Model Representation

Let's see an example:

 

To establish notation for future use, we'll use $x^{(i)}$ to denote the "input" variables (living area in this example), also called inpute features, and $y^{(i)}$ to denote the "output" or target variable that we are trying to predict (price). A pair ($x^{(i)}$, $y^{(i)}$) is called a training example, and the dataset that we'll be using to learn '''a list of m tarining examples ($x^{(i)}, y^{(i)}$); i=1, ..., m is called a training set. Note that the superscript "$(i)$" in the notation is simply an index into the training set, and has nothing to do with exponentiation. We will also use X to denote the space of input values, and Y to denote the space of output values. In this example, X=Y=$\mathbb{R}$.

 

To describe the supervised learning problem slightly more formally, our goal is, given a training set, to learn a fuction $h : X → Y$ so that $h(x)$ is a "good" predictor for the corresponding value of $y$. For historical reasons, this function $h$ is called a hypothesis.

 

Seen pictorially, the process is therefore like this:

 

hypothesis $h = h_\theta(x) = h(x)$

When the target variable that we're trying to predict is continuous, such as in our housing example, we call the learning problem a regression problem. When y can take on only a small number of discrete values(such as if, given the living area, we wanted to predict if a dwelling is a house or an apartment, say), we call it a classification problem.

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Q. Consider the training set shown below. $(x^{(i)}, y^{(i)})$ is the $i^{th}$ training example, What is $y^{(i)}$?

 

Ans. 315