for a data with 1000 samples and 700 dimensions, how would you find a line that best fits the data, to be able to extrapolate? this is not a supervised ML problem, there's no target. and how would you do it, if you want to treat this as a supervised ML problem? how would you pick the column to use as target? what are the potential problems to treating this as a supervised problem?

Question

Accepted Answer

To find a line that best fits the data with 1000 samples and 700 dimensions, we can use linear regression.

For unsupervised ML approach, we can use Principal Component Analysis (PCA) to reduce dimensions and then fit a line using linear regression.
For supervised ML approach, we need to select a target column. We can choose any of the 700 dimensions as the target and treat it as a regression problem.
Potential problems of treating this as a supervised problem include: lack of interpretability, overfitting due to high dimensionality, and difficulty in selecting the most relevant target column.

Accepted Answer

When no target variable is available, the problem can be approached using dimensionality reduction techniques to find a line (or principal direction) that best represents the data.

Principal Component Analysis (PCA):

Why PCA?
PCA identifies the principal axes in the data that maximize variance. The first principal component (PC1) represents the "line of best fit" in the data.
How?
Perform PCA on the data (700 dimensions, 1000 samples).
The first principal component is the direction of maximum variance in the data. This line can be used for extrapolation by projecting new data onto this axis.
Steps:
Center the data (subtract mean from each feature).
Compute the covariance matrix of the data.
Compute eigenvalues and eigenvectors of the covariance matrix.
Select the eigenvector corresponding to the largest eigenvalue (PC1).
Limitations of PCA:

Assumes linear relationships in the data.
Might lose important information if the data has non-linear patterns.
Does not inherently support extrapolation beyond the observed range.
Supervised Approach
In a supervised approach, we must choose one of the 700 columns as the target variable. The remaining columns act as predictors. Here's how you can proceed:

Choosing the Target Column:

Use domain knowledge: Select the column most relevant to the task or objective.
Data-driven approach: Look for features with higher variability or correlation to other columns. For instance, compute the variance of each column and choose one with high variance as the target.
Random selection: If there’s no clear guidance, pick a column arbitrarily.
Modeling:

Train a regression model, such as Linear Regression, on the data.
Fit the model using the selected target and predictors.
Potential Problems in Treating This as a Supervised Problem:

Arbitrary Target Selection: The choice of the target can significantly impact the model’s interpretability and performance.
Overfitting: With 700 dimensions, there is a high risk of overfitting the model to noise.
Collinearity: Many features might be correlated, leading to unstable regression coefficients.
Loss of Context: Treating one column as a target might ignore the relationships among other features.
Bias in Extrapolation: Extrapolation is inherently risky in supervised learning, especially in high-dimensional spaces, as the learned relationships may not hold outside the observed range.
Suggestion: 
Start with unsupervised learning using PCA to understand the dominant patterns in the data.
If treating the problem as supervised:
Use domain knowledge to guide target selection.
Apply dimensionality reduction (e.g., PCA) to reduce predictors before modeling to avoid overfitting.
Evaluate the model carefully and understand its limitations for extrapolation.

Accepted Answer

while we can treat this problem as a supervised one by creating a synthetic target variable, we should exercise caution and carefully consider the implications and limitations of doing so. Additionally, using PCA for dimensionality reduction to find the line of best fit without a target can provide valuable insights without making assumptions about relationships between variables.

Accepted Answer

For the unsupervised approach, you could use Principal Component Analysis (PCA) to find the line that best fits the data. PCA is a technique that identifies the directions of maximum variance in the data and projects the data onto those directions. In this case, you could use PCA to project the data onto a lower-dimensional subspace that captures most of the variation in the data. The first principal component would represent the best line that fits the data. You could then extrapolate by projecting new data onto that line.

For the supervised approach, you could use a regression model to predict one of the dimensions based on the others. However, since there is no clear target column in the data, you would need to select one of the dimensions to be the target. You could try each dimension as the target and evaluate the performance of the regression model on a held-out set. The potential problem with treating this as a supervised problem is that there may not be a clear relationship between the columns, making it difficult to predict one column based on the others. Additionally, selecting an arbitrary target column could lead to biased or incorrect extrapolations.

Accepted Answer

We first perform eda and would find the dependent variable among 700 and also discard the redundant columns and then clean the data, after completing all this stuff, now we would do training the model, but as there not mention about target variable then we should findout , is it classfication or regression porblem after checking the type of problem then we train the model on the testing dataset and validate it on validation dataset , as there are lot more features so overfitting problem most probably arise so we should also apply regularization to overcome on this overfitting problem . After training the model we would test the model on test dateset and check accuracy ,if accuracy is not satisfying then apply another machine learning algorithm and which algo would give better accuracy then we would accept that model and check its accuracy on unseen data which will be not part of the given dataset.