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statistical learning | Journals in statistical learning / machine learning | https://stats.stackexchange.com/questions/88330/journals-in-statistical-learning-machine-learning | <p>Can you please name some major and minor journals publishing articles in the field of statistical learning / machine learning. </p>
| <p>Some influential journals in machine learning:</p>
<ul>
<li><a href="http://www.computer.org/portal/web/tpami" rel="nofollow">IEEE TPAMI</a></li>
<li><a href="http://jmlr.org/" rel="nofollow">Journal of Machine Learning Research</a></li>
<li><a href="http://www.mitpressjournals.org/loi/neco" rel="nofollow">Neural C... | 400 |
statistical learning | Machine Learning VS Statistical Learning vs Statistics | https://stats.stackexchange.com/questions/442128/machine-learning-vs-statistical-learning-vs-statistics | <p>I have seen posts about the difference between ML and Statistics. And I have also seen posts explaining that Statistical Learning is a statistical approach to ML. But then, this is confusing because what is the difference between Statistics and Statistical Learning anyways?</p>
<p>To finally resolve this confusion,... | <p>Statistics is a mathematical science that studies the collection, analysis, interpretation, and presentation of data.</p>
<p>Statistical/Machine Learning is the application of statistical methods (<em>mostly</em> <a href="https://en.wikipedia.org/wiki/Regression_analysis" rel="noreferrer">regression</a>) to make pr... | 401 |
statistical learning | Easier than Element of statistical Learning and harder than Introduction to statistical learning | https://stats.stackexchange.com/questions/392814/easier-than-element-of-statistical-learning-and-harder-than-introduction-to-stat | <p>I'm majoring industrial engineering on a master's course. Recently, I've realized that I need to study statistical perspective on M.L.</p>
<p>So I'm studying the book Introduction to statistical learning with myself. But it seems to lack of mathematical background. On the other hand, Element of statistical learning... | <p>I like <em>Learning From Data</em> by Abu-Mustafa, et al., which should be enjoyed with the You-Tubed lecture series. It accurately describes itself as a short course, but not a hurried course. Neural nets get half of one lecture, which makes perfect sense when you get there. </p>
<p>ESL is a long book and a hurrie... | 402 |
statistical learning | Statistical Learning book with theoretical content | https://stats.stackexchange.com/questions/468321/statistical-learning-book-with-theoretical-content | <p>I'm currently reading the book 'An Introduction to Statistical Learning with application in R(ISLR)', it is very helpful for learning the applications of statistical model, but less complement of theoreotical content or mathematical proof/derivation of formulas. I'm often confused with some conclusion/formulas provi... | <p>You can try elements of statistical learning. It is freely available <a href="https://web.stanford.edu/~hastie/Papers/ESLII.pdf" rel="nofollow noreferrer">here</a>. It has quite a bit of theoretical content, but not many proofs. Most of the derivations and/or proofs are published in separate papers by Hastie and Tib... | 403 |
statistical learning | What is the difference between Statistical Learning and Machine Learning? | https://stats.stackexchange.com/questions/617771/what-is-the-difference-between-statistical-learning-and-machine-learning | <p><a href="https://hastie.su.domains/ISLR2/ISLRv2_website.pdf" rel="nofollow noreferrer">An Introduction to Statistical Learning
with Applications in R</a> 2nd edition by Hastie et al. says that</p>
<blockquote>
<p><em>Statistical learning</em> refers to a set of tools for making sense of complex datasets.</p>
</block... | <p>I think any answers to this question will be verging on opinion-based, but I would say there is a gradient from</p>
<ul>
<li><em>theoretical</em> or <em>pure</em> statistics, focused on rigorous proofs of the properties of various statistical procedures or tests;</li>
<li><em>applied</em> statistics, more interested... | 404 |
statistical learning | Statistical learning theory VS computational learning theory? | https://stats.stackexchange.com/questions/63077/statistical-learning-theory-vs-computational-learning-theory | <p>What relations and differences are between <a href="http://en.wikipedia.org/wiki/Statistical_learning_theory" rel="nofollow noreferrer">statistical learning theory</a> and <a href="http://en.wikipedia.org/wiki/Computational_learning_theory" rel="nofollow noreferrer">computational learning theory</a>?</p>
<p>Are the... | <p>Computational learning, more concretely the probably approximately correct (<a href="http://www.cs.iastate.edu/~honavar/pac.pdf" rel="noreferrer">PAC</a>) framework, answers questions like: how many training examples are needed for a learner to learn with high probability a good hypothesis? how much computational ef... | 405 |
statistical learning | Elements of Statistical Learning alternatives | https://stats.stackexchange.com/questions/154788/elements-of-statistical-learning-alternatives | <p>Elements of Statistical Learning (ESL) is a book that has fantastic breadth and depth. It covers the essentials to the very modern methods by citing the papers where these original studies come about. However, I really find the language of the book very very prohibitive. I believe there is an easier way to discuss c... | <p>I agree that <em>An Intro to Statistical Learning</em> has a very accommodating tone. You may want to look at <em>Learning From Data, A Short Course</em> by Yaser Abu-Mostafa et al. I found this book and the accompanying youtube videos to be great. </p>
<p>Lastly, spdrnl's comment about <em>Applied Predictive Model... | 406 |
statistical learning | Introduction to Statistical Learning | https://stats.stackexchange.com/questions/438143/introduction-to-statistical-learning | <p>For those that have read the book <a href="http://faculty.marshall.usc.edu/gareth-james/ISL/ISLR%20Seventh%20Printing.pdf" rel="nofollow noreferrer">Introduction to Statistical Learning</a>, I'm having a problem understanding a certain line: </p>
<p><a href="https://i.sstatic.net/2eTms.png" rel="nofollow noreferrer... | 407 | |
statistical learning | A good literature for statistical learning theory? | https://stats.stackexchange.com/questions/441192/a-good-literature-for-statistical-learning-theory | <p>Any recommendations for good literature on statistical learning theory? I mean, something what goes into more details than Elements of Statistical Learning, in terms of losses, empirical error estimation etc.</p>
| 408 | |
statistical learning | Motivations for experiment design in statistical learning? | https://stats.stackexchange.com/questions/422186/motivations-for-experiment-design-in-statistical-learning | <p>My interests in statistics centre around statistical learning, including Bayesian inference, inference in combinatorial spaces, Monte Carlo methods, Markov decision processes, modeling stochastic processes and so on. It is mandatory at my university’s program that we take an experiment design course, which seems rat... | <p>This is an interesting question. The following <a href="https://www.google.no/search?q=experimental+design+in+machine+learning&oq=experimental+design+in+machine+learning&aqs=chrome..69i57.14387j0j7&sourceid=chrome&ie=UTF-8" rel="noreferrer">stored google search</a> gives many interesting hits, and bo... | 409 |
statistical learning | Solutions to 'Statistical Learning with Sparsity' | https://stats.stackexchange.com/questions/583137/solutions-to-statistical-learning-with-sparsity | <p>I've recently been working through <em>Statistical Learning with Sparsity</em> (SLS) by Hastie, Tibshirani and Hastie.</p>
<p>I found some exercises very hard, and think I found some mistakes. A set of solutions would be very helpful.</p>
<p>I've recently discovered that <em>Elements of Statistical Learning</em> now... | 410 | |
statistical learning | Statistical learning and expected value | https://stats.stackexchange.com/questions/521720/statistical-learning-and-expected-value | <p>I'm studying some statistical learning theory. If i have <span class="math-container">$X$</span>, <span class="math-container">$Y$</span> as random variables representing the data-labels samples drawn from a certain distribution and a loss function, it is right to say that:
<span class="math-container">$$ E[loss(Y, ... | <p>In statistical learning theory we pretty much always are considering the joint distribution of <span class="math-container">$(X,Y)$</span> and the expected value here is with respect to that distribution, not just the marginal distribution of <span class="math-container">$X$</span>. In general
<span class="math-cont... | 411 |
statistical learning | How are statistical decision theory and statistical learning theory related? | https://stats.stackexchange.com/questions/135923/how-are-statistical-decision-theory-and-statistical-learning-theory-related | <p><a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.115.9500&rep=rep1&type=pdf" rel="nofollow">This</a> paper attempts to contrast the basic elements of statistical learning theory and statistical decision theory, but I'm still confused about how the two are related.</p>
| <p>Have you read the Wikipedia articles? Decision theory is a subset of or problem in statistical learning, in my view; both driven by statistics -- data. It concerns the optimal making of decisions such as choosing between alternatives, once or over a period of time (possibly without termination), or deciding when to... | 412 |
statistical learning | Relation between Nonparametric Statistics and Statistical Learning Theory | https://stats.stackexchange.com/questions/258417/relation-between-nonparametric-statistics-and-statistical-learning-theory | <p>I used to hear some Statistics professor complaining about Machine Learning theories: "It is just Non-parametric Statistics". And, when I read Vapnik's book "Statistical Learning Theory", it seems he has been influenced a lot by non-parametric statistics. So, would anybody explain the similarity and difference betwe... | 413 | |
statistical learning | Is there a difference between the terms statistical learning and machine learning? | https://stats.stackexchange.com/questions/271027/is-there-a-difference-between-the-terms-statistical-learning-and-machine-learnin | <p>Quick question I guess, but is there a perceivable difference between the terms <em>Statistical Learning</em> and <em>Machine Learning</em>, or is it simply area jargon? I gather the computer scientists like to refer to machine learning while statisticians might refer to statistical learning (no less influenced by t... | 414 | |
statistical learning | Valid references on origins of Machine Learning, Statistical Learning and Data Mining | https://stats.stackexchange.com/questions/179021/valid-references-on-origins-of-machine-learning-statistical-learning-and-data-m | <p>I know it's a rather debated question on Stack Exchange communities but let me explain the points of this question.</p>
<p>I'm writing my capstone on Machine Learning and I need to clarify deeply, giving valid references, the differences among Data Mining, Big Data, Statistical Learning and Machine Learning. As far... | <p>The issues you're noting are definitional ones where standard, widely accepted meanings for each term have yet to be agreed upon -- different authors and practitioners use them differently. I think nearly everyone would agree that there is a high degree of overlap in their use. This is frequently the case during the... | 415 |
statistical learning | Statistical Learning Theory - Loss Function | https://stats.stackexchange.com/questions/247373/statistical-learning-theory-loss-function | <p>I am reading Vapnik's "Statistical Learning Theory" and I am confused about his use of Q(z,alpha). On page 23 he explains that Q is the loss function which takes as argument a function g, the function used by the learning algorithm to generate predictions:</p>
<p>Q(z,alpha) = L(z,g(z,alpha))</p>
<p>However, later ... | 416 | |
statistical learning | Books Similar to Introduction to Statistical learning | https://stats.stackexchange.com/questions/476270/books-similar-to-introduction-to-statistical-learning | <p>I'm looking for books similar to Introduction to Statistical Learning with Applications in R (ISLR), which is not too rigorous in terms of the mathematical treatment, but still able to provide you the intuition about the methods? I'm particularly looking at this topics:</p>
<ul>
<li>Generalized Linear Models</li>
<l... | <ul>
<li>For time series analysis: "<a href="https://otexts.com/fpp2/" rel="nofollow noreferrer">Forecasting Principles and Practices</a>" by Hyndman and Athanasopoulos is absolutely excellent and is roughly on the same order of mathematical complexity as ISLR (i.e. enough, but not too much). It has the addit... | 417 |
statistical learning | Elements of Statistical Learning -2.4 Statistical Decision Theory：how to prove formula (2.12) | https://stats.stackexchange.com/questions/618307/elements-of-statistical-learning-2-4-statistical-decision-theory-how-to-prove-f | <p>I have a question, how to prove formula (2.12) in book 《Elements of Statistical Learning》</p>
<p><a href="https://i.sstatic.net/XfiOD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XfiOD.png" alt="enter image description here" /></a></p>
| 418 | |
statistical learning | Question about notation in Introduction to Statistical Learning | https://stats.stackexchange.com/questions/135468/question-about-notation-in-introduction-to-statistical-learning | <p>I've been working my way through the problems in the book "Introduction to Statistical Learning". I have a question about the notation in Question 5 from Chapter 3 (screenshot below). What does $x_{i'}$ mean exactly, in comparison to $x_{i}$.</p>
<p><img src="https://i.sstatic.net/KnJdH.png" alt="enter image descri... | <p>First, it is useless to use a new index $i'$ in the expression $\hat\beta$ is useless: you can write
$$\hat\beta=\frac{\sum_{i=1}^nx_iy_i}{\sum_{i=1}^nx_i^2}.$$
But now, it is ambiguous to strictly replace $\hat\beta$ with this expression in $\hat y_i=x_i\hat\beta$:
$$
\hat y_i=x_i\frac{\sum_{i=1}^nx_iy_i}{\sum_{i=... | 419 |
statistical learning | Data Modelling/Statistical Learning - Interview Questions | https://stats.stackexchange.com/questions/197044/data-modelling-statistical-learning-interview-questions | <p>I was asked the following 2 questions in an interview. I wasn't selected which means my answers were wrong. </p>
<p>Now I need to learn from my mistakes. I have though quite a bit since the interview and still not got anywhere concrete so need your help..</p>
<p>Question 1 : It is raining during evening and you ca... | <p>For the <strong>second question</strong>, I don't think people want to know specifically which model that you going to use because it is normally decided after some cross-validation.
In this case, salary is a good feature to be included but there might also be: years of experience, department, age, degree, gender, m... | 420 |
statistical learning | Elements of Statistical Learning training set | https://stats.stackexchange.com/questions/372525/elements-of-statistical-learning-training-set | <p>I am trying to read the Elements of Statistical Learning Tibshirani, Hastie and Friedman, however I have a problem with understanding the expected (squared) prediction error (<span class="math-container">$EPE$</span>) formula that they provide on page <span class="math-container">$26$</span>:</p>
<p>The start they ... | <p><span class="math-container">$E_T$</span> is the expectation taken over the training set. <span class="math-container">$\hat y_0$</span> is a deterministic function of <span class="math-container">$x_0,$</span> i.e., <span class="math-container">$\hat y_0 = \hat \beta^Tx_0.$</span> The
training set consists of rows ... | 421 |
statistical learning | Elements of Statistical Learning - Statistical Decision Theory : formula 2.10 EPE | https://stats.stackexchange.com/questions/618301/elements-of-statistical-learning-statistical-decision-theory-formula-2-10-ep | <p>Recently, I have been reading the 《Elements of Statistical Learning》book . Now,I have three question in chapter 2 formula(2.10).
(1)What does <span class="math-container">$pr(dx,dy)$</span> in Formula 2.10 mean?</p>
<p>(2)how to derivate this (2.10)formula?</p>
<p>(3)Can <span class="math-container">$pr(dx,dy)$</spa... | 422 | |
statistical learning | A Fundamental Question about Statistical Learning | https://stats.stackexchange.com/questions/189583/a-fundamental-question-about-statistical-learning | <p>In statistical learning (many textbooks), we assume that the data $Y$ is generated by $Y=f(X)+\epsilon$, where $X$ are predictors and $\epsilon$ is some random noise. Then the problem becomes: using various methods to find an estimates of $f$, i.e., $\hat{f}$ such that the expected mean square error on the test set ... | <p>There are several issues here: the space your friend shall minimize in is uncountably. Hence, if he is not really lucky and guesses the correct function, there is no chance in minimizing the expression. Thus, one chooses a certain class in that one seeks to minimize. However, typically the class of function is biase... | 423 |
statistical learning | AIC formula in Introduction to Statistical Learning | https://stats.stackexchange.com/questions/181539/aic-formula-in-introduction-to-statistical-learning | <p>I'm a little puzzled by a formula presented in Hastie's "Introduction to Statistical Learning". In Chapter 6, page 212 (sixth printing, available <a href="http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Sixth%20Printing.pdf">here</a>), it is stated that:</p>
<p>$AIC = \frac{RSS}{n\hat\sigma^2} + \frac{2d}{n} $</p>
<p>Fo... | <p>I think that you are confusing the two residual sum of squares that you have. You have one RSS to estimate the $\hat{\sigma}^2$ in the formula, this RSS is in some sense independent of the number of parameters, $p$. This $\hat{\sigma}^2$ should be estimated using all your covariates, giving you a <strong>baseline un... | 424 |
statistical learning | Book for reading before Elements of Statistical Learning? | https://stats.stackexchange.com/questions/18973/book-for-reading-before-elements-of-statistical-learning | <p>Based on <a href="https://quant.stackexchange.com/questions/111/how-can-i-go-about-applying-machine-learning-algorithms-to-stock-markets">this post</a>, I want to digest Elements of Statistical Learning. Fortunately it is available for free and I started reading it.</p>
<p>I don't have enough knowledge to understan... | <p>I bought, but have not yet read, </p>
<blockquote>
<p>S. Marsland, <em><a href="http://rads.stackoverflow.com/amzn/click/1420067184">Machine Learning: An Algorithmic Perspective</a></em>, Chapman & Hall, 2009. </p>
</blockquote>
<p>However, the reviews are favorable and state that it is more suitable for beg... | 425 |
statistical learning | Statistical learning - How to determine the irreducible error? | https://stats.stackexchange.com/questions/285750/statistical-learning-how-to-determine-the-irreducible-error | <p>I'm reading <em>Introduction to Statistical Learning</em>, currently Chapter 2 about the Bias-Variance trade-off. </p>
<p>In all examples the irreducible error is 1, i.e. $Var(\epsilon) = 1$. I read in <a href="https://stats.stackexchange.com/questions/228896/why-is-the-variance-of-the-error-term-a-k-a-the-irreduci... | 426 | |
statistical learning | Statistical Learning. Contradictions? | https://stats.stackexchange.com/questions/493564/statistical-learning-contradictions | <p>Currently I am re-reading some chapters of: <em>An Introduction to Statistical Learning with Applications in R</em> by Gareth James, Daniela Witten, Trevor Hastie and Robert Tibshirani (Springer, 2015). Now, I have some doubts about what is said there.</p>
<p>Above all it seems to me relevant to note that in chapter... | <p>There’s no contradiction. The fact that something is easy to interpret has nothing to do with how accurate is it. The most interpretable model you could imagine is to predict constant, independently of the data. In such case, you would always be able to explain why your model made the prediction it made, but the pre... | 427 |
statistical learning | The error-rate in "The elements of statistical learning" | https://stats.stackexchange.com/questions/645621/the-error-rate-in-the-elements-of-statistical-learning | <p>This picture is from the book "the elements of statistical learning":</p>
<p><a href="https://i.sstatic.net/3gmgr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/3gmgr.png" alt="enter image description here" /></a></p>
<p>I am wondering how the test-error rate is calculated based on how the ... | <p>Error rate is calculated as the number of errors divided by the number of attempts. This equals one minus the proportion classified correctly (accuracy expressed as a decimal or fraction).</p>
| 428 |
statistical learning | What is alpha in Vapnik's statistical learning theory? | https://stats.stackexchange.com/questions/478351/what-is-alpha-in-vapniks-statistical-learning-theory | <p>I'm currently studying Vapnik's theory of statistical learning. I rely on <a href="https://statisticalsupportandresearch.files.wordpress.com/2017/05/vladimir-vapnik-the-nature-of-statistical-learning-springer-2010.pdf" rel="nofollow noreferrer">Vapnik (1995)</a> and some secondary literature that is more accessible ... | <h2>Short Answer</h2>
<p><span class="math-container">$\alpha$</span> is the parameter or vector of parameters, including all so-called "hyperparameters," of a set of functions <span class="math-container">$V$</span>, and has nothing to do with the VC dimension.</p>
<h2>Long Answer: What is <span class="math-... | 429 |
statistical learning | What is next after finished reading Elements of Statistical Learning? | https://stats.stackexchange.com/questions/468483/what-is-next-after-finished-reading-elements-of-statistical-learning | <p>I am a Pure Maths PhD student specialising in functional analysis.</p>
<p>I would like to work as a data scientist after my PhD graduation, particularly in the field of machine learning, deep learning and artificial intelligence. </p>
<p>I have some backgrounds on machine learning such as linear regression, logist... | <p>Well the answer to the question is probably a matter of preference and depends on whether you want to specialize in some specific field (e.g. reinforcement learning) or you're aiming at having a more in-depth (but not limited to one subfield) view on machine learning.</p>
<p>If it's the latter I would recommend you... | 430 |
statistical learning | Intro to Statistical Learning - Solutions for 2.1 | https://stats.stackexchange.com/questions/539362/intro-to-statistical-learning-solutions-for-2-1 | <p>I am reading An Introduction to Statistical Learning with Applications in R (ISLR) and I wonder what would be the answer for exercise 2.1 part (d). The question is, If the variance of the error terms <span class="math-container">$$\sigma^2 = \mathrm{Var}(\epsilon)$$</span> is extremely high, a more flexible method w... | <p>It matters what you choose because a more flexible method may fit to the noise very easily, and you'll have to battle with it. As you mentioned, this is irreducible error, but an overfitted model will make much larger errors on the holdout set. Its aim is never reducing the irreducible error.</p>
| 431 |
statistical learning | what machine learning isn't statistical? | https://stats.stackexchange.com/questions/497986/what-machine-learning-isnt-statistical | <p>If I understand correctly, statistical learning theory is just <em>one</em> approach to machine learning,</p>
<p>What machine learning isn't statistical?</p>
<p>Based on my very limited understanding, I thought that things like PAC-learning or Empirical Risk minimization pretty much cover everything. Isn't statistic... | 432 | |
statistical learning | Decomposition of average squared bias (in Elements of Statistical Learning) | https://stats.stackexchange.com/questions/201779/decomposition-of-average-squared-bias-in-elements-of-statistical-learning | <p>I can't figure out how formula 7.14 on page 224 of <em>The Elements of Statistical Learning</em> is derived. Can anyone help me figure it out? </p>
<p>$$\textrm{Average squared bias} = \textrm{Average}[\textrm{model bias}]^2 + \textrm{Average}[\textrm{estimation bias}]^2$$</p>
<p><a href="https://i.sstatic.net/bi... | <p>The result is basically due to the property of best linear estimator. Note that we don't assume <span class="math-container">$f(X)$</span> is linear here. Nevertheless we can find the linear predictor that approximates <span class="math-container">$f$</span> the best. </p>
<p>Recall the definition of <span class="m... | 433 |
statistical learning | Hastie "statistical learning" 2.28. Least squares and covariance | https://stats.stackexchange.com/questions/641539/hastie-statistical-learning-2-28-least-squares-and-covariance | <p>In Hasties book "statistical learning", just above equation 2.28, it says that <span class="math-container">$\mathbf{X}^T\mathbf{X} \rightarrow NCov(X)$</span> (when <span class="math-container">$N$</span> is large and <span class="math-container">$E(X)=0$</span>).</p>
<p>Why is this true?</p>
<p><span cla... | <p><span class="math-container">$X = (X_1, \ldots, X_p)$</span> is a random vector with <span class="math-container">$p$</span> entries (not a scalar random variable).</p>
<p>The expectation of this random vector is denoted <span class="math-container">$E[X] = (E[X_1], \ldots, E[X_p])$</span>.</p>
<p>The <a href="https... | 434 |
statistical learning | Reproducing table 3.3 from Elements of Statistical Learning | https://stats.stackexchange.com/questions/161517/reproducing-table-3-3-from-elements-of-statistical-learning | <p>I am trying to reproduce a table 3.3 in Elements of statistical learning. Specifically, I am trying to get the coefficient estimates for ridge regression and lasso. I know that the estimate can be a bit off depending on the seeding value, but I personally think it is significantly off. The code is below, and would a... | 435 | |
statistical learning | Alternative to The Elements of Statistical Learning: Data Mining, Inference, and Prediction | https://stats.stackexchange.com/questions/332842/alternative-to-the-elements-of-statistical-learning-data-mining-inference-and | <p>I am taking a class in statistics which uses The Elements of Statistical Learning: Data Mining, Inference, and Prediction as a textbook. However, I find this book very terse. </p>
<p>Could anyone please recommend a book which has similar topic coverage but contains more examples and detailed explanations and does n... | 436 | |
statistical learning | Books to read on ML after ESL (Elements of Statistical Learning)? | https://stats.stackexchange.com/questions/460411/books-to-read-on-ml-after-esl-elements-of-statistical-learning | <p>I am almost finished reading ESL; Elements of Statistical Learning.
I come from a strong mathematical and statistical background, and that was my first book about Machine Learning.</p>
<p><strong>What other books would be good to go over now?</strong></p>
<p>I am aware of books such as:</p>
<ul>
<li>Machine Learn... | <p>You're going in the right way! In my journey towards machine learning, I found <strong>Python Machine Learning</strong> by Sebastian Raschka very helpful for the start. Though I read a lot books then, this one seriously helped me to kick start the journey.</p>
<p>Beside this one, you could check out some more books... | 437 |
statistical learning | How validation set in statistical learning works? | https://stats.stackexchange.com/questions/500236/how-validation-set-in-statistical-learning-works | <p>In statistical learning, we split the data into three parts for training, validation, and test, separately. With training data we can get a model <span class="math-container">$T$</span>, then we seem to optimize or change the model by validation data. How does that happen (since the model <span class="math-container... | <p>Imagine a multiple linear regression with a penalty on the magnitude of the coefficients (otherwise Lasso).</p>
<p>On the Training data, you will fit your regression coefficients <span class="math-container">$\underline{w}$</span> by minimizing the loss function <span class="math-container">$L$</span>.</p>
<p>On the... | 438 |
statistical learning | Measure-theoretically rigorous treatment of statistical learning theory | https://stats.stackexchange.com/questions/552709/measure-theoretically-rigorous-treatment-of-statistical-learning-theory | <p>My main source on statistical learning theory has been <a href="https://www.cs.huji.ac.il/%7Eshais/UnderstandingMachineLearning/" rel="nofollow noreferrer">Shwartz/Ben-David</a>. This is a good book but it's a little vague from a measure-theoretic point of view. For example, in the definition of PAC learnability (De... | <p>I will try to explain systematically, so apologies for going over things you probably already know.</p>
<p>To construct a probability space, <span class="math-container">$\Omega$</span>, we first choose some elementary subsets/events appropriate to the situation; for example, in a coin toss the events might be <em>h... | 439 |
statistical learning | Is it allowed to refer to Artificial Neural Networks as Statistical learning? | https://stats.stackexchange.com/questions/524205/is-it-allowed-to-refer-to-artificial-neural-networks-as-statistical-learning | <p>I am producing a research statement to be sent to a statistics department and I was trying to avoid the term Machine learning in favour of the more friendly one of Statistical learning. Probably I could not avoid such use.</p>
| <p>The classic <a href="https://web.stanford.edu/%7Ehastie/ElemStatLearn/" rel="noreferrer">The Elements of
Statistical Learning</a> handbook by Hastie et al discusses neural networks among other algorithms, so it needs to be a “statistical learning” algorithm.</p>
<p>Depending whom you’d ask, neural networks are eithe... | 440 |
statistical learning | Free PDF for Bayes with R, similar to Elements of Statistical Learning | https://stats.stackexchange.com/questions/47442/free-pdf-for-bayes-with-r-similar-to-elements-of-statistical-learning | <p>Is there a good book/pdf similar to "Elements of Statistical Learning" that's available for free online, that deals with Bayesian statistics, ideally with code for <code>R</code>?</p>
| <p>Well, given that you asked for something similar to the elements, I'm going to assume that you are of a machine learning bent.</p>
<p>Therefore, I would suggest the following:</p>
<p><a href="http://web4.cs.ucl.ac.uk/staff/D.Barber/textbook/090310.pdf" rel="noreferrer">Bayesian Reasoning and Machine Learning</a></... | 441 |
statistical learning | Understanding the Bootstrap method in *Introduction to Statistical Learning* | https://stats.stackexchange.com/questions/475693/understanding-the-bootstrap-method-in-introduction-to-statistical-learning | <p>I am having a hard time understanding Bootstrap method. In the book <a href="http://faculty.marshall.usc.edu/gareth-james/ISL/ISLR%20Seventh%20Printing.pdf" rel="nofollow noreferrer"><em>Introduction to Statistical Learning</em></a> (pp. 187-190) the Bootstrap method is explained by first using "simulated data... | <p>Bootstrapping is introduced as a method to <em>estimate</em> the variance of a <em>statistics</em> <span class="math-container">$S$</span>, given a <em>sample</em> <span class="math-container">$X=\{X_1, X_2, \ldots, X_2\}$</span>.</p>
<p>Usually, you can have two different scenarios. One scenario is when you know th... | 442 |
statistical learning | Justifying an early equation from *Introduction to Statistical Learning* | https://stats.stackexchange.com/questions/206645/justifying-an-early-equation-from-introduction-to-statistical-learning | <p>I'm self-studying <em>Introduction to Statistical Learning</em>. Page 19 of the book states the following:</p>
<blockquote>
<p>Consider a given estimate $\hat{f}$ and a set of predictors $X$, which yields the prediction $\hat{Y} = \hat{f}(X)$. Assume for a moment that both $\hat{f}$ and $X$ are fixed. Then, it is... | 443 | |
statistical learning | Introduction to Statistical Learning with R Equation 2.7 | https://stats.stackexchange.com/questions/451021/introduction-to-statistical-learning-with-r-equation-2-7 | <p>I'm really confused about equation 2.7 on page 34 in the Introduction to Statistical Learning with R text book found here: <a href="http://faculty.marshall.usc.edu/gareth-james/ISL/ISLR%20Seventh%20Printing.pdf" rel="nofollow noreferrer">http://faculty.marshall.usc.edu/gareth-james/ISL/ISLR%20Seventh%20Printing.pdf<... | <p><span class="math-container">$x_0$</span> is any value in the test set, in contrast to <span class="math-container">$x_1,x_2,\ldots, x_n$</span> in the training set. Similarly the pair <span class="math-container">$(x_0,y_0)$</span> is any such pair in the test set. The aim is to minimise the expected square of th... | 444 |
statistical learning | Where do artificial neural networks belong in the 'taxonomy' of statistical learning methods? | https://stats.stackexchange.com/questions/320664/where-do-artificial-neural-networks-belong-in-the-taxonomy-of-statistical-lear | <p>I'm a non-stats person trying to learn more about statistical learning methods, and to organize my thinking I am trying to construct a mental taxonomy of the methods I'm learning about. For instance: </p>
<blockquote>
<p>Statistical learning methods can be divided into supervised and unsupervised categories. </p... | <p>Neural nets form a broad class of models, and cover many parts of the taxonomy you're describing (and even extend outside it). An individual neural net (or subclass of them) could be placed into the taxonomy, but the entire class of neural nets cannot.</p>
<p>For example, neural nets can be used for both supervised... | 445 |
statistical learning | Derivation of EPE in “The elements of statistical learning ” | https://stats.stackexchange.com/questions/314517/derivation-of-epe-in-the-elements-of-statistical-learning | <p>I am currently trying to read the "<a href="http://amzn.to/2yXDxdf" rel="nofollow noreferrer">Elements of Statistical Learning</a>", by Efron, Hastie, and Tibshirani, and already at the beginning there is a bit above my level in mathematics.
I have 3 questions regarding the move from (2.9) to (2.10):</p>
<... | <blockquote>
<ol>
<li>what is the meaning of integrating with respect to Pr(dx,dy) instead of with respect to dx,dy by themselves?</li>
</ol>
</blockquote>
<p>You are missing the notion that this is an expectation, that is, the average value of $(Y-f(X))^2$ under the joint distribution of $(X,Y)$. (Using <em>Pr<... | 446 |
statistical learning | Explanation on a Minsky's critique on statistical learning related to XOR | https://stats.stackexchange.com/questions/134857/explanation-on-a-minskys-critique-on-statistical-learning-related-to-xor | <p>I was listening to the first session of society of Minds by Minsky (2011) and he mentions at some point around minute 48 the following:</p>
<p>"...lots of statistical learning tools is good for lots of applications, but they won't cut it to solve hard problems, where the hypothesis more complicated than seven or ei... | <p>Minksy is famous for criticizing neural networks for their inability to solve the XOR problem. It's possible that is what he's referring to here. Linear statistical relationships are not enough to detect patterns that resemble the XOR function.</p>
<p><a href="http://www.ucs.louisiana.edu/~isb9112/dept/phil341/hist... | 447 |
statistical learning | Statistical Learning With Sparsity: Direct Inspection of the LASSO function | https://stats.stackexchange.com/questions/432403/statistical-learning-with-sparsity-direct-inspection-of-the-lasso-function | <p>In the book Statistical Learning with Sparsity: The Lasso and Generalizations, in section 2.4.1, they mention that the absolute value of <span class="math-container">$\beta$</span> has no derivative at <span class="math-container">$\beta=0$</span>, therefore they proceed by direct inspection to determine the value o... | 448 | |
statistical learning | Simulated annealing for deep learning: Why is gradient free statistical learning not in the main stream? | https://stats.stackexchange.com/questions/559251/simulated-annealing-for-deep-learning-why-is-gradient-free-statistical-learning | <p>In order to define what <a href="https://www.deeplearningbook.org" rel="nofollow noreferrer">deep learning</a> is, the learning portion is often listed with <a href="https://en.wikipedia.org/wiki/Backpropagation" rel="nofollow noreferrer">backpropagation</a> as a requirement without alternatives in the main stream s... | <p>Gradient-free learning is in the mainstream very heavily, but not used heavily in deep learning. Methods used for training neural networks that don't involve derivatives are typically called "metaheuristics." In computer science and pattern recognition (which largely originated in electrical engineering)... | 449 |
statistical learning | Sparsity in Lasso and advantage over ridge (Statistical Learning) | https://stats.stackexchange.com/questions/151954/sparsity-in-lasso-and-advantage-over-ridge-statistical-learning | <p>I'm learning about the Statistical learning and in the section comparing Lasso and Ridge Regression it shows that the main difference between these two problems is the way the constraint/penalty is formulated. </p>
<p>In Lasso, the penalty is $\ell_1$ norm: $\lambda \sum |\beta_j|$, while in regression, the penalty... | <p>The lasso penalty will force some of the coefficients quickly to zero. This means that variables are removed from the model, hence the sparsity. </p>
<p>Ridge regression will more or less compress the coefficients to become smaller. This does not necessarily result in 0 coefficients and removal of variables.</p>
<... | 450 |
statistical learning | Recreating figure 3.6 from Elements of Statistical Learning | https://stats.stackexchange.com/questions/411327/recreating-figure-3-6-from-elements-of-statistical-learning | <p>I am trying to recreate FIGURE 3.6 from Elements of Statistical Learning. The only information about the figure is included in the caption.
<a href="https://i.sstatic.net/XiK3l.png" rel="noreferrer"><img src="https://i.sstatic.net/XiK3l.png" alt=""></a></p>
<p>To recreate the forward stepwise line my process is as... |
<p>There are probably some numbers wrong in the caption in the graph and/or the rendering of the graph.</p>
<p>An interesting anomaly is this graph on the version of chapter 3 on Tibshirani's website: <a href="http://statweb.stanford.edu/%7Etibs/book/" rel="nofollow noreferrer">http://statweb.stanford.edu/~tibs/book/<... | 451 |
statistical learning | Are there any good references regarding non-learnability in statistical learning theory? | https://stats.stackexchange.com/questions/463034/are-there-any-good-references-regarding-non-learnability-in-statistical-learning | <p>For example, I don't see a lot come up in my search for "non-learnability" ... is there another term? Specifically I'm imagining that similar to the Learning Guarantees from statistical learning theory there might be some results concerning non-learnability guarantees. Or are these equivalent in some simple way that... | 452 | |
statistical learning | Reproducing table 18.1 from "Elements of Statistical Learning" | https://stats.stackexchange.com/questions/12360/reproducing-table-18-1-from-elements-of-statistical-learning | <p>Table 18.1 in the <a href="http://www-stat.stanford.edu/~tibs/ElemStatLearn/" rel="noreferrer">Elements of Statistical Learning</a> summarizes the performance of several classifiers on a 14 class data set. I am comparing a new algorithm with the lasso and elastic net for such multiclass classification problems. </p>... | <p>have you checked the R package of the <a href="http://cran.r-project.org/web/packages/ElemStatLearn" rel="nofollow">book?</a>
it contains all the datasets, function and most of the scripts used
in there...</p>
| 453 |
statistical learning | Computation of LDA in Elements of Statistical Learning 4.3.2 | https://stats.stackexchange.com/questions/405541/computation-of-lda-in-elements-of-statistical-learning-4-3-2 | <p>Elements of Statistical Learning 4.3.2 elaborates on computation for Linear Discriminant Analysis. <a href="https://web.stanford.edu/~hastie/Papers/ESLII.pdf" rel="nofollow noreferrer">https://web.stanford.edu/~hastie/Papers/ESLII.pdf</a></p>
<p>Procedure is said to be </p>
<blockquote>
<p>• Sphere the data with... | <p>Sphering ( or whitening ) the data (<span class="math-container">$X$</span>) means applying a transformation so that in the new basis, the covariance for sphered data (<span class="math-container">$X^{*}$</span>) is the identity matrix, i.e. <span class="math-container">$E[X^{*T}X^{*}]=I_{n}$</span> .</p>
<p>We oper... | 454 |
statistical learning | Least Squares Definition in Elements of Statistical Learning | https://stats.stackexchange.com/questions/181648/least-squares-definition-in-elements-of-statistical-learning | <p>In <em>Elements of Statistical Learning</em>, they state on p. 11 that all vectors are column vectors and start developing the least squares idea.</p>
<p>So if we have
$$\mathbf{X} = \begin{bmatrix}
1 \\
X_1 \\
X_2 \\
\vdots \\
X_p\end{bmatrix}$$
and $$\hat{\boldsymbol{\beta}} = \begin{bmatrix}
\hat{\beta}_0 \\
\ha... | <p>I think you confuse yourself a bit. In page 10 the authors say: </p>
<p><em>"a set of $N$ input $p$-vectors $x_i$ , $i = 1, \dots, N$ would be represented by the $N \times p$ matrix <strong>$X$</strong>."</em> </p>
<p>This means that the $p$ feature vectors/regressors they use will be represented as column vectors... | 455 |
statistical learning | Elements of Statistical Learning - Statistical Decision Theory : Doubt regarding Minimization of EPE | https://stats.stackexchange.com/questions/286290/elements-of-statistical-learning-statistical-decision-theory-doubt-regarding | <p>With reference to Expected Prediction Error derivation - page 18, section 2.4 in Elements of Statistical Learning. Please refer text below: </p>
<p><a href="https://i.sstatic.net/ZJbWz.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZJbWz.gif" alt="Please Refer below:"></a></p>
<p>I have been able t... | <p>Let $H$ be any set of functions of $x$. Then, for each $h\in H$,
$\int h(x)\,dx \ge \int \inf_{g\in H} g(x)\,dx$.
Sometimes, as in the current situation, the function $\lambda$, given by $\lambda(x)=\inf_{g\in H}g(x)$, is already in $H$, in which case the least value of the integral of $h$, as $h$ varies over $H$, i... | 456 |
statistical learning | What does this figure in “Introduction to statistical learning” mean? | https://stats.stackexchange.com/questions/300097/what-does-this-figure-in-introduction-to-statistical-learning-mean | <p>I am currently reading the book "Introduction to statistical learning" and on <a href="https://books.google.co.in/books?id=qcI_AAAAQBAJ&lpg=PR2&pg=PA3#v=onepage&q&f=false" rel="nofollow noreferrer">Page 3</a> there is a problem statement regarding the SMarket data (stock exchange data) fig 2.2 whose ... | <blockquote>
<p>The left-handed panel of Figure 1.2 displays two boxplots of the previous day's percentage changes in the stock index: one for the 648 days for with the market increased on subsequent days, and one for the 602 days for which the market decreased. </p>
</blockquote>
<p>The blue boxplot of the left-mos... | 457 |
statistical learning | Interpreting exercise in Elements of Statistical Learning | https://stats.stackexchange.com/questions/465439/interpreting-exercise-in-elements-of-statistical-learning | <p>I am reading exercise 6.4 from <a href="https://web.stanford.edu/~hastie/Papers/ESLII.pdf" rel="nofollow noreferrer">The Elements of Statistical Learning</a> (Hastie, Tibshirani and Friedman) and I am having difficulty interpreting exactly what is being asked in the following question</p>
<blockquote>
<p>Ex. 6.4 ... | 458 | |
statistical learning | Generating pseudodata as in "Elements of Statistical Learning" | https://stats.stackexchange.com/questions/317961/generating-pseudodata-as-in-elements-of-statistical-learning | <p>I am trying to implement a Simulation from the book "Elements of Statistical Learning" by Hastie et al. </p>
<p>My Problem is that I don't understand how to generate the pseudodata as they did.
The book says </p>
<blockquote>
<p>For each of N = 100 Samples, we generated p standard Gaussian features X with pairw... | <blockquote>
<p>The standard deviation sigma was chosen in each case so that the signal-to-noise-ratio $Var(E[Y|X]) / \sigma^2$ equaled 2.</p>
</blockquote>
<p>Because $\epsilon$ has mean 0, we know that:</p>
<p>$$E[Y \mid X] = \sum_{j=1}^p X_j \beta_j = \beta^T X$$</p>
<p>So, using the $X$ and $\beta$ you generat... | 459 |
statistical learning | Bayes decision boundary of Figure 2.5 in Elements of Statistical Learning | https://stats.stackexchange.com/questions/35728/bayes-decision-boundary-of-figure-2-5-in-elements-of-statistical-learning | <p>When I read "Elements of Statistical Learning", I met some difficulty in calculating the Bayes decision boundary of Figure 2.5. In the package <code>ElemStatLearn</code>, it already calculated the probability at each point and used contours to draw the boundary. Can any one tell me how to calculate the probability? ... | <p>I asked the authors this question, and apparently they no longer are in possession of the code that created the data. So there is no real way to reconstruct the Bayes rule for this particular data set. Otherwise, it would be based on the ratio of the densities that would have been known for the Gaussian mixture dist... | 460 |
statistical learning | Hints for exercise 7.3 from The elements of statistical learning | https://stats.stackexchange.com/questions/306777/hints-for-exercise-7-3-from-the-elements-of-statistical-learning | <p>I am stuck on problem 7.3 from the book 'The Elements of Statistical Learning'. This is the problem:
<a href="https://i.sstatic.net/CoQ3s.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/CoQ3s.jpg" alt="enter image description here"></a></p>
<p>Here is my attempt to a solution for least-squares projec... | <p>Some hints: You correctly note
$$ X_{-i}^TX_{-i}=X^TX-\vec{x}_i\vec{x}_i^T$$($\vec{x_i}$ is a column vector),
and that you need to find $$\hat{\vec{\beta}}_{-i} = (X_{-i}^TX_{-i})^{-1}X_{-i}^T\vec{y}_{-i},$$ the estimated coefficients obtained by leaving out sample $i$. This will lead you to the new predicted value... | 461 |
statistical learning | Understanding linear projection in "The Elements of Statistical Learning" | https://stats.stackexchange.com/questions/185634/understanding-linear-projection-in-the-elements-of-statistical-learning | <p>In the book "The Elements of Statistical Learning" in chapter 2 ("Linear models and least squares; page no: 12"), it is written that </p>
<blockquote>
<p>In the (p+1)-dimensional input-output space, (X,Y) represent a hyperplane. If the constant is included in X, then the hyperplane includes the origin and is a su... | <p>Including the constant <code>1</code> in the input vector is a common trick to include a bias (think about Y-intercept) but keeping all the terms of the expression symmetrical: you can write $\beta X$ instead of $\beta_0 + \beta X$ everywhere.</p>
<p>If you do this, it is then correct that the hyperplane $Y = \beta... | 462 |
statistical learning | QDA - Missing term in quadratic discriminant function in 'Introduction to Statistical Learning' | https://stats.stackexchange.com/questions/490651/qda-missing-term-in-quadratic-discriminant-function-in-introduction-to-statis | <p>Some resources such as <a href="https://online.stat.psu.edu/stat508/book/export/html/696" rel="nofollow noreferrer">https://online.stat.psu.edu/stat508/book/export/html/696</a>, give the following for the quadratic discriminant function; <span class="math-container">$$ln(\pi_k)-\frac{1}{2}(x-\mu_k)^T\Sigma_k^{-1}(x-... | <p>This is listed as errata for an early edition of the book (1st edition prior to the 4th printing) <a href="http://faculty.marshall.usc.edu/gareth-james/ISL/errata.html" rel="nofollow noreferrer">http://faculty.marshall.usc.edu/gareth-james/ISL/errata.html</a></p>
| 463 |
statistical learning | Derivation of equation 6.15 of Introduction to Statistical Learning - 2nd ed | https://stats.stackexchange.com/questions/590667/derivation-of-equation-6-15-of-introduction-to-statistical-learning-2nd-ed | <p>I was reading the book "Introduction to Statistical Learning - 2nd ed" and I can't understand the derivation of equation 6.15 on the page 247.</p>
<p><a href="https://i.sstatic.net/9m9pt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9m9pt.png" alt="Page 247 of ISLR" /></a></p>
<p>This ques... | 464 | |
statistical learning | Trying to understand splines according to "The elements of statistical learning" | https://stats.stackexchange.com/questions/441574/trying-to-understand-splines-according-to-the-elements-of-statistical-learning | <p>I'm working on Chapter 5 from <a href="https://web.stanford.edu/%7Ehastie/Papers/ESLII.pdf" rel="nofollow noreferrer">The elements of statistical learning</a> which describes the more general linear models that is splines. The fragment (chapter 5, page 143) below describes the alternative and more direct method of b... | <p>The positive part of a function <span class="math-container">$f(x)$</span>, noted <span class="math-container">$f^+(x)$</span> or <span class="math-container">$f(x)_+$</span>, is defined as:
<span class="math-container">$$
f(x)_+ = \max(f(x),0)
$$</span></p>
<p>For example the function <span class="math-container">... | 465 |
statistical learning | Introduction to statistical learning Ch. 3 Pages 65-66 | https://stats.stackexchange.com/questions/534570/introduction-to-statistical-learning-ch-3-pages-65-66 | <p>In the textbook <a href="https://static1.squarespace.com/static/5ff2adbe3fe4fe33db902812/t/6062a083acbfe82c7195b27d/1617076404560/ISLR%2BSeventh%2BPrinting.pdf" rel="nofollow noreferrer"><em>Introduction to Statistical Learning with Applications in R</em></a> by James et al. (2014), the authors give the following fo... | <blockquote>
<p>I can't wrap my head around this, each <span class="math-container">$y_i$</span> has an exact value, how can
it have a standard deviation?</p>
</blockquote>
<p>Because each <span class="math-container">$y_i$</span> is the realization of a random variable. Look at page 61: <span class="math-container">$Y... | 466 |
statistical learning | Understanding Vapnik's Regression Function in Statistical Learning Theory | https://stats.stackexchange.com/questions/605801/understanding-vapniks-regression-function-in-statistical-learning-theory | <p>I am struggling to understand what is meant by Equation 1.8 in Vapnik's Statistical Learning Theory. Vapnik introduces the equation below as the <em>regression</em> function.</p>
<p><span class="math-container">$ r(x) = \int y\ dF(y|x) $</span></p>
<p>I know that <span class="math-container">$x$</span> and <span cla... | 467 | |
statistical learning | Question about an equation on bagging in Elements of Statistical Learning book | https://stats.stackexchange.com/questions/578939/question-about-an-equation-on-bagging-in-elements-of-statistical-learning-book | <p>This should be a simple question but I must have missed something.</p>
<p>Equation (8.52) of Section 8.7 Bagging on page 285 of Trevor Hastie, <em>The Elements of Statistical Learning: Data Mining, Inference, and Prediction</em> is equivalent to
<span class="math-container">$$\mathbf E_{\mathcal P}[(Y-f_{\text{ag}}(... | 468 | |
statistical learning | Can you explain this description of tree pruning in Intro to Statistical Learning? | https://stats.stackexchange.com/questions/643222/can-you-explain-this-description-of-tree-pruning-in-intro-to-statistical-learnin | <p>The underlined sentences below from p. 331 in <a href="https://www.statlearning.com/" rel="nofollow noreferrer">An Introduction to Statistical Learning</a> have me scratching my head: Given that the splitting algorithm always finds the best next split in terms of error reduction, how could it be possible for there t... | <p>It's quite possible for a bad split to be followed by a good split. Consider two predictor variables centered on (0,0), where the outcome variable <span class="math-container">$Y$</span> is 1 if the variables have the same sign and 0 if they have different signs. No single split will predict usefully, but a split o... | 469 |
statistical learning | Deriving the in-sample error for linear model from the elements of statistical learning | https://stats.stackexchange.com/questions/91766/deriving-the-in-sample-error-for-linear-model-from-the-elements-of-statistical-l | <p>From the elements of statistical learning, it was claimed that
$$
\frac{1}{N}\sum_{i=1}^N ||h(x_i) ||^2 \sigma^2_\varepsilon= \frac{p}{N}\sigma^2_\varepsilon$$</p>
<p>where $h(x_i) = X(X^TX)^{-1}x_i$. Can someone show me how to prove this ? Thanks</p>
<p>This came from the image below</p>
<p><img src="https://i.... | <p>$$
\sum_{i=1}^{N}\|\textbf{h}(x_i)\|^2
=\sum_{i=1}^{N}\|\textbf{X}(\textbf{X}^T\textbf{X})^{-1}x_i\|^2
=tr\{(\textbf{X}(\textbf{X}^T\textbf{X})^{-1}\textbf{X}^T)^T(\textbf{X}(\textbf{X}^T\textbf{X})^{-1}\textbf{X}^T)\}
=tr\{(\textbf{X}(\textbf{X}^T\textbf{X})^{-1}\textbf{X}^T)(\textbf{X}(\textbf{X}^T\textbf{X})^{-1}... | 470 |
statistical learning | Question on loss function notation in Elements of Statistical Learning II | https://stats.stackexchange.com/questions/517082/question-on-loss-function-notation-in-elements-of-statistical-learning-ii | <p>In <a href="https://web.stanford.edu/%7Ehastie/Papers/ESLII.pdf" rel="nofollow noreferrer">Elements of Statistical Learning II</a> on page 349, the multinomial deviance loss function is given by <span class="math-container">$L(y,p(x))=-\sum_{k=1}^KI(y=G_k)f_k(x)+\log(\sum_{\ell=1}^Ke^{f_\ell(x)})$</span>, but there ... | <p>It's done to prevent confusion, because the second summation is inside the first one. If you've two nested summations or for loops, you wouldn't use <span class="math-container">$i$</span> to index both, right?</p>
<p>Also, since the deviance loss is a function of <span class="math-container">$p_k(x)$</span>, that <... | 471 |
statistical learning | On the importance of the i.i.d. assumption in statistical learning | https://stats.stackexchange.com/questions/213464/on-the-importance-of-the-i-i-d-assumption-in-statistical-learning | <p>In statistical learning, implicitly or explicitly, one <em>always</em> assumes that the training set $\mathcal{D} = \{ \bf {X}, \bf{y} \}$ is composed of $N$ input/response tuples $({\bf{X}}_i,y_i)$ that are <em>independently drawn from the same joint distribution</em> $\mathbb{P}({\bf{X}},y)$ with</p>
<p>$$ p({\bf... | <p>The i.i.d. assumption about the pairs <span class="math-container">$(\mathbf{X}_i, y_i)$</span>, <span class="math-container">$i = 1, \ldots, N$</span>, is often made in statistics and in machine learning. Sometimes for a good reason, sometimes out of convenience and sometimes just because we usually make this assum... | 472 |
statistical learning | The Intercept terms for Ridge regression, lasso , pcr and PLS (elements of statistical learning) | https://stats.stackexchange.com/questions/92590/the-intercept-terms-for-ridge-regression-lasso-pcr-and-pls-elements-of-stati | <p>In table 3.3 (page 63) of the elements of statistical learning book, the intercept terms for Ridge regression, lasso , pcr and PLS differ. </p>
<p>However, according to the theory in the book, these models should all have the same $\hat{\beta_0} = \bar{y}$. How are the intercepts estimated in the table ? </p>
<p>N... | 473 | |
statistical learning | Show $\hat{f}(x) \to E[Y|X=x]$ elements of statistical learning | https://stats.stackexchange.com/questions/629745/show-hatfx-to-eyx-x-elements-of-statistical-learning | <p>I was reading elements of statistical learning and it mentions let <span class="math-container">$\hat{f}(x)=Ave(y_i|x_i \in N_k(x))$</span> where <span class="math-container">$N_k(x)$</span> is the neighborhood containing the k points closest to x .</p>
<p>Then it says "under mild regularity conditions on the j... | 474 | |
statistical learning | Derivation of the conditional median for linear regression in “The elements of statistical learning ” | https://stats.stackexchange.com/questions/344920/derivation-of-the-conditional-median-for-linear-regression-in-the-elements-of-s | <p>My question is about "The elements of statistical learning" book. I would like to know how to prove that the use of the $L_1$ loss $$L_1: E\bigg[|Y-f(X)|\bigg]$$ leads to have conditional median $\hat{f}(x)=median(Y|X=x)$ as solution to the $EPF(f)$ criterion minimisation in eq(2.11): $$EPF(f)= E_X\bigg[E_{Y|X}\big[... | <p>First, I think you misspelled something in the question. In your case it should be
$$
EPE(f)=\mathbb{E}(\vert Y-f(X)\vert).
$$
What you want to show is that
$$
\text{argmin}_{f \text{ measurable}}EPE(f)=\left(X\mapsto\text{median}(Y\vert X)\right)
$$
This is in fact equivalent to showing that the median is the best ... | 475 |
statistical learning | Book recommendations needed - building foundational knowledge for ISL - Introduction to Statistical Learning (by Gareth James) | https://stats.stackexchange.com/questions/486182/book-recommendations-needed-building-foundational-knowledge-for-isl-introduc | <p>I'm trying to build a data science base from scratch. I started a book called Introduction to Statistical Learning by Gareth James and found that there are many mathematical & statistical concepts that I'm unfamiliar with. I want to bridge this gap in my knowledge. Please recommend some books that will help me d... | <p>For Mathematics:</p>
<ol>
<li>James Stewart's Calculus</li>
</ol>
<p>For Statistics:</p>
<ol>
<li>Basic Business Statistics (7e) - Beenson, Levine, Szabat, ...</li>
</ol>
| 476 |
statistical learning | Loss functions in statistical decision theory vs. machine learning? | https://stats.stackexchange.com/questions/485964/loss-functions-in-statistical-decision-theory-vs-machine-learning | <p>I'm quite familiar with loss functions in machine learning, but am struggling to connect them to loss functions in statistical decision theory [1].</p>
<p>In machine learning, a loss function is usually only considered at <strong>training time</strong>. It's a differentiable function of two variables, <code>loss(tru... | <p>The loss that is of ultimate interest is the <strong>prediction loss</strong> (or <strong>decision loss</strong>). It represents real (financial/material/...) consequences of any given decision for the decision maker. It is this and only this loss that we want to minimize for its own sake rather than as an intermedi... | 477 |
statistical learning | Statistical learning when observations are not iid | https://stats.stackexchange.com/questions/563419/statistical-learning-when-observations-are-not-iid | <p>As far as I am concerned, statistical/machine learning algorithms always suppose that data are independent and identically distributed (<span class="math-container">$iid$</span>).</p>
<p>My question is: what can we do when this assumption is clearly unsatisfied? For instance, suppose that we have a data set whith re... | <p>There is nothing in the theory of statistical learning or machine learning that requires samples to be i.i.d.</p>
<p>When samples are i.i.d, you can write the joint probability of the samples given some model as a product, namely <span class="math-container">$P(\{x\}) = \Pi_{i} P_i(x_i)$</span> which makes the log-l... | 478 |
statistical learning | Undirected graphical models with for discrete variables with hidden nodes - loglikelihood (The elements of statistical learning) | https://stats.stackexchange.com/questions/271376/undirected-graphical-models-with-for-discrete-variables-with-hidden-nodes-logl | <p>I don't understand the equation of loglikelihood of the observed data in graphical models with hidden nodes that appears in "The Elements of Statistical Learning" (Hastie, Tibshirani, Friedmann, chapter 17.4.2)</p>
<p><a href="https://i.sstatic.net/OXyKc.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.ne... | <p>This is a special case of the <a href="https://en.wikipedia.org/wiki/Law_of_total_probability" rel="nofollow noreferrer">law of total probability</a>. (See also the <a href="http://people.reed.edu/~jones/Courses/P02.pdf" rel="nofollow noreferrer">second equation of slide 5</a>.)</p>
<p>Specifically the lower-case $... | 479 |
statistical learning | How to plot decision boundary of a k-nearest neighbor classifier from Elements of Statistical Learning? | https://stats.stackexchange.com/questions/21572/how-to-plot-decision-boundary-of-a-k-nearest-neighbor-classifier-from-elements-o | <p>I want to generate the plot described in the book ElemStatLearn "The Elements of
Statistical Learning: Data Mining, Inference, and Prediction. Second Edition" by Trevor Hastie
& Robert Tibshirani& Jerome Friedman. The plot is:</p>
<p><img src="https://i.sstatic.net/oY7hr.png" alt="enter image description he... | <p>To reproduce this figure, you need to have the <a href="http://cran.r-project.org/web/packages/ElemStatLearn/index.html" rel="noreferrer">ElemStatLearn</a> package installed on you system. The artificial dataset was generated with <code>mixture.example()</code> as pointed out by @StasK.</p>
<pre><code>library(ElemS... | 480 |
statistical learning | Parameter estimation for basis function model in Elements of Statistical Learning (ESL) | https://stats.stackexchange.com/questions/487702/parameter-estimation-for-basis-function-model-in-elements-of-statistical-learnin | <p>In the book <em>Elements of Statistical Learning</em>, section 2.8.3 describes Basis Functions, citing an example of a radial basis function as <span class="math-container">$f_{\theta}(x) = \sum_{m=1}^M \beta_M \sigma(\alpha_m'x + b_m)$</span>, with <span class="math-container">$\sigma(x)$</span> as the activation f... | 481 | |
statistical learning | Does learning thorough statistical theory require learning analysis? | https://stats.stackexchange.com/questions/631739/does-learning-thorough-statistical-theory-require-learning-analysis | <p>Does learning thorough statistical theory requires learning analysis before that?
I looked at the textbook for statistical theory. So far I don't know if analysis is required, but I think I have heard analysis is a prerequisite. Should I learn analysis beforehand?</p>
| <p>No, you do not need to know real analysis to learn statistics. In fact, in many universities (intro level) statistics courses are not even in the math department.</p>
<p>One can make a lot progress in statistics by letting the computer do all the math and you worrying only in how the statistical methods are being ap... | 482 |
statistical learning | Questions regarding the Bayes Classifier in *Introduction to Statistical Learning* | https://stats.stackexchange.com/questions/208795/questions-regarding-the-bayes-classifier-in-introduction-to-statistical-learnin | <p>I am having trouble grokking some very elementary material regarding Bayesian Classification in <a href="http://www.stat.berkeley.edu/~rabbee/s154/ISLR_First_Printing.pdf" rel="nofollow"><em>Introduction to Statistical Learning</em></a> at the end of pg. 37 to the very top of pg. 39 (i.e., the section entitled "The ... | <ol>
<li><p>We make an assumption that all objects are realizations of random variables, which are independent and identically distributed from some distribution. Very often we have one more assumption that this distribution is from exact parametric family of distributions. In this case training data is used to evaluat... | 483 |
statistical learning | Derivation of EPE for linear regression in "The elements of statistical learning " | https://stats.stackexchange.com/questions/257124/derivation-of-epe-for-linear-regression-in-the-elements-of-statistical-learning | <p>My question is about "The elements of statistical learning" book (yup, the one).
Right now I am kinda stuck on second chapter at part, where they derive EPE for linear regression (Somewhat related to <a href="https://stats.stackexchange.com/questions/253101/confusion-about-derivation-of-regression-function">Confusio... | <p>I am trying to answer the first question
Suppose $x_0$ and $y_0$ are both in $R^p$.
$T$ is the space of training parameters which are sets of pairs $(x_0,y_0)$ such that $y_0=x_0^T\beta + \epsilon$. These sets define exactly the training data and starting from there an linear estimation of $y$ as a function of $x$ ... | 484 |
statistical learning | What supplemental resource do you recommend in order to fully comprehend The Elements of Statistical Learning | https://stats.stackexchange.com/questions/204830/what-supplemental-resource-do-you-recommend-in-order-to-fully-comprehend-the-ele | <p>I am learning the book "Elements of Statistical Learning," but it is very hard because it requires very heavy knowledge about statistics, which I have some, but apparently not enough to understand the derivations in the book. For example,
<a href="https://i.sstatic.net/8yZ06.jpg" rel="nofollow noreferrer"><img src=... | <p>The same authors have a more introductory book called Introduction to Statistical Learning. There is a free PDF version online <a href="http://www-bcf.usc.edu/~gareth/ISL/" rel="nofollow">http://www-bcf.usc.edu/~gareth/ISL/</a></p>
| 485 |
statistical learning | Loss function in Supervised Learning vs Statistical Decision Theory | https://stats.stackexchange.com/questions/543535/loss-function-in-supervised-learning-vs-statistical-decision-theory | <p>I am confused by the different definitions of Loss Function in statistical decision theory vs machine learning.</p>
<p>In statistical decision theory, a loss function is typically defined as <span class="math-container">$L(\theta, \delta(X))$</span>, where <span class="math-container">$\theta$</span> is the true, un... | <p>I would say this is more a difference in the form of the <em>decision</em> than the loss. The loss function in both cases is Loss(true state of nature, your decision), but it simplifies differently depending on the form of the decision</p>
<p>In point prediction settings (such as a lot of ML), the decision is a pote... | 486 |
statistical learning | Understanding notation in Bias-Variance decomposition in Elements of Statistical Learning | https://stats.stackexchange.com/questions/458620/understanding-notation-in-bias-variance-decomposition-in-elements-of-statistical | <p>I'm going through Elements of Statistical Learning and I'm having a bit of trouble understanding this bit of notation from Chapter 2 (this example is (2.27))</p>
<p><span class="math-container">$$EPE(x_0) = E_{y_o|x_o}E_T(y_0 - \hat{y}_0)^2$$</span></p>
<p>Here, <span class="math-container">$T$</span> is the set o... | 487 | |
statistical learning | Is there an error Section 6.6.2 of the book An Introduction to Statistical Learning? | https://stats.stackexchange.com/questions/490725/is-there-an-error-section-6-6-2-of-the-book-an-introduction-to-statistical-learn | <p>In Section 6.6.2 of An Introduction to Statistical Learning, the authors do the following:</p>
<p>A) Fit a lasso model</p>
<pre><code>lasso.mod=glmnet(x[train ,],y[ train],alpha=1, lambda =grid)
</code></pre>
<p>B) Perform cross-validation</p>
<pre><code>set.seed(1)
cv.out=cv.glmnet(x[train ,],y[ train],alpha=1)
plo... | <p>Step A doesn't provide a single model; it provides a set of models, one for each value of <span class="math-container">$\lambda$</span>, developed on all of <code>x[train,]</code> and <code>y[train]</code>. There is no single model in Step B <em>even for a single value of</em> <span class="math-container">$\lambda$<... | 488 |
statistical learning | Proof/Derivation of Residual Sum of Squares (Based on Introduction to Statistical Learning) | https://stats.stackexchange.com/questions/110190/proof-derivation-of-residual-sum-of-squares-based-on-introduction-to-statistica | <p>On page 19 of the textbook <a href="http://www-bcf.usc.edu/%7Egareth/ISL/" rel="noreferrer">Introduction to Statistical Learning</a> (by James, Witten, Hastie and Tibshirani--it is freely downloadable on the web, and very good), the following is stated:</p>
<blockquote>
<p>Consider a given estimate <span class="math... | <p>Simply expand the square ...</p>
<p>$$[f(X)- \hat{f}(X) + \epsilon ]^2=[f(X)- \hat{f}(X)]^2 +2 [f(X)- \hat{f}(X)]\epsilon+ \epsilon^2$$</p>
<p>... and use linearity of expectations:</p>
<p>$$\mathrm{E}[f(X)- \hat{f}(X) + \epsilon ]^2=E[f(X)- \hat{f}(X)]^2 +2 E[(f(X)- \hat{f}(X))\epsilon]+ E[\epsilon^2]$$</p>
<p>... | 489 |
statistical learning | Cross-validation scheme used in the Introduction to Statistical Learning, Chapter 6, Lab 3 | https://stats.stackexchange.com/questions/223623/cross-validation-scheme-used-in-the-introduction-to-statistical-learning-chapte | <p>I've been really enjoying the <em><a href="http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Fourth%20Printing.pdf" rel="nofollow">Introduction to Statistical Learning</a></em> textbook so far, and I'm currently working my way through chapter 6. I realize that I am very confused by the process used in lab 3 of this chapter ... | <p>It is indeed not very clearly explained in the text, but here is what I think is going on.</p>
<p><strong>First, they perform cross-validation on the whole dataset.</strong> They say that "the smallest cross-validation error occurs when $M = 16$ components
are used", but also remark that the difference between diff... | 490 |
statistical learning | statistical regression vs machine learning regression | https://stats.stackexchange.com/questions/514247/statistical-regression-vs-machine-learning-regression | <p>I was trying to understand the difference between statistical regression VS machine learning regression. My background is from Economics and learned regression from statistical point of view for the first time. I learned machine learning later on and it also had regression. There might not be clear distinction betwe... | <p>There really isn't much of a difference. A strained distinction between the two might be consideration of the data generating process (what statisticians call the likelihood). Statisticians care about this because different likelihoods lead to different types of inference. A hotly debated example of this would be ... | 491 |
statistical learning | What is meant by the variance of *functions* in *Introduction to Statistical Learning*? | https://stats.stackexchange.com/questions/208672/what-is-meant-by-the-variance-of-functions-in-introduction-to-statistical-lea | <p>On pg. 34 of <em>Introduction to Statistical Learning</em>: $\newcommand{\Var}{{\rm Var}}$</p>
<blockquote>
<p>Though the mathematical proof is beyond the scope of this book, it is possible to show that the expected test MSE, for a given value $x_0$, can always be decomposed into the sum of three fundamental quan... | <p>Your correspondence with @whuber is correct.</p>
<p>A learning algorithm $\mathcal{A}$ can be viewed as a higher level function, mapping training sets to functions.</p>
<p>$$ \mathcal{A} : \mathcal{T} \rightarrow \{f \mid f: X \rightarrow \mathbb{R} \} $$</p>
<p>where $\mathcal{T}$ is the space of possible traini... | 492 |
statistical learning | What is the relationship between Online Learning and Statistical Learning? | https://stats.stackexchange.com/questions/392301/what-is-the-relationship-between-online-learning-and-statistical-learning | <p>Online Learning also known as Online Convex Optimization has famous algorithms like Follow-the-Leader and Online Gradient Descent (See <a href="http://ocobook.cs.princeton.edu/OCObook.pdf" rel="nofollow noreferrer">OCO Book)</a></p>
<p>Now stochastic programming has algorithms like Sample Average Approximation and S... | 493 | |
statistical learning | Finding optimal subspace for Linear Discriminant Analysis - Elements of Statistical Learning 4.3.3 | https://stats.stackexchange.com/questions/405607/finding-optimal-subspace-for-linear-discriminant-analysis-elements-of-statisti | <p>Linear Discriminant Analysis (LDA) possibly operates a dimension reduction. Section 4.3.3 in Elements of Statistical Learning explicits this notion as well as a method for computing the "optimal subspace for LDA".</p>
<p><a href="https://web.stanford.edu/~hastie/Papers/ESLII.pdf" rel="nofollow noreferrer">https://w... | <p><strong><em>Within-class, between-class covariance matrices</em></strong></p>
<p>• Assuming common covariance matrix <span class="math-container">$\hat{\Sigma}=\hat{\Sigma}_{k}$</span> for all classes <span class="math-container">$k$</span> we write</p>
<p><span class="math-container">$\hat{\Sigma}=\sum_{k=1}^{K}\... | 494 |
statistical learning | Is there a textbook / handbook with full derivations for statistical / machine learning concepts? | https://stats.stackexchange.com/questions/74908/is-there-a-textbook-handbook-with-full-derivations-for-statistical-machine-l | <p>In particular, I am looking for a textbook which will go over the details of derivations (including all calculus and linear algebra) for learning models and concepts such as logistic regression, Gaussian Discriminant Analysis, with full proofs for variants like Gaussian Naive Bayes.</p>
<p>Books such as "Elements o... | <p>Have a look at <a href="http://rads.stackoverflow.com/amzn/click/1439824142" rel="nofollow">'A First Course in Machine Learning,' Simon Rogers and Mark Girolami</a>.
There are many easy to follow step by step derivations of concepts that include calculus and linear algebra. Also, you can look at google book preview ... | 495 |
statistical learning | Bias-variance docomposition of linear model fit in 'The Elements of Statistical Learning' | https://stats.stackexchange.com/questions/307110/bias-variance-docomposition-of-linear-model-fit-in-the-elements-of-statistical | <p>In section 7.3 of 'The Elements of Statistical Learning', the authors have shown the expression for bias-variance decomposition of linear model fit:
<a href="https://i.sstatic.net/tvwY3.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tvwY3.jpg" alt="enter image description here"></a></p>
<p><a href="... | <p>You should drop the outside <span class="math-container">$E_X$</span>, and only use <span class="math-container">$E_{Y|X}$</span>, because here they are computing the test error (definition is at the beginning of this chapter eq 7.2), and the training sample X is fixed(you add <span class="math-container">$E_X$</spa... | 496 |
statistical learning | Why doesn't test error increase for a high number of boosting iterations? Figure 10.13 of The Elements of Statistical Learning | https://stats.stackexchange.com/questions/548557/why-doesnt-test-error-increase-for-a-high-number-of-boosting-iterations-figure | <p>My question refers to the figure 10.13 of <a href="https://web.stanford.edu/%7Ehastie/Papers/ESLII.pdf" rel="nofollow noreferrer">The Elements of Statistical Learning</a>. Test error decreases monotonically with the increase in tree iterations. However, I don't understand why the test error does not raise for the hi... | <p>Note the rest of the paragraph you quoted:</p>
<blockquote>
<p>This tends to be the case in
many applications. The shrinkage strategy (10.41) tends to eliminate the
problem of overfitting, especially for larger data sets.</p>
</blockquote>
<p>Shrinkage slows down overfitting, but it does happen.
Using basically the ... | 497 |
statistical learning | Statistical Significance of a learning Model | https://stats.stackexchange.com/questions/79677/statistical-significance-of-a-learning-model | <p>I built a learning model (for classification) based on a Random Forest classifier and i am asked to assess the statistical significance of its performances. </p>
<p>Up to now, i trained and tested it on two different datasets A and B, respectively.</p>
<p>What kind of test can i use?</p>
| <p>You can get an un-biased estimate of the classification error with the out-of-bag error estimate. See explanation here: <a href="http://www.stat.berkeley.edu/~breiman/RandomForests/cc_home.htm#ooberr" rel="nofollow noreferrer">http://www.stat.berkeley.edu/~breiman/RandomForests/cc_home.htm#ooberr</a></p>
<p>I suppo... | 498 |
statistical learning | Equation 10.2 from The Elements of Statistical Learning. Median of a chi-squared distribution | https://stats.stackexchange.com/questions/545107/equation-10-2-from-the-elements-of-statistical-learning-median-of-a-chi-squared | <p>I'm reading about AdaBoost in the <em>The Elements of Statistical Learning</em> and I don't understand the equation 10.2. Below is an excerpt from the book.</p>
<blockquote>
<p>The power of AdaBoost to dramatically increase the performance of even a very weak classifier is illustrated in Figure 10.2. The features
<s... | 499 |
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