# Questions tagged [linear-algebra]

A field of mathematics concerned with the study of finite dimensional vector spaces, including matrices and their manipulation, which are important in statistics.

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### How to have a constant error[b-a] for a range of values [1d array] in decending order that are not linear but are semi-linear?

I am trying to generate n numbers in decending order (for the purpose of decreasing order of weights) such that their sum should be 100 and more importantly the difference between anytwo numbers ...
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### Optimal truncation in SVD

I am working with SVD on a matrix $$Y_{m,n} = T_{m,m} \Sigma D^T_{n,n}$$ where $T$ and $D$ describe the row and the column entities of Y, respectively. The truncated SVD takes the first $r$ ...
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### What is the meaning of the inner product between two regression variables?

I have been analyzing the effect of design matrix columns on the contour line of the least squares regression. These contours obviously are ellipses when only two columns $\phi_1$ and $\phi_2$ are ...
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### How to compare row entries in a sparse table with lots of missing values?

I have a dataset with ~1000 laptops and performance results across ~100 different benchmarks. Using the benchmark results, I want to give each laptop a single composite performance score, and rank the ...
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### Quantifying how underdetermined a system of equations or optimization problem is

In linear systems we have an exact solution when we have as many equations as unknowns and the equations are linearly independent, e.g., $$x_0 + 2 x_1 = 5 \\ x_0 + 3 x_1 = 7 \\$$ has the unique ...
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### On distributions over orthonormal sets: existing families, construction, and simulation

Have families of distributions over orthonormal sets been defined and studied in the literature? What are a couple examples and/or references? Are there known methods for constructing distributions ...
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### minimum determinant covariance matrix and covariance

I am trying to understand minimum determinant covariance. I gather from this stack exchange post that it tries to select a subset of data that is tightly distributed to exclude anomalies, and it does ...
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### How to calculate the bias b in support vector machine when the dual coefficient alpha is obtained?

For my example, I have two data points x = {(54001.988, 19999), (30021.983, 15000} and their labels are y = {1, -1}. I calculated the dual coefficient(Lagrange multipliers) alpha = {10000, 10000}. The ...
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### Do SVM's find the minimal number of support vectors in the case of redundancy?

Consider the case where more than the minimal number of vectors lie on the lines (or hyperplanes in higher dimensions) defined by the margin found by the SVM algorithm. For example, see the image ...
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### distance between regression models

Consider two multivariate linear regression models (vector inputs and outputs) with the same domain observations. Namely, let: $X \in \mathcal{R}^{a \times N}$ be a matrix of domain observations (...
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### Finite non-zero determinant constraint?

Assume I want to train a multivariable normal distribution on given data set T. One definition of multivariable normal distribution is this: A random vector X has a (multivariate) normal distribution ...
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### kernel PCA similarity matrix analogy

The standard explanation to linear PCA begins with the covariance matrix. That is, for a dataset $D$ of dimension $N \times d$, the covariance matrix is given as $\sum = \frac{D^{T}D}{N}$ where the ...
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### Show inequality for $var(\tilde{\beta})$ for a linear mixed model

Consider the usual linear mixed model: $$Y=X \beta+ZB+\epsilon$$ where Y and $\epsilon$ are $n$-dimensional random variables and $B$ is a $q$-dimensional random variable independent of $\epsilon$ so ...
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### How to prove Mean Squarred Error (MSE)

I would like to prove this equation of Mean Squared Error (MSE): m is the number of training instances. X is a m × n matrix containing all the feature values (excluding labels) of all instances in ...
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### Help with polynomial long division equations in "Three Model Representation for an ARMA Model" section by Tsay

I have 2 follow up questions to the same titled question posted here 7-years ago. Relating to Tsay's book section "2.6.5 Three Model Representations for an ARMA Model": Question 1 relates to ...
1 vote
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### Fitting of a sum of vectors $Y \alpha = X \beta + \epsilon$ [duplicate]

Related question / Motivation This question is related to the question here. But the difference in this question is that the weights can be different on both sides of the equation. This question This ...
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### Question regarding matrix notation

I'm trying to get my head around a statistical topic where I look at dental measurements on eleven girls and sixteen boys at four different ages. One matrix that shows up is the following one: \begin{...
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