Linear Regression Basics

Questions and Answers

What is a primary disadvantage of replacing missing values with estimated values such as mean or median?

It enhances statistical power.

It increases the size of the dataset.

It may introduce bias or reduce variance. (correct)

It eliminates all missing data.

Which advanced method is designed to predict missing values based on feature relationships?

Univariate Analysis

K-Nearest Neighbors (KNN) (correct)

Linear Regression

Simple Imputation

What is a significant consideration when using advanced methods for imputing missing data?

They simplify data interpretation.

They are always easy to implement.

They can be computationally intensive. (correct)

They require minimal computational resources.

Which strategy is illustrated by the code snippet provided for filling missing data in sklearn?

Applying median value imputation Signup and view all the answers

How does feature scaling affect data modeling?

It can lead to features dominating others. Signup and view all the answers

What is one major advantage of using K-Nearest Neighbors (KNN) over basic imputation methods?

Considers relationships between features Signup and view all the answers

What is the primary purpose of feature scaling in data preparation?

To standardize data for uniformity Signup and view all the answers

Which of the following is NOT a pro of using imputation techniques?

Always provides accurate predictions Signup and view all the answers

Which imputation method might require significant computational resources and expertise for accurate implementation?

K-Nearest Neighbors (KNN) Signup and view all the answers

What is a common consequence of not scaling features in a dataset?

Inconsistent model performance Signup and view all the answers

Study Notes

Linear Regression Principles

Parameters (𝜃1, 𝜃2, …, 𝜃n) represent feature contributions to house pricing, while 𝜃0 is the bias term, setting a base price for predictions.
The linear regression model prediction formula combines parameters and features:
( y_{predicted} = \theta_0 + \theta_1 \times \text{size} + \theta_2 \times \text{no.bedrooms} + \theta_3 \times \text{other features} ).

Optimisation in Linear Regression

Optimisation involves adjusting the parameter vector (𝜃) to minimize a cost function, with Mean Squared Error (MSE) commonly used for evaluation.
For N features, 𝑥 and 𝜃 are vectors with dimensions (N+1), indicating N+1 parameters need to be fitted.

Methods for Finding Optimal Solutions

Analytical Solution: Utilizes direct mathematical formulas (e.g., Normal Equation). Efficient for smaller datasets.
- Example formula: ( \theta = (X^T X)^{-1} X^T y ).
Iterative Methods (Gradient Descent): Suitable for large datasets where analytical solutions are impractical.
- Involves starting with an initial guess for 𝜃 and updating iteratively to minimize the cost function.

Gradient Descent Mechanics

Calculate the gradient (slope) of the cost function at a point.
Update parameters by adjusting each ( x_i ), "moving downhill" along the gradient based on partial derivatives.

Challenges with Gradient Descent

Learning Rate: A critical parameter controlling step sizes toward the cost function's minimum.
- High Learning Rate: Can overshoot the minimum and cause oscillation or divergence.
- Low Learning Rate: Leads to slow convergence, making the process inefficient.

Polynomial Regression Insights

Linear regression is versatile and can model polynomial relationships (e.g., quadratic terms), while remaining linear concerning the parameters.
Example polynomial regression equation:
( y = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_2^2 ).

Handling Missing Data

Replacing Missing Values: Logic involves filling missing values with estimates (e.g., mean, median) to maintain dataset size and statistical power.
- Pros: Preserves dataset integrity.
- Cons: May introduce bias if imputed values are inaccurate.

Advanced Missing Data Techniques

Using methods like K-Nearest Neighbors (KNN) or Multivariate Imputation by Chained Equations (MICE) can predict missing values considering feature relationships.
- Pros: Offers more nuanced and accurate imputations.
- Cons: Computationally intensive and complex to implement.

Sklearn Imputation Example

sklearn allows various imputation strategies, such as replacing missing values with the median:

from sklearn.impute import SimpleImputer
imputer = SimpleImputer(strategy="median")
imputer.fit(training_features)
filled_features = imputer.transform(training_features)

Importance of Scaling Data

Consistent units and ranges among features are crucial; otherwise, some features may dominate the model due to differing scales.

Description

Explore the foundational concepts of linear regression, including how parameters and the bias term influence predictions of house prices. This quiz covers the mathematical formulation and optimization techniques used in linear regression models.