MATLAB MATLAB Multidimensional Array

    Hello, I'd like to access a column of values from the 4th dimension of a 4D Double array in MATLAB and then save them to a new matrix. For example: A = rand(3,3,3,3); A(1,1,1,:) gives me: ans(:,:,1,1) = 0.7077 ans(:,:,1,2) = 0.0669 ans(:,:,1,3) = 0.7794 I...
  39. djeitnstine

    Multidimensional arrays in C

    Hi guys I just learned M-D arrays and was trying to execute this piece of code: //initial value of y printf("\nTime of initial displacement - y(t): t="); fflush(stdin); scanf("%lf", &IVP[1][1]); printf("\nInitial displacement - y(%.4lf)=",&IVP[1][1]); fflush(stdin)...
  40. K

    Visualizing intersecting multidimensional objects.

    If we look at 2 intersecting orthogonal planes in 3D, the intersection forms a line if you are "living" on either plane. How would the intersection look if there are 2D planes in 4D where the planes do not share a dimension? For example plane 1 exists on X and Y, and plane 2 exists on Z and T...
  41. M

    Multidimensional regresion

    i have this problem, given a series of values (x_{i} , y_{j} , z_{k}) =U could we find a plane 0=Ax+By+Cz+D so the distance from the set of points given in 'U' and the plane with normal vector N=(A,B,C) is a minimum, or in more general case the distance from the set of points 'U' and...
  42. T

    Exploring the Multidimensional Virasoro Algebra and its Significance for Physics

    There is place one where it does not arise: QFT. The Virasoro extension is a diff anomaly, defined in any dimension, and the analogous Kac-Moody-like extension is a gauge anomaly, proportional to the second Casimir. However, diff and gauge anomalies in QFT were classified many years ago...
  43. G

    Understanding Multidimensional Time

    Let's see if I get this thought right! In accordance with Uncertainty Principle, if we know the location of the particle (space), we cannot know the direction (rate of change/time). But, if we know X-pos of a particle, then Y-pos and Z-pos cannot be know either since the first axis of...
  44. M

    Multidimensional limits

    I'm trying to understand how to interpret multidemensional limits. For example, suppose you have the following: \lim\limits_{x \to \infty}\lim\limits_{y \to \infty} x\frac{1}{y} Would this be infinity, 0, or 1? This is really a more general version of the question I'm working with...
  45. T

    Random questions + Geometric multidimensional patterns.

    I was reading through various books about black holes, time warps, and multiple dimensions, and now I am simply asking for clarification on a few things, and previous discoveries on something I noticed while looking at geometric patterns via different dimensions. Last part first. Let me...
  46. V

    Multidimensional Space Time

    Hi, I just wanted to know that do u guys think that Multidimensional Space Time can exist. I know the theories can permit this but i am curious to what it would be like in a place having 2/3 D Time matrix. How will onw relate to an event. I also wanted to know that whether any of u guys can...
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