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-rw-r--r--docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html10
-rw-r--r--docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html90
-rw-r--r--docs/posts/index.html11
3 files changed, 104 insertions, 7 deletions
diff --git a/docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html b/docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html
index 7a25daf..ab46ec7 100644
--- a/docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html
+++ b/docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html
@@ -107,9 +107,7 @@
<script src="https://cdn.jsdelivr.net/npm/mathjax@4.0.0-beta.4/input/tex/extensions/noerrors.js" charset="UTF-8"></script>
-<p>$$
-y = ax^3 + bx^2 + cx + d
-$$</p>
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>y</mi><mo>&#x0003D;</mo><mi>a</mi><msup><mi>x</mi><mn>3</mn></msup><mo>&#x0002B;</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mi>c</mi><mi>x</mi><mo>&#x0002B;</mo><mi>d</mi></mrow></math>
<h3>Optimizer Selection &amp; Training</h3>
@@ -134,9 +132,7 @@ $$</p>
<p>Our loss function is Mean Squared Error (MSE):</p>
-<p>$$
-= \frac{1}{n} \sum_{i=1}^{n}{(Y_i - \hat{Y_i})^2}
-$$</p>
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>&#x0003D;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><msubsup><mo>&#x02211;</mo><mrow><mi>i</mi><mo>&#x0003D;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo stretchy="false">&#x00028;</mo><mi>Y</mi><mi>&#x0005F;</mi><mi>i</mi><mo>&#x02212;</mo><mover><mrow><mi>Y</mi><mi>&#x0005F;</mi><mi>i</mi></mrow><mo stretchy="false">&#x0005E;</mo></mover><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup></mrow></mrow></math>
<p>Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false" style="math-style:normal;math-depth:0;">^</mo></mover></math> is the predicted value and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub></math> is the actual value</p>
@@ -276,7 +272,7 @@ $$</p>
<p>How would you modify this code to use another type of nonlinear regression? Say, </p>
-<p>$$ y = ab^x $$</p>
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>y</mi><mo>&#x0003D;</mo><mi>a</mi><msup><mi>b</mi><mi>x</mi></msup></mrow></math>
<p>Hint: Your loss calculation would be similar to:</p>
diff --git a/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html b/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html
new file mode 100644
index 0000000..6f02f7c
--- /dev/null
+++ b/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html
@@ -0,0 +1,90 @@
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+ <title>Quadratic Formula Derivation</title>
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+ <h1>Quadratic Formula Derivation</h1>
+
+<p>The standard form of a quadratic equation is:</p>
+
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mi>b</mi><mi>x</mi><mo>&#x0002B;</mo><mi>c</mi><mo>&#x0003D;</mo><mn>0</mn></mrow></math>
+
+<p>Here, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>&#x0002C;</mo><mi>b</mi><mo>&#x0002C;</mo><mi>c</mi><mo>&#x02208;</mo><mi>&#x0211D;</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>&#x02260;</mo><mn>0</mn></mrow></math></p>
+
+<p>We begin by first dividing both sides by the coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi></mrow></math></p>
+
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>&#x027F9;</mi><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>&#x0002B;</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>&#x0003D;</mo><mn>0</mn></mrow></math>
+
+<p>We can rearrange the equation:</p>
+
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>&#x0003D;</mo><mo>&#x02212;</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac></mrow></math>
+
+<p>We can then use the method of completing the square. (<a rel="noopener" target="_blank" href="https://www.mathsisfun.com/algebra/completing-square.html">Maths is Fun</a> has a really good explanation for this technique)</p>
+
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>&#x0002B;</mo><mo stretchy="false">&#x00028;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>&#x0002B;</mo><mo stretchy="false">&#x00028;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup></mrow></math>
+
+<p>On our LHS, we can clearly recognize that it is the expanded form of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">&#x00028;</mo><mi>x</mi><mo>&#x0002B;</mo><mi>d</mi><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup></mrow></math> i.e <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mn>2</mn><mi>x</mi><mi>&#x000B7;</mi><mi>d</mi><mo>&#x0002B;</mo><msup><mi>d</mi><mn>2</mn></msup></mrow></math></p>
+
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>&#x027F9;</mi><mo stretchy="false">&#x00028;</mo><mi>x</mi><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>&#x0002B;</mo><mfrac><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>&#x0002B;</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></mrow></math>
+
+<p>Taking the square root of both sides</p>
+
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable displaystyle="true" rowspacing="3pt" columnspacing="0em 2em"><mtr><mtd columnalign="right"><mi>x</mi><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd><mtd columnalign="left"><mi /><mo>&#x0003D;</mo><mfrac><mrow><msqrt><mrow><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>&#x0002B;</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mi /><mo>&#x0003D;</mo><mfrac><mrow><mi>&#x000B1;</mi><msqrt><mrow><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>&#x0002B;</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt><mo>&#x02212;</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><mi /><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mi>b</mi><mi>&#x000B1;</mi><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr></mtable></mrow></math>
+
+<p>This gives you the world famous quadratic formula:</p>
+
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>x</mi><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mi>b</mi><mi>&#x000B1;</mi><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></math>
+
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diff --git a/docs/posts/index.html b/docs/posts/index.html
index d886b19..40b6a92 100644
--- a/docs/posts/index.html
+++ b/docs/posts/index.html
@@ -52,6 +52,17 @@
<ul>
+ <li><a href="/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html">Quadratic Formula Derivation</a></li>
+ <ul>
+ <li>Quick derivation of the quadratic equation by completing the square</li>
+ <li>Published On: 2024-03-26 15:36</li>
+ <li>Tags:
+
+ <a href='/tags/mathematics.html'>mathematics</a>
+
+ </ul>
+
+
<li><a href="/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html">Polynomial Regression Using TensorFlow 2.x</a></li>
<ul>
<li>Predicting n-th degree polynomials using TensorFlow 2.x</li>