# Questions tagged [underdetermined]

Analyses are underdetermined when the number of parameters to be estimated is greater than the number of data. This problem is also referred to as 'p >> n'.

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### What is the intuition behind the idea that for linear regression, the number of observations should exceed the number of parameters?

If a population model has k independent variables and 1 intercept, why are k+1 observations required to perform OLS estimates? What is the intuition behind this?
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### IV estimation for when instrument is independent of the outcome

So I came across this post and have some use for it in my own work. Recap of Original Question: In the canonical IV setting, suppose we have a data-generating model $Z\rightarrow X\rightarrow Y$ (...
1 vote
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### Is Modelling of an Under-determined System possible?

We have a dataset with around 20,000 variables and only 200 observations. Our Naive Modelling: We split it into train set (=150 observations) and validation set (=50 observations) and fit Linear ...
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### Gauss-Markov with $p>n$

Let $p$ be the number of parameters in a linear regression model, let $n$ be the number of observations, and let $p>n$. $$\mathbb E[Y\vert X] = \beta_0 +\beta_1X_1 +...+\beta_pX_p$$ Does the Gauss-...
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1 vote
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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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### What is scikit-learn's LinearRegression doing when there are more features than observations? [duplicate]

I'm trying to understand what sklearn's LinearRegression (which should be using ordinary least squares) is doing when there are more features than observations. ...
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### Why do I not get a p-value and F value from ANOVA in R?

I am asked to determine using an appropriate test whether the internal concentration of isoleucine is statistically different between appropriate experiments. So I constructed a data frame in R as ...
1 vote
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### How to interpret a specific data transformation?

I came across this specific data transformation in the context of a physics application, which by itself is rather complex and hence out of the scope of this question. However since this ...
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1 vote
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### Appropriate dimensionality reduction technique for a small, but high-dimensional sample

I am attempting to conduct some multivariate analysis on a dataset I've been given with a sample size (n) of 23 and a feature number (p) of ~800. I would like to use dimensionality reduction, but ...
281 views

### Best estimate of underdetermined system using prior

I have measured two variables which depend on the same set of four parameters. I want to know the parameters which best explain my measurements. Of course, I cannot solve for four unknowns from just ...
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### Linear discriminant analysis with $p\gg n$

I am studying Linear Discriminant Analysis (LDA). According to the formula for LDA, we are supposed to get the inverse of within group covariance. However, if $p\gg n$ (i.e., the dimension is much ...
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### Can one (theoretically) train a neural network with fewer training samples than weights?

First of all: I know, there is no general number of sample size required to train a neural network. It depends on way too many factors like complexity of the task, noise in the data and so on. And the ...
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### Why is $n < p$ a problem for OLS regression?

I realize I can't invert the $X'X$ matrix but I can use gradient descent on the quadratic loss function and get a solution. I can then use those estimates to calculate standard errors and residuals. ...
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### Fitting least squares when number of predictors are larger than instances

A statement from the book Introduction to Statistical learning with applications in R, didn't quite make sense to me. It says, "In cases when number of predictors are greater than the instances we ...
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### How to identify a SEM with formative dependent variable (with R's lavaan package)?

I have a formative construct in a structural equation model (SEM) which I would like to estimate with the function sem in the ...
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### Do I have too many variables and not enough data points for cluster analysis?

I have 75 observations and 152 variables. I want to perform cluster analysis. If I perform cluster analysis and this data will the results be meaningful? Do I need to reduce the number of variables ...
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### LASSO prediction model question

I am trying to create a prediction model with 33 predictors (brain metabolite levels in various regions) and 8 observations (cognitive test scores) with p>>n problem using LASSO in MATLAB (...
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### Solving a practical machine learning problem

I am currently doing my Phd in computational biology at Stanford. I get the data I need to answer the questions I am interested in. The data sets are sometimes "large" and these large problems take ...
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### Dataset for Least Angle Regression

I have read that least angle regression is good for high dimensional data. I didn't actually understand the meaning of high dimensional data, so does this mean $p>>n$ case? And does anyone know ...
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### Applying ridge regression for an underdetermined system of equations?

When $y = X\beta + e$, the least squares problem which imposes a spherical restriction $\delta$ on the value of $\beta$ can be written as \begin{array} &\operatorname{min}\ \| y - ...
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1 vote
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### Interpretation of regression coefficients obtained from applying left inverse of regressor matrix in an underdetermined system?

If $X^\dagger$ is the pseudo-inverse of $X$, $\beta = X^\dagger y$ is the least squares solution for $\beta$ when $y=X\beta$. In the overdetermined case, applying $X^{\dagger,L} = (X^TX)^{-1}X^T$ ...
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### Dealing with underdetermination in Bayesian models

Bayesian models are supposedly well equipped to deal with high-dimensionality problems, and can handle sparse data well, too. But suppose I've created a model that estimate more parameters than there ...
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### Feature selection & model with glmnet on Methylation data (p>>N)

I would like to use GLM and Elastic Net to select those relevant features + build a linear regression model (i.e., both prediction and understanding, so it would be better to be left with relatively ...
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### SVM has relatively low classification rate for high-dimensional data even though 2-D projections show they are separable

I have another problem with 14000 features and 500 training samples. It is a binary classification problem and approximately in the form of an ellipse. My classification accuracy using the 2nd degree ...
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### Why is it bad if number of dimensions / factors > sample size?

I've been told (read) this many times, but I never understood why it's bad for the number of dimensions in your data, or the number of explanatory variables in your model to be higher than your ...
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In a dataset of two non-overlapping populations (patients & healthy, total $n=60$) I would like to find (out of $300$ independent variables) significant predictors for a continuous dependent ...