# Questions tagged [condition-number]

The condition number (also known as condition index) is a diagnostic tool for collinearity in regression models. A regression model has as many condition numbers as it has independent variables. Each is defined as the square root of the ratio of the largest eigenvalue to the eigenvalue for the respective variable. Condition numbers over 30 are considered to be signs of problematic collinearity.

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### Does the correlation matrix always have a smaller condition number than the covariance matrix?

In my experience and the experience of others (for example: https://stats.stackexchange.com/a/287737/193216), a covariance matrix $\textrm{COV}=\langle {\bf x}_t{\bf x}_t'\rangle$ has a larger ...
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### Explaining conditioning number in statistics to non-statisticians

I work these days as a statistician and a lot of what I do is evaluating design of experiments; I started this job less than a year ago, after getting a PhD in mathematical statistics. I remember once ...
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### What is the meaning of the regressor characteristic root?

As described by Greene's Econometric Analysis (7th Edition), the regressor matrix's condition number measures how singular the matrix is. Therefore, the condition number is a measure of ...
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### Multicollinearity negating $\beta$

In Regression coefficients that flip sign after including other predictors, ars's answer states that "Basically, if your variables are positively correlated, then the coefficients will be ...
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### Comparing numerical stability and computing bounds on the condition number of learned weights

I have an empirical risk minimization problem with two equivalent losses that solves it, $f_1(x; \theta_1)$ and $f_2(x ; \theta_2)$, where $x$ is the data and $\theta$ are the model parameters (in ...
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### QR decomposition used for estimating condition numbers

I have read that the QR decomposition is often used to estimate the condition number of a matrix but I don't understand... what is the benefit of using the QR decomposition for this? Is it purely ...
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### Kappa condition number in R

I have read that the kappa function in R does not always explicitly calculate the condition number of a matrix, but rather, estimates the 2 norm of a matrix or a QR ...
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### How are the condition numbers of a design matrix and its correlation matrix related?

Given a design matrix $X$ for a linear regression model, what is the relationship between the condition number of $X$ and its correlation matrix $R$? I would be interested in the case of a centered ...
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### Why does matrix condition number change drastically when a constant is added?

If I create a regression model design matrix with 3 uncorrelated variables, I get a small condition number as expected. MWE: ...
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### Choosing the basis functions in a linear regression

I have two random variables $X$ and $Y$ and I'm trying to model $\mathbb{E}[Y|X]$. To this end, I'd like to pick a collection of functions $f_1, f_2 \dots f_n : \mathbb{R} \to \mathbb{R}$ and then ...
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### Deep Learning: Condition Number and Poor Conditioning

I am reading the following section of the book Deep Learning. Can you provide an intuitive explanation of the above section? I don't quite understand the statement "When this number is large, matrix ...
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### Proximal Gradient Descent and Proximal Coordinate descent for Lasso Problem

Why is proximal coordinate descent much less affected by bad conditioning than proximal gradient descent? For example, we can consider this problem : $\min_x \frac{1}{2}\|Ax-b\|^2_2 + \lambda\|x\|_1$ ...
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### Is condition number important when fitting splines?

I am fitting data using p-splines. The authors of p-splines, Eilers and Marx, remark that there is no technical requirement to have a small number of knots and in fact you can have many more knots ...
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### Multicolinearity and Condition number of logistic regresison

It seems to be common to take a "high" condition number as a sign for multicolinearity in regression analysis. For linear models I'm totally convinced that this is a good idea, but is there any ...
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### Cause of a high condition number in a python statsmodels regression?

I'm pretty new to regression analysis, and I'm using python's statsmodels to look at the relationship between GDP/health/social services spending and health outcomes (DALYs) across the OECD. Just to ...
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### Condition number calculation in R

If I understood correctly, the condition number should be a product of Frobenious norms of a matrix and its inverse. In R if I do the following: ...
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### Wrong choice of covariance function?

I am applying Gaussian process regression (GPR) model on some data assuming covSEiso (a.k.a. RBF) covariance function. I made sure there are no identical data points (but there are similar data ...
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### When is it appropriate to override the default reciprocal condition number tolerance for solve() in R?

I am estimating a GMM IV model, where I'm creating a weighting matrix by taking the inverse of Z'Z, where Z is a matrix of instruments. For certain combinations of instruments, when I try to compute ...
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### Should the intercept be included when you check the condition index?

Many sources state that a condition index >30 constitutes a multicollinearity problem. When I've tried to implement this check in practice, I've realized that the condition index (and VIFs) change ...
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### Problems with ill-conditioned matrix for Dantzig Selector

I'm using the Dantzig Selector on high dimensional data and my matrix is ill conditioned (condition number on the order of magnitude of 10^17). I know there are ways to improve the conditioning, ...
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### How to interpret differences in VIF and condition number?

In my present data, the Variance Inflation Factors suggest lack of substential multicolliniearity (<1,7). However, the condition number of 28 is almost at the critical value of 30. How do I ...
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### How do you interpret the condition number of a correlation matrix

I have two correlation matrices, one with a condition number of 9 and the other with a condition number of 70. From what i have read, it will appear that the first matrix is better conditioned than ...
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### Automatically fixing ill-conditioning or collinearity

I'm backtesting a regression model, which entails running it on a bunch of bootstrap samples of a "rewound" version of our data set. Unfortunately, in some of these resamplings, I end up getting some "...
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