Questions tagged [projection]

For on-topic questions involving the mathematical concept projection, a linear transformation $P$ such that $P=P^2$. Please include also a more statistical methods tag. For purely mathematical questions about projections it is better to ask on math SE https://math.stackexchange.com/

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What is the geometric meaning of correlation matrix

I recently read this article explaining the geometric meaning of covariance matrix. http://www.visiondummy.com/2014/04/geometric-interpretation-covariance-matrix/ My question is : is there an ...
21 views

Best linear prediction as a projection in a Hilbert space $L^2$

Consider two random variables $Y$ and $X$. In the context of the best linear prediction, if we would like to predict $Y$ given $X$ known, we derive the solution solving the following minimize problem ...
26 views

The joint distribution of Y=AX and Z=BX given a projection matrix A and residual maker matrix B, and a random vector X with known pdf?

This question follows on from a previous question I asked which was answered. It turns out my question lacked some important details, which was revealed by the answer posted on that thread. This is ...
19 views

How can I recover full dimensional VAR model coefficients after fitting a VAR model to a dimensionality reduced (via PCA) dataset?

I am using PCA to reduce dimensionality prior to fitting a multivariate time-series dataset to a VAR (vector autoregressive) model. Is there any way to convert a PCA-derived VAR model to a full ...
54 views

Least Squares Fit Covariance Matrix

How do you perform a least squares fit for $\Sigma$ in the equation $v = u^T\Sigma u$? The term $v$ is a vector of observed variances of a projected Gaussian distribution. The matrix $u$ is made of ...
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Projection of a vector on unit sphere

I am currently reading the paper G.Salton et al, A Vector Space Model for Automatic Indexing, November 1975, Volume 18 It is about indexing documents. Representing them in vectors to find ...
97 views

Vertically translated depreciation curve: Update the exponential regression coefficient

I have an exponential regression equation that I use to predict the condition of roads. The equation can be found on page 53 of the original master's thesis: Development of a Flexible Framework for ...
87 views

Road condition: Project future condition by appending part of a deterioration curve?

I inspect a road network's condition every three years: ...
125 views

What is wrong with my computation of projections on the first principal component?

Following the links in parenthesis (link1, link2) I wrote a bit of code in MATLAB to simulate a PCA. After running the code and plotting the results I obtain this ...
201 views

Markov chain which is also a projection

Let $P$ be the transition matrix of a Markov chain, and assume that $P^2=P$. One immediate conclusion is that $P=P^\infty$. Furthermore, assume that there is a state $i$ such as each state $j$ (...
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What model should I use for retirement forecasting?

I am a HR professional looking to self learn statistical modeling for new responsibilities at work. I need to forecast no. of employees who may retire next 10 years. What would be simple way to ...
202 views

Projection Matrix and linear model(Question about notation)

I just want to ask a question about notation in this exercise. In Equation $X^*_{.,i} =M_{X.,-i} * X_i$ ; $X^*_{.,i}$ means ith column of original matrix $M_{X.,-i}$ means orthogonal projection ...
638 views

Sampling from marginal distribution using joint sample? [duplicate]

I have a sample of a multivariate distribution, and I am interested in obtaining a sample from the marginals. I know the right way to do so is by simply taking the corresponding entries. What is the ...
205 views

Temporal Multi Dimensional Scaling

Let's say I apply a multidimensional scaling(MDS) to a dynamic dataset of $n$ points (eg, time series). At each step I will obtain a projection (in 2/3D) of the $n$ points. If nothing meaningful ...
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Since a projection matrix is idempotent, symmetric and square, why isn't it just the identity matrix?

I was working on a question on projection matrix. Since, projection matrix is idempotent, symmetric and square matrix, it must always be equal to $I$ (Identity matrix). This can be shown by ...
694 views

Projecting data on a sphere

I am used to working with PCA, tSNE, LLEs... They all do a great job projecting the data on a plane (or on linear subspaces of $\mathbb{R}^n$). Is there any other embedding technique that projects the ...
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Changing the basis of random variables

Let $X_1$ and $X_2$ be two independent (1-dimensional) random variables and let $Y_1 = f^1(X_1, X_2)$ ($f^1$ is a deterministic function) be a (1-dimensional) random variable too. Question Does ...
550 views

Find projection matrix using partitioned matrices

If $X$ is a ($n$, $p+1$) design matrix, partition $X$ to be $X$=[$J$ $X$*] where $J$ is a ($n$,$1$) vector of all $1$'s, and $X$* is a ($n$,$p$) matrix. Let $H_X$ be a projection matrix, where $H_X$...
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How to project data onto a model (specifically, GMM)?

I'm using data to train a Gaussian mixture model (GMM). I then take a sample and would like to see its projection on the GMM 'space'. I can think of an optimization problem such as this: consider $y$ ...
956 views

Relationship between Linear Projection and OLS Regression

In Wooldridge's Econometric Analysis of Cross Section and Panel Data, he defines linear projection of $y$ on $1,\mathbf{x}$, in the following way: Let's assume that $Var(\mathbf{x})$ is positive-...
126 views

Independence of random projection of Gaussian random vector

If $v$, $y$ and $h$ are independent Gaussian random vectors, are $|v^H y|^2$ and $|v^H h|^2$ independent?
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Transforming data on the Principal Components axis?

x y 2.5 2.4 0.5 0.7 2.2 2.9 1.9 2.2 3.1 3.0 2.3 2.7 2 1.6 1 1.1 1.5 1.6 1.1 0.9 Eigenvalues: ...
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Utility of the Frisch-Waugh theorem

I am supposed to teach the Frish Waugh theorem in econometrics, which I have not studied. I have understood the maths behind it and I hope the idea too "the coefficient you get for a particular ...
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The projection matrix and proof of an unbiased estimator for sigma-squared

Hi, given this information we are meant to prove that the above estimator is unbiased. I understand the proof for the most part (below). What I do not understand is the intuitive reason why the ...
151 views

$E(u x)=0$ and endogeneity

I am confused as to how Endogeneity arises. I understand all the examples but something basic just doesn't fit: The usual assumption is that $E(xu) \ne 0$. But, we can always find $\beta$'s that ...
245 views

Proving Orthogonal Properties of Projection Matrix

I am working through a multi-part proof of how orthogonal projection matrices give specific results from their properties. I read through the Gauss-Markov model theory to get a start. This is only a ...