From d6b3bfeb8f4ccc1a921cd976e40935bcfec5565e Mon Sep 17 00:00:00 2001 From: Pawel Date: Sun, 15 Feb 2026 10:18:57 -0500 Subject: [PATCH] spacing --- README.md | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/README.md b/README.md index 289cecf..01cdb8c 100644 --- a/README.md +++ b/README.md @@ -102,6 +102,7 @@ If the system $\tilde{X}\tilde{\beta} = y$ is consistent, then we can find a sol $$ e = y - \tilde{X}\tilde{\beta} $$ 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) The structure of this sections is as follows. @@ -339,7 +340,9 @@ df.plot( ) plt.show() ``` + ![house_price_vs_square_ft.png](./figures/house_price_vs_square_ft.png) + ![house_price_vs_bedrooms.png](./figures/house_price_vs_bedrooms.png) 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. For example, suppose our data looked like the following. + ![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... @@ -1366,6 +1370,7 @@ We will perform the following steps. ### Loading and Preprocessing the Image Let's start with this picture of my beautiful dog Bella. Here it is! + ![bella.jpg](./pictures/bella.jpg) Let's first convert it to grayscale. @@ -1386,6 +1391,7 @@ plt.show() ``` Here is the result. + ![bella_grayscale.jpg](./pictures/bella_grayscale.jpg) ### Adding Noise @@ -1404,6 +1410,7 @@ plt.axis("off") ``` This gives the following image. + ![bella_grayscale_noisy.jpg](./pictures/bella_grayscale_noisy.jpg)