title

Support Vector Machines Part 3: The Radial (RBF) Kernel (Part 3 of 3)

description

Support Vector Machines use kernel functions to do all the hard work and this StatQuest dives deep into one of the most popular: The Radial (RBF) Kernel. We talk about the parameter values, how they calculate high-dimensional coordinates and then we'll figure out, step-by-step, how the Radial Kernel works in infinite dimensions.
NOTE: This StatQuest assumes you already know about...
Support Vector Machines: https://youtu.be/efR1C6CvhmE
Cross Validation: https://youtu.be/fSytzGwwBVw
The Polynomial Kernel: https://youtu.be/Toet3EiSFcM
ALSO NOTE: This StatQuest is based on...
1) The description of Kernel Functions, and associated concepts on pages 352 to 353 of the Introduction to Statistical Learning in R: http://faculty.marshall.usc.edu/gareth-james/ISL/
2) The derivation of the of the infinite dot product is based on Matthew Bernstein's notes: http://pages.cs.wisc.edu/~matthewb/pages/notes/pdf/svms/RBFKernel.pdf
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detail

{'title': 'Support Vector Machines Part 3: The Radial (RBF) Kernel (Part 3 of 3)', 'heatmap': [{'end': 203.542, 'start': 172.076, 'weight': 0.788}, {'end': 489.51, 'start': 459.555, 'weight': 0.708}, {'end': 924.049, 'start': 893.849, 'weight': 0.884}], 'summary': 'Covers support vector machines part 3, explaining the radial kernel in infinite dimensions, its parameters, and its effectiveness using a drug dosage dataset. it also discusses challenges in using a support vector classifier and proposes the use of a support vector machine with a radial basis function (rbf) kernel, along with the taylor series expansion of e to the x and its relationship to the radial kernel.', 'chapters': [{'end': 89.879, 'segs': [{'end': 64.385, 'src': 'embed', 'start': 0.829, 'weight': 0, 'content': [{'end': 6.732, 'text': 'The radial kernel works in infinite dimensions.', 'start': 0.829, 'duration': 5.903}, {'end': 12.994, 'text': "I know that sounds kind of crazy, but it's actually not that bad.", 'start': 7.272, 'duration': 5.722}, {'end': 20.177, 'text': "StatQuest Hello, I'm Josh Starmer and welcome to StatQuest.", 'start': 13.555, 'duration': 6.622}, {'end': 26.08, 'text': "Today we're going to talk about Support Vector Machines Part 3, the radial kernel.", 'start': 20.558, 'duration': 5.522}, {'end': 37.287, 'text': "Specifically, we're going to talk about the radial kernel's parameters, how the radial kernel calculates high-dimensional relationships,", 'start': 27.859, 'duration': 9.428}, {'end': 41.771, 'text': 'and then show you how the radial kernel works in infinite dimensions.', 'start': 37.287, 'duration': 4.484}, {'end': 50.538, 'text': 'this StatQuest assumes that you are already familiar with support vector machines and the polynomial kernel.', 'start': 43.732, 'duration': 6.806}, {'end': 52.68, 'text': 'If not, check out the quests.', 'start': 51.079, 'duration': 1.601}, {'end': 54.902, 'text': 'The links are in the description below.', 'start': 53.2, 'duration': 1.702}, {'end': 64.385, 'text': 'In the StatQuest on support vector machines, we had a training dataset based on drug dosages measured in a bunch of patients.', 'start': 56.641, 'duration': 7.744}], 'summary': 'Statquest explains the radial kernel in support vector machines part 3.', 'duration': 63.556, 'max_score': 0.829, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns829.jpg'}], 'start': 0.829, 'title': 'Support vector machines part 3', 'summary': 'Explains the working of the radial kernel in infinite dimensions, its parameters, and how it calculates high-dimensional relationships, using a drug dosage dataset to illustrate its effectiveness.', 'chapters': [{'end': 89.879, 'start': 0.829, 'title': 'Support vector machines part 3', 'summary': 'Explains the working of the radial kernel in infinite dimensions, its parameters, and how it calculates high-dimensional relationships, using a drug dosage dataset to illustrate its effectiveness.', 'duration': 89.05, 'highlights': ['The radial kernel works in infinite dimensions, explaining high-dimensional relationships (quantifiable data: infinite dimensions).', 'The chapter uses a drug dosage dataset to show the effectiveness of the radial kernel (quantifiable data: drug dosages measured in patients).', "The training data set had significant overlap, illustrating the importance of the radial kernel's effectiveness in complex classification tasks (quantifiable data: significant overlap in the training dataset).", 'The chapter assumes familiarity with support vector machines and the polynomial kernel, providing links for further exploration (quantifiable data: assumes prior knowledge).']}], 'duration': 89.05, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns829.jpg', 'highlights': ['The radial kernel works in infinite dimensions, explaining high-dimensional relationships (quantifiable data: infinite dimensions).', 'The chapter uses a drug dosage dataset to show the effectiveness of the radial kernel (quantifiable data: drug dosages measured in patients).', "The training data set had significant overlap, illustrating the importance of the radial kernel's effectiveness in complex classification tasks (quantifiable data: significant overlap in the training dataset).", 'The chapter assumes familiarity with support vector machines and the polynomial kernel, providing links for further exploration (quantifiable data: assumes prior knowledge).']}, {'end': 582.008, 'segs': [{'end': 141.068, 'src': 'embed', 'start': 89.879, 'weight': 0, 'content': [{'end': 98.023, 'text': 'we were unable to find a satisfying support vector classifier to separate the patients that were cured from the patients that were not cured.', 'start': 89.879, 'duration': 8.144}, {'end': 110.488, 'text': 'One way to deal with overlapping data is to use a support vector machine with a radial kernel, aka the radial basis function, RBF.', 'start': 99.683, 'duration': 10.805}, {'end': 121.139, 'text': "Because the radial kernel finds support vector classifiers in infinite dimensions, it's not possible to visualize what it does.", 'start': 113.035, 'duration': 8.104}, {'end': 131.163, 'text': 'However, when using it on a new observation like this, the radial kernel behaves like a weighted nearest neighbor model.', 'start': 122.739, 'duration': 8.424}, {'end': 141.068, 'text': 'In other words, the closest observations, aka the nearest neighbors, have a lot of influence on how we classify the new observation.', 'start': 132.524, 'duration': 8.544}], 'summary': 'Challenges in finding support vector classifier for patient outcome, using radial kernel in svm to handle overlapping data and influence of nearest neighbors on classification.', 'duration': 51.189, 'max_score': 89.879, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns89879.jpg'}, {'end': 228.066, 'src': 'heatmap', 'start': 172.076, 'weight': 2, 'content': [{'end': 177.66, 'text': 'Just like with the polynomial kernel, A and B refer to two different dosage measurements.', 'start': 172.076, 'duration': 5.584}, {'end': 185.806, 'text': 'The difference between the measurements is then squared, giving us the squared distance between the two observations.', 'start': 179.321, 'duration': 6.485}, {'end': 193.912, 'text': 'Thus, the amount of influence one observation has on another is a function of the squared distance.', 'start': 187.367, 'duration': 6.545}, {'end': 203.542, 'text': 'Gamma, which is determined by cross-validation, scales the squared distance, and thus, it scales the influence.', 'start': 195.52, 'duration': 8.022}, {'end': 217.406, 'text': 'For example, if we set gamma equal to 1, and plug in the dosages from two observations that are relatively close to each other and do the math,', 'start': 205.123, 'duration': 12.283}, {'end': 225.464, 'text': 'we get 0.11 when gamma equals 1..', 'start': 217.406, 'duration': 8.058}, {'end': 228.066, 'text': "So let's put 0.11 here.", 'start': 225.464, 'duration': 2.602}], 'summary': 'The influence of observations is determined by squared distance and scaled by gamma, e.g., gamma=1 results in 0.11 influence.', 'duration': 32.546, 'max_score': 172.076, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns172076.jpg'}, {'end': 348.269, 'src': 'embed', 'start': 319.51, 'weight': 4, 'content': [{'end': 328.858, 'text': 'Medium BAM! Now, before we move on, I want to simplify the training dataset to just two observations.', 'start': 319.51, 'duration': 9.348}, {'end': 337.241, 'text': 'and use the polynomial kernel to give us intuition into how the radial kernel works in infinite dimensions.', 'start': 330.256, 'duration': 6.985}, {'end': 348.269, 'text': 'When r equals zero, the polynomial kernel simplifies to a single term, and that gives us a dot product with a single coordinate.', 'start': 338.902, 'duration': 9.367}], 'summary': 'Simplify training dataset to two observations, use polynomial kernel to understand radial kernel in infinite dimensions.', 'duration': 28.759, 'max_score': 319.51, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns319510.jpg'}, {'end': 489.51, 'src': 'heatmap', 'start': 459.555, 'weight': 0.708, 'content': [{'end': 471.144, 'text': "Going back to the original training data set, let's talk about what happens if we take a polynomial kernel with r equals 0 and d equals 1,", 'start': 459.555, 'duration': 11.589}, {'end': 477.406, 'text': 'and add another polynomial kernel with r equals 0 and d equals 2..', 'start': 471.144, 'duration': 6.262}, {'end': 480.967, 'text': 'This gives us a dot product with coordinates for two dimensions.', 'start': 477.406, 'duration': 3.561}, {'end': 489.51, 'text': 'The first coordinate is the original dosage, and the second coordinate is dosage squared.', 'start': 482.668, 'duration': 6.842}], 'summary': 'Using polynomial kernels to create a two-dimensional dot product.', 'duration': 29.955, 'max_score': 459.555, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns459555.jpg'}, {'end': 548.697, 'src': 'embed', 'start': 518.746, 'weight': 3, 'content': [{'end': 526.15, 'text': 'then the dot product has coordinates for three dimensions and we can plot the transformed data on x, y,', 'start': 518.746, 'duration': 7.404}, {'end': 531.926, 'text': 'z axes and find a support vector classifier to separate the data.', 'start': 526.15, 'duration': 5.776}, {'end': 542.533, 'text': 'Now, what if we just kept adding polynomial kernels with r equals zero and increasing d until d equals infinity?', 'start': 533.607, 'duration': 8.926}, {'end': 548.697, 'text': 'That would give us a dot product with coordinates for an infinite number of dimensions.', 'start': 544.034, 'duration': 4.663}], 'summary': 'Using polynomial kernels with r=0 and increasing d to infinity yields a dot product with coordinates for an infinite number of dimensions.', 'duration': 29.951, 'max_score': 518.746, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns518746.jpg'}], 'start': 89.879, 'title': 'Support vector classifier with radial kernel', 'summary': 'Discusses challenges in using a support vector classifier to separate cured and non-cured patients and proposes the use of a support vector machine with a radial basis function (rbf) kernel, which behaves like a weighted nearest neighbor model. it also explains the influence of gamma on the squared distance between observations, the high-dimensional relationship between observations, and the transformation of data using polynomial kernels in support vector classification.', 'chapters': [{'end': 170.451, 'start': 89.879, 'title': 'Support vector classifier with radial kernel', 'summary': 'Discusses the challenges in using a support vector classifier to separate cured and non-cured patients and proposes the use of a support vector machine with a radial basis function (rbf) kernel, which behaves like a weighted nearest neighbor model.', 'duration': 80.572, 'highlights': ['The radial kernel in support vector machines behaves like a weighted nearest neighbor model, where the closest observations have a significant influence on classifying new observations.', 'The support vector classifier encountered challenges in separating cured and non-cured patients, leading to the proposal of using a support vector machine with a radial basis function (RBF) kernel.']}, {'end': 582.008, 'start': 172.076, 'title': 'Radial kernel and polynomial kernel', 'summary': 'Explains the influence of gamma on the squared distance between observations, the high-dimensional relationship between observations, and the transformation of data using polynomial kernels in support vector classification.', 'duration': 409.932, 'highlights': ['The influence of gamma on the squared distance between observations is demonstrated by setting gamma to different values and calculating the resulting distances, showing how gamma scales the influence (e.g., with gamma equal to 1, the distance is 0.11, and with gamma equal to 2, the distance is 0.01).', 'The chapter provides insight into the transformation of data using polynomial kernels in support vector classification, demonstrating how setting r equals 0 and increasing d results in a dot product with coordinates for an infinite number of dimensions, which is exactly what the radial kernel does.', "The explanation of how to simplify the training dataset to two observations and use the polynomial kernel to gain intuition into the radial kernel's workings in infinite dimensions is provided, showcasing how different values of r and d affect the resulting dot products and the transformation of data on x-y axes and x, y, z axes."]}], 'duration': 492.129, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns89879.jpg', 'highlights': ['The radial kernel in support vector machines behaves like a weighted nearest neighbor model, where the closest observations have a significant influence on classifying new observations.', 'The support vector classifier encountered challenges in separating cured and non-cured patients, leading to the proposal of using a support vector machine with a radial basis function (RBF) kernel.', 'The influence of gamma on the squared distance between observations is demonstrated by setting gamma to different values and calculating the resulting distances, showing how gamma scales the influence (e.g., with gamma equal to 1, the distance is 0.11, and with gamma equal to 2, the distance is 0.01).', 'The chapter provides insight into the transformation of data using polynomial kernels in support vector classification, demonstrating how setting r equals 0 and increasing d results in a dot product with coordinates for an infinite number of dimensions, which is exactly what the radial kernel does.', "The explanation of how to simplify the training dataset to two observations and use the polynomial kernel to gain intuition into the radial kernel's workings in infinite dimensions is provided, showcasing how different values of r and d affect the resulting dot products and the transformation of data on x-y axes and x, y, z axes."]}, {'end': 951.751, 'segs': [{'end': 614.41, 'src': 'embed', 'start': 583.768, 'weight': 0, 'content': [{'end': 587.63, 'text': "Now let's create the Taylor series expansion of this last term.", 'start': 583.768, 'duration': 3.862}, {'end': 597.138, 'text': "Wait a minute, what's a Taylor series expansion? This big thing is a Taylor series.", 'start': 589.332, 'duration': 7.806}, {'end': 606.404, 'text': 'Although there are exceptions, the main idea is that a function f of x can be split into an infinite sum.', 'start': 598.298, 'duration': 8.106}, {'end': 614.41, 'text': "Since this is very abstract, let's walk through how to convert e to the x into an infinite sum.", 'start': 608.205, 'duration': 6.205}], 'summary': 'Taylor series expands functions into infinite sums.', 'duration': 30.642, 'max_score': 583.768, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns583768.jpg'}, {'end': 748.955, 'src': 'embed', 'start': 689.471, 'weight': 1, 'content': [{'end': 694.073, 'text': 'And since e to the 0 equals 1, e to the 0 exists.', 'start': 689.471, 'duration': 4.602}, {'end': 698.656, 'text': 'So we will set a equal to 0.', 'start': 694.754, 'duration': 3.902}, {'end': 699.696, 'text': 'And simplify.', 'start': 698.656, 'duration': 1.04}, {'end': 712.803, 'text': 'Thus, if we can accept that the Taylor series expansion does what it says it does, e to the x is equal to this infinite sum.', 'start': 703.599, 'duration': 9.204}, {'end': 724.554, 'text': 'Large BAM! Going back to the radial kernel, we can now create the Taylor series expansion of this last term.', 'start': 714.044, 'duration': 10.51}, {'end': 734.159, 'text': 'To create the Taylor series expansion of E to the AB, we plug in AB for X.', 'start': 726.135, 'duration': 8.024}, {'end': 738.801, 'text': 'Now we have the Taylor series expansion of the last part of the radial kernel.', 'start': 734.159, 'duration': 4.642}, {'end': 742.723, 'text': 'Okay, time to take a deep breath.', 'start': 740.542, 'duration': 2.181}, {'end': 748.955, 'text': "We've done a lot, but we still have a few more steps before we are done and get to eat snacks.", 'start': 743.271, 'duration': 5.684}], 'summary': 'Taylor series expansion simplifies e to the x, leading to infinite sum', 'duration': 59.484, 'max_score': 689.471, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns689471.jpg'}, {'end': 822.428, 'src': 'embed', 'start': 784.202, 'weight': 3, 'content': [{'end': 793.366, 'text': 'and a polynomial kernel with r equals 0 and d equals 2 is equal to a times b squared, etc.', 'start': 784.202, 'duration': 9.164}, {'end': 794.366, 'text': ', etc , etc.', 'start': 793.366, 'duration': 1}, {'end': 807.356, 'text': 'Thus, each term in this Taylor series expansion contains a polynomial kernel with r equals 0 and d going from 0 to infinity.', 'start': 797.608, 'duration': 9.748}, {'end': 815.322, 'text': 'Now, just to remind you, converting this sum to a dot product was easy,', 'start': 809.257, 'duration': 6.065}, {'end': 822.428, 'text': 'because the dot product tells us to multiply each term together and then add up all the terms.', 'start': 815.322, 'duration': 7.106}], 'summary': 'Polynomial kernel with r=0, d=2 equals a times b squared. taylor series has terms with r=0, d=0 to infinity.', 'duration': 38.226, 'max_score': 784.202, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns784202.jpg'}, {'end': 924.049, 'src': 'heatmap', 'start': 893.849, 'weight': 4, 'content': [{'end': 902.317, 'text': 'And at long last, we see that the radial kernel is equal to a dot product that has coordinates for an infinite number of dimensions.', 'start': 893.849, 'duration': 8.468}, {'end': 911.76, 'text': 'That means when we plug numbers into the radial kernel and do the math,', 'start': 904.434, 'duration': 7.326}, {'end': 917.744, 'text': 'the value we get at the end is the relationship between the two points in infinite dimensions.', 'start': 911.76, 'duration': 5.984}, {'end': 924.049, 'text': 'Triple bam! Now we can go eat snacks.', 'start': 919.125, 'duration': 4.924}], 'summary': 'Radial kernel equals dot product in infinite dimensions, revealing relationship between points.', 'duration': 30.2, 'max_score': 893.849, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns893849.jpg'}], 'start': 583.768, 'title': 'Taylor series and radial kernel', 'summary': 'Covers the taylor series expansion of e to the x and its conversion into an infinite sum, along with the representation of radial kernel as a dot product, providing insights into infinite-dimensional space and relationship between points.', 'chapters': [{'end': 748.955, 'start': 583.768, 'title': 'Taylor series expansion of e to the x', 'summary': 'Discusses the taylor series expansion of e to the x, illustrating the process of converting e to the x into an infinite sum and determining the value of a to simplify the expansion, ultimately leading to the expansion of e to the ab in the radial kernel.', 'duration': 165.187, 'highlights': ['The Taylor series expansion of e to the x is explored, demonstrating the process of converting e to the x into an infinite sum and determining the value of a to simplify the expansion. The Taylor series expansion of e to the x is explained, showcasing the process of converting e to the x into an infinite sum and determining the value of a to simplify the expansion.', 'The concept of setting a in the Taylor series expansion and simplifying the expansion by setting a equal to 0 is elucidated. The concept of setting a in the Taylor series expansion and simplifying the expansion by setting a equal to 0 is clarified.', 'The process of creating the Taylor series expansion of E to the AB in the radial kernel is discussed, demonstrating the steps involved in the expansion. The process of creating the Taylor series expansion of E to the AB in the radial kernel is explained, illustrating the steps involved in the expansion.']}, {'end': 951.751, 'start': 750.836, 'title': 'Radial kernel and dot product', 'summary': 'Explains how the dot product converts the taylor series expansion to an infinite-dimensional space, and how the radial kernel can be represented as a dot product, providing insights into the relationship between two points in infinite dimensions.', 'duration': 200.915, 'highlights': ['The dot product converts the Taylor series expansion to an infinite-dimensional space, with each term containing a polynomial kernel with r equals 0 and d going from 0 to infinity.', 'The radial kernel can be represented as a dot product, revealing the relationship between two points in infinite dimensions.', 'Support for StatQuest can be provided through various means such as subscribing, contributing to the Patreon campaign, becoming a channel member, or purchasing original songs, t-shirts, or hoodies.']}], 'duration': 367.983, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qc5IyLW_hns/pics/Qc5IyLW_hns583768.jpg', 'highlights': ['The Taylor series expansion of e to the x is explored, demonstrating the process of converting e to the x into an infinite sum and determining the value of a to simplify the expansion.', 'The process of creating the Taylor series expansion of E to the AB in the radial kernel is discussed, demonstrating the steps involved in the expansion.', 'The concept of setting a in the Taylor series expansion and simplifying the expansion by setting a equal to 0 is elucidated.', 'The dot product converts the Taylor series expansion to an infinite-dimensional space, with each term containing a polynomial kernel with r equals 0 and d going from 0 to infinity.', 'The radial kernel can be represented as a dot product, revealing the relationship between two points in infinite dimensions.']}], 'highlights': ['The radial kernel works in infinite dimensions, explaining high-dimensional relationships (quantifiable data: infinite dimensions).', 'The chapter uses a drug dosage dataset to show the effectiveness of the radial kernel (quantifiable data: drug dosages measured in patients).', "The training data set had significant overlap, illustrating the importance of the radial kernel's effectiveness in complex classification tasks (quantifiable data: significant overlap in the training dataset).", 'The radial kernel in support vector machines behaves like a weighted nearest neighbor model, where the closest observations have a significant influence on classifying new observations.', 'The support vector classifier encountered challenges in separating cured and non-cured patients, leading to the proposal of using a support vector machine with a radial basis function (RBF) kernel.', 'The influence of gamma on the squared distance between observations is demonstrated by setting gamma to different values and calculating the resulting distances, showing how gamma scales the influence (e.g., with gamma equal to 1, the distance is 0.11, and with gamma equal to 2, the distance is 0.01).', 'The Taylor series expansion of e to the x is explored, demonstrating the process of converting e to the x into an infinite sum and determining the value of a to simplify the expansion.', 'The process of creating the Taylor series expansion of E to the AB in the radial kernel is discussed, demonstrating the steps involved in the expansion.', 'The concept of setting a in the Taylor series expansion and simplifying the expansion by setting a equal to 0 is elucidated.', 'The dot product converts the Taylor series expansion to an infinite-dimensional space, with each term containing a polynomial kernel with r equals 0 and d going from 0 to infinity.']}