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fp2024:hw4 [2024/05/09 18:43] pdmatei |
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| ====== Objective: house price prediction ===== | ====== Objective: house price prediction ===== | ||
| - | 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 $math[x] you need to **predict** its price $math[y(x)]. More generally, we need to find a suitable function $math[y], which is usually called a **hypothesis**. | + | 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 $math[x] you need to **predict** its price $math[y(x)]. More generally, we need to find a suitable price prediction function $math[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 $math[y = a*x + b]. We now need to find the values for $math[a] and $math[b] in such a way that the $math[y] line **fits** best our dataset. One example of a line $math[y] is given by two of its points, shown in green in the dataset. We simply examining the figure, we can see that this line is a good fit for the dataset. | + | 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 $math[y = a*x + b]. We now need to find the values for $math[a] and $math[b] in such a way that the $math[y] line **fits** best our dataset. One example of a line $math[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 $math[a] and $math[b] we need a formal condition to express //**best fit**//. Let be a point $math[ (x_i,y_i) ] from the dataset. Then, the value $math[ \mid y_i - a*x_i + b \mid ] is the error between our price estimation $math[a*x_i + b] and the real price $math[y_i]. We will choose $math[a] and $math[b] in such a way as to **minimise** the **sum** of all such errors, for the **entire** dataset. | + | In order to find $math[a] and $math[b] we need a formal condition to express //**best fit**//. Let $math[ (x_i,y_i) ] be a point from the dataset. Then, the value $math[ \mid y_i - (a*x_i + b) \mid ] is the error between our price estimation $math[a*x_i + b] and the real price $math[y_i]. We will choose $math[a] and $math[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 [[https://mubaris.com/posts/linear-regression/|details]] about linear regression to get a better perspective of this homework. | In order to solve this homework, it is not necessary to understand how minimisation is performed, but you can read more [[https://mubaris.com/posts/linear-regression/|details]] about linear regression to get a better perspective of this homework. | ||
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| 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: $math[y = a*x_1 + b*x_2 + c] where $math[x_1] represents the surface area of the house and $math[x_2] represents the garage area. Now, our hypothesis has three **parameters** ($math[a,b,c]) which must be computed in such as way at to achieve a **best fit**. | 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: $math[y = a*x_1 + b*x_2 + c] where $math[x_1] represents the surface area of the house and $math[x_2] represents the garage area. Now, our hypothesis has three **parameters** ($math[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 $math[x]), it is much more convenient to use a matrix representation. Suppose we also add a third feature $math[x_3] in the dataset, and have that it is allways equal to 1: | + | When using more than one feature (more than one $math[x]), it is much more convenient to use a matrix representation. Suppose we also add a third feature $math[x_3] in the dataset, and have it be always equal to 1. |
| $$ y = a * x_1 + b * x_2 + c * x_3 $$ | $$ y = a * x_1 + b * x_2 + c * x_3 $$ | ||
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| $$ y = \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \cdot \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$ | $$ y = \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \cdot \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$ | ||
| - | If $math[X] is a matrix with three columns (for the two features of the model), and $math[n] lines, one for each entry in the dataset, then | + | If $math[X] is a matrix with three columns (for the two features of the model, //plus// $math[x_3 = 1]), and $math[n] lines, one for each entry in the dataset, then |
| evaluating our price estimations for each **multi-dimensional point** in the dataset is given by the vector ($math[n] lines, one column): | evaluating our price estimations for each **multi-dimensional point** in the dataset is given by the vector ($math[n] lines, one column): | ||
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| def split(percentage: Double): (Dataset, Dataset) = ??? | def split(percentage: Double): (Dataset, Dataset) = ??? | ||
| </code> | </code> | ||
| - | 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 evaluated 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 $math[(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: $math[(x_1,y_1), (x_5,y_5), (x_{10}, y_{10}) (x_{15}, y_{15})]. | + | 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 $math[(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: $math[(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. | 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. | ||