# What is Multivariate: Definition and 135 Discussions

Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied.
In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both

how these can be used to represent the distributions of observed data;
how they can be used as part of statistical inference, particularly where several different quantities are of interest to the same analysis.Certain types of problems involving multivariate data, for example simple linear regression and multiple regression, are not usually considered to be special cases of multivariate statistics because the analysis is dealt with by considering the (univariate) conditional distribution of a single outcome variable given the other variables.

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1. ### Calculus Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...
2. ### Probability: Multivariate distribution change of variables

Hi, I was attempting the problem above and got stuck along the way. Problem: Suppose that ## Y_1 ## and ## Y_2 ## are random variables with joint pdf: f_{y_1, y_2} (y_1, y_2) = 8y_1 y_2 for ## 0 < y_1 < y_2 < 1 ## and 0 otherwise. Let ## U_1 = Y_1/Y_2 ##. Find the probability distribution ##...
3. ### B Limitations of Multivariate Linear Regression

Hello, With multivariate linear regression, there is a single dependent variable ##y## and multiple independent variables ##x_1##, ##x_2##, ##x_3##, etc. There is a linear, weighted relationship between ##y## and the various ##x## variables: $$y = c_1 x_1 + c_2 x_2 + c_3 x_3$$ The...
4. ### Find the composition of a multivariate function with itself

Solution attempt: $$F(F(x,y))=(z,w)$$ is the map given by $$x=z$$ $$y=w$$
5. ### MHB Multivariate gaussian integral

I am looking for the solution of a multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are p $\times$ 1 and T is a positive definite symmetric p $\times$ p matrix. The integral is the...
6. ### Minimum or Maximum value of a multivariate function

First thank you for taking your time to take a look at this simple question. And sorry for the informal math language and equations, I hope you guys can understand it. So, depending on the case, I have 2 or 8 simple quadratic functions f(a), f(b), f(c),… f(z). Each a,b,c,…,z have a different...
7. ### Finding the image of a multivariate function

Hi all, I am trying to figure out a way to simplify this problem to give the image of the map. I have not seen this function before and I am having trouble figuring out what the image should come out as. I have tried graphing u and v separately as a function of x and y in R3 and both surfaces...
8. ### I Self-Study General Relativity: After Multivariate Calc in HS

I just finished multivariate calculus (without any linear algebra experience yet) and I am seeking out a path to understanding General Relativity. I am wondering what are the mathematical fields after multivariate calculus that I need to master before beginning to understand GR, and what...
9. ### A multivariate function of Toruses - tangent vectors

Thank you to all those who helped me solve my last question. This week, I've been assigned an interesting problem about toruses. I think I've solved most of this problem on my own, but I'd like to hear a few suggestions for part c. I think this map multiplies tangent vectors by a factor of...
10. ### Inverse of a multivariate function

Any help with this introductory differential geometry HW would be greatly appreciated. My attempt at solving the first problem: y=4x^3-3x has the derivative 12x^2-3, which is 0 when x^2=(1/4). {x^2=(1/4)} is the singular set, and the inverse is defined for everywhere except F({x^2<(1/4)}). the...
11. ### MHB Finding support for multivariate transformation

For the given joint pdf of X and Y $$f(x,y) = 12xy(1 - y); 0 < x < 1; 0 < y < 1$$ Let $Z = XY^2$ and $W = Y$ be a joint transformation of (X,Y) Sketch the graph of the support of $(Z,W)$ and describe it mathematically. I'm not very sure how to describe (Z,W). First, I draw the graph of the...
12. ### Multivariate calculus problem: Calculating the gradient vector

1. We find the partial derivatives of ##f## with respect to ##x## and ##y## to get ##f_x = \frac{2\ln{(x)}}{x}## and ##f_y = \frac{2\ln{(y)}}{y}.## This makes the gradient vector \nabla{f} = \begin{bmatrix} f_x \\ f_y \end{bmatrix} = \begin{bmatrix} \frac{2\ln{(x)}}{x} \\ \frac{2\ln{(y)}}{y}...
13. ### Solving Multivariate Problem: Critical Points for ##y=-x##

There are sets of the form ##\left\{(x,y)\in \mathbb{R}^2: f(x,y) = \ln{\left(3+(x+y)^2\right)} = c\right\}## where ##c## is some fixed number ##> 1##. Let's see what happens for a few values of ##c##. Suppose ##c = 2##, then ##\ln{\left(3+(x+y)^2\right)} = 2 \Longleftrightarrow (x+y)^2 =...
14. ### I Variable transformation for a multivariate normal distribution

Hello. I would like to draw (sample) several random vectors x from a n-dimensional multivariate normal distribution. For this purpose I want to use C++ and the GNU Scientific Library function gsl_ran_multivariate_gaussian ...
15. ### Divergence operator for multi-dimensional neutron diffusion

Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...
16. ### MHB Solve f(x,y,z): Min & Max Values Subject to Constraint x^2+2y^2+6z^2=81

Afternoon, I have been working on this problem for awhile now but have been stuck on a certain point, and once I set everything equal to each other I end up with the same thing for example: x=x, y=y, & z=z Find the minimum and maximum values of the function f(x,y,z)=3x+2y+4z subject to the...
17. ### MHB Solve Lagrange Multipliers Problem w/ e^(9x) - Find (a,b)

I have been stumped on this problem and I am probably overthinking it as usual. The problem deals with the Lagrange Multipliers: Use Lagrange multipliers to find the point (a,b) on the graph of y=e^(9x), where the value (a,b) is as small as possible. I have found the gradient for both when...
18. ### I Multivariate Calculus for a Game World

Okay, I never took calculus 3 so I don't know how to write/solve an integral for a power I made up. I want to flesh out my game world with math because it's cool and I can. Like Chester wrote some of the physics for the Bizarro universe. Applying physics and math to a fictitious universe is...
19. ### Expected bounds of a continuous bi-variate distribution

Homework Statement [/B] ##-1\leq\alpha\leq 1## ##f(y_1,y_2)=[1-\alpha\{(1-2e^{-y_1})(1-2e^{-y_2})\}]e^{-y_1-y_2}, 0\leq y_1, 0\leq y_2## and ##0## otherwise. Find ##V(Y_1-Y_2)##. Within what limits would you expect ##Y_1-Y_2## to fall? Homework Equations N/A The Attempt at a Solution...
20. ### Applied Looking for a book on Multivariate Statistics

Hey all, Joined a Material Science program and I have a course on Multivariate Stats. I have pretty minimal background on the topic so I thought I would pop back here asking for recommendations for a good start on the topic. Here's the course description -
21. ### I Sampling from a multivariate Gaussian distribution

I was watching a lecture on youtube about linear regression and there's a section where it had the statement below (written in purple). Does multiplying by sigma rotate the distribution to make it look like x - N(mew, sigma^2)? Mew in this case is 0 so it doesn't shift the distribution.
22. T

### I Proving a multivariate normal distribution by the moment generating function

I have proved (8.1). However I am trying to prove that ##\bar{X},X_i-\bar{X},i=1,...,n## has a joint distribution that is multivariate normal. I am trying to prove it by looking at the moment generating function: ##E(e^{t(X_i-\bar{X})}=E(e^{tX_i})E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})## I am...
23. ### A Multivariate Regression vs Stratification for confounding

So say we want to see if child birth order causes down syndrome. Say we get that child birth order is statisitcally associated down syndrome. We know that this association could be explained by a third variable: age. Women that give birth to say their third child, is of course going to be...
24. ### A Adding Multivariate Normal Distributions

So ##\vec Y##~MVN(X##\vec\beta##, ##\sigma^2##I) and ##\hat {\vec Y}##~MVN(X##\vec\beta##, ##\sigma^2##H) and I want to show ##\hat{\vec e}##~MVN(##\vec 0##, ##\sigma^2##(I-H)) Where ##\hat{\vec e}## is the vector of observed residuals( ##\vec e=\vec Y- \hat{\vec Y}=(I-H) \vec Y ##). And...
25. ### Multivariate Gaussian - Normalization factor via diagnolization

Homework Statement Hi, I am trying to follow my book's hint that to find the normalization factor one should "Diagnoalize ##\Sigma^{-1}## to get ##n## Gaussian which will have variance given by the eigenvalues of ##\Sigma## . Then integrate gives ##\sqrt{2\pi}\Lambda_i##, then use that the...
26. ### A Applying group theory to multivariate eqs

Are there any good examples of how group theory can be applied to solve multivariate algebraic equations? The type of equations I have in mind are those that set a "multilinear" polynomial (e.g. ## xyz + 3xy + z##) equal to a monomial (e.g. ##x^3##). However, I'd like to hear about any sort...
27. ### Show that a multivariate function is bijective

Homework Statement Show that ##f: \mathbb{Z}^{+} \times \mathbb{Z}^{+} \longrightarrow \mathbb{Z}^{+} ##where ##\displaystyle f(m,n) = \frac{(m+n-2)(m+n-1)}{2}+m## is bijective Homework EquationsThe Attempt at a Solution First, we show that ##f(a,b) = f(c,d) \implies a=c \land b=d##. ##f(a,b)...
28. ### A PDF formed from a multivariate function

Homework Statement A probability density function, ##p\left(x\right)=\int _{-a}^a\left(f\left(x,\left(a-y\right)\right)-f\left(y,ax\right)\right)dy##, can be formed from the bivariate function ##f\left(x,y\right)=y\sqrt{\left(x-2\right)^2-y}## over the domain ##[0,b]## - where ##b## is the...
29. ### MHB A question about 'properties of multivariate normal distributions'.

hello guys, i have a question and looking for an answer quickly... question; prove that, Z is a vector.thank you.
30. ### Calculus Hubbard and Hubbard for multivariate calculus

I've taken the single and multivariate calculus classes at my school, (college class offered in my high school for accelerated students). I'm currently a junior, but in the summer of my senior year, I plan to read a book on proofs, brush up on math glossed over by the american education system...
31. ### A Discrete Multivariate Probability Distribution

Homework Statement A fair coin has a ##1## painted upon one side and a ##2## painted upon the other side. The coin is tossed ##3## times. Write down a sample space for this experiment. Let ##X_1## be the sum of the numbers obtained on the first ##2## tosses and ##X_2## be the sum of the numbers...
32. ### A Continuous Multivariate Distribution

Homework Statement The random variable ##(x,y)## has density ##f(x,y) = ce^{-(ax+by)}## for ##0\leq y\leq x\leq 1##, with given constants ##a > 0##, ##b > 0##. 1. Compute the constant ##c##. 2. Find the conditional probability density ##f_y(y|x)##. 3. Compute the regression curve of ##Y## on...
33. ### Expected value and variance of multivariate exponential distr.

Homework Statement https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202016-02-02%20kl.%2007.35.26.png [Broken] I want to find variance matrix and expected variance vector of Y=(Y1,Y2). Y1 and Y2 are independant. Γ is the gamma function and ϒ is a known parameter. λ1>0 λ2>0 and...
34. ### Biot-Savart law from Ampère's (with multivariate calculus)

Let us assume the validity of Ampère's circuital law\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}where ##\mathbf{B}## is the magnetic field, ##\gamma## a closed path linking the current of intensity ##I_{\text{linked}}##. All the derivations of the Biot-Savart law for a...
35. ### Is DEHP a kind of Multivariate hard problem?

Please correct me if I am wrong. To my understanding , given a '##m##' multivariate set of equations in '##n##' variables in a integer field '##F##' is hard to solve, even in case of ##MQ(multiquadratic)## usually with field having characteristic as '##2##'. Where in case of ##DEHP(Diophatine...
36. ### Looking for a multivariate function with certain properties

As a undergrad future scientist, in my free time I like to do some programming to get some practice as my education programme doesn't include it, but it is a useful skill to have. I am programming some type of roguelike game, so I can get away from graphics and focus on procedural generation...
37. ### Continous limit of a multivariate normal distribution

Hello everyone, I am currently considering a set of random variables, \vec{x} = [x_1,x_2,...x_N] which are know to follow a multivariate normal distribution, P(\vec{x}) \propto \mathrm{exp}(-\frac{1}{2}(\vec{x}-\vec{\mu})^\mathrm{T}\Sigma^{-1}(\vec{x}-\vec{\mu})) The covariance matrix Σ and...
38. ### Function Composition of Multivariate Functions

Homework Statement This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of...
39. ### ROOT multivariate reweighting

Suppose I have a programme. In there I have the creation of histograms: signal: histo_s_Pt histo_s_v1 histo_s_v2 background: histo_b_Pt, histo_b_v1 histo_b_v2 iteration to fill signal/bckg from events j. So at this point I'm having the 6 above histograms filled. Now if I wanted to compare...
40. ### Understanding multivariate linear regression

I am trying to understand multivariate linear regression. I have a list of time that it took running processes based on several params, like % of cpu usage, and data read. Eg, I have a process that took 50 seconds to run, with a cpu usage of 70%, and the process read 10bytes of data. I have...
41. ### Outlier identification of multivariate data

Dear all, I have more than 1500 matrices that are the occurency frequency of a bivariate dataset, something like the following: [0.1, 0.3,0.1;0.4,0.05,0.05]. I need to extract the few of them that are really different from the others. Instead of visually analyze each of them and trying to...
42. ### Multivariate piecewise fxn continuity and partial derivative

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...
43. ### Jacobian when there's a multivariate function inside it

Homework Statement differentiate the function F(x,y) = f( g(x)k(y) ; g(x)+h(y) ) Homework Equations Standard rules for partial differentiation The Attempt at a Solution The Jacobian will have two columns because of the variables x and y. But what then? f is a multivariate function inside...
44. ### Max/min of multivariate function

Homework Statement max/min of f(x,y) = x + y constraint xy = 16 The Attempt at a Solution With lagrange multipliers I did ## \nabla f = (1,1) ## ## \nabla g = (y,x) ## ## \nabla f = \lambda \nabla g ## ## 1 = \lambda y ## ## 1 = \lambda x ## Since y=0, x=0 aren't a part of xy = 16 I can isolate...
45. ### Please suggest books on multivariate analysis

Please suggest, book on 'Partial Least Square.'
46. ### Multivariate probability distribution

"In multivariate probability distribution higher-order cumulants contain information of decreasing significance, unlike higher-order moments".
47. ### Derivative of multivariate integral

Homework Statement Trying to figure our how to solve the following: \frac{dW}{dσ} where W(σ) = 2π\int_0^∞y(H(x,σ))x,dx Homework Equations both y and H(x,y) are continuous functions from 0 to Infinity The Attempt at a Solution Tried using the leibniz rule but it's not really...
48. ### Roots of multivariate polynomials

What is the maximum number of roots of a multivariate polynomial over a field? Is there a multivariate version of the fundamental theorem of algebra?
49. ### Question about multivariate statistics on proportions

Hi all, I have no experience with multivariate statistics but I'm trying to figure out a type of problem that seems like it should be easy, but I don't know where to begin. I ran into the problem in my research, and I don't have a strong statistics background. I'll try to explain it with an...
50. ### Navigating a Math Education: Pre-Calc to Multivariate Calc

Hey all! I have just finished a course in Calc II and have started a course in multivariate calculus. Is this right course I should have taken? Also, what does a typical order for math look like? I am in my last semester of high school; so far my career in math looks like this: MiddleSchool...