From 84d30838d93c3a815b4c6e6af16c5cdb5e0c5235 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Felix=20Ramnel=C3=B6v?= <felra653@student.liu.se>
Date: Tue, 19 Nov 2024 08:36:31 +0100
Subject: [PATCH] Lab 1: Updated plot titles and labels to include correlation
 coefficient (r=0.5)

---
 lab1/assignment3.R             |   4 +--
 lab1/figures/assignment3-2.eps |   4 +--
 lab1/figures/assignment3-2.png | Bin 8674 -> 8890 bytes
 lab1/figures/assignment3-3.eps |   4 +--
 lab1/figures/assignment3-3.png | Bin 8960 -> 9174 bytes
 lab1/figures/assignment3-5.eps |   4 +--
 lab1/figures/assignment3-5.png | Bin 8585 -> 8804 bytes
 lab1/lab-notes.md              |  44 +++++++++++++++++++++++----------
 8 files changed, 39 insertions(+), 21 deletions(-)

diff --git a/lab1/assignment3.R b/lab1/assignment3.R
index 40a5cd5..a2fb635 100644
--- a/lab1/assignment3.R
+++ b/lab1/assignment3.R
@@ -39,7 +39,7 @@ missclass_rate = missclass(data$V9, predict_reg)
 print(missclass_rate)
 
 plot(
-  main = "Plasma Glucose Concentration vs Age",
+  main = "Plasma Glucose Concentration vs Age (r=0.5)",
   data$V2,
   data$V8,
   xlab = "Plasma Glucose Concentration",
@@ -92,7 +92,7 @@ predict_reg2 <- ifelse(predict_reg2 > r, 1, 0)
 theta2 <- coefficients(model2)
 
 plot(
-  main = "Plasma Glucose Concentration vs Age",
+  main = "Plasma Glucose Concentration vs Age (r=0.5)",
   data$V2,
   data$V8,
   xlab = "Plasma Glucose Concentration",
diff --git a/lab1/figures/assignment3-2.eps b/lab1/figures/assignment3-2.eps
index 6742697..c3dcf68 100644
--- a/lab1/figures/assignment3-2.eps
+++ b/lab1/figures/assignment3-2.eps
@@ -1963,8 +1963,8 @@ cp p1
 0.00 0.00 352.50 282.00 cl
 /Font2 findfont 14 s
 0 0 0 srgb
-62.28 247.45 (Plasma Glucose Concentration vs Ag) 0 ta
-0.140 (e) tb gr
+39.13 247.45 (Plasma Glucose Concentration vs Ag) 0 ta
+0.140 (e \(r=0.5\)) tb gr
 /Font1 findfont 12 s
 107.95 18.72 (Plasma Glucose Concentr) 0 ta
 -0.120 (ation) tb gr
diff --git a/lab1/figures/assignment3-2.png b/lab1/figures/assignment3-2.png
index 63c84180c7ac25eb8bdc60dd87477602ee8e3fbf..b32550c32d0a54edefbeed1ec246e3f6dc4a3f42 100644
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diff --git a/lab1/lab-notes.md b/lab1/lab-notes.md
index 6314f64..7c5dc75 100644
--- a/lab1/lab-notes.md
+++ b/lab1/lab-notes.md
@@ -130,41 +130,59 @@ Confusion matrix o misclassification error e framtana.
 
 ## Assignment 3
 
-1. Probably not very easy to classify due to a lot of overlapping. We can not separate the two classes with a signle line.
+1. Probably not very easy to classify due to a lot of overlapping. We can not separate the two classes with a single line.
 
    ![Plasma Glukos Concentration vs Age](./figures/assignment3-1.png)
 
-2. The probibalistic equation:
+2. The probibalistic equation for diabetes:
 
    $$p(y = 1 \mid \mathbf{x}^*) = g(\mathbf{x}^*, \boldsymbol{\theta}) = \frac{1}{1 + e^{-\boldsymbol{\theta}^\top \mathbf{x}^*}}$$
 
-   Transform into decision:
+   Consequently the equation for no diabetes becomes:
+
+   $$p(y = 0 \mid \mathbf{x}^*) = 1 - p(y = 1 \mid \mathbf{x}^*)$$
+
+   We transform the probability into a decision:
 
    $$
    \hat{y} =
    \begin{cases}
-   1 & \text{if } p(y = 1 \mid \mathbf{x}^*) > t \\
+   1 & \text{if } p(y = 1 \mid \mathbf{x}^*) > r \\
    0 & \text{otherwise}
    \end{cases}
    $$
 
-   Normally, \( t = 0.5 \).
+   ![Plasma Glukos Concentration vs Age](./figures/assignment3-2.png)
+
+   $E_{\text{mis}}(r = 0.5) = 0.2630208$
+
+   Comment: Not a very good model since we missclassify every fourth descision. Note: $r = 0.5$ gives the lowest possible missclassification rate.
+
+3. The equation of the decision boundary between the two classes
+
+   $$r = sigm(\mathbf{\theta}^T\mathbf{x}) = \frac{1}{1+e^{-\mathbf{\theta}^T\mathbf{x}}}$$
+
+   which results in
+
+   $$\ln(\frac{r}{1-r}) = \mathbf{\theta}^T\mathbf{x}$$
+
+   We get the following boundry:
 
-   Wrong 1/4 of times, not very good.
+   ![Plasma Glukos Concentration vs Age](./figures/assignment3-3.png)
 
-   0.2630208
+   This does not capture the original data distibution very well.
 
-3. a. theta0 + theta1* C1 + theta2*C2 = 0
+4. Lower $r$ gives lower risk of missing patient with diabetes as shown in the plots bellow.
 
-   b.
+   ![Plasma Glukos Concentration vs Age](./figures/assignment3-4-1.png)
 
-   Does not capture data distibution very well.
+   ![Plasma Glukos Concentration vs Age](./figures/assignment3-4-2.png)
 
-4. Lower r gives lower risk of missing patient with diabetes.
+5. The model becomes more complex with lower missclassification rate, which would mean that it is better.
 
-5. Model becomes more complex with lower missclassification rate.
+   ![Plasma Glukos Concentration vs Age](./figures/assignment3-5.png)
 
-0.2447917
+   $E_{\text{miss}}(r=0.5) = 0.2447917$
 
 ## Assignment 4
 
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