From ab86c828346a8e85acf49da81144d6a3842a777a Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Thu, 6 Jan 2022 21:29:26 +0330 Subject: [PATCH 01/19] Add main structure --- notebooks/14_regression/index.md | 11 ++++++++ notebooks/14_regression/metadata.yml | 40 ++++++++++++++++++++++++++++ 2 files changed, 51 insertions(+) create mode 100644 notebooks/14_regression/index.md create mode 100644 notebooks/14_regression/metadata.yml diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md new file mode 100644 index 00000000..f3631617 --- /dev/null +++ b/notebooks/14_regression/index.md @@ -0,0 +1,11 @@ +# Regression + +# Contents + +# Introduction + + + +# Conclusion + +# Refrences \ No newline at end of file diff --git a/notebooks/14_regression/metadata.yml b/notebooks/14_regression/metadata.yml new file mode 100644 index 00000000..2fa2b019 --- /dev/null +++ b/notebooks/14_regression/metadata.yml @@ -0,0 +1,40 @@ +title: Regression + +header: + title: Regression + +authors: + label: + position: top + text: Authors + kind: people + content: + - name: Mehrshad Mirmohammadi + role: Author + contact: + - icon: fab fa-github + link: https://github.com/Helium-5 + - icon: fas fa-envelope + link: mailto:mehrshad.mirmohammadi@gmail.com + + - name: Nazanin Azarian + role: Author + contact: + - icon: fab fa-github + link: https://github.com/Nazhixx + - icon: fas fa-envelope + link: mailto:badi.mojgan@gmail.com + + - name: Pouya Esmaili + role: Author + contact: + - icon: fab fa-github + link: https://github.com/PouyaEsmaili + - icon: fas fa-envelope + link: mailto:poesmaili@gmail.com + + - name: Nima Jamali + role: Supervisor + contact: + - icon: fas fa-envelope + link: mailto:nimxj4141@gmail.com From fd438fefc44e29b913e8ac2b5bfea41e39c23945 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 7 Jan 2022 12:42:28 +0330 Subject: [PATCH 02/19] Add introduction --- notebooks/14_regression/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index f3631617..db7a9df0 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -3,7 +3,7 @@ # Contents # Introduction - +We have some data points and a number or a label is assigned to each data point. Our goal is to predict the number or label of an unseen data point after learning from the data we already have. We assume each data point is a vector $x$ and we want to predict $f(x)$. The first idea is to use interpolation. By using interpolation, we will have a high degree polynomial which fits our training data perfectly. But the problem is that, interpolation leads to overfitting. So the error for unseen data will be too large. In regression, we aim to find the best curve with lower degree. Although there will be some training error here, our test error will decrease since we are avoiding overfitting. # Conclusion From 511d89f2f9be983cee619e1cdef1c21989cfb044 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 7 Jan 2022 13:52:23 +0330 Subject: [PATCH 03/19] Add linear regression --- notebooks/14_regression/index.md | 69 +++++++++++++++++++++++++++++++- 1 file changed, 68 insertions(+), 1 deletion(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index db7a9df0..3553a379 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -5,7 +5,74 @@ # Introduction We have some data points and a number or a label is assigned to each data point. Our goal is to predict the number or label of an unseen data point after learning from the data we already have. We assume each data point is a vector $x$ and we want to predict $f(x)$. The first idea is to use interpolation. By using interpolation, we will have a high degree polynomial which fits our training data perfectly. But the problem is that, interpolation leads to overfitting. So the error for unseen data will be too large. In regression, we aim to find the best curve with lower degree. Although there will be some training error here, our test error will decrease since we are avoiding overfitting. +# Linear Regression +Here, we want to assign $f(x)$ to each data point $x$. In linear regression, we assume $f$ is a linear function. We can define $f$ as + +$$ + f_w(x) = w^T x = \begin{bmatrix} + w_0 & w_1 & \cdots & w_n + \end{bmatrix} . \begin{bmatrix} + 1 \\ x_1 \\ \vdots \\ x_n + \end{bmatrix}. +$$ +We assumed $x_0 = 1$ in $x$ to have bias in our function. So the data points are in an n-dimentional space. + +We also define $y$ as + +$$ + y_w = \begin{bmatrix} + f_w(x^{(1)}) \\ f_w(x^{(2)}) \\ \vdots \\ f_w(x^{(m)}) + \end{bmatrix} +$$ +where $x^{(i)}$ is the i'th data point. + +## Loss Function +After defining $f$, we need to find the best function. By defining a loss function, we can try to minimize loss by changing $w$ in the main function. Assuming $L$ as our loss function, best $f$ will be + +$$ + \hat{w} = argmin_w L(y_w, \hat{y}) \rightarrow f_{best}(x) = f_{\hat{w}}(x) +$$ + +where $\hat{y}$ is the given number for each data point. + +### Mean Squared Error (MSE) +The main loss function we use is mean squared error. It's defined as + +$$ + MSE(y_w, \hat{y}) = \frac{1}{2} \Sigma_{i=1}^m \left[ y_w^{(i)} - \hat{y}^{(i)} \right]^2 . +$$ +The main reason for using this function is that we can calculate gradient easily. So we can use gradient descent to find $\hat{w}$. + +## Finding $\hat{w}$ +We want to use gradient descent to find $\hat{w}$. First, we need to calculate $\nabla_w L(y_w, \hat{y})$ because it's used in the gradient descent method. The partial derivitives for MSE are: + +$$ + \frac{\partial MSE}{\partial w_j} = - \Sigma_{i = 1}^m x_j^{(i)} \left[ y_w^{(i)} - \hat{y}^{(i)} \right] +$$ + +So for gradinet, we have: + +$$ + \nabla_w MSE = \begin{bmatrix} + \frac{\partial MSE}{\partial w_0} \\ \vdots \\ \frac{\partial MSE}{\partial w_n} + \end{bmatrix} +$$ + +Now, we can find $\hat{w}$ by the gradient descent algorithm as + +$$ + w^{(i+1)} = w^{(i)} - \eta \nabla_{w^{(i)}} L +$$ + +where $\eta$ is the learning rate. + +# N'th Order Polynomial + + +# Logistic Regression # Conclusion -# Refrences \ No newline at end of file +# Refrences +- https://en.wikipedia.org/wiki/Linear_regression +- https://math.stackexchange.com/questions/1962877/compute-the-gradient-of-mean-square-error \ No newline at end of file From e70aa0207611eb56cd662b34f824c37643e23d82 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 7 Jan 2022 13:56:20 +0330 Subject: [PATCH 04/19] Update index.yml --- notebooks/index.yml | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 1b2133c0..943336a5 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -54,4 +54,6 @@ notebooks: kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ - kind: S2021, LN, Notebook \ No newline at end of file + kind: S2021, LN, Notebook + - md: notebooks/14_regression + test: Regression \ No newline at end of file From 2dd3978fec05fc3a776f159a7e663d7470ef58d3 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 7 Jan 2022 13:57:40 +0330 Subject: [PATCH 05/19] Update index.yml --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 943336a5..29b5bd4a 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -56,4 +56,4 @@ notebooks: - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - md: notebooks/14_regression - test: Regression \ No newline at end of file + text: Regression \ No newline at end of file From a7a279cf79c8b61ce7cb1a915b52691b36000e1a Mon Sep 17 00:00:00 2001 From: mirmohamadi Date: Fri, 7 Jan 2022 14:36:07 +0330 Subject: [PATCH 06/19] Adds general contents structure --- notebooks/14_regression/index.md | 31 +++++++++++++++++++++++++++++++ 1 file changed, 31 insertions(+) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index 3553a379..5d7d92d5 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -68,9 +68,40 @@ where $\eta$ is the learning rate. # N'th Order Polynomial +# -------------------------------------------------------------------------- # Logistic Regression +# 1 - What is regression ? + +## Problem definition + +# 2 - Linear Regression + +## definitions + +## closed form equation + +## Solving using GD + +# 3 - Learning Curves using Polynomials + +reduction to Linear regression + +## overfitting + +# 4 - Regularization + +## ridge (OPTIONAL) + +## lasso (OPTIONAL) + +## elastic (OPTIONAL) + +# 5 - Logistic Regression + + + # Conclusion # Refrences From de6b19ebe44cddb78c2cd33847e99353ab3937a4 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 7 Jan 2022 19:18:33 +0330 Subject: [PATCH 07/19] Finish linear regression --- notebooks/14_regression/index.md | 36 ++++++++++++++++++-------------- 1 file changed, 20 insertions(+), 16 deletions(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index 5d7d92d5..a85fd578 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -5,7 +5,12 @@ # Introduction We have some data points and a number or a label is assigned to each data point. Our goal is to predict the number or label of an unseen data point after learning from the data we already have. We assume each data point is a vector $x$ and we want to predict $f(x)$. The first idea is to use interpolation. By using interpolation, we will have a high degree polynomial which fits our training data perfectly. But the problem is that, interpolation leads to overfitting. So the error for unseen data will be too large. In regression, we aim to find the best curve with lower degree. Although there will be some training error here, our test error will decrease since we are avoiding overfitting. -# Linear Regression +# 1 - What is regression ? + +## Problem definition + +# 2 - Linear Regression + Here, we want to assign $f(x)$ to each data point $x$. In linear regression, we assume $f$ is a linear function. We can define $f$ as $$ @@ -44,6 +49,8 @@ $$ The main reason for using this function is that we can calculate gradient easily. So we can use gradient descent to find $\hat{w}$. ## Finding $\hat{w}$ + +### Gradient Descent We want to use gradient descent to find $\hat{w}$. First, we need to calculate $\nabla_w L(y_w, \hat{y})$ because it's used in the gradient descent method. The partial derivitives for MSE are: $$ @@ -66,23 +73,20 @@ $$ where $\eta$ is the learning rate. -# N'th Order Polynomial - -# -------------------------------------------------------------------------- - -# Logistic Regression - -# 1 - What is regression ? +### Normal Equation +If we define -## Problem definition - -# 2 - Linear Regression - -## definitions - -## closed form equation +$$ + X = \begin{bmatrix} + x^{(1)} \\ \vdots \\ x^{(m)} + \end{bmatrix} +$$ -## Solving using GD +and solve the equation $\nabla_w MSE = 0$, we get the normal equation. It's defined as +$$ + \hat{w} = (X^TX)^{-1} X^T \hat{y}. +$$ +However, for using this equation, our features must be linearly independent. Otherwise, $(X^TX)^{-1}$ is not defined. In that case we can use pseudo inverse of $X^TX$ instead of the $(X^TX)^{-1}$. Since the calculation of inverse is computationaly inefficient, we usually prefer using gradient descent. # 3 - Learning Curves using Polynomials From ed943d376509734d1fd71dcd5de2fbcff594ebd1 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 7 Jan 2022 19:41:22 +0330 Subject: [PATCH 08/19] Add polynomials --- notebooks/14_regression/index.md | 39 ++++++++++++++++++++++++++++---- 1 file changed, 35 insertions(+), 4 deletions(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index a85fd578..59f10d24 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -88,11 +88,41 @@ $$ $$ However, for using this equation, our features must be linearly independent. Otherwise, $(X^TX)^{-1}$ is not defined. In that case we can use pseudo inverse of $X^TX$ instead of the $(X^TX)^{-1}$. Since the calculation of inverse is computationaly inefficient, we usually prefer using gradient descent. -# 3 - Learning Curves using Polynomials +# 3 - Learning Curves Using Polynomials + +In this section, we want to find the best $P(x)$ where x is a real number and $P$ is an n'th degree polynomial. + +## Reduction to Linear Regression + +We can define +$$ + z = \begin{bmatrix} + x^0 \\ x^1 \\ \vdots \\ x^n + \end{bmatrix}, +$$ +then use linear regression to find $f_w(z) = w^T z$. From definition, we know that $P(x) = f_w(z)$. However, if $n$ is too large, overfitting might happen since we are getting closer and closer to interpolation. + +## Overfitting +To prevent overfitting, we must try to define $n$ optimaly. Since $n$ is a hyperparameter here, we can use validation set. + +### Using Validation Set (Held-Out Data) +We split a part of the training data, and don't use it for training. Then, by calculating loss function over this data, we can optimize $n$. Since the model hasn't seen these data points, we can be sure that overfitting will decrease. + +### Regularization +Although using validation set is a good way to prevent overfitting, We still might be trying to find the curve which best fits the data, not the curve which best matches it. Regularization tries to make $w$ smaller. The intuition is that large coefficients in $w$ happen because it tries to fit the points. It means that the curve will only get closer to each point. However, decreasing coeffiecients will make a better curve which might be further from each point, but does a better job at predicting the unseen data. + +We only define $l_2-\text{regularization}$ since it's easier to derive and understand. The loss function is + +$$ + \tilde{E}(y_w, \hat{y}) = \frac{1}{2} \Sigma_{i=1}^m \left[ y_w^{(i)} - \hat{y}^{(i)} \right]^2 + \frac{\lambda}{2}||w||^2 +$$ +where $\lambda$ is hyperparameter which controls how small the $w$ should be. + + + +********* Examples ************* -reduction to Linear regression -## overfitting # 4 - Regularization @@ -110,4 +140,5 @@ reduction to Linear regression # Refrences - https://en.wikipedia.org/wiki/Linear_regression -- https://math.stackexchange.com/questions/1962877/compute-the-gradient-of-mean-square-error \ No newline at end of file +- https://math.stackexchange.com/questions/1962877/compute-the-gradient-of-mean-square-error +- https://towardsdatascience.com/normal-equation-in-python-the-closed-form-solution-for-linear-regression-13df33f9ad71 \ No newline at end of file From 032ea314f6406c2b04044599e63e3382fe6b63a2 Mon Sep 17 00:00:00 2001 From: mirmohamadi Date: Fri, 7 Jan 2022 20:44:32 +0330 Subject: [PATCH 09/19] Adds Logistic Regression v1.0 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be. # 5 - Logistic Regression -There are some regression algorithms that can be used for classifiction (And vise versa). _Logistic Regression_ is one of the these algorithms. It can calculated the probability that an instance belongs to a particular class. we can use this probability for classification : if the probability is higher thatn 0.5 then that instance belongs to the particular class. +There are some regression algorithms that can be used for classification (And vise versa). _Logistic Regression_ is one of these algorithms. It can calculate the probability that an instance belongs to a particular class. we can use this probability for classification: if the probability is higher than 0.5 then that instance belongs to the particular class. ## How to calculate probabilities: -Let's start with a simple example. We want to classify flowers that are _Iris Virginica_ from those that are _not Iris Virginica_. This classification is based on Petal Width and Petal Height. Also consider that there is some line _l_(black line in the image below) that seperates these 2 types of flowers. +Let's start with a simple example. We want to classify flowers that are _Iris Virginica_ from those that are _not Iris Virginica_. This classification is based on Petal Width and Petal Height. Also, consider that there is some line _l_ (black line in the image below) that separates these 2 types of flowers. ![image info](./imgs/Logistic%20Regression.jpg) -Now, to calculate the probabilities, we use the intiation that, the further we are from _l_, the heigher is the probability of belonging to that particular class. -To be more exact, consider we want to find the probability of a given flower being _Iris Virginica_. Let's call the signed distance between the flower and the line, _t_. Signed distance means if we are in the _Iris Virginica_ region, the distance is positive, other wise if we are in the _not Iris Virginica_ region, the distance is negetive. The probability of this flower being _Iris Virginica_ is related to the value of _t_, the heigher the value of _t_, the heigher the probability. So we need a function $\sigma(.)$ that when given the singed distance _t_, it returns the probability _p_. -Logistic function is a commen choise for this purpose. It is a sigmoid function (i.e., S-shaped) that outputs a number between 0 and 1. +Now, to calculate the probabilities, we use the intuition that, the further we are from _l_, the higher is the probability of belonging to that particular class. +To be more exact, consider we want to find the probability of a given flower being _Iris Virginica_. Let's call the signed distance between the flower and the line, _t_. Signed-distance means if we are in the _Iris Virginica_ region, the distance is positive, otherwise, if we are in the _not Iris Virginica_ region, the distance is negative. The probability of this flower being _Iris Virginica_ is related to the value of _t_, the higher the value of _t_, the higher the probability. So we need a function $\sigma(.)$ that when given the singed distance _t_, returns the probability _p_. +The logistic function is a common choice for this purpose. It is a sigmoid function (i.e., S-shaped) that outputs a number between 0 and 1. Logistic function: $$ \sigma(t) = \frac{1}{1 + \exp(-t)} $$ +In the image above, you can see the probability of colored lines. For example, all the points on the green line have a probability of 0.9. These probabilities are calculated using the Logistic function. +Also in the image below, you can see the value of the Logistic function for different inputs: + ![image info](./imgs/Logistic%20function.jpg) ## defining line _l_ and cost function $J(\theta)$: -Now we know how to calculate the probabilities, the problem is reduced to finding the best line _l_ that seperates 2 classes. For this First, we need to define the line and Second, we need to define a coss function $J(\theta)$ and then we should find the line _l_ such that it minimizes $J(\theta)$. +Now we know how to calculate the probabilities, the problem is reduced to finding the best line _l_ that separates 2 classes. For this First, we need to define the line and Second, we need to define a coss function $J(\theta)$ and then we should find the line _l_ such that it minimizes $J(\theta)$. We define line _l_ with parameters $\theta$ like before. With this definition, the signed distance between x and the _l_ can be calculated as follows: $$ @@ -180,11 +183,11 @@ And for _N_ data points, we can define cost function as: $$ J(\theta) = -\frac{1}{N}\sum_{i = 1}^{N}log(\hat{P}^{i})y^{i} -log(1 - \hat{P}^{i})(1 - y^{i}) $$ -where $\hat{P}^{i}$ is the probability predicted and $y^{i}$ is the label for the i-th data point given the line _l_ with parameters $\theta$. +where $\hat{P}^{i}$ is the probability predicted and $y^{i}$ is the label for the $i^{th}$ data point given the line _l_ with parameters $\theta$. ## Training: -Unfortunatly, there is no known closed-form equation to compute the value of $\theta$ that minimizes this cost function. But the good news is that this cost function is convex! So we use Gradient Descent to find the minimum. Because of convexity, Gradient Descent is guaranteed to find the global minimum (with the right learning rate). +Unfortunately, there is no known closed-form equation to compute the value of $\theta$ that minimizes this cost function. But the good news is that this cost function is convex! So we can use Gradient Descent to find the minimum. Because of convexity, Gradient Descent is guaranteed to find the global minimum (with the right learning rate). We can compute the partial derivatives for the $j^{th}$ parameter of $\theta$ as follows: $$ @@ -196,4 +199,5 @@ $$ # Refrences - https://en.wikipedia.org/wiki/Linear_regression - https://math.stackexchange.com/questions/1962877/compute-the-gradient-of-mean-square-error -- https://towardsdatascience.com/normal-equation-in-python-the-closed-form-solution-for-linear-regression-13df33f9ad71 \ No newline at end of file +- https://towardsdatascience.com/normal-equation-in-python-the-closed-form-solution-for-linear-regression-13df33f9ad71 +- Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems - 2th Edition \ No newline at end of file From 34c3cd057e9a0ca951b4811f189f3906683a0c35 Mon Sep 17 00:00:00 2001 From: mirmohamadi Date: Fri, 7 Jan 2022 21:17:56 +0330 Subject: [PATCH 11/19] Adds table of contents v1.0 --- notebooks/14_regression/index.md | 29 ++++++++++++++++++++++++++++- 1 file changed, 28 insertions(+), 1 deletion(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index cc854718..3deb0328 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -1,6 +1,33 @@ # Regression -# Contents +# Table of Contents + +- [Regression](#regression) +- [Table of Contents](#table-of-contents) +- [Introduction](#introduction) +- [1 - What is regression ?](#1---what-is-regression-) + - [Problem definition](#problem-definition) +- [2 - Linear Regression](#2---linear-regression) + - [Loss Function](#loss-function) + - [Mean Squared Error (MSE)](#mean-squared-error-mse) + - [Finding $\hat{w}$](#finding-hatw) + - [Gradient Descent](#gradient-descent) + - [Normal Equation](#normal-equation) +- [3 - Learning Curves Using Polynomials](#3---learning-curves-using-polynomials) + - [Reduction to Linear Regression](#reduction-to-linear-regression) + - [Overfitting](#overfitting) + - [Using Validation Set (Held-Out Data)](#using-validation-set-held-out-data) + - [Regularization](#regularization) +- [4 - Regularization](#4---regularization) + - [ridge (OPTIONAL)](#ridge-optional) + - [lasso (OPTIONAL)](#lasso-optional) + - [elastic (OPTIONAL)](#elastic-optional) +- [5 - Logistic Regression](#5---logistic-regression) + - [How to calculate probabilities:](#how-to-calculate-probabilities) + - [defining line _l_ and cost function $J(\theta)$:](#defining-line-l-and-cost-function-jtheta) + - [Training:](#training) +- [Conclusion](#conclusion) +- [Refrences](#refrences) # Introduction We have some data points and a number or a label is assigned to each data point. Our goal is to predict the number or label of an unseen data point after learning from the data we already have. We assume each data point is a vector $x$ and we want to predict $f(x)$. The first idea is to use interpolation. By using interpolation, we will have a high degree polynomial which fits our training data perfectly. But the problem is that, interpolation leads to overfitting. So the error for unseen data will be too large. In regression, we aim to find the best curve with lower degree. Although there will be some training error here, our test error will decrease since we are avoiding overfitting. 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z%SM%8??AErQU-4Cx`2(2xoOr>1NY@MihpPPn2W02jgZd@PH$adxfQpq+PLGir#k=U zTgnCV`YHc0Qi0VZm)&b63oPO#Gd;LZ#Qm4GIvNZ4j&uYu@$&r~r9dKk!Sw!z^zDDm h`~G(iazeY#$$Dz5-*5~k2mqrbQG%$yECIg`_zzzyci{j4 literal 0 HcmV?d00001 diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index 3deb0328..d5f2f9c0 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -5,38 +5,28 @@ - [Regression](#regression) - [Table of Contents](#table-of-contents) - [Introduction](#introduction) -- [1 - What is regression ?](#1---what-is-regression-) - - [Problem definition](#problem-definition) -- [2 - Linear Regression](#2---linear-regression) +- [1 - Linear Regression](#2---linear-regression) - [Loss Function](#loss-function) - [Mean Squared Error (MSE)](#mean-squared-error-mse) - [Finding $\hat{w}$](#finding-hatw) - [Gradient Descent](#gradient-descent) - [Normal Equation](#normal-equation) -- [3 - Learning Curves Using Polynomials](#3---learning-curves-using-polynomials) +- [2 - Learning Curves Using Polynomials](#3---learning-curves-using-polynomials) - [Reduction to Linear Regression](#reduction-to-linear-regression) - [Overfitting](#overfitting) - [Using Validation Set (Held-Out Data)](#using-validation-set-held-out-data) - [Regularization](#regularization) -- [4 - Regularization](#4---regularization) - - [ridge (OPTIONAL)](#ridge-optional) - - [lasso (OPTIONAL)](#lasso-optional) - - [elastic (OPTIONAL)](#elastic-optional) -- [5 - Logistic Regression](#5---logistic-regression) - - [How to calculate probabilities:](#how-to-calculate-probabilities) - - [defining line _l_ and cost function $J(\theta)$:](#defining-line-l-and-cost-function-jtheta) - - [Training:](#training) +- [3 - Logistic Regression](#5---logistic-regression) + - [How to calculate probabilities](#how-to-calculate-probabilities) + - [defining line _l_ and cost function $J(\theta)$](#defining-line-l-and-cost-function-jtheta) + - [Training](#training) - [Conclusion](#conclusion) - [Refrences](#refrences) # Introduction We have some data points and a number or a label is assigned to each data point. Our goal is to predict the number or label of an unseen data point after learning from the data we already have. We assume each data point is a vector $x$ and we want to predict $f(x)$. The first idea is to use interpolation. By using interpolation, we will have a high degree polynomial which fits our training data perfectly. But the problem is that, interpolation leads to overfitting. So the error for unseen data will be too large. In regression, we aim to find the best curve with lower degree. Although there will be some training error here, our test error will decrease since we are avoiding overfitting. -# 1 - What is regression ? - -## Problem definition - -# 2 - Linear Regression +# 1 - Linear Regression Here, we want to assign $f(x)$ to each data point $x$. In linear regression, we assume $f$ is a linear function. We can define $f$ as @@ -115,7 +105,7 @@ $$ $$ However, for using this equation, our features must be linearly independent. Otherwise, $(X^TX)^{-1}$ is not defined. In that case we can use pseudo inverse of $X^TX$ instead of the $(X^TX)^{-1}$. Since the calculation of inverse is computationaly inefficient, we usually prefer using gradient descent. -# 3 - Learning Curves Using Polynomials +# 2 - Learning Curves Using Polynomials In this section, we want to find the best $P(x)$ where x is a real number and $P$ is an n'th degree polynomial. @@ -132,9 +122,17 @@ then use linear regression to find $f_w(z) = w^T z$. From definition, we know th ## Overfitting To prevent overfitting, we must try to define $n$ optimaly. Since $n$ is a hyperparameter here, we can use validation set. +The example below demonstrates how $n$ can change curve we find and when overfitting happens. The green curve is our goal. The red curve is the M'th degree polynomial we using this method. + +![M = 1](images/1.png) +![M = 3](images/3.png) +![M = 9](images/9.png) + ### Using Validation Set (Held-Out Data) We split a part of the training data, and don't use it for training. Then, by calculating loss function over this data, we can optimize $n$. Since the model hasn't seen these data points, we can be sure that overfitting will decrease. +We can see in the above examples that $M = 3$ gives us the best curve. If we compare the validation loss for $M=3$ and $M=9$, the loss will be less for $M=3$ since it's closer to the green curve. + ### Regularization Although using validation set is a good way to prevent overfitting, We still might be trying to find the curve which best fits the data, not the curve which best matches it. Regularization tries to make $w$ smaller. The intuition is that large coefficients in $w$ happen because it tries to fit the points. It means that the curve will only get closer to each point. However, decreasing coeffiecients will make a better curve which might be further from each point, but does a better job at predicting the unseen data. @@ -145,29 +143,20 @@ $$ $$ where $\lambda$ is hyperparameter which controls how small the $w$ should be. +In the following examples, we use regularization with 9'th degree polynomials. We can compare these with the overfitted example above. Also, we can see how imoprtant the choice of $\lambda$ is. So, using validation set might be still usefull to optimize $\lambda$. +![Reguralization 18](images/r18.png) +![Reguralization 0](images/r0.png) -********* Examples ************* - - - -# 4 - Regularization - -## ridge (OPTIONAL) - -## lasso (OPTIONAL) - -## elastic (OPTIONAL) - -# 5 - Logistic Regression +# 3 - Logistic Regression There are some regression algorithms that can be used for classification (And vise versa). _Logistic Regression_ is one of these algorithms. It can calculate the probability that an instance belongs to a particular class. we can use this probability for classification: if the probability is higher than 0.5 then that instance belongs to the particular class. -## How to calculate probabilities: +## How to calculate probabilities Let's start with a simple example. We want to classify flowers that are _Iris Virginica_ from those that are _not Iris Virginica_. This classification is based on Petal Width and Petal Height. Also, consider that there is some line _l_ (black line in the image below) that separates these 2 types of flowers. -![image info](./imgs/Logistic%20Regression.jpg) +![image info](./images/Logistic%20Regression.jpg) Now, to calculate the probabilities, we use the intuition that, the further we are from _l_, the higher is the probability of belonging to that particular class. To be more exact, consider we want to find the probability of a given flower being _Iris Virginica_. Let's call the signed distance between the flower and the line, _t_. Signed-distance means if we are in the _Iris Virginica_ region, the distance is positive, otherwise, if we are in the _not Iris Virginica_ region, the distance is negative. The probability of this flower being _Iris Virginica_ is related to the value of _t_, the higher the value of _t_, the higher the probability. So we need a function $\sigma(.)$ that when given the singed distance _t_, returns the probability _p_. @@ -181,9 +170,9 @@ $$ In the image above, you can see the probability of colored lines. For example, all the points on the green line have a probability of 0.9. These probabilities are calculated using the Logistic function. Also in the image below, you can see the value of the Logistic function for different inputs: -![image info](./imgs/Logistic%20function.jpg) +![image info](./images/Logistic%20function.jpg) -## defining line _l_ and cost function $J(\theta)$: +## defining line _l_ and cost function $J(\theta)$ Now we know how to calculate the probabilities, the problem is reduced to finding the best line _l_ that separates 2 classes. For this First, we need to define the line and Second, we need to define a coss function $J(\theta)$ and then we should find the line _l_ such that it minimizes $J(\theta)$. @@ -212,7 +201,7 @@ $$ $$ where $\hat{P}^{i}$ is the probability predicted and $y^{i}$ is the label for the $i^{th}$ data point given the line _l_ with parameters $\theta$. -## Training: +## Training Unfortunately, there is no known closed-form equation to compute the value of $\theta$ that minimizes this cost function. But the good news is that this cost function is convex! So we can use Gradient Descent to find the minimum. Because of convexity, Gradient Descent is guaranteed to find the global minimum (with the right learning rate). We can compute the partial derivatives for the $j^{th}$ parameter of $\theta$ as follows: @@ -224,7 +213,8 @@ $$ # Conclusion # Refrences +- Artificial Inteligence Course at Sharif University of Technology (Fall, 2021) +- Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems - 2th Edition - https://en.wikipedia.org/wiki/Linear_regression - https://math.stackexchange.com/questions/1962877/compute-the-gradient-of-mean-square-error -- https://towardsdatascience.com/normal-equation-in-python-the-closed-form-solution-for-linear-regression-13df33f9ad71 -- Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems - 2th Edition \ No newline at end of file +- https://towardsdatascience.com/normal-equation-in-python-the-closed-form-solution-for-linear-regression-13df33f9ad71 \ No newline at end of file From aa33bdf5a5f607ea7bdee212cafed91c35daf284 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 7 Jan 2022 22:03:39 +0330 Subject: [PATCH 13/19] Add conclusion --- notebooks/14_regression/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index d5f2f9c0..303dcf42 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -212,6 +212,8 @@ $$ # Conclusion +Regression is used in a variaty of AI problems. We can use it to find the best line or curve to predict a function, or to find the best hyperplane dividing 2 classes in a classification problem. In this lecture note, we discussed how to solve these problems and how to prevent overfitting. However, for more complex functions or more classes, you might want to take a look at neural networks. + # Refrences - Artificial Inteligence Course at Sharif University of Technology (Fall, 2021) - Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems - 2th Edition From f4a5e1c4da549d013e8d8ab37855b07ea9580ba3 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 7 Jan 2022 23:15:47 +0330 Subject: [PATCH 14/19] Fix dictations --- notebooks/14_regression/index.md | 24 ++++++++++++------------ 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index 303dcf42..8ae4e71e 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -24,7 +24,7 @@ - [Refrences](#refrences) # Introduction -We have some data points and a number or a label is assigned to each data point. Our goal is to predict the number or label of an unseen data point after learning from the data we already have. We assume each data point is a vector $x$ and we want to predict $f(x)$. The first idea is to use interpolation. By using interpolation, we will have a high degree polynomial which fits our training data perfectly. But the problem is that, interpolation leads to overfitting. So the error for unseen data will be too large. In regression, we aim to find the best curve with lower degree. Although there will be some training error here, our test error will decrease since we are avoiding overfitting. +In regression, we have some data points and a number or a label is assigned to each data point. Our goal is to predict the number or label of an unseen data point after learning from the data we already have. We assume each data point is a vector $x$ and we want to predict $f(x)$. The first idea is to use interpolation. By using interpolation, we will have a high degree polynomial that fits our training data perfectly. But the problem is that interpolation leads to overfitting. So the error for unseen data will be too large. In regression, we aim to find the best curve with a lower degree. Although there will be some training error here, our test error will decrease since we are avoiding overfitting. # 1 - Linear Regression @@ -37,7 +37,7 @@ $$ 1 \\ x_1 \\ \vdots \\ x_n \end{bmatrix}. $$ -We assumed $x_0 = 1$ in $x$ to have bias in our function. So the data points are in an n-dimentional space. +We assumed $x_0 = 1$ in $x$ to have bias in our function. So the data points are in an n-dimensional space. We also define $y$ as @@ -63,12 +63,12 @@ The main loss function we use is mean squared error. It's defined as $$ MSE(y_w, \hat{y}) = \frac{1}{2} \Sigma_{i=1}^m \left[ y_w^{(i)} - \hat{y}^{(i)} \right]^2 . $$ -The main reason for using this function is that we can calculate gradient easily. So we can use gradient descent to find $\hat{w}$. +The main reason for using this function is that we can calculate gradients easily. So we can use gradient descent to find $\hat{w}$. ## Finding $\hat{w}$ ### Gradient Descent -We want to use gradient descent to find $\hat{w}$. First, we need to calculate $\nabla_w L(y_w, \hat{y})$ because it's used in the gradient descent method. The partial derivitives for MSE are: +We want to use gradient descent to find $\hat{w}$. First, we need to calculate $\nabla_w L(y_w, \hat{y})$ because it's used in the gradient descent method. The partial derivatives for MSE are: $$ \frac{\partial MSE}{\partial w_j} = - \Sigma_{i = 1}^m x_j^{(i)} \left[ y_w^{(i)} - \hat{y}^{(i)} \right] @@ -103,7 +103,7 @@ and solve the equation $\nabla_w MSE = 0$, we get the normal equation. It's defi $$ \hat{w} = (X^TX)^{-1} X^T \hat{y}. $$ -However, for using this equation, our features must be linearly independent. Otherwise, $(X^TX)^{-1}$ is not defined. In that case we can use pseudo inverse of $X^TX$ instead of the $(X^TX)^{-1}$. Since the calculation of inverse is computationaly inefficient, we usually prefer using gradient descent. +However, for using this equation, our features must be linearly independent. Otherwise, $(X^TX)^{-1}$ is not defined. In that case, we can use the pseudo inverse of $X^TX$ instead of the $(X^TX)^{-1}$. Since the calculation of inverse is computationally inefficient, we usually prefer using gradient descent. # 2 - Learning Curves Using Polynomials @@ -120,30 +120,30 @@ $$ then use linear regression to find $f_w(z) = w^T z$. From definition, we know that $P(x) = f_w(z)$. However, if $n$ is too large, overfitting might happen since we are getting closer and closer to interpolation. ## Overfitting -To prevent overfitting, we must try to define $n$ optimaly. Since $n$ is a hyperparameter here, we can use validation set. +To prevent overfitting, we must try to define $n$ optimally. Since $n$ is a hyperparameter here, we can use validation set. -The example below demonstrates how $n$ can change curve we find and when overfitting happens. The green curve is our goal. The red curve is the M'th degree polynomial we using this method. +The example below demonstrates how $n$ can change the curve we find and when overfitting happens. The green curve is our goal. The red curve is the M'th degree polynomial using this method. ![M = 1](images/1.png) ![M = 3](images/3.png) ![M = 9](images/9.png) ### Using Validation Set (Held-Out Data) -We split a part of the training data, and don't use it for training. Then, by calculating loss function over this data, we can optimize $n$. Since the model hasn't seen these data points, we can be sure that overfitting will decrease. +We split a part of the training data, and don't use it for training. Then, by calculating the loss function over this data, we can optimize $n$. Since the model hasn't seen these data points, we can be sure that overfitting will decrease. We can see in the above examples that $M = 3$ gives us the best curve. If we compare the validation loss for $M=3$ and $M=9$, the loss will be less for $M=3$ since it's closer to the green curve. ### Regularization -Although using validation set is a good way to prevent overfitting, We still might be trying to find the curve which best fits the data, not the curve which best matches it. Regularization tries to make $w$ smaller. The intuition is that large coefficients in $w$ happen because it tries to fit the points. It means that the curve will only get closer to each point. However, decreasing coeffiecients will make a better curve which might be further from each point, but does a better job at predicting the unseen data. +Although using a validation set is a good way to prevent overfitting, We still might be trying to find the curve which best fits the data, not the curve which best matches it. Regularization tries to make $w$ smaller. The intuition is that large coefficients in $w$ happen because it tries to fit the points. It means that the curve will only get closer to each point. However, decreasing coefficients will make a better curve which might be further from each point, but does a better job at predicting the unseen data. We only define $l_2-\text{regularization}$ since it's easier to derive and understand. The loss function is $$ \tilde{E}(y_w, \hat{y}) = \frac{1}{2} \Sigma_{i=1}^m \left[ y_w^{(i)} - \hat{y}^{(i)} \right]^2 + \frac{\lambda}{2}||w||^2 $$ -where $\lambda$ is hyperparameter which controls how small the $w$ should be. +where $\lambda$ is a hyperparameter that controls how small the $w$ should be. -In the following examples, we use regularization with 9'th degree polynomials. We can compare these with the overfitted example above. Also, we can see how imoprtant the choice of $\lambda$ is. So, using validation set might be still usefull to optimize $\lambda$. +In the following examples, we use regularization with 9'th degree polynomials. We can compare these with the overfitted example above. Also, we can see how important the choice of $\lambda$ is. So, using the validation set might be still useful to optimize $\lambda$. ![Reguralization 18](images/r18.png) ![Reguralization 0](images/r0.png) @@ -212,7 +212,7 @@ $$ # Conclusion -Regression is used in a variaty of AI problems. We can use it to find the best line or curve to predict a function, or to find the best hyperplane dividing 2 classes in a classification problem. In this lecture note, we discussed how to solve these problems and how to prevent overfitting. However, for more complex functions or more classes, you might want to take a look at neural networks. +Regression is used in a variety of AI problems. We can use it to find the best line or curve to predict a function, or to find the best hyperplane dividing 2 classes in a classification problem. In this lecture note, we discussed how to solve these problems and how to prevent overfitting. However, for more complex functions or more classes, you might want to take a look at neural networks. # Refrences - Artificial Inteligence Course at Sharif University of Technology (Fall, 2021) From e0da212b21ba40d2e90f128e91446be2dc74f657 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Sat, 8 Jan 2022 01:06:59 +0330 Subject: [PATCH 15/19] Update config files --- .github/workflows/main.yml | 2 +- notebooks/index.yml | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/.github/workflows/main.yml b/.github/workflows/main.yml index 7e8ef31a..b0d9f5c0 100644 --- a/.github/workflows/main.yml +++ b/.github/workflows/main.yml @@ -24,5 +24,5 @@ jobs: github_token: ${{ secrets.GITHUB_TOKEN }} enable_jekyll: true allow_empty_commit: true - publish_dir: . + publish_dir: ./webified/ exclude_assets: '.github' diff --git a/notebooks/index.yml b/notebooks/index.yml index 29b5bd4a..c2451db3 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -56,4 +56,4 @@ notebooks: - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - md: notebooks/14_regression - text: Regression \ No newline at end of file + kind: F2021, LN \ No newline at end of file From ca7af34f0e56427fc71b8d2686eb483fdb01446d Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Fri, 28 Jan 2022 17:05:04 +0330 Subject: [PATCH 16/19] Fix review requests --- .DS_Store | Bin 0 -> 6148 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 .DS_Store diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..9a70ace7fc197cf20514e8ae9e53171a6efffcbf GIT binary patch literal 6148 zcmeHKJx>Bb5S=|vL}`ezu)JSjptC&4Sy^a8W2uM%BX}f2VtGHr_zP@^SXgUgUv z(%)-O*M{J(j8s}5*QGWMsNq&83ijqBJP!A-V-7R->81{orBN;)v;I*Y8g%ItTAb6h z$0k?(1RKxqayfjQaG8mjGFcYm@yH#qE^FX%MNP1{@mNrh_q{HMhmQ#dC*Rd#afDZ& z^n2Mr^_2Rk-vFj`>8lYG%TM63Aj;)498G>B3nQn>td4Z~1zTDl)=2E<3SiG>D-;b% zFA9hPqQFD}ULPVf#@J$JP%j6njf zTx>BjXwpg9!-uk!l|7*-T^;w2bU3Nlp!A}EDBvrQGnXaa|M!=l|NSJn5(Pwof2Dv5 zlX_CaBiX%m?%{Z^jnGce*f=gTs8i6{?N|?ZE8aoV24BPjU~Dlnh#8pt5zsP7Ckp(k F0^d&+xI+K{ literal 0 HcmV?d00001 From 7787d7b2cc7ae7900b73d0f53a0fb3d36503c038 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Sat, 29 Jan 2022 01:49:06 +0330 Subject: [PATCH 17/19] Fix review requests --- notebooks/14_regression/index.md | 22 ++++++++++------------ 1 file changed, 10 insertions(+), 12 deletions(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index 8ae4e71e..5a41b758 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -2,8 +2,6 @@ # Table of Contents -- [Regression](#regression) -- [Table of Contents](#table-of-contents) - [Introduction](#introduction) - [1 - Linear Regression](#2---linear-regression) - [Loss Function](#loss-function) @@ -17,11 +15,11 @@ - [Using Validation Set (Held-Out Data)](#using-validation-set-held-out-data) - [Regularization](#regularization) - [3 - Logistic Regression](#5---logistic-regression) - - [How to calculate probabilities](#how-to-calculate-probabilities) - - [defining line _l_ and cost function $J(\theta)$](#defining-line-l-and-cost-function-jtheta) + - [How to Calculate Probabilities](#how-to-calculate-probabilities) + - [Defining Line _l_ and Cost Function $J(\theta)$](#defining-line-l-and-cost-function-jtheta) - [Training](#training) - [Conclusion](#conclusion) -- [Refrences](#refrences) +- [References](#references) # Introduction In regression, we have some data points and a number or a label is assigned to each data point. Our goal is to predict the number or label of an unseen data point after learning from the data we already have. We assume each data point is a vector $x$ and we want to predict $f(x)$. The first idea is to use interpolation. By using interpolation, we will have a high degree polynomial that fits our training data perfectly. But the problem is that interpolation leads to overfitting. So the error for unseen data will be too large. In regression, we aim to find the best curve with a lower degree. Although there will be some training error here, our test error will decrease since we are avoiding overfitting. @@ -134,7 +132,7 @@ We split a part of the training data, and don't use it for training. Then, by ca We can see in the above examples that $M = 3$ gives us the best curve. If we compare the validation loss for $M=3$ and $M=9$, the loss will be less for $M=3$ since it's closer to the green curve. ### Regularization -Although using a validation set is a good way to prevent overfitting, We still might be trying to find the curve which best fits the data, not the curve which best matches it. Regularization tries to make $w$ smaller. The intuition is that large coefficients in $w$ happen because it tries to fit the points. It means that the curve will only get closer to each point. However, decreasing coefficients will make a better curve which might be further from each point, but does a better job at predicting the unseen data. +Although using a validation set is a good way to prevent overfitting, We still might try to find the curve which best fits the data, not the curve which best matches it. Regularization tries to make $w$ smaller. The intuition is that large coefficients in $w$ happen because it tries to fit the points. It means that the curve will only get closer to each point. However, decreasing coefficients will make a better curve which might be further from each point, but does a better job at predicting the unseen data. We only define $l_2-\text{regularization}$ since it's easier to derive and understand. The loss function is @@ -150,9 +148,9 @@ In the following examples, we use regularization with 9'th degree polynomials. W # 3 - Logistic Regression -There are some regression algorithms that can be used for classification (And vise versa). _Logistic Regression_ is one of these algorithms. It can calculate the probability that an instance belongs to a particular class. we can use this probability for classification: if the probability is higher than 0.5 then that instance belongs to the particular class. +There are some regression algorithms that can be used for classification (And vise versa). _Logistic Regression_ is one of these algorithms. It can calculate the probability that an instance belongs to a particular class. We can use this probability for classification: if the probability is higher than 0.5 then that instance belongs to the particular class. -## How to calculate probabilities +## How to Calculate Probabilities Let's start with a simple example. We want to classify flowers that are _Iris Virginica_ from those that are _not Iris Virginica_. This classification is based on Petal Width and Petal Height. Also, consider that there is some line _l_ (black line in the image below) that separates these 2 types of flowers. @@ -172,9 +170,9 @@ Also in the image below, you can see the value of the Logistic function for diff ![image info](./images/Logistic%20function.jpg) -## defining line _l_ and cost function $J(\theta)$ +## Defining Line _l_ and Cost Function $J(\theta)$ -Now we know how to calculate the probabilities, the problem is reduced to finding the best line _l_ that separates 2 classes. For this First, we need to define the line and Second, we need to define a coss function $J(\theta)$ and then we should find the line _l_ such that it minimizes $J(\theta)$. +Now that we know how to calculate the probabilities, the problem is reduced to finding the best line _l_ that separates 2 classes. For this, first, we need to define the line and second, we need to define a cost function $J(\theta)$ and then we should find the line _l_ such that it minimizes $J(\theta)$. We define line _l_ with parameters $\theta$ like before. With this definition, the signed distance between x and the _l_ can be calculated as follows: $$ @@ -203,7 +201,7 @@ where $\hat{P}^{i}$ is the probability predicted and $y^{i}$ is the label for th ## Training -Unfortunately, there is no known closed-form equation to compute the value of $\theta$ that minimizes this cost function. But the good news is that this cost function is convex! So we can use Gradient Descent to find the minimum. Because of convexity, Gradient Descent is guaranteed to find the global minimum (with the right learning rate). +Unfortunately, there is no known closed-form equation to compute the value of $\theta$ that minimizes this cost function. But the good news is that this cost function is convex! So we can use gradient descent to find the minimum. Because of convexity, Gradient Descent is guaranteed to find the global minimum (with the right learning rate). We can compute the partial derivatives for the $j^{th}$ parameter of $\theta$ as follows: $$ @@ -214,7 +212,7 @@ $$ Regression is used in a variety of AI problems. We can use it to find the best line or curve to predict a function, or to find the best hyperplane dividing 2 classes in a classification problem. In this lecture note, we discussed how to solve these problems and how to prevent overfitting. However, for more complex functions or more classes, you might want to take a look at neural networks. -# Refrences +# References - Artificial Inteligence Course at Sharif University of Technology (Fall, 2021) - Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems - 2th Edition - https://en.wikipedia.org/wiki/Linear_regression From 9e4a90ff2da3f67a4f0cf8e72e819605ffa3bb77 Mon Sep 17 00:00:00 2001 From: mirmohamadi Date: Sun, 30 Jan 2022 23:43:29 +0330 Subject: [PATCH 18/19] Added hyperlinks to References --- notebooks/14_regression/index.md | 14 ++++++++------ 1 file changed, 8 insertions(+), 6 deletions(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index 5a41b758..0efaf1dd 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -2,19 +2,21 @@ # Table of Contents +- [Regression](#regression) +- [Table of Contents](#table-of-contents) - [Introduction](#introduction) -- [1 - Linear Regression](#2---linear-regression) +- [1 - Linear Regression](#1---linear-regression) - [Loss Function](#loss-function) - [Mean Squared Error (MSE)](#mean-squared-error-mse) - [Finding $\hat{w}$](#finding-hatw) - [Gradient Descent](#gradient-descent) - [Normal Equation](#normal-equation) -- [2 - Learning Curves Using Polynomials](#3---learning-curves-using-polynomials) +- [2 - Learning Curves Using Polynomials](#2---learning-curves-using-polynomials) - [Reduction to Linear Regression](#reduction-to-linear-regression) - [Overfitting](#overfitting) - [Using Validation Set (Held-Out Data)](#using-validation-set-held-out-data) - [Regularization](#regularization) -- [3 - Logistic Regression](#5---logistic-regression) +- [3 - Logistic Regression](#3---logistic-regression) - [How to Calculate Probabilities](#how-to-calculate-probabilities) - [Defining Line _l_ and Cost Function $J(\theta)$](#defining-line-l-and-cost-function-jtheta) - [Training](#training) @@ -215,6 +217,6 @@ Regression is used in a variety of AI problems. We can use it to find the best l # References - Artificial Inteligence Course at Sharif University of Technology (Fall, 2021) - Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems - 2th Edition -- https://en.wikipedia.org/wiki/Linear_regression -- https://math.stackexchange.com/questions/1962877/compute-the-gradient-of-mean-square-error -- https://towardsdatascience.com/normal-equation-in-python-the-closed-form-solution-for-linear-regression-13df33f9ad71 \ No newline at end of file +- [*Linear Regression* from Wikipedia.org](https://en.wikipedia.org/wiki/Linear_regression) +- [*Compute the gradient of mean square error* from math.stackexchange.com](https://math.stackexchange.com/questions/1962877/compute-the-gradient-of-mean-square-error) +- [*Normal Equation in Python: The Closed-Form Solution for Linear Regression* from towardsdatascience.com](https://towardsdatascience.com/normal-equation-in-python-the-closed-form-solution-for-linear-regression-13df33f9ad71) \ No newline at end of file From 646112c7ac78e0a5f72061cb3f336b3b40363ae9 Mon Sep 17 00:00:00 2001 From: Pouya Esmaili Date: Sat, 5 Feb 2022 16:53:21 +0330 Subject: [PATCH 19/19] Revert table of contents to the correct form --- notebooks/14_regression/index.md | 8 +++----- 1 file changed, 3 insertions(+), 5 deletions(-) diff --git a/notebooks/14_regression/index.md b/notebooks/14_regression/index.md index 0efaf1dd..6983b173 100644 --- a/notebooks/14_regression/index.md +++ b/notebooks/14_regression/index.md @@ -2,21 +2,19 @@ # Table of Contents -- [Regression](#regression) -- [Table of Contents](#table-of-contents) - [Introduction](#introduction) -- [1 - Linear Regression](#1---linear-regression) +- [1 - Linear Regression](#2---linear-regression) - [Loss Function](#loss-function) - [Mean Squared Error (MSE)](#mean-squared-error-mse) - [Finding $\hat{w}$](#finding-hatw) - [Gradient Descent](#gradient-descent) - [Normal Equation](#normal-equation) -- [2 - Learning Curves Using Polynomials](#2---learning-curves-using-polynomials) +- [2 - Learning Curves Using Polynomials](#3---learning-curves-using-polynomials) - [Reduction to Linear Regression](#reduction-to-linear-regression) - [Overfitting](#overfitting) - [Using Validation Set (Held-Out Data)](#using-validation-set-held-out-data) - [Regularization](#regularization) -- [3 - Logistic Regression](#3---logistic-regression) +- [3 - Logistic Regression](#5---logistic-regression) - [How to Calculate Probabilities](#how-to-calculate-probabilities) - [Defining Line _l_ and Cost Function $J(\theta)$](#defining-line-l-and-cost-function-jtheta) - [Training](#training)

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