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Pawel
2026-02-15 10:18:57 -05:00
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@@ -102,6 +102,7 @@ If the system $\tilde{X}\tilde{\beta} = y$ is consistent, then we can find a sol
$$ e = y - \tilde{X}\tilde{\beta} $$ $$ e = y - \tilde{X}\tilde{\beta} $$
is small. By small, we often mean that $e$ is small in $L^2$ norm; i.e., we are minimizing the the sums of the squares of the differences between the components of $y$ and the components of $\tilde{X}\tilde{\beta}$. This is known as a **least squares solution**. Assuming that our data points live in the Euclidean plane, this precisely describes finding a line of best fit. is small. By small, we often mean that $e$ is small in $L^2$ norm; i.e., we are minimizing the the sums of the squares of the differences between the components of $y$ and the components of $\tilde{X}\tilde{\beta}$. This is known as a **least squares solution**. Assuming that our data points live in the Euclidean plane, this precisely describes finding a line of best fit.
![line_of_best_fit_generated1.png](./figures/line_of_best_fit_generated1.png) ![line_of_best_fit_generated1.png](./figures/line_of_best_fit_generated1.png)
The structure of this sections is as follows. The structure of this sections is as follows.
@@ -339,7 +340,9 @@ df.plot(
) )
plt.show() plt.show()
``` ```
![house_price_vs_square_ft.png](./figures/house_price_vs_square_ft.png) ![house_price_vs_square_ft.png](./figures/house_price_vs_square_ft.png)
![house_price_vs_bedrooms.png](./figures/house_price_vs_bedrooms.png) ![house_price_vs_bedrooms.png](./figures/house_price_vs_bedrooms.png)
We can even do square footage vs bedrooms. We can even do square footage vs bedrooms.
@@ -949,6 +952,7 @@ where $B$ will be a $p \times k$ matrix of parameters and $Y$ will be the $p \ti
Sometimes fitting a line to a set of $n$ data points clearly isn't the right thing to do. To emphasize the limitations of linear models, we generate data from a purely quadratic relationship. In this setting, the space of linear functions is not rich enough to capture the underlying structure, and the linear least-squares solution exhibits systematic error. Expanding the feature space to include quadratic terms resolves this issue. Sometimes fitting a line to a set of $n$ data points clearly isn't the right thing to do. To emphasize the limitations of linear models, we generate data from a purely quadratic relationship. In this setting, the space of linear functions is not rich enough to capture the underlying structure, and the linear least-squares solution exhibits systematic error. Expanding the feature space to include quadratic terms resolves this issue.
For example, suppose our data looked like the following. For example, suppose our data looked like the following.
![quadratic_data.png](./figures/quadratic_data.png) ![quadratic_data.png](./figures/quadratic_data.png)
If we try to find a line of best fit, we get something that doesn't really describe or approximate our data at all... If we try to find a line of best fit, we get something that doesn't really describe or approximate our data at all...
@@ -1366,6 +1370,7 @@ We will perform the following steps.
### Loading and Preprocessing the Image ### Loading and Preprocessing the Image
Let's start with this picture of my beautiful dog Bella. Here it is! Let's start with this picture of my beautiful dog Bella. Here it is!
![bella.jpg](./pictures/bella.jpg) ![bella.jpg](./pictures/bella.jpg)
Let's first convert it to grayscale. Let's first convert it to grayscale.
@@ -1386,6 +1391,7 @@ plt.show()
``` ```
Here is the result. Here is the result.
![bella_grayscale.jpg](./pictures/bella_grayscale.jpg) ![bella_grayscale.jpg](./pictures/bella_grayscale.jpg)
### Adding Noise ### Adding Noise
@@ -1404,6 +1410,7 @@ plt.axis("off")
``` ```
This gives the following image. This gives the following image.
![bella_grayscale_noisy.jpg](./pictures/bella_grayscale_noisy.jpg) ![bella_grayscale_noisy.jpg](./pictures/bella_grayscale_noisy.jpg)