# 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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From asset returns of $N$ stocks, the symmetric covariance matrix sized $N\times N$ is constructed, which treats the asset returns as variables. When the number of variables $N$ is fairly low like $N=... 0answers 21 views ### Condition number and leverage Condition number is defined using the eigenvalues of$X^\top X$while leverage values are the diagonal elements of the projection matrix. How are the two related? How can rescaling help, because aren'... 0answers 9 views ### Variance inflation factor vs condition number [duplicate] We know the Variance inflation factor and condition number both help to measure multicollinearity. Which one should we use when? 0answers 12 views ### What is the meaning of the regressor characteristic root? As described by Greene's Econometric Analysis (7th Edition), the regressor matrix's condition number measures how singular the matrix is. Therefore, the condition number is a measure of ... 0answers 19 views ### Condition numbers, invertibility and multicollinearity The following is an excerpt from Greene's Regression Analysis (Seventh Edition): a) What does it mean to be "difficult" to invert a matrix accurately? Shouldn't all matrixes be either ... 0answers 18 views ### Does Ridge Regression generally improve the condition number? While learning about Ridge regression and its applications I found a test question about impact of Ridge regression on the condition number. As far as I understand, Ridge regression can decrease the ... 0answers 17 views ### Multicollinearity negating$\beta$In Regression coefficients that flip sign after including other predictors, ars's answer states that "Basically, if your variables are positively correlated, then the coefficients will be ... 0answers 35 views ### Comparing numerical stability and computing bounds on the condition number of learned weights I have an empirical risk minimization problem with two equivalent losses that solves it,$f_1(x; \theta_1)$and$f_2(x ; \theta_2)$, where$x$is the data and$\theta$are the model parameters (in ... 0answers 21 views ### QR decomposition used for estimating condition numbers I have read that the QR decomposition is often used to estimate the condition number of a matrix but I don't understand... what is the benefit of using the QR decomposition for this? Is it purely ... 1answer 239 views ### Kappa condition number in R I have read that the kappa function in R does not always explicitly calculate the condition number of a matrix, but rather, estimates the 2 norm of a matrix or a QR ... 1answer 167 views ### How are the condition numbers of a design matrix and its correlation matrix related? Given a design matrix$X$for a linear regression model, what is the relationship between the condition number of$X$and its correlation matrix$R$? I would be interested in the case of a centered ... 1answer 1k views ### Why does matrix condition number change drastically when a constant is added? If I create a regression model design matrix with 3 uncorrelated variables, I get a small condition number as expected. MWE: ... 0answers 574 views ### Choosing the basis functions in a linear regression I have two random variables$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}$and then ... 2answers 2k views ### Deep Learning: Condition Number and Poor Conditioning I am reading the following section of the book Deep Learning. Can you provide an intuitive explanation of the above section? I don't quite understand the statement "When this number is large, matrix ... 0answers 329 views ### Proximal Gradient Descent and Proximal Coordinate descent for Lasso Problem Why is proximal coordinate descent much less affected by bad conditioning than proximal gradient descent? For example, we can consider this problem :$\min_x \frac{1}{2}\|Ax-b\|^2_2 + \lambda\|x\|_1$... 0answers 82 views ### 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 ... 2answers 3k views ### 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 ... 0answers 8k views ### 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 ... 1answer 1k views ### 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: ... 1answer 229 views ### 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 ... 2answers 891 views ### 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 ... 0answers 413 views ### 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 ... 0answers 44 views ### 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, ... 0answers 527 views ### 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 ... 2answers 11k views ### 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 ... 1answer 174 views ### 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 "... 0answers 880 views ### Propogation of error in a matrix inversion I'm trying to find the deterministic error bounds for some parameters calculated through distance geometry. The equation can be simplified to the following form:$ \left[\begin{matrix} x_1 \\ x_2 \\ ...
I have a multivariate regression model $Y=X\beta ' + \epsilon$. The variables in the $X$ matrix have very different scales and hence the condition number of $X'X$ is huge (order of trillions). I ...