CS-GY 6923: Machine Learning
1.0.0
1.0.0
  • Introduction
  • What is Machine Learning?
  • Types of Machine Learning
    • Supervised Learning
      • Notations
      • Probabilistic Modeling
        • Naive Bayes Classifier
      • Linear Regression
      • Nearest Neighbor
      • Evaluating a Classifier
      • Parametric Estimation
        • Bayesian Approach to Parameter Estimation
        • Parametric Estimation for Simple Linear Regression
        • Parametric Estimation for Multivariate Linear Regression
        • Parametric Estimation for Simple Polynomial Regression
        • Parametric Estimation for Multivariate Polynomial Regression
      • Bias and Variance of an Estimator
      • Bias and Variance of a Regression Algorithm
        • Model Selection
      • Logistic Regression
      • Decision Trees
        • Using Decision Trees for Regression
        • Bias and Variance
      • Dimensionality Reduction
      • Neural Networks
        • Training a Neuron
        • MLP
          • Regression with Multiple Outputs
          • Advice/Tricks and Issues to Train a Neural Network
        • Deep Learning
      • Support Vector Machines
      • Ensemble Learning
    • Unsupervised Learning
      • K-Means Clustering
      • Probabilistic Clustering
    • Reinforcement Learning
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  1. Types of Machine Learning
  2. Supervised Learning
  3. Parametric Estimation

Parametric Estimation for Multivariate Polynomial Regression

Say we have a dataset with 2 features and it cannot be fit using Linear Regression.

We may want to use Multivariate Polynomial Regression. The equation is given by:

g(x2,x1∣w5,w4,w3,w2,w1,w0)=w5x22+w4x12+w3x1x2+w2x2+w1x1+w0g(x_2,x_1|w_5,w_4,w_3,w_2,w_1,w_0) = w_5x_2^2+w_4x_1^2+w_3x_1x_2+w_2x_2+w_1x_1+w_0g(x2​,x1​∣w5​,w4​,w3​,w2​,w1​,w0​)=w5​x22​+w4​x12​+w3​x1​x2​+w2​x2​+w1​x1​+w0​

This is a degree 2 polynomial and we need to estimate w5,w4,w3,w2,w1,w0w_5,w_4,w_3,w_2,w_1,w_0w5​,w4​,w3​,w2​,w1​,w0​ that minimize the squared error.

Note: In some cases, this can be simplified to a Linear Regression (as shown in Simple Polynomial Regression) by simply adding appropriate features to the dataset.

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