In the figure below, we have plotted a dataset containing, on the X axis, the overall surface of houses (in square feet) for sale, as well as, on the Y axis, their sale price. These house prices are part of a real dataset from a certain region of the US.

Given this dataset, your task is to predict prices of other houses, outside of the dataset, based on their surface. In other words, for a given house surface $ x$ you need to predict its price $ y(x)$ . More generally, we need to find a suitable price prediction function $ y$ , which is usually called a hypothesis.

By examining the distribution of points in the figure, one can see that there is a linear dependency between house area and house price. This dependency is not perfect but it is strong. Therefore, the hypothesis we will use in this homework is $ y = a*x + b$ . We now need to find the values for $ a$ and $ b$ in such a way that the $ y$ line fits best our dataset. One example of a line $ y$ is given by two of its points, shown in green in the dataset. By simply examining the figure, we can see that this line is a good fit for the dataset.

In order to find $ a$ and $ b$ we need a formal condition to express best fit. Let $ (x_i,y_i) $ be a point from the dataset. Then, the value $ \mid y_i - (a*x_i + b) \mid $ is the error between our price estimation $ a*x_i + b$ and the real price $ y_i$ . We will choose $ a$ and $ b$ in such a way as to minimise the sum of all such errors, for the entire dataset.

In order to solve this homework, it is not necessary to understand how minimisation is performed, but you can read more details about linear regression to get a better perspective of this homework.

In the previous figure, you might have already seen that a linear hypothesis starts to work poorly for surfaces over 2000 square feet. As it happens, there are also other features of such properties that influence its price. One such feature is the Garage Area, which is also expressed in square feet. Hence, we can improve our hypothesis into: $ y = a*x_1 + b*x_2 + c$ where $ x_1$ represents the surface area of the house and $ x_2$ represents the garage area. Now, our hypothesis has three parameters ($ a,b,c$ ) which must be computed in such as way at to achieve a best fit.

When using more than one feature (more than one $ x$ ), it is much more convenient to use a matrix representation. Suppose we also add a third feature $ x_3$ in the dataset, and have it be always equal to 1. $$ y = a * x_1 + b * x_2 + c * x_3 $$

We can now express our hypothesis as:

$$ y = \begin{pmatrix} a & b & c \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$$

or equivalently as: $$ y = \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \cdot \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$

If $ X$ is a matrix with three columns (for the two features of the model, plus $ x_3 = 1$ ), and $ n$ lines, one for each entry in the dataset, then evaluating our price estimations for each multi-dimensional point in the dataset is given by the vector ($ n$ lines, one column):

$$ Y = X \cdot \begin{pmatrix} a \\ b \\ c \end{pmatrix} $$

The dataset houseds.csv (from the datasets folder) contains a big dataset with over 60 features (columns) for properties as well as their sale price. In order to implement regression we may use some, or all of these features. In this homework we will use only two of these features.

The class Dataset from the Dataset.scala file already contains a companion object and apply method which extracts a table (as a List[List[String]] object) from a csv file.

Your first task is to extract a subset of features (columns) from a dataset, by providing a list of columns to be extracted. For this task:

1.1. Implement the function zipWith from Helpers.scala

def zipWith[A, B, C](op: (A, B) => C)(l1: List[A], l2: List[B]): List[C] = ???

which takes an operation, two lists, and uses to operation to zip the lists together. For instance, zipWith(_ + _)(List(1,2,3), List(4,5,6)) will produce List(5,7,9).

1.2. Use zipWith to implement the member method selectColumns which creates a new dataset by extracting those columns specified as a list of strings in the function parameter.

def selectColumns(cols: List[String]): Dataset = ???

Be careful to preserve the first line which contains the column names of the dataset.

1.3. Use selectColumns to implement the helper function selectColumn.

1.4. Implement the function split, which splits the dataset into training and evaluation.

def split(percentage: Double): (Dataset, Dataset) = ???

Generally, when a dataset is being used to implement linear regression, we need to put aside part of the dataset (usually 20%) for evaluation. It is essential that this part is not used in the training process, in order to faithfully evaluate how the hypothesis behaves on unseen data. At the same time, it is important that this 20% data is representative for the entire dataset (hence it cannot be the first or last 20% part of the dataset), but a representative sample. For instance if $ (x_1, y_1), (x_2,y_2), \ldots, (x_{20},y_{20})$ is the set of surface-to-price points, sorted after surfaces, and we decide to keep 20% (or 0.2) for evaluation, then the points that will be put aside might be: $ (x_1,y_1), (x_5,y_5), (x_{10}, y_{10}) (x_{15}, y_{15})$ . This is very similar to sampling the dataset.

In the split function, percentage is expressed as a value between 0 and 1, and represents the amount of evaluation data to be put aside from the entire dataset. In the returned pair, the first component is the training part of the dataset, and the second - the evaluation.

Be careful to preserve the first line which contains the column names of the dataset.

Matrices are represented as boxes storing an inner object m: Option[List[List[Double]]. The object can be a value Some(mp) where mp is a matrix of doubles, or it can be Nothing. This latter value will be produced whenever the current matrix operation fails (for instance, in addition, if the dimensions of the matrices do not match).

2.1. Implement the apply methods in the companion object Matrix, which allows creating matrices in various ways:

  def apply(): Matrix = ??? // should return the 'Nothing' matrix when no parameter is supplied.
  def apply(raw: List[List[Double]]): Matrix = ???
  def apply(dataset: Dataset): Matrix = ???
  def apply(s: String): Matrix = ??? // it should parse the string as a matrix, with whitespace as the column delimiter and newline '\n' as the line delimiter.

2.2. Implement method transpose which returns the transposition of a matrix. If the current matrix is a Nothing, the returned value should also be Nothing.

def transpose: Matrix = ???

2.3. Implement matrix multiplication. If the dimensions of the two matrices do not match for multiplication, the resulting matrix should be Nothing.

def *(other: Matrix): Matrix = ???

2.4. Implement matrix subtraction. If the dimensions of the two matrices do not match, the result should be a Nothing matrix.

def -(other: Matrix): Matrix = ???

2.5. Implement the operation normalize which takes a matrix $ m$ and returns the matrix $ 1/n \cdot m$ where $ n$ is the number of entries in the dataset. Normalization will be used in regression:

def normalize: Matrix = ???

2.6. Implement the operation map which applies a Double ⇒ Double transformation on each element of the matrix:

def map(f: Double => Double): Matrix = ???

2.7. Implement the operations which adds another column at the end of a matrix, whose values are equal to the constant x:

def ++(x: Double): Matrix = ???

2.8. Implement the method dimensions that returns a string formatted as a pair “(n,m)” where n is the number of lines and m - the number of columns in the matrix. This function is useful for troubleshooting when performing matrix operations.

def dimensions: String = ???

2.9. Implement the method meanedSquaredError, which, given a n x 1 matrix: $ \begin{pmatrix} x_1 & x_2 & \ldots & x_n \end{pmatrix}^T$ (n lines, one column), it will compute the meaned squared error: $ \sqrt{1/n \cdot (x_1^2 + x_2^2 + \ldots + x_n^n)}$ . If the matrix has other dimensions or is Nothing the result should also be Nothing.

def meanSquaredError: Option[Double] = ???

There are several ways to implement linear regression. We will opt for the gradient descent approach. In a given number of iterations we will take an estimate vector of parameters $ a,b$ (or $ a,b,c$ ), and produce better ones, i.e. ones closer to the ideal ones which minimise the squared root error.

3.1. Implement the function rootMeanSquareError in the executable object Regression, which takes a matrix of features $ X$ as well as one of real prices $ Y$ , together with a n-line, one column vector of parameters (e.g. for n=2, $ \begin{pmatrix} a & b \end{pmatrix}^T$ ), and computes the root mean square error of the vector of errors (values $ y_i - (a*x_i + b)$ for two parameters). Use the function meanSquaredError from the class Matrix.

def rootMeanSquareError(X: Matrix, Y: Matrix, parameters: Matrix): Option[Double] = ???

3.2. Implement the function gradientDescentStep which, given $ X$ and $ Y$ , takes a vector $ parameters$ of parameters and produces an improved version of these parameters. These are the steps you should follow in your implementation:

• Compute current estimation of prices given current parameters parameters and feature matrix $ X$ . Let $ h$ be this matrix (what dimensionality should it have?)
• Compute the loss i.e. the error of the current estimation, by difference with real prices from $ Y$ . (Again, take care of matrix dimensionality)
• Compute the gradient. For a single entry $ x_i$ whose loss is $ e_i$ , the gradient is $ (x_i * e_i)/n$ where $ n$ is the size of the dataset. To understand this formula, please read this post. Generalize this formula to matrices and take care with matrix dimensionality. For the division by $ n$ use the method normalize.
• Compute the improved version of the parameters by combining (adding or subtracting) the gradient. For a single parameter $ p$ , the update is $ p_{new} = p_{old} - (x_i * e_i)\cdot \alpha/n$ , where $\math[\alpha] is a very small constant which prevents the gradient from shifting too aggressively the current optimum. This value is a constant in the code stub. Generalise this formula to matrices.

Ultimately, this function should return the updated parameters.

def gradientDescentStep(X: Matrix, Y: Matrix, parameters: Matrix): Matrix = ???

3.3. Implement the function linearRegression which performs linear regression over dataset dataset (defined as constant in the Regression object), with initial parameters parameters that are also defined as constants, and with the features to be used given as a list. Proceed as follows:

• Create a dataset from the dataset file and select the desired features columns
• Split the dataset into training and evaluation (80% training, 20% evaluation). However, in this homework we will not evaluate your prediction.
• Create a matrix $ X$ of features from the training dataset.
• Add a feature of 1s at the end of the matrix.
• Create a matrix $ Y$ of prices (predictions) from the training dataset.
• Repeat gradientDescentStep for steps times over parameters to get your model parameters.
• Return a pair consisting in your model (the matrix parameters) and the rootMeanSquareError over the trained dataset.

Beyond our tests, you can run the linearRegression function by yourself to see your prediction.

3.4. To test your model, use this online plotly app to test your results. Use the app (or your Scala code) to extract the features from the dataset, and plot two predictions of your own (e.g. for $ x_1 = 1000$ and $ x_2 = 2000$ ), to see how your model behaves. Add a screenshot similar to the one below to your project to validate your work:

• Please follow the Submission guidelines which are the same for all homework.
• To solve your homework, download the Project template, import it in IntellIJ, and you are all set. Do not rename the project manually, as this may cause problems with IntellIJ.