Summary of Week 2
This week, we have introduced a general framework of supervised learning and focused on linear regression.
We start with linear regression as an example to go through the key components of the supersized learning framework, i.e. data, model, loss function, optimization, prediction and evaluation. We then proceed with discussing the pros and cons of linear regression. We introduce the three main regularization methods to overcome the potential overfitting issue. Lastly, we extend the regression framework to tackle the classification problems.
Supervised Learning Framework
- Dataset:(mathcal{D} = {(x_{i}, y_{i})}_{i = 1}^{N})
- Model: ({color{blue}f_{theta}}(x) approx mathbb{E}[y vert x] = f(x),~forall x in mathbb{R}^{d})
- Empirical Loss: (L({color{blue}theta} vert mathcal{D}) = frac{1}{N}sum_{i =1}^{N}d(f_{theta}(x_{i}), y_{i}) rightarrow) Minimize
- Optimization: ({color{blue}theta^{*}} = argmin_{theta} (L({color{blue}theta} vert mathcal{D})))
- Prediction: (hat{y}_{*} = {color{blue}f_{theta^{*}}}(x_{*}))
- Validation: Compute the indicators for the goodness of fitting.
Linear Regression and Regularization
Linear Regression (Ordinary Least Squares)
- (text{Model:} y = f_{color{blue}theta}(x) + varepsilon = x theta + varepsilon.)
- ({text{Loss Function: }}L({color{blue}theta} vert X, Y) = (Y- X {color{blue}theta})^{T} (Y- {Xcolor{blue}theta}) rightarrow min;)
- ({text{Optimization:}}{color{blue}{hat{theta}}} = (X^{T}X)^{-1}X^{T}y.)
- (text{Prediction:} hat{y}_{*} = x_{*}{color{blue}{hat{theta}}}.)
- (text{Validation:}text{Compute RMSE, }R^{2}, text{ the adjusted }R^{2}, ptext{-value}.)
Regularization – Lasso Regression: (L(beta vert X, Y) = (Y- Xbeta)^{T} (Y- Xbeta) + lambda vertvert beta vert vert_{1} 😉 – Ridge Regression: (L(beta vert X, Y) = (Y- Xbeta)^{T} (Y- Xbeta) + lambda vertvert beta vertvert_{2}^{2};) – Elastic Net: (L(beta vert X, Y) = (Y- Xbeta)^{T} (Y- Xbeta) + lambda left( frac{1-alpha}{2}vertvert beta vertvert_{2}^{2}+ alpha vert vert beta vertvert_{1}right);)
From Regression to Classification
| Regression | Classification | |
|---|---|---|
| Dataset: (mathcal{D} = {(x_{i}, y_{i})}_{i = 1}^{N}, (x_i, y_i) in mathcal{X} times mathcal{Y}) | ||
| (color{blue}{mathcal{Y} = mathbb{R}^d}) | ({color{blue}{mathcal{Y}}}) is a finite set. | |
| Model: (y = f_{theta}(x) + varepsilon) | (f_{theta}(x)(approx {color{blue}{mathbb{E}[y vert x]}})) | (f_{theta}(x)(approx {color{blue}{mathbb{P}[y vert x]}})) |
| Empirical Loss: (L(theta vert mathcal{D})rightarrow) Minimize | (e.g. ({color{blue}{text{MSE}}})) | (e.g. ({color{blue}{text{Cross entropy}}})) |
| Optimization: (theta^{*} = argmin_{theta} (L(theta vert mathcal{D}))) | ||
| Prediction: | (hat{y}_{*} = {color{blue}f_{theta^{*}}(x_{*})}.) | (hat{y}_{*} = {color{blue}{text{arg}max_{i in mathcal{Y}} f^{i}}_{theta^{*}}(x_{*})}.) |
| Validation: Compute the test metrics | e.g.({color{blue}{text{MSE}}}). | e.g. ({color{blue}{text{Accuracy}}}). |
Table: Summary of the framework of regression and classification. The difference between them is highlighted in ({color{blue}{blue}}).
An Introduction to Machine Learning in Quantitative Finance
An Introduction to Machine Learning in Quantitative Finance
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