Full cross-validation and generating learning curves for time-series models

Kindly reposted to KDnuggets by Gregory Piatetsky-Shapiro

Preamble

Time-series analysis is needed almost in any quantitative field and real-life systems that collects data over time, i.e., temporal datasets. Building predictive models on temporal datasets for future evolution of systems in consideration are usually called forecasting. The validation of such models deviates from the standard holdout method of having random disjoint splits of train, test and validation sets used in supervised learning. This stems from the fact that time-series are ordered and order induces all sorts of statistical properties that should be retained. For this reason, applying direct cross-validation to time-series model building is not possible and only restricted to out-of-sample (OOS) validation, using the end-tail of a temporal set as a single test set. A recent work proposed an approach that overcomes the known limitation achieving full cross-validation for time-series. The approach opens up for a possibility to produce learning curves for the time-series models as well, which is usually also not possible due to similar reasons.

Reconstructive Cross-validation (rCV) : A meta-algorithm design principles

rCV is proposed recently in the paper titled Generalised learning of time-series: Ornstein-Uhlenbeck processes. The design principles of rCV for time-series aims at the following principles:

   Figure 1 : rCV meta-algorithm for time-series
cross-validation and learning curves.

1. Logically close to standard cross-validation: Arbitrary test-set size and number of folds. 
2. Preserve correlations and data order. 
3. Does not create absurdity of predicting past from the future data. 
4. Applicable in generic fashion regardless of learning algorithm. 
5. Applicable to multi-dimensional time-series. 
6. Evaluation metric agnostic.

Idea of introducing missing-data  : Temporal cross-validation and learning curves

The key idea of rCV is to create cross-validation sets via creating missing-data sets K-times, as in K-fold, with a given degree of missing ratio, i.e., random data point removal. Each fold will have disjoint set of missing data points. By an imputation method, we would fill out the K-disjoint missing data sets and generate K-different training datasets.  This would allow us to have K-different models and we could measure the generalised performance of the modelling approach by testing the primary models prediction on the Out-of-sample (OOS) test set. To avoid confusion about what is a model?, what we are trying to achieve is to find out hypothesis, i.e., the modelling approach.  By changing the ratio of missing data and repeating the cross-validation exercise will yield to set of ratio of missing-missing data introduced and their corresponding rCV errors, the plot is nothing but a learning-curve from supervised learning perspective.  Note that the imputation and prediction models are different models. The primary model we are trying to build is the prediction model we used for producing OOS predictions. The procedure is summarised in Figure 1. 

    Figure 2 : Synthetic data and reconstructions. 

Show-case with Gaussian process models on Ornstein-Uhlenbeck processes

To demonstrate the utility of rCV, the mentioned paper uses a synthetic data generated by Ornstein-Uhlenbeck process, i.e., Gaussian process with certain parameter setting.  Figure 2, shows the synthetic data and example locations of generated missing-data sets's reconstruction errors. Figure 3 shows learning curves depending on the different ratios of missing data setting. 

    Figure 3: Learning curves for the Gaussian Process model
generated by rCV.

Conclusion

rCV provides logically consistent way of practicing cross-validation in time-series. It is usually not possible to produce learning-curves on the same time-window for time-series model: by using rCV with different ratio missing data achieves this as well. rCV paves way to do generalised learning for time-series.

Further Reading

Apart from the paper Generalised learning of time-series: Ornstein-Uhlenbeck processes. the results can be reproduced with the Python prototype implementation,  here.

Gems of data science: 1, 2, infinity

 Summary

Problem-solving is the core activity of data science using scientific principles and evidence. On our side, there is an irresistible urge to solve the most generic form of the problem. We do this almost always from programming to formulation of the problem. But, don't try to solve a generalised version of the problem. Solve it for N=1 if N is 1 in your setting, not for any integer: Save time and resources and try to embed this culture to your teams and management. Extent later when needed on demand.

Solving for N=1 is sufficient if it is the setting

This generalisation phenomenon manifests itself as an algorithmic design: From programming to problem formulation, strategy and policy setting. The core idea can be expressed as mapping, let's say the solution to a problem  is a function, mapping from one domain to a range 

$$ f : \mathbb{R} \to \mathbb{R} $$

Trying to solve for the most generic setting of the problem, namely multivariate setting

$$ f : \mathbb{R}^{m} \to \mathbb{R}^{n} $$

where $m, n$ are the integers generalising the problem.  

Conclusion

It is elegant to solve a generic version of a problem. But is it really needed? Does it reflect reality and would be used? If N=1 is sufficient, then try to implement that solution first before generalising the problem. An exception to this basic pattern would be if you don't have a solution at N=1 but once you move larger N that there is a solution: you might think this is absurd, but SVM works exactly in this setting by solving classification problem for disconnected regions.

Postscripts

• The title intentionally omits three, while it is a reference to Physics's inability to solve, or rather a mathematical issue of the three-body problem.

Conjugacy and Equivalence for Deep Neural Networks: Architecture compression to selection

Preamble

A recently shown phenomenon can classify deep learning architectures with only using the knowledge gained by trained weights [suezen20a]. The classification produces a measure of equivalence between two trained neural network and astonishingly captures a family of closely related architectures as equivalent within a given accuracy. In this post, we will look into this from a conceptual perspective. 

Figure 1: VGG architecture spectral difference in the long
positive tail [suezen20a]. 

The concept of conjugate matrix ensembles and equivalence

Conjugacy is a mathematical construct reflecting different approaches to the same system should yield to the same outcome: It is reflected in the statistical mechanic's concept of ensembles. However, for matrix ensembles, like the ones offered in Random Matrix Theory, the conjugacy is not well defined in the literature. One possible resolution is to look at the cumulative spectral difference between two ensembles in the long positive tail part of the spectrum [suezen20a]. If this is vanishing we can say that two matrix ensembles are conjugate to each other. We observe this with matrix ensembles VGG vs. circular ensembles. 

 Conjugacy is the first step in building equivalence among different architectures.  If two architectures are conjugate to the same third matrix ensemble and their fluctuations on the spectral difference are very close over the spectral locations, they are equivalant in a given accuracy [suezen20a].


Outlook: Where to use equivalence in practice?

The equivalence can be used in selecting or compressing an architecture or classify different neural network architectures. Python notebook to demonstrate this with different vision architecture in PyTorch is provided, here.

Reference

[suezen20a] Equivalence in Deep Neural Networks via Conjugate Matrix Ensembles, Mehmet Suezen, arXiv:2006.13687 (2020)

Book review: A tutorial introduction to the mathematics of deep learning

Preamble

Artificial Intelligence Engines:
An introduction to the Mathematics
of Deep Learning by Dr James V. Stone
the book and Github repository.
(c) 2019 Sebtel Press

Deep learning and associated connectionist approaches are now applied routinely in industry and academic research from image analysis to natural language processing and areas as cool as reinforcement learning. As practitioners, we use these techniques and utilise them from well designed and tested reliable libraries like Tensorflow or Pytorch as shipped black-boxed algorithms. However, most practitioners lack mathematical foundational knowledge and core algorithmic understanding. Unfortunately, many academic books and papers try to make an impression of superiority show subliminally and avoid a simple pedagogical approach. In this post we review, a unique book trying to fill this gap with a pedagogical approach to the mathematics of deep learning avoiding showing of mathematical complexity but aiming at conveying the understanding of how things work from the ground up. Moreover, the book provides pseudo-codes that one can be used to implement things from scratch along with a supporting implementation in Github repo. Author Dr James V. Stone, a trained cognitive scientist and researcher in mathematical neuroscience provides such approaches with other books many years now, writing for students, not for his peers to show off. One important note that this is not a cookbook or practice tutorial but an upper-intermediate level academic book.

Building associations and classify with a network

The logical and conceptual separation of associations and classification tasks are introduced in the initial chapters. It is ideal to start with from learning one association with one connection to many via gentle introduction to Gradient descent in learning the weights before going to 2 associations and 2 connections. This reminds me of George Gamow's term 1, 2 and infinity as a pedagogical principle. Perceptron is introduced later on how classification rules can be generated via a network and the problems it encounters with XOR problem.

Backpropagation, Hopfield networks and Boltzmann machines

Detail implementation of backpropagation is provided from scratch without too many cluttering index notation in such clarity. Probably this is the best explanation I have ever encountered. Following chapters introduced Hopfield networks and Boltzmann machines from the ground up to applied level. Unfortunately, many modern deep learning books skip these two great models but Dr Stone makes these two models implementable for a practitioner by reading his chapters.  It is very impressive. Even though I am a bit biased in Hopfield networks as I see them as an extension to Ising models and its stochastic counterparts, but I have not seen anywhere else such explanations on how to use Hopfield networks in learning and in a pseudo-code algorithm to use in a real task.

Advanced topics

Personally, I see the remaining chapters as advanced topics: Deep Boltzmann machines, variational encoders, GANs and introduction to reinforcement learning. Probably exception of deep backpropagation in Chapter 9. I would say what is now known as deep learning now was the inception of the architectures mentioned in sections 9.1 till 9.7.

Glossary, basic linear algebra and statistics

Appendices provide a fantastic conceptual introduction to jargon and basics to main mathematical techniques. Of course, this isn't a replacement to fully-fledged linear algebra and statistics book but it provides immediate concise explanations.

Not a cookbook: Not import tensorflow as tf book

One other crucial advantage of this book is that it is definitely not a cookbook. Unfortunately, almost all books related to deep learning are written in a cookbook style. This book is not. However, it is supplemented by full implementation in a repository supporting each chapter, URL here.

Conclusion

This little book archives so much with down to earth approach with introducing basic concepts with a respectful attitude, assuming the reader is very smart but inexperience in the field. If you are a beginner or even experienced research scientist this is a must-have book.  I still see this book as an academic book and can be used in upper-undergraduate class as the main book in an elective such as
"Mathematics of Deep Learning".

Enjoy reading and learning from this book. Thank you, Dr Stone, for your efforts on making academic books more accessible.

Disclosure: I received a review copy of the book but I have bought another copy for a family member. 

A simple and interpretable performance measure for a binary classifier

Kindly reposted to KDnuggets by Gregory Piatetsky-Shapiro 

Preamble

The core application of machine learning models is a binary classification task. This appears in polyhedra of areas from medicine for diagnostic tests to credit risk decision making for consumers.  Techniques in building classifiers vary from simple decision trees to logistic regression and lately super cool deep learning models that leverage multilayered neural networks. However, they are mathematically different in construction and training methodology, when it comes to their performance measure, things get tricky. In this post, we propose a simple and interpretable performance measure for a binary classifier in practice. Some background in classification is assumed. 

Why ROC-AUC is not intepretable? 

PCC in Python (Author). 

De-facto standard in reporting classifier performance is to use Receiver Operating Characteristic (ROC) - Area Under Curve (AUC) measure. It originates from the 1940s during the development of Radar by the US Navy, in measuring the performance of detection.  There are at least 5 different definitions of what does ROC-AUC means and even if you have a PhD in Machine Learning, people have an excessively difficult time to explain what does AUC means as a performance measure. As AUC functionality is available in almost all libraries and it becomes almost like a religious ritual to report in Machine Learning papers as a classification performance. However, its interpretation is not easy, apart from its absurd comparison issues, see hmeasure.  

AUC measures the area under the True Positive Rate (TPR) curve as a function of the False Positive Rate (FPR) that are extracted from confusion matrices with different thresholds, $f(x)=y$ and  $\int_{0}^{1} f(x) dx = AUC$ whereby, y is TPR and x is FPR. Apart from a multitude of interpretations and easy to have confusions, there is no clear purpose of taking the integral over FPR. Obviously, we would like to have perfect classification by having FPR zero, but the area is not mathematically clear. Meaning that what is it as a mathematical object is not clear.

Probability of correct classification (PCC)

A simple and interpretable performance measure for a binary classifier would be great for both highly technical data scientist and non-technical stakeholders. The basic tenant in this direction is that the purpose of a classifier technology is the ability to differentiate two classes. This boils down to a probability value, Probability of correct classification (PCC). An obvious choice is so-called balanced accuracy (BA). This is usually recommended for unbalanced problems, even by SAS; though they used multiplication of probabilities. Here we will call BA as PCC and use addition instead, due to statistical dependence:
$PCC = \alpha TPR+ (1.0-\alpha TNR$

$TPR=TP/(ConditionPositive)=TP/(TP+FN)$
$TNR=TN/(ConditionNegative)=TN/(TN+FP)$
PCC tells us how good the classifier in detecting either of the class and it is a probability value, $[0,1]$. Note that, using total accuracy over both positive and negative cases are misleading, even if our training data is balanced in production, batches we measure the performance may not be balanced so accuracy alone is not a good measure. 

Attentive readers might note that this is about predictive values, Positive and Negative Predictive values, PPV and NPV match with the TPR and TNR respectively.

Production issues
Immediate question would be, how to choose the threshold in generating confusion matrix? One option would be to chose a threshold that maximizes PCC for production on the test set. To improve the estimation of PCC, resampling on the test set can be performed to get a good uncertainty.

Conclusion

We try to circumvent in reporting AUCs by introducing PCC, or balanced accuracy as a simple and interpretable performance measure for a binary classifier. This is easy to explain to a non-technical audience. An improved PCC, that takes into account better estimation properties can be introduced, but the main interpretation remains the same as probability of correct classification.  

Explore the notebook : pcc.ipynb

Please cite as follows:

 @misc{suezen20pcc, 

     title = {A simple and interpretable performance measure for a binary classifier}, 

     howpublished = {\url{https://memosisland.blogspot.com/2020/02/a-simple-and-interpretable-performance.html}, 

     author = {Mehmet Süzen},

     year = {2020}

}  

Bringing back Occam's razor to modern connectionist machine learning: A simple complexity measure based on statistical physics Cascading Periodic Spectral Ergodicity (cPSE)

Kindly reposted to KDnuggets by Gregory Piatetsky-Shapiro with the title Applying Occam's razor to Deep Learning 

Kindly reviewed by Cornelius Weber

Preamble: Changing concepts in machine learning due to deep learning

Occam's razor or principle of parsimony has been the guiding principle in statistical model selection. In comparing two models, which they provide similar predictions or description of reality, we would vouch for the one which is less complex. This boils down to the problem of how to measure the complexity of a statistical model and model selection. What constitutes a model, as discussed by McCullagh (2002) in statistical models context is a different discussion, but here we assume a machine learning algorithms are considered as a statistical model.  Classically, the complexity of statistical models usually measured with Akaike information criterion (AIC) or similar. Using a complexity measure, one would choose a less complex model to use in practice, other things fixed.

Figure 1: Arbitrary architecture, each node represents
a layer for a given deep neural network, such as
convolutions or set of units.  Süzen-Cerdà-Weber (2019)

The surge in interest in using complex neural network architectures, i.e., deep learning due to their unprecedented success in certain tasks,  pushes the boundaries of "standard" statistical concepts such as overfitting/overtraining and regularisation. 

Now Overfitting/overtraining is often used as an umbrella term to describe any unwanted performance drop off a machine learning model Roelofs et. al. (2019) and nearly anything that improves generalization is called regularization, Martin and Mahoney (2019).

A complexity measure for connectionist machine learning

Deep learning practitioners rely on choosing the best performing model and do not practice Occam's razor.  The advent of Neural Architecture Search and new complexity measures  

that take the structure of the network into account gives rise the possibility of practising Occam's razor in deep learning. Here, we would cover one of the very practical and simple measures called cPSE, i.e., cascading periodic spectral ergodicity. This measure takes into account the depth of the neural network and computes fluctuations of the weight structure over the entire network,  Süzen-Cerdà-Weber (2019) Figure 1. It is shown that the measure is correlated with the generalisation performance almost perfectly, see Figure 2.

Practical usage of cPSE

Figure 2: Evolution of PSE, periodic spectral ergodicity,
it is shown that complexity measure cPSE saturates
after a certain depth,  Süzen-Cerdà-Weber (2019)

The cPSE measure is implemented in Bristol python package, starting from version 0.2.6. If a trained network wrapped into a PyTorch model object, cPSE can be used to compare two different architectures. If two architectures give similar test performance, we would select the one with higher cPSE value. Lower the cPSE, more complex the model.

An example of usage requires a couple of lines, example measurements for VGG and ResNet are given in Süzen-Cerdà-Weber (2019):

from bristol import cPSE
import torchvision.models as models
netname = 'vgg11'
pmodel = getattr(models, netname)(pretrained=True)
(d_layers, cpse) = cPSE.cpse_measure(pmodel)

Conclusion and take-home message

Using a less complex deep neural network that would give similar performance is not practised by the deep learning community due to the complexity of training and designing new architectures. However, quantifying the complexity of similarly performing neural network architecture would bring the advantage of using less computing power to train and deploy such less complex models into production. Bringing back the Occam's razor to modern connectionist machine learning is not only a theoretical and philosophical satisfaction but the practical advantages for environment and computing time is immense.

Postscript : Appendix : Vanilla computation of cPSE

Recipe is quite easy to implement in practice: Get all weight matrices of your trained architecture as list of 2D matrices, special layers and biases subject to mapping, then:

from bristol import cPSE

spectrum_set = cPSE.get_eigenvals_layer_matrix_set(layer_matrices)

periodic_spectrum_set = cPSE.eigenvals_set_to_periodic(spectrum_set)

spectral_ergodicity_over_layers = cPSE.d_layers_pse(periodic_spectrum_set)

One should stop adding more layers where spectral_ergodicity_over_layers saturates to a constant value, i.e., decreasing curve.