Linear Regression Basics

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Questions and Answers

What is the primary purpose of gradient descent in linear regression?

  • To determine the optimal values for the parameters θ0 and θ1. (correct)
  • To update the learning rate α based on the cost function.
  • To adjust the size of the training dataset for improved accuracy.
  • To calculate the difference between predicted and actual values.

What is the main function of the learning rate (α) in gradient descent?

  • It represents the difference between the predicted and actual values.
  • It controls the step size during the parameter updates. (correct)
  • It determines the direction of the gradient descent.
  • It measures the accuracy of the linear model.

What is the significance of the partial derivative ∂J(θ0, θ1)/∂θj in the gradient descent update rule?

  • It represents the slope of the cost function at the current parameter values. (correct)
  • It indicates the size of the training dataset.
  • It measures the difference between the predicted and actual values.
  • It determines the learning rate α for the update.

Why is it essential to update both θ0 and θ1 simultaneously during gradient descent?

<p>To guarantee that the update is based on the same cost function evaluation. (D)</p> Signup and view all the answers

What is the consequence of updating θ0 before updating θ1 in gradient descent?

<p>It leads to a mismatch in the calculated cost function gradient. (C)</p> Signup and view all the answers

Which of these describes the correct method for updating parameters in gradient descent?

<p>Updating θ0 and θ1 simultaneously using current parameter values. (C)</p> Signup and view all the answers

What is the objective of linear regression in the provided context?

<p>To predict real-valued outputs based on input data. (D)</p> Signup and view all the answers

What is the goal of minimizing the cost function J(θ0, θ1) in gradient descent?

<p>To improve the accuracy of the predicted values. (C)</p> Signup and view all the answers

What is the main advantage of using small mini-batches compared to batch gradient descent?

<p>They are less likely to get stuck in local minima. (B)</p> Signup and view all the answers

Which of the following is a potential disadvantage of using large mini-batches?

<p>They may miss out on the benefits of faster updates. (C)</p> Signup and view all the answers

What is the typical range for mini-batch sizes in practice?

<p>32 to 256 (A)</p> Signup and view all the answers

What is the purpose of the forward pass in mini-batch gradient descent?

<p>To compute the model’s predictions for the mini-batch. (B)</p> Signup and view all the answers

What does the cost function in mini-batch gradient descent measure?

<p>The difference between the predicted and actual outputs. (A)</p> Signup and view all the answers

What does the gradient in mini-batch gradient descent indicate?

<p>The direction in which each parameter should be adjusted. (B)</p> Signup and view all the answers

How is the gradient calculated in mini-batch gradient descent?

<p>By finding the derivative of the cost function with respect to each parameter. (B)</p> Signup and view all the answers

What is meant by 'real-valued output' in the context of this model?

<p>The output is a continuous value. (C)</p> Signup and view all the answers

In the provided scenario, what does the size of the house represent?

<p>The primary feature used for predictions. (A)</p> Signup and view all the answers

What is the role of the cost function in the learning algorithm?

<p>To quantify the difference between predicted outputs and actual target values. (C)</p> Signup and view all the answers

Which of the following best describes the relationship defined by the hypothesis function in linear regression?

<p>It is a linear equation of the form h(x) = θ0 + θ1 x. (A)</p> Signup and view all the answers

What does the training set consist of in this supervised learning problem?

<p>Pairs of input features and target values. (C)</p> Signup and view all the answers

What does the term 'features' refer to in the context of the provided content?

<p>The input variables used for predictions. (A)</p> Signup and view all the answers

What is indicated by the number of training examples (m) in the dataset?

<p>The total count of samples used for training. (D)</p> Signup and view all the answers

What is the expected outcome when applying the hypothesis function after training?

<p>It predicts a house's price based on its size. (B)</p> Signup and view all the answers

What does the slope (β1) in a simple linear regression represent?

<p>The expected change in Y for a unit increase in X. (B)</p> Signup and view all the answers

Which assumption is NOT necessary for simple linear regression?

<p>The dependent variable Y must be categorical. (A)</p> Signup and view all the answers

In the equation Y = β0 + β1 X + ϵ, what does β0 represent?

<p>The value of Y when X is zero. (C)</p> Signup and view all the answers

What is implied by the term 'homoscedasticity' in the context of regression?

<p>The variance of the errors is constant across all values of X. (D)</p> Signup and view all the answers

What is the purpose of the error term (ϵ) in the regression model?

<p>To represent the difference between observed and predicted values. (B)</p> Signup and view all the answers

In multiple linear regression, how many independent variables are being considered?

<p>At least two independent variables. (D)</p> Signup and view all the answers

Which of the following statements is true regarding the intercept in simple linear regression?

<p>It serves as a reference point when X = 0. (A)</p> Signup and view all the answers

What does the intercept term θ0 represent in the hypothesis function?

<p>The value of the predicted output when the house size is zero (B)</p> Signup and view all the answers

The independent variable in a regression model is also referred to as which of the following?

<p>Explanatory variable (D)</p> Signup and view all the answers

Which of the following statements is true regarding the hypothesis function hθ (x)?

<p>It consists of parameters that define both the position and orientation of a prediction line (D)</p> Signup and view all the answers

What is the role of the slope term θ1 in the hypothesis function?

<p>It controls how the output price changes with respect to an increase in house size (D)</p> Signup and view all the answers

What is the main goal when using the training set in linear regression?

<p>To minimize the error between predicted and actual house prices (D)</p> Signup and view all the answers

How does the hypothesis function hθ (x) visually appear on a graph?

<p>As a straight line demonstrating a linear relationship (A)</p> Signup and view all the answers

Which statement accurately describes the cost function in linear regression?

<p>It measures the fit of the model by calculating the difference between predicted and actual values (A)</p> Signup and view all the answers

In the context of the hypothesis function, what does 'x' represent?

<p>The variable feature, indicating house size (B)</p> Signup and view all the answers

What happens to the position of the prediction line if θ0 is increased?

<p>The line shifts vertically upwards (C)</p> Signup and view all the answers

Flashcards

Supervised Learning

A machine learning type where models learn from labeled data to make predictions.

Linear Regression

A method to predict real-valued outputs by finding the relationship between inputs and outputs.

Cost Function

A measure of how well a model's predictions match the actual data.

Gradient Descent

An optimization algorithm used to minimize the cost function by adjusting parameters iteratively.

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Updating Parameters

Adjusting parameters (θ0, θ1) using the update rule to minimize the cost function.

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Learning Rate (α)

A hyperparameter that determines the step size in the update process of gradient descent.

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Simultaneous Update

The method of updating all model parameters at the same time to maintain consistency.

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Incorrect Update Method

Updating parameters sequentially causing inaccuracies in subsequent updates.

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Mean of Size

The average size of all given properties measured in sq. ft.

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Range of Size

The difference between the maximum and minimum size of properties.

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Mean of Bedrooms

The average number of bedrooms across the dataset.

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Range of Bedrooms

The difference between the maximum and minimum number of bedrooms.

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Mean Normalization Formula

A method to scale features by subtracting mean and dividing by range.

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Mini-Batch Gradient Descent

An optimization algorithm that updates model parameters using subsets of the training data.

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Small Mini-Batches

Mini-batches close to size 1 that introduce noise in updates but can escape local minima.

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Large Mini-Batches

Mini-batches close to the full dataset that provide stable updates but are computationally expensive.

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Balanced Mini-Batch Size

A mini-batch size between 32 and 256 chosen to optimize SGD and batch gradient descent benefits.

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Forward Pass

Step in mini-batch gradient descent where the model makes predictions for each example.

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Mean Squared Error (MSE)

A common cost function that computes the average squared difference between predicted and actual values.

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Backward Pass

The step that computes gradients of the cost function with respect to the model parameters.

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Real-Valued Output

A continuous value, such as a house price, rather than a category.

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Independent Variable

The primary feature used to predict the output, e.g., size of the house.

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Training Set

The dataset used for training, includes input features and their corresponding outputs.

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Learning Algorithm

The model that learns the relationship between features and targets from the training data.

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Hypothesis Function (h)

Represents the predicted relationship between input and output after training.

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Size in feet² (x)

The independent variable representing the size of the house in the training set.

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Price ($) in 1000’s (y)

The dependent variable representing the predicted price of the house in the training set.

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Simple Linear Regression

A regression model with one dependent and one independent variable.

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Regression Equation

The formula Y = β0 + β1 X + ϵ represents a linear relationship in regression.

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Intercept (β0)

The predicted value of Y when the independent variable X is zero.

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Slope (β1)

The change in Y for a one-unit increase in X, describing the relationship strength.

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Multiple Linear Regression

A regression model that uses multiple independent variables to predict a dependent variable.

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Hypothesis Function hθ(x)

A mathematical function predicting output from input data.

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Minimizing Error

The process of adjusting parameters to reduce prediction errors.

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Linear Relationship

A direct correlation where predictions can be modeled by a straight line.

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Prediction Line

The graphical representation of the model's predictions based on inputs.

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Study Notes

Linear Regression

  • Linear regression is a statistical method used to model the relationship between a dependent variable (target/response) and one or more independent variables (predictors/explanatory variables) by fitting a linear equation to observed data.

Simple Linear Regression

  • In simple linear regression, there is one dependent variable (Y) and one independent variable (X).
  • The goal is to model the relationship between X and Y using a linear function of X.
  • The model equation is: Y = β₀ + β₁X + ε
    • Y: Dependent variable (response variable)
    • X: Independent variable (explanatory variable)
    • β₀: Intercept, represents the value of Y when X = 0.
    • β₁: Slope, represents the change in Y for a one-unit change in X.
    • ε: Error term (residual), represents the difference between the observed value of Y and the value predicted by the model.

Interpretation of Parameters

  • β₀ (Intercept): Predicted value of Y when X = 0. May not always be meaningful.
  • β₁ (Slope): Describes the relationship between X and Y. It quantifies the expected change in Y for a unit increase in X.

Assumptions of Simple Linear Regression

  • Linearity: The relationship between the dependent variable (Y) and the independent variable (X) is linear.
  • Independence: The residuals (errors) ε are independent.
  • Homoscedasticity: The residuals have constant variance (the variance of errors is the same across all values of X).
  • Normality: The residuals are normally distributed.

Multiple Linear Regression

  • Multiple linear regression models the relationship between a dependent variable (Y) and multiple independent variables (X₁, X₂, ..., Xp).
  • Model equation: Y = β₀ + β₁X₁ + β₂X₂ +...+ βpXp + ε
    • β₀: Intercept
    • β₁, β₂, ..., βp: Coefficients (slopes) associated with each independent variable.

Interpretation of Parameters in Multiple Linear Regression

  • β₀: The predicted value of Y when all independent variables (X₁, X₂, ..., Xp) are equal to 0.
  • βr : The expected change in Y for a one-unit increase in Xi, holding all other independent variables constant.

Assumptions of Multiple Linear Regression

  • Linearity: The relationship between each independent variable (X;) and the dependent variable (Y) is linear.
  • Independence: The residuals are independent.
  • Homoscedasticity: The residuals have constant variance.
  • Normality: The residuals are normally distributed.
  • No Multicollinearity: The independent variables (X₁, X₂, ..., Xp) are not too highly correlated with each other.

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