What does this mean?:
Ridge penalizes the l2 norm of the coefficients, or the Euclidean length of w.
It is from the textbook : Introduction to Machine Learning with Python
Ridge regression penalizes the model coefficients in order to correct for overfitting. As you have probably learned, overfitting occurs when you have too many parameters in your model which causes it to memorize the data (which leads to poor generalization). In ridge regression you add a bias to the regression estimates in order to reduce the variance. This works by adding a penalty term to the loss function in order to decrease the contribution of each feature to the model outcome. I hope this helps!
Related
I'm trying to construct a multivariate likelihood function in c++ code with the aim of comparing multiple temperature simulations for consistency with observations but taking into account autocorrelation between the time steps. I am inexperienced in c++ and so have been struggling to understand how to write the equation in c++ form. I have the covariance matrix, the simulations I wish to judge and the observations to compare to. The equation is as follows:
f(x,μ,Σ) = (1/√(∣Σ∣(2π)^d))*exp(−1/2(x-μ)Σ^(-1)(x-μ)')
So I need to find the determinant and the inverse of the covariance matrix. Does anyone know how to do that in c++ if x,μ and Σ are all specified?
I have found a few examples and resources to follow
https://github.com/dirkschumacher/rcppglm
https://www.youtube.com/watch?v=y8Kq0xfFF3U&t=953s
https://www.codeproject.com/Articles/25335/An-Algorithm-for-Weighted-Linear-Regression
https://www.geeksforgeeks.org/regression-analysis-and-the-best-fitting-line-using-c/
https://cppsecrets.com/users/489510710111510497118107979811497495464103109971051084699111109/C00-MLPACK-LinearRegression.php
https://stats.stackexchange.com/questions/146230/how-to-implement-glm-computationally-in-c-or-other-languages
I am doing machine learning with python (scikit-learn) using the same data but with different classifiers. When I use 500k of data, LR and SVM (linear kernel) take about the same time, SVM (with polynomial kernel) takes forever. But using 5 million data, it seems LR is faster than SVM (linear) by a lot, I wonder if this is what people normally find?
Faster is a bit of a weird question, in part because it is hard to compare apples to apples on this, and it depends on context. LR and SVM are very similar in the linear case. The TLDR for the linear case is that Logistic Regression and SVMs are both very fast and the speed difference shouldn't normally be too large, and both could be faster/slower in certain cases.
From a mathematical perspective, Logistic regression is strictly convex [its loss is also smoother] where SVMs are only convex, so that helps LR be "faster" from an optimization perspective, but that doesn't always translate to faster in terms of how long you wait.
Part of this is because, computationally, SVMs are simpler. Logistic Regression requires computing the exp function, which is a good bit more expensive than just the max function used in SVMs, but computing these doesn't make the majority of the work in most cases. SVMs also have hard zeros in the dual space, so a common optimization is to perform "shrinkage", where you assume (often correctly) that a data point's contribution to the solution won't change in the near future and stop visiting it / checking its optimality. The hard zero of the SVM loss and the C regularization term in the soft margin form allow for this, where LR has no hard zeros to exploit like that.
However, when you want something to be fast, you usually don't use an exact solver. In this case, the issues above mostly disappear, and both tend to learn just as quick as the other in this scenario.
In my own experience, I've found Dual Coordinate Descent based solvers to be the fastest for getting exact solutions to both, with Logistic Regression usually being faster in wall clock time than SVMs, but not always (and never by more than a 2x factor). However, if you try and compare different solver methods for LRs and SVMs you may get very different numbers on which is "faster", and those comparisons won't necessarily be fair. For example, the SMO solver for SVMs can be used in the linear case, but will be orders of magnitude slower because it is not exploiting the fact that you only care are Linear solutions.
I'm reading the article "Ridge Estimators in Logistic Regression" by le Cessie and van Houwelingen, which is cited in Weka's documentation on the logistic regression function. I have to say, my maths are shaky in this area (it's been a while). In particular, I'm trying to work out the logic behind how the ridge parameter works, and what it's main purpose is.
The authors say that ridge estimators improve parameter estimates and reduce error in future predictions (this is in the abstract). I'm not exactly sure what an "ill posed problem" is, but as I understand it, the ridge estimator is meant to be a method of regularisation for this type of problem.
What do different values of the ridge parameter in Weka's Logistic regression do to change the performance of the logistic regression?
Does the ridge paramter involve the computation a Tikhonov Matrix that favours minimum residuals?
I'm sorry if I combined too many questions into one post. I think I understand what the ridge parameter is meant to do, but not how it does it.
I am doing a little research on text mining and data mining. I need more help in understanding cosine similarity. I have read about it and notice that all of the given examples on the internet is using tf-idf before computing it through cosine-similarity.
My question
Is it possible to calculate cosine similarity just by using highest frequency distribution from a text file which will be the dataset. Most of the videos and tutorials that i go through has tf-idf ran before inputting it's data into cosine similarity, if no, what other types of equation/algorithm that can be input into cosine similarity?
2.Why is normalization used with tf-idf to compute cosine similarity? (can i do it without normalization?) Cosine similarity are computed from normalization of tf-idf output. Why is normalization needed?
3.What cosine similarity actually does to the weights of tf-idf?
I do not understand question 1.
TF-IDF weighting is a weighting scheme that worked well for lots of people on real data (think Lucene search). But the theoretical foundations of it are a bit weak. In particular, everybody seems to be using a slightly different version of it... and yes, it is weights + cosine similarity. In practise, you may want to try e.g. Okapi BM25 weighting instead, though.
I do not undestand this question either. Angular similarity is beneficial because the length of the text has less influence than with other distances. Furthermore, sparsity can be nicely exploitet. As for the weights, IDF is a heuristic with only loose statistical arguments: frequent words are more likely to occur at random, and thus should have less weight.
Maybe you can try to rephrase your questions so I can fully understand them. Also search for related questions such as these: Cosine similarity and tf-idf and
Better text documents clustering than tf/idf and cosine similarity?
Im trying to understand the difference between each of these.
What is the difference between empirical naive bayes classifiers and parametric bayes classifiers?
The emperical part means that the distribution is estimated from the data, rather than being fixed before analysis begins
Empirical Bayes methods are procedures for statistical inference in which the prior distribution is estimated from the data. This approach stands in contrast to standard Bayesian methods, for which the prior distribution is fixed before any data are observed. Despite this difference in perspective, empirical Bayes may be viewed as an approximation to a fully Bayesian treatment of a hierarchical model wherein the parameters at the highest level of the hierarchy are set to their most likely values, instead of being integrated out. Empirical Bayes, also known as maximum marginal likelihood,[1] represents one approach for setting hyperparameters.
http://en.wikipedia.org/wiki/Empirical_Bayes_method
Naive means that the value of features being analyzed are independent of each other
A naive Bayes classifier is a simple probabilistic classifier based on applying Bayes' theorem with strong (naive) independence assumptions. A more descriptive term for the underlying probability model would be "independent feature model".
http://en.wikipedia.org/wiki/Naive_Bayes_classifier