title

3.2: Linear Regression with Ordinary Least Squares Part 1 - Intelligence and Learning

description

In this video, part of my series on "Machine Learning", I explain how to perform Linear Regression for a 2D dataset using the Ordinary Least Squares method. In Part 2, I demonstrate how to code the algorithm in JavaScript, using the p5.js library
This video is part of session 3 of my Spring 2017 ITP "Intelligence and Learning" course (https://github.com/shiffman/NOC-S17-2-Intelligence-Learning/tree/master/week3-classification-regression)
Link to Part 2: https://youtu.be/_cXuvTQl090
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Links discussed in this video:
Session 3 of Intelligence and Learning: https://github.com/shiffman/NOC-S17-2-Intelligence-Learning/tree/master/week3-classification-regression
Nature of Code: http://natureofcode.com/
kwichmann's Linear Regression Diagnostics: https://kwichmann.github.io/ml_sandbox/linear_regression_diagnostics/
Linear Regression on Wikipedia: https://en.wikipedia.org/wiki/Linear_regression
Anscombe's quartet on Wikipedia: https://en.wikipedia.org/wiki/Anscombe%27s_quartet
Source Code for the all Video Lessons: https://github.com/CodingTrain/Rainbow-Code
p5.js: https://p5js.org/
Processing: https://processing.org
For More Coding Challenges: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
For More Intelligence and Learning: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6YJ3XfHhT2Mm4Y5I99nrIKX
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{'title': '3.2: Linear Regression with Ordinary Least Squares Part 1 - Intelligence and Learning', 'heatmap': [{'end': 523.812, 'start': 489.585, 'weight': 0.745}, {'end': 784.057, 'start': 771.127, 'weight': 0.713}, {'end': 946.369, 'start': 921.933, 'weight': 0.866}], 'summary': 'Series explores linear regression in machine learning, covering topics such as using two-dimensional data sets for sales prediction, ordinary least squares method for fitting lines, and the application of linear equations to regression, providing a comprehensive foundation for understanding and utilizing linear regression in machine learning.', 'chapters': [{'end': 86.06, 'segs': [{'end': 86.06, 'src': 'embed', 'start': 18.107, 'weight': 0, 'content': [{'end': 22.971, 'text': "So what I'm leading towards and what I'm going to get to if you keep watching these videos they don't exist yet,", 'start': 18.107, 'duration': 4.864}, {'end': 26.975, 'text': "but I'm going to keep making them so eventually you might keep watching them is I'm going to get to neural networks.", 'start': 22.971, 'duration': 4.004}, {'end': 37.035, 'text': 'Neural networks are useful and powerful in the case of large data sets with many, many variables, many, many inputs,', 'start': 29.088, 'duration': 7.947}, {'end': 40.918, 'text': "parameters that we almost can't figure out mathematically how to make sense of it.", 'start': 37.035, 'duration': 3.883}, {'end': 44.22, 'text': 'Maybe a neural network can do that in some almost magical way.', 'start': 41.178, 'duration': 3.042}, {'end': 45.842, 'text': "We're going to get into all the details of that.", 'start': 44.32, 'duration': 1.522}, {'end': 54.151, 'text': 'But there are machine learning scenarios where we can actually just calculate, precisely using a statistical method,', 'start': 46.542, 'duration': 7.609}, {'end': 56.474, 'text': 'the relationship between inputs and outputs.', 'start': 54.151, 'duration': 2.323}, {'end': 61.419, 'text': 'So if we were to review, we have this idea of a machine learning recipe.', 'start': 56.654, 'duration': 4.765}, {'end': 70.45, 'text': 'Previously, I looked at k nearest neighbor as a possible algorithm to make sense of input data and predict some sort of output,', 'start': 62.625, 'duration': 7.825}, {'end': 73.632, 'text': "whether it's classifying or predicting a price, that type of thing.", 'start': 70.45, 'duration': 3.182}, {'end': 76.754, 'text': 'So we have some sort of input.', 'start': 74.352, 'duration': 2.402}, {'end': 79.275, 'text': 'We get some sort of output.', 'start': 77.855, 'duration': 1.42}, {'end': 86.06, 'text': "So let's take the simplest scenario of inputs related to outputs.", 'start': 79.916, 'duration': 6.144}], 'summary': 'Neural networks useful for large datasets, but statistical methods also effective in some cases.', 'duration': 67.953, 'max_score': 18.107, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk18107.jpg'}], 'start': 1.152, 'title': 'Linear regression in machine learning', 'summary': 'Introduces linear regression as a method to calculate the relationship between inputs and outputs in machine learning. it leads towards the understanding of neural networks for large data sets and compares it with other machine learning algorithms.', 'chapters': [{'end': 86.06, 'start': 1.152, 'title': 'Linear regression in machine learning', 'summary': 'Introduces linear regression as a method to calculate the relationship between inputs and outputs in machine learning, leading towards the understanding of neural networks for large data sets, and compares it with other machine learning algorithms.', 'duration': 84.908, 'highlights': ['Linear regression is a method to precisely calculate the relationship between inputs and outputs in machine learning scenarios, before delving into the complexities of neural networks.', 'Neural networks are highlighted as powerful for large data sets with numerous variables and inputs, where mathematical comprehension becomes challenging.', 'The video series aims to progressively cover topics, including neural networks, to further understand their capability in handling complex datasets.', 'The chapter compares linear regression with other machine learning algorithms, such as k nearest neighbor, for making sense of input data and predicting outputs.']}], 'duration': 84.908, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk1152.jpg', 'highlights': ['Neural networks are highlighted as powerful for large data sets with numerous variables and inputs, where mathematical comprehension becomes challenging.', 'The chapter compares linear regression with other machine learning algorithms, such as k nearest neighbor, for making sense of input data and predicting outputs.', 'Linear regression is a method to precisely calculate the relationship between inputs and outputs in machine learning scenarios, before delving into the complexities of neural networks.', 'The video series aims to progressively cover topics, including neural networks, to further understand their capability in handling complex datasets.']}, {'end': 266.474, 'segs': [{'end': 114.64, 'src': 'embed', 'start': 86.4, 'weight': 0, 'content': [{'end': 91.143, 'text': 'And a simple scenario for this would be something like a two-dimensional data set.', 'start': 86.4, 'duration': 4.743}, {'end': 106.218, 'text': "Okay, so we could graph, using something called a scatter, plot a data set and we're going to make the data set bear with me here a temperature,", 'start': 93.095, 'duration': 13.123}, {'end': 109.939, 'text': 'the x-axis I want to think of as temperature.', 'start': 106.218, 'duration': 3.721}, {'end': 112.039, 'text': "so maybe I'm really sorry.", 'start': 109.939, 'duration': 2.1}, {'end': 113.099, 'text': "I'm going to do this in Fahrenheit.", 'start': 112.039, 'duration': 1.06}, {'end': 114.64, 'text': 'I just have to apologize for that.', 'start': 113.099, 'duration': 1.541}], 'summary': 'Analyzing a two-dimensional data set using a scatter plot to represent temperature in fahrenheit.', 'duration': 28.24, 'max_score': 86.4, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk86400.jpg'}, {'end': 250.102, 'src': 'embed', 'start': 196.717, 'weight': 1, 'content': [{'end': 204.583, 'text': "could you make a guess as to how many ice creams you're going to sell? And so we could look and say, here's 50 degrees.", 'start': 196.717, 'duration': 7.866}, {'end': 208.686, 'text': 'Well, there was some other day where I sold this much when it was 50, some other day, some other day.', 'start': 204.924, 'duration': 3.762}, {'end': 222.753, 'text': 'How could we make a prediction? Well, this is a scenario where it appears there is a linear relationship between temperature and sales.', 'start': 209.207, 'duration': 13.546}, {'end': 226.134, 'text': 'The higher the temperature, the more the sales.', 'start': 223.733, 'duration': 2.401}, {'end': 228.475, 'text': 'The lower the temperature, the less.', 'start': 226.375, 'duration': 2.1}, {'end': 231.057, 'text': 'The fewer, the fewer the sales.', 'start': 229.376, 'duration': 1.681}, {'end': 241.816, 'text': 'So the idea of a linear regression is to figure out how can we fit a line best fit aligned to this data.', 'start': 232.837, 'duration': 8.979}, {'end': 246.88, 'text': "And I could look at this and, hold on, I'm going to be back in a second for some magic.", 'start': 241.956, 'duration': 4.924}, {'end': 247.8, 'text': "I'm back.", 'start': 247.54, 'duration': 0.26}, {'end': 250.102, 'text': 'By the way, this is like a historic moment.', 'start': 248, 'duration': 2.102}], 'summary': 'Linear regression predicts ice cream sales based on temperature.', 'duration': 53.385, 'max_score': 196.717, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk196717.jpg'}], 'start': 86.4, 'title': 'Two-dimensional data set example and linear regression for sales prediction', 'summary': 'Discusses using a two-dimensional data set to graph temperature against ice cream and sorbet sales, with examples of ice cream sales at different temperatures. it also covers using linear regression to predict ice cream sales based on temperature, emphasizing the importance of fitting a line to the data for accurate predictions.', 'chapters': [{'end': 171.095, 'start': 86.4, 'title': 'Two-dimensional data set example', 'summary': 'Discusses using a two-dimensional data set to graph temperature against ice cream and sorbet sales, with examples of ice cream sales at different temperatures.', 'duration': 84.695, 'highlights': ['Graphing a two-dimensional data set with temperature on the x-axis and ice cream sales on the y-axis.', 'Example of ice cream sales at 24 degrees and 90 degrees, with 3 and 18 units sold respectively.']}, {'end': 266.474, 'start': 171.095, 'title': 'Linear regression for sales prediction', 'summary': 'Discusses using linear regression to predict ice cream sales based on temperature, with the concept of a linear relationship between temperature and sales, and the importance of fitting a line to the data for accurate predictions.', 'duration': 95.379, 'highlights': ['The concept of a linear relationship between temperature and ice cream sales is discussed, with the higher temperature leading to more sales and the lower temperature leading to fewer sales.', 'The importance of fitting a line to the data for accurate predictions is highlighted, emphasizing the need to use linear regression to determine the best-fit line for sales prediction.', 'The scenario of using historical sales data to make predictions about future sales based on temperature is mentioned, illustrating the practical application of linear regression for sales forecasting.', 'The humorous interlude about using a marker with a different color is included, providing a lighthearted moment in the discussion.']}], 'duration': 180.074, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk86400.jpg', 'highlights': ['Graphing a two-dimensional data set with temperature on the x-axis and ice cream sales on the y-axis.', 'Example of ice cream sales at 24 degrees and 90 degrees, with 3 and 18 units sold respectively.', 'The importance of fitting a line to the data for accurate predictions is highlighted, emphasizing the need to use linear regression to determine the best-fit line for sales prediction.', 'The concept of a linear relationship between temperature and ice cream sales is discussed, with the higher temperature leading to more sales and the lower temperature leading to fewer sales.', 'The scenario of using historical sales data to make predictions about future sales based on temperature is mentioned, illustrating the practical application of linear regression for sales forecasting.']}, {'end': 466.185, 'segs': [{'end': 323.023, 'src': 'embed', 'start': 267.597, 'weight': 0, 'content': [{'end': 271.887, 'text': 'And I could say, look, that looks like a line that kind of fits the data.', 'start': 267.597, 'duration': 4.29}, {'end': 276.518, 'text': "That's me just as a human being kind of eyeballing it.", 'start': 273.717, 'duration': 2.801}, {'end': 284.74, 'text': 'So now, if I wanted to say when the temperature is 95 degrees, I could just look at 95 degrees,', 'start': 276.598, 'duration': 8.142}, {'end': 291.142, 'text': 'find this and find the corresponding 200 ice creams or whatever sold.', 'start': 284.74, 'duration': 6.402}, {'end': 297.784, 'text': 'So this is the idea of linear regression, looking at a data set and fitting a line to that data set.', 'start': 291.542, 'duration': 6.242}, {'end': 304.611, 'text': "Now, how do you do this? There are many different methods, and I'm going to look at multiple methods in different videos.", 'start': 297.844, 'duration': 6.767}, {'end': 313.557, 'text': 'In this video, I would like to discuss the method called ordinary least squares.', 'start': 304.791, 'duration': 8.766}, {'end': 314.898, 'text': "I'm going to go over here, I'm going to write this down.", 'start': 313.577, 'duration': 1.321}, {'end': 316.979, 'text': 'Ordinary, oh boy, I got a little dizzy.', 'start': 315.178, 'duration': 1.801}, {'end': 318.54, 'text': "Everything's going to be okay.", 'start': 317.64, 'duration': 0.9}, {'end': 320.682, 'text': 'Least squares.', 'start': 318.66, 'duration': 2.022}, {'end': 323.023, 'text': 'What does that mean?', 'start': 322.283, 'duration': 0.74}], 'summary': 'Introduction to linear regression and ordinary least squares method for data analysis.', 'duration': 55.426, 'max_score': 267.597, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk267597.jpg'}, {'end': 383.871, 'src': 'embed', 'start': 349.252, 'weight': 2, 'content': [{'end': 361.159, 'text': 'And the idea of ordinary least squares is the least squares method is we want to find the line that minimizes all of these distances.', 'start': 349.252, 'duration': 11.907}, {'end': 383.871, 'text': 'So if we could think of all of these as data points like x0, x1, x2, x3, x4, we could think of all of these distances as like d0, d1, d2, d3, d4.', 'start': 361.579, 'duration': 22.292}], 'summary': 'Ordinary least squares minimizes distances to find the line.', 'duration': 34.619, 'max_score': 349.252, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk349252.jpg'}, {'end': 466.185, 'src': 'embed', 'start': 413.957, 'weight': 3, 'content': [{'end': 415.818, 'text': 'Well, this is a common technique.', 'start': 413.957, 'duration': 1.861}, {'end': 419.621, 'text': "You'll notice that some points are below the line and some points are above the line.", 'start': 416.078, 'duration': 3.543}, {'end': 422.123, 'text': 'So the difference could be positive or negative.', 'start': 419.881, 'duration': 2.242}, {'end': 424.745, 'text': 'Squaring it gets rid of that difference.', 'start': 422.583, 'duration': 2.162}, {'end': 427.553, 'text': 'Okay, so how do we do this?', 'start': 425.892, 'duration': 1.661}, {'end': 440.277, 'text': "Like I said, there are a variety of methods, and what I'm going to do is I'm going to show you a formula for this, which I have written down.", 'start': 428.973, 'duration': 11.304}, {'end': 443.058, 'text': "It's another historic moment.", 'start': 440.297, 'duration': 2.761}, {'end': 445.379, 'text': "I prepared for today's video.", 'start': 443.118, 'duration': 2.261}, {'end': 446.9, 'text': 'This was by preparation.', 'start': 445.959, 'duration': 0.941}, {'end': 447.96, 'text': 'I wrote down the formula.', 'start': 446.98, 'duration': 0.98}, {'end': 453.001, 'text': 'I could pretend that I memorized it by editing this out.', 'start': 450, 'duration': 3.001}, {'end': 456.502, 'text': 'If this is still in this video, then I did not pretend.', 'start': 453.681, 'duration': 2.821}, {'end': 459.683, 'text': "OK, so let's look at.", 'start': 457.563, 'duration': 2.12}, {'end': 466.185, 'text': 'so, first of all, how do we represent mathematically this pinkish reddish line?', 'start': 459.683, 'duration': 6.502}], 'summary': 'Explaining a formula for representing a line mathematically with preparations and historic moment.', 'duration': 52.228, 'max_score': 413.957, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk413957.jpg'}], 'start': 267.597, 'title': 'Linear regression', 'summary': 'Covers the concept of linear regression, with a focus on the ordinary least squares method for fitting a line to predict outcomes, and explores mathematical representation and plans for further methods in subsequent videos.', 'chapters': [{'end': 323.023, 'start': 267.597, 'title': 'Linear regression in data analysis', 'summary': 'Discusses the concept of linear regression, particularly focusing on the method of ordinary least squares for fitting a line to a dataset to predict outcomes, with plans to cover multiple methods in different videos.', 'duration': 55.426, 'highlights': ['The concept of linear regression involves fitting a line to a dataset to analyze and predict outcomes, illustrated with the example of predicting ice cream sales based on temperature.', 'The method of ordinary least squares is introduced as a technique for fitting the line to the dataset, with plans to cover multiple methods in different videos.']}, {'end': 466.185, 'start': 323.343, 'title': 'Ordinary least squares method', 'summary': 'Discusses the concept of ordinary least squares method for finding the line that minimizes the sum of squared differences between data points and the line, employing a common technique of squaring differences to eliminate positive and negative values, and explores the mathematical representation of the line.', 'duration': 142.842, 'highlights': ['The ordinary least squares method aims to find the line that minimizes the sum of squared differences between data points and the line, with the goal of minimizing the value obtained by adding up the squared differences, using a common technique of squaring differences to eliminate positive and negative values.', 'By squaring the differences between data points and the line, positive and negative differences are eliminated, allowing for a clearer representation of the overall differences and enabling the minimization of the sum of squared differences.', 'The chapter presents a discussion on the mathematical representation of the line using a formula, emphasizing the significance of preparation and providing insight into the process of memorizing key components.', 'Comparison of data points with the line involves examining the distance of each data point from the line, with the ultimate goal of finding a line that minimizes these distances.']}], 'duration': 198.588, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk267597.jpg', 'highlights': ['The concept of linear regression involves fitting a line to a dataset to analyze and predict outcomes, illustrated with the example of predicting ice cream sales based on temperature.', 'The method of ordinary least squares is introduced as a technique for fitting the line to the dataset, with plans to cover multiple methods in different videos.', 'The ordinary least squares method aims to find the line that minimizes the sum of squared differences between data points and the line, with the goal of minimizing the value obtained by adding up the squared differences, using a common technique of squaring differences to eliminate positive and negative values.', 'By squaring the differences between data points and the line, positive and negative differences are eliminated, allowing for a clearer representation of the overall differences and enabling the minimization of the sum of squared differences.', 'Comparison of data points with the line involves examining the distance of each data point from the line, with the ultimate goal of finding a line that minimizes these distances.', 'The chapter presents a discussion on the mathematical representation of the line using a formula, emphasizing the significance of preparation and providing insight into the process of memorizing key components.']}, {'end': 989.117, 'segs': [{'end': 523.812, 'src': 'heatmap', 'start': 466.526, 'weight': 0, 'content': [{'end': 473.328, 'text': 'So the formula for a line is typically written as y equals mx plus b.', 'start': 466.526, 'duration': 6.802}, {'end': 487.125, 'text': 'I will point out, however, that if you look in the statistics textbook, you might see something like y equals b0 plus b1 times x.', 'start': 474.522, 'duration': 12.603}, {'end': 488.905, 'text': 'This is the same exact formula.', 'start': 487.125, 'duration': 1.78}, {'end': 498.847, 'text': 'm refers to our b1 here as the slope, and b0 or b here is the quote unquote y-intercept.', 'start': 489.585, 'duration': 9.262}, {'end': 505.232, 'text': 'which is the value where the line intersects the y-axis.', 'start': 500.069, 'duration': 5.163}, {'end': 506.513, 'text': 'So the slope.', 'start': 505.453, 'duration': 1.06}, {'end': 510.716, 'text': 'this m value determines which way does the line point?', 'start': 506.513, 'duration': 4.203}, {'end': 515.219, 'text': 'and then the y-intercept is how high or low?', 'start': 510.716, 'duration': 4.503}, {'end': 518.26, 'text': 'where is that line relative to the x-axis?', 'start': 515.219, 'duration': 3.041}, {'end': 523.812, 'text': 'So all we need to do is we need to both calculate N and B.', 'start': 519.687, 'duration': 4.125}], 'summary': "The formula for a line y=mx+b is equivalent to y=b0+b1x in statistics, with m as slope and b as y-intercept, determining the line's direction and position relative to the x-axis.", 'duration': 39.987, 'max_score': 466.526, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk466526.jpg'}, {'end': 564.834, 'src': 'embed', 'start': 532.944, 'weight': 1, 'content': [{'end': 539.192, 'text': "There might be, there's temperature, there's population of the city that the store is in.", 'start': 532.944, 'duration': 6.248}, {'end': 541.875, 'text': "Maybe there's the hours that it's open.", 'start': 539.232, 'duration': 2.643}, {'end': 542.476, 'text': "I don't know.", 'start': 541.895, 'duration': 0.581}, {'end': 550.786, 'text': 'You could think of all sorts of other data inputs that might relate to the sale of ice cream.', 'start': 542.516, 'duration': 8.27}, {'end': 561.873, 'text': 'And this can actually be generalized much This could be y equals b0 plus b1 times x1 plus b2 times x2.', 'start': 551.267, 'duration': 10.606}, {'end': 564.834, 'text': 'So there could actually be multiple linear.', 'start': 562.093, 'duration': 2.741}], 'summary': 'Factors affecting ice cream sales include temperature, city population, store hours, and other related data inputs.', 'duration': 31.89, 'max_score': 532.944, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk532944.jpg'}, {'end': 802.572, 'src': 'heatmap', 'start': 771.127, 'weight': 0.713, 'content': [{'end': 773.489, 'text': 'Right? The slope would be 1 if y equals x.', 'start': 771.127, 'duration': 2.362}, {'end': 774.109, 'text': 'Well, look at this.', 'start': 773.489, 'duration': 0.62}, {'end': 780.474, 'text': 'If y equals x, then x minus x bar times x minus x bar would be like that squared.', 'start': 774.349, 'duration': 6.125}, {'end': 784.057, 'text': 'So you could see how m would equal 1 if y equals x.', 'start': 780.714, 'duration': 3.343}, {'end': 787.019, 'text': 'And then you could sort of see the numerator is essentially the correlation.', 'start': 784.057, 'duration': 2.962}, {'end': 791.903, 'text': "Thank you to Kay Weekbaugh in the chat for typing it out like that, because I think that's a good way of thinking about it.", 'start': 787.56, 'duration': 4.343}, {'end': 793.084, 'text': 'You could see that.', 'start': 792.143, 'duration': 0.941}, {'end': 802.572, 'text': 'is y growing more as x grows or is y growing less as x grows??', 'start': 795.005, 'duration': 7.567}], 'summary': 'The slope would be 1 if y equals x, and the numerator is essentially the correlation.', 'duration': 31.445, 'max_score': 771.127, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk771127.jpg'}, {'end': 913.205, 'src': 'embed', 'start': 866.793, 'weight': 3, 'content': [{'end': 870.954, 'text': 'The idea of a model is you try to fit it to the data.', 'start': 866.793, 'duration': 4.161}, {'end': 877.015, 'text': 'We have known data training, data temperature with actual sales.', 'start': 871.214, 'duration': 5.801}, {'end': 878.776, 'text': 'we want to fit our model.', 'start': 877.015, 'duration': 1.761}, {'end': 887.638, 'text': 'our model just has two parameters the slope of the line and the y-intercept, And once we solve for those parameters, we can make new predictions.', 'start': 878.776, 'duration': 8.862}, {'end': 896.22, 'text': "And even though it's kind of overly simplistic here, this is the exact same process that I'll employ again and again.", 'start': 888.098, 'duration': 8.122}, {'end': 903.941, 'text': 'Once we look at a simple perceptron, then a multi-layered perceptron and then things like convolutional networks or recurrent networks.', 'start': 896.88, 'duration': 7.061}, {'end': 910.843, 'text': 'all of this this is laying the foundation for more sophisticated, robust, machine learning-based systems.', 'start': 903.941, 'duration': 6.902}, {'end': 913.205, 'text': 'Oops, I forgot.', 'start': 911.483, 'duration': 1.722}], 'summary': 'Fitting a model to known data with 2 parameters allows for making new predictions, laying foundation for more sophisticated machine learning systems.', 'duration': 46.412, 'max_score': 866.793, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk866793.jpg'}, {'end': 951.872, 'src': 'heatmap', 'start': 921.933, 'weight': 0.866, 'content': [{'end': 926.216, 'text': "Once we have that slope, it's pretty easy to calculate the y-intercept.", 'start': 921.933, 'duration': 4.283}, {'end': 934.083, 'text': 'Where is b? And the formula for that is b equals y bar minus m times x bar.', 'start': 926.357, 'duration': 7.726}, {'end': 937.165, 'text': 'kind of see why this is the case.', 'start': 934.744, 'duration': 2.421}, {'end': 940.807, 'text': 'Because remember, y equals mx plus b.', 'start': 937.385, 'duration': 3.422}, {'end': 946.369, 'text': 'So all I need to do is say b equals y minus mx.', 'start': 940.807, 'duration': 5.562}, {'end': 951.872, 'text': 'And we could just use the average of all the x and all the y to figure out where should that line be shifted.', 'start': 946.529, 'duration': 5.343}], 'summary': 'Calculate y-intercept using b = y bar - m * x bar.', 'duration': 29.939, 'max_score': 921.933, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk921933.jpg'}], 'start': 466.526, 'title': 'Linear equations and regression', 'summary': 'Covers the formula for a line (y = mx + b), extension to multiple variables, calculation of slope and y-intercept in linear regression using the least squares method, and its application in fitting a model to known data for predictions.', 'chapters': [{'end': 564.834, 'start': 466.526, 'title': 'Linear equations and multiple variables', 'summary': 'Explains the formula for a line (y = mx + b), where m represents the slope and b represents the y-intercept. it also discusses the extension of the formula to include multiple variables in a linear equation.', 'duration': 98.308, 'highlights': ['The formula for a line is typically written as y = mx + b, where m represents the slope and b represents the y-intercept, which determines the direction and position of the line relative to the axes.', 'The formula can be extended to include multiple variables, such as y = b0 + b1x1 + b2x2, to accommodate more complex 2D data sets involving various factors like temperature, population, and store hours.']}, {'end': 989.117, 'start': 564.994, 'title': 'Linear regression and least squares', 'summary': 'Covers the calculation of slope and y-intercept in linear regression using the least squares method, and its application in fitting a model to known data for making predictions, laying the foundation for more sophisticated machine learning-based systems.', 'duration': 424.123, 'highlights': ['The chapter covers the calculation of slope and y-intercept in linear regression using the least squares method The formula for calculating the slope (m) involves summing the product of the differences between each x and its mean and each y and its mean, divided by the sum of the squared differences between each x and its mean.', 'Application in fitting a model to known data for making predictions Linear regression with the least squares method is used to fit the model to known data, with the aim of making predictions based on the calculated slope and y-intercept.', 'Laying the foundation for more sophisticated machine learning-based systems The chapter emphasizes that the concept of linear regression and the least squares method serves as the foundation for more sophisticated and robust machine learning-based systems, including perceptrons, multi-layered perceptrons, convolutional networks, and recurrent networks.']}], 'duration': 522.591, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/szXbuO3bVRk/pics/szXbuO3bVRk466526.jpg', 'highlights': ['The formula for a line is typically written as y = mx + b, where m represents the slope and b represents the y-intercept, which determines the direction and position of the line relative to the axes.', 'The formula can be extended to include multiple variables, such as y = b0 + b1x1 + b2x2, to accommodate more complex 2D data sets involving various factors like temperature, population, and store hours.', 'The chapter covers the calculation of slope and y-intercept in linear regression using the least squares method The formula for calculating the slope (m) involves summing the product of the differences between each x and its mean and each y and its mean, divided by the sum of the squared differences between each x and its mean.', 'Application in fitting a model to known data for making predictions Linear regression with the least squares method is used to fit the model to known data, with the aim of making predictions based on the calculated slope and y-intercept.', 'Laying the foundation for more sophisticated machine learning-based systems The chapter emphasizes that the concept of linear regression and the least squares method serves as the foundation for more sophisticated and robust machine learning-based systems, including perceptrons, multi-layered perceptrons, convolutional networks, and recurrent networks.']}], 'highlights': ['Linear regression is a method to precisely calculate the relationship between inputs and outputs in machine learning scenarios, before delving into the complexities of neural networks.', 'The video series aims to progressively cover topics, including neural networks, to further understand their capability in handling complex datasets.', 'The scenario of using historical sales data to make predictions about future sales based on temperature is mentioned, illustrating the practical application of linear regression for sales forecasting.', 'The ordinary least squares method aims to find the line that minimizes the sum of squared differences between data points and the line, with the goal of minimizing the value obtained by adding up the squared differences, using a common technique of squaring differences to eliminate positive and negative values.', 'The formula for a line is typically written as y = mx + b, where m represents the slope and b represents the y-intercept, which determines the direction and position of the line relative to the axes.', 'The formula can be extended to include multiple variables, such as y = b0 + b1x1 + b2x2, to accommodate more complex 2D data sets involving various factors like temperature, population, and store hours.', 'The chapter covers the calculation of slope and y-intercept in linear regression using the least squares method The formula for calculating the slope (m) involves summing the product of the differences between each x and its mean and each y and its mean, divided by the sum of the squared differences between each x and its mean.', 'Application in fitting a model to known data for making predictions Linear regression with the least squares method is used to fit the model to known data, with the aim of making predictions based on the calculated slope and y-intercept.', 'Laying the foundation for more sophisticated machine learning-based systems The chapter emphasizes that the concept of linear regression and the least squares method serves as the foundation for more sophisticated and robust machine learning-based systems, including perceptrons, multi-layered perceptrons, convolutional networks, and recurrent networks.', 'Graphing a two-dimensional data set with temperature on the x-axis and ice cream sales on the y-axis.', 'Example of ice cream sales at 24 degrees and 90 degrees, with 3 and 18 units sold respectively.', 'The importance of fitting a line to the data for accurate predictions is highlighted, emphasizing the need to use linear regression to determine the best-fit line for sales prediction.', 'The concept of a linear relationship between temperature and ice cream sales is discussed, with the higher temperature leading to more sales and the lower temperature leading to fewer sales.', 'The concept of linear regression involves fitting a line to a dataset to analyze and predict outcomes, illustrated with the example of predicting ice cream sales based on temperature.', 'The method of ordinary least squares is introduced as a technique for fitting the line to the dataset, with plans to cover multiple methods in different videos.', 'By squaring the differences between data points and the line, positive and negative differences are eliminated, allowing for a clearer representation of the overall differences and enabling the minimization of the sum of squared differences.', 'Comparison of data points with the line involves examining the distance of each data point from the line, with the ultimate goal of finding a line that minimizes these distances.', 'The chapter presents a discussion on the mathematical representation of the line using a formula, emphasizing the significance of preparation and providing insight into the process of memorizing key components.']}