Can anyone please explain the difference between generalized linear model and logistic regression model table with statsmodels. Why do I get different results with both the models while performing logistic regression??
GLM is a generalized linear model and Logit Model is specific to models with binary classification. While using GLM model you have to mention the parameter family which can be binomial (logit model), Poisson etc. This parameter is not required in Logit model as its only for binary output. The difference in the output can be because of regularization parameters.
GLM with Binomial family and Logit link and the discrete Logit model represent the same underlying model and both fit by maximum likelihood estimation
Implementation, default optimization method, options for extensions and the availability of some results statistics differs between GLM and their discrete counterparts.
For regular, well defined cases and well behaved data, both model produce the same results, up to convergence tolerance in the optimization and numerical noise.
However, the behavior of the estimation methods can differ in corner cases and for singular or near-singular data.
Related aside:
For models with non-canonical links using GLM there can be differences in the definitions used across optimization methods, e.g. whether standard errors are based on expected or observed information matrix. With canonical links like logit those two are the same.
Related
There are lots of posts here about the "Perfect Separation Error" in statsmodels when running a logisitc regression. But I'm not doing logistic regression. I'm doing GLM with frequency weights and gaussian distribution. So basically OLS.
All of my independent variables are categorical with lots of categories. So high dimensional binary coded feature set.
But I'm very frequently getting the "perfectseperationerror" from statsmodels
I'm running many many models. I think I'm getting this error when my data is too thin for that many variables. However, With freq weights, in theory, I actually have many more features then the dataframe holds because the observations should be multiplied by the freq.
Any guidance on how to proceed?
reg = sm.GLM(dep, Indies, freq_weights = freq)
<p>Error: class 'statsmodels.tools.sm_exceptions.PerfectSeparationError'>
The check is on perfect prediction and is used independently of the family.
Currently, there is now workaround when using irls. Using scipy optimizers, e.g. method="bfgs", avoids the perfect prediction/separation check.
https://github.com/statsmodels/statsmodels/issues/2680
Perfect separation is only defined for the binary case, i.e. family binomial in GLM, and could be extended to other discrete models.
However, there can be other problems with inference if the residual variance is zero, i.e. we have a perfect fit.
Here is an issue with perfect prediction in OLS
https://github.com/statsmodels/statsmodels/issues/1459
I'm using OneVsRestClassifier on a multiclass problem with svm.SVC as the base estimator.
The argmax from the predict_proba() does not match the predicted class:
Is there some normalization going on in the background? How do I get predict_proba() and predict()
to match?
According to the scikit learn's SVC documentation on multi-class classification, there can be discrepancies between the output of predict and the argmax of predict_proba (emphasis mine):
The decision_function method of SVC and NuSVC gives per-class scores for each sample (or a single score per sample in the binary case). When the constructor option probability is set to True, class membership probability estimates (from the methods predict_proba and predict_log_proba) are enabled. In the binary case, the probabilities are calibrated using Platt scaling: logistic regression on the SVM’s scores, fit by an additional cross-validation on the training data. In the multiclass case, this is extended as per Wu et al. (2004).
Needless to say, the cross-validation involved in Platt scaling is an expensive operation for large datasets. In addition, the probability estimates may be inconsistent with the scores, in the sense that the “argmax” of the scores may not be the argmax of the probabilities. (E.g., in binary classification, a sample may be labeled by predict as belonging to a class that has probability <½ according to predict_proba.) Platt’s method is also known to have theoretical issues. If confidence scores are required, but these do not have to be probabilities, then it is advisable to set probability=False and use decision_function instead of predict_proba.
You cannot get them to match using a SVC. You can try another model if you need the probabilities. If you do not need probabilities, as stated in the documentation, you can use decision_function (see here for more details.)
scikit-learn has two logistic regression functions:
sklearn.linear_model.LogisticRegression
sklearn.linear_model.LogisticRegressionCV
I'm just curious what the CV stands for in the second one. The only acronym I know in ML that matches "CV" is cross-validation, but I'm guessing that's not it, since that would be achieved in scikit-learn with a wrapper function, not as part of the logistic regression function itself (I think).
You are right in guessing that the latter allows the user to perform cross validation. The user can pass the number of folds as an argument cv of the function to perform k-fold cross-validation (default is 10 folds with StratifiedKFold).
I would recommend reading the documentation for the functions LogisticRegression and LogisticRegressionCV
Yes, it's cross-validation. Excerpt from the docs:
For the grid of Cs values (that are set by default to be ten values in a logarithmic scale between 1e-4 and 1e4), the best hyperparameter is selected by the cross-validator StratifiedKFold, but it can be changed using the cv parameter.
The point here is the following:
yes: sklearn has general model-selection wrappers providing CV-functionality for all those classifiers/regressors
but: when the classifier/regressor is known/fixed a-priori (to some extent) or sometimes even some CV-model, one can gain advantages using these facts with specialized code bound to one classifier/regressor resulting in improved performance!
Typically:
CV already embedded in optimization-algorithm
Efficient warm-starting (instead of full re-optimization after just the change of one parameter like alpha)
It seems, at least the latter idea is used in sklearn's LogisticRegressionCV, as seen in this excerpt:
In the case of newton-cg and lbfgs solvers, we warm start along the path i.e guess the initial coefficients of the present fit to be the coefficients got after convergence in the previous fit, so it is supposed to be faster for high-dimensional dense data.
May I also refer you to this section in scikit-learn documentation which I beleive explains it well:
Some models can fit data for a range of values of some parameter
almost as efficiently as fitting the estimator for a single value of
the parameter. This feature can be leveraged to perform a more
efficient cross-validation used for model selection of this parameter.
The most common parameter amenable to this strategy is the parameter
encoding the strength of the regularizer. In this case we say that we
compute the regularization path of the estimator.
And logistic regression is one such model. That's why scikit-learn has the dedicated LogisticRegressionCV class that does this.
There are some things left out on other answers, e.g. about gridsearch functionality. See the docs:
cross-validation estimator
An estimator that has built-in cross-validation capabilities to automatically select the best hyper-parameters (see the User Guide). Some example of cross-validation estimators are ElasticNetCV and LogisticRegressionCV. Cross-validation estimators are named EstimatorCV and tend to be roughly equivalent to GridSearchCV(Estimator(), ...). The advantage of using a cross-validation estimator over the canonical estimator class along with grid search is that they can take advantage of warm-starting by reusing precomputed results in the previous steps of the cross-validation process. This generally leads to speed improvements. An exception is the RidgeCV class, which can instead perform efficient Leave-One-Out CV.
https://scikit-learn.org/stable/glossary.html#term-cross-validation-estimator
https://github.com/amueller/talks_odt/blob/master/2015/nyc-open-data-2015-andvanced-sklearn.pdf
I would like to fit a regression model to probabilities. I am aware that linear regression is often used for this purpose, but I have several probabilities at or near 0.0 and 1.0 and would like to fit a regression model where the output is constrained to lie between 0.0 and 1.0. I want to be able to specify a regularization norm and strength for the model and ideally do this in python (but an R implementation would be helpful as well). All the logistic regression packages I've found seem to be only suited for classification whereas this is a regression problem (albeit one where I want to use the logit link function). I use scikits-learn for my classification and regression needs so if this regression model can be implemented in scikits-learn, that would be fantastic (it seemed to me that this is not possible), but I'd be happy about any solution in python and/or R.
The question has two issues, penalized estimation and fractional or proportions data as dependent variable. I worked on each separately but never tried the combination.
Penalization
Statsmodels has had L1 regularized Logit and other discrete models like Poisson for some time. In recent months there has been a lot of effort to support more penalization but it is not in statsmodels yet. Elastic net for linear and Generalized Linear Model (GLM) is in a pull request and will be merged soon. More penalized GLM like L2 penalization for GAM and splines or SCAD penalization will follow over the next months based on pull requests that still need work.
Two examples for the current L1 fit_regularized for Logit are here
Difference in SGD classifier results and statsmodels results for logistic with l1 and https://github.com/statsmodels/statsmodels/blob/master/statsmodels/examples/l1_demo/short_demo.py
Note, the penalization weight alpha can be a vector with zeros for coefficients like the constant if they should not be penalized.
http://www.statsmodels.org/dev/generated/statsmodels.discrete.discrete_model.Logit.fit_regularized.html
Fractional models
Binary and binomial models in statsmodels do not impose that the dependent variable is binary and work as long as the dependent variable is in the [0,1] interval.
Fractions or proportions can be estimated with Logit as Quasi-maximum likelihood estimator. The estimates are consistent if the mean function, logistic, cumulative normal or similar link function, is correctly specified but we should use robust sandwich covariance for proper inference. Robust standard errors can be obtained in statsmodels through a fit keyword cov_type='HC0'.
Best documentation is for Stata http://www.stata.com/manuals14/rfracreg.pdf and the references therein. I went through those references before Stata had fracreg, and it works correctly with at least Logit and Probit which were my test cases. (I don't find my scripts or test cases right now.)
The bad news for inference is that robust covariance matrices have not been added to fit_regularized, so the correct sandwich covariance is not directly available. The standard covariance matrix and standard errors of the parameter estimates are derived under the assumption that the model, i.e. the likelihood function, is correctly specified, which will not be the case if the data are fractions and not binary.
Besides using Quasi-Maximum Likelihood with binary models, it is also possible to use a likelihood that is defined for fractional data in (0, 1). A popular model is Beta regression, which is also waiting in a pull request for statsmodels and is expected to be merged within the next months.
How should one decide between using a linear regression model or non-linear regression model?
My goal is to predict Y.
In case of simple x and y dataset I could easily decide which regression model should be used by plotting a scatter plot.
In case of multi-variant like x1,x2,...,xn and y. How can I decide which regression model has to be used? That is, How will I decide about going with simple linear model or non linear models such as quadric, cubic etc.
Is there any technique or statistical approach or graphical plots to infer and decide which regression model has to be used? Please advise.
That is a pretty complex question.
You start visually first: if the data is normally distributed, and satisfy conditions for classical linear model, you use linear model. I normally start by making a scatter plot matrix to observe the relationships. If it is obvious that the relationship is non linear then you use non-linear model. But, a lot of times, I visually inspect, assuming that the number of factors are just not too many.
For example, this would be a non linear model:
However, if you want to use data mining (and computationally demanding methods), I suggest starting with stepwise regression. What you do is set a model evaluation criteria first: could be R^2 for example. You start a model with nothing and sequentially add predictors or permutations of them until your model evaluation criteria is "maximized". However, adding new predictor almost always increases R^2, a type of over-fitting.
The solution is to split the data into training and testing. You should make model based on the training and evaluate the mean error on testing. The best model will be the one that that minimized mean error on the testing set.
If your data is sparse, try integrating ridge or lasso regression in model evaluation.
Again, this is a kind of a complex question. The answer also kind of depends on whether you are building descriptive or explanatory model.