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<p>We developed an cross-platform, integrated, and graphical Java app
based on JavaFX for BNDSer to practice AP CSA FRQs.</p>
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<h3 id="前言">1. 前言</h3>
<p>在Coursera上吴恩达的Machine Learning中，Logistic
Regression作为初学者们接触到的第二个模型，其本身是非常重要的。但是在课程当中其损失函数的梯度公式则被一笔带过。而很多人都对其为什么与线性回归的损失函数梯度是一样的表示好奇，因此，我特地从头推导了这一式子。</p>
<h3 id="逻辑斯蒂回归损失函数梯度推导">2.
逻辑斯蒂回归损失函数梯度推导</h3>
<h4 id="introduction">2.1. Introduction</h4>
<p>The "Machine Learning" course on Coursera is a one of the most
popular course over all of MOOC. This course is famous for its
simplified teaching step and other bright spots. Logistics regression,
as the most important model for beginner, lacks appropriate detailed
derivation of formula. In this blog, I am going to expound this section
step by step.</p>
<h4 id="logistic-regression">2.2. Logistic Regression</h4>
<p>I think most of you have already got familiar with logistic
regression.</p>
<h4 id="cost-function">2.3. Cost Function</h4>
<p>As we all know, the cost function of the logistic regression is
modified from the one from linear regression model. <span
class="math display">\[
Cost\left( h_{\theta}\left( x \right) ,y \right) =\left\{
\begin{aligned}
-\log \left( h_{\theta}\left( x \right) \right)
\,\, if\,\,y&=1\\
-\log \left( 1-h_{\theta}\left( x \right) \right) \,\,
if\,\,y&=0\\
\end{aligned} \right.
\]</span></p>
<p>To simplify this function, we can write it as</p>
<p><span class="math display">\[
J\left( \theta \right) =\frac{1}{2m}\sum_{i=1}^m{Cost\left(
h_{\theta}\left( x^{\left( i \right)} \right) ,y^{\left( i \right)}
\right)}
\]</span></p>
<p>which</p>
<p><span class="math display">\[
Cost\left( h_{\theta}\left( x^{\left( i \right)} \right) ,y^{\left( i
\right)} \right) =-y^{\left( i \right)}\log \left( h_{\theta}\left( x
\right) \right) -\left( 1-y^{\left( i \right)} \right) \log \left(
1-h_{\theta}\left( x^{\left( i \right)} \right) \right)
\]</span></p>
<h4 id="gradient-of-cost-function">2.4. Gradient of Cost Function</h4>
<p>And next, is the part I want to expound. Andrew, in this MOOC, just
briefly introduce that the gradient of the new cost function is the same
as the one for linear regression. There is no explicit derivation
here.</p>
<p>For cost function <span class="math inline">\(J\)</span>, we can
write it as</p>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
J\left( \theta \right) &=-\frac{1}{m}\left[ \sum_{i=1}^m{-y^{\left(
i \right)}\log \left( h_{\theta}\left( x \right) \right) -\left(
1-y^{\left( i \right)} \right) \log \left( 1-h_{\theta}\left( x^{\left(
i \right)} \right) \right)} \right]
\\
\frac{\partial}{\partial \theta _j}J\left( \theta \right)
&=\frac{\partial}{\partial \theta _j}\left[ -\frac{1}{m}\left[
\sum_{i=1}^m{-y^{\left( i \right)}\log \left( h_{\theta}\left( x \right)
\right) -\left( 1-y^{\left( i \right)} \right) \log \left(
1-h_{\theta}\left( x^{\left( i \right)} \right) \right)} \right] \right]
\\
&=-\frac{1}{m}\left[ \sum_{i=1}^m{\left( y^{\left( i
\right)}\frac{1}{h_{\theta}\left( x^{\left( i \right)} \right)}\cdot
\frac{\partial}{\partial \theta _j}h_{\theta}\left( x^{\left( i \right)}
\right) +\left( 1-y^{\left( i \right)} \right) \cdot
\frac{1}{1-h_{\theta}\left( x^{\left( i \right)} \right)}\cdot
\frac{\partial}{\partial \theta _j}\left( -h_{\theta}\left( x^{\left( i
\right)} \right) \right) \right)} \right]
\\
&=-\frac{1}{m}\left[ \sum_{i=1}^m{\left( y^{\left( i
\right)}\frac{1}{h_{\theta}\left( x^{\left( i \right)} \right)}-\left(
1-y^{\left( i \right)} \right) \cdot \frac{1}{1-h_{\theta}\left(
x^{\left( i \right)} \right)} \right) \cdot \frac{\partial}{\partial
\theta _j}\left( h_{\theta}\left( x^{\left( i \right)} \right) \right)}
\right]
\\
&=-\frac{1}{m}\left[ \sum_{i=1}^m{\left( y^{\left( i
\right)}\frac{1}{g\left( \theta ^Tx \right)}-\left( 1-y^{\left( i
\right)} \right) \cdot \frac{1}{1-g\left( \theta ^Tx \right)} \right)
\cdot \frac{\partial}{\partial \theta _j}\left( g\left( \theta ^Tx
\right) \right)} \right]
\end{aligned}
\end{equation}\]</span></p>
<p>In logistic regression, we use the logistic function as our decision
function. Therefore, (Mentioned: <span class="math inline">\(T\)</span>
refers to transpose)</p>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
\frac{\partial}{\partial \theta _j}g\left( \theta ^Tx \right)
&=\frac{\partial}{\partial \theta _j}\cdot \frac{1}{1+e^{-\theta
^Tx}}
\\
&=\frac{\partial}{\partial \theta _j}\left( 1+e^{-\theta ^Tx}
\right) ^{-1}
\\
&=-\left( 1+e^{-\theta ^Tx} \right) ^{-2}\cdot e^{e^{-\theta
^Tx}}\cdot x
\\
&=-\frac{e^{e^{-\theta ^Tx}}\cdot -x}{\left( 1+e^{e^{-\theta ^Tx}}
\right) ^2}
\end{aligned}
\end{equation}\]</span></p>
<p>We set <span class="math inline">\(k=e^{-\theta ^Tx}\)</span>, plug
it in,</p>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
&=\frac{k}{\left( 1+k \right) ^2}\cdot x
\\
&=\left( \frac{1}{1+k}\cdot \frac{1+k-1}{1+k} \right) \cdot x
\\
&=\left[ \frac{1}{1+k}\cdot \left( 1-\frac{1}{1+k} \right) \right]
\cdot x
\end{aligned}
\end{equation}\]</span></p>
<p>Bring it back, we can see we construct the logistic function
itself,</p>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
&=\left[ \frac{1}{1+e^{-\theta ^Tx}}\cdot \left(
1-\frac{1}{1+e^{-\theta ^Tx}} \right) \right] \cdot x
\end{aligned}
\end{equation}\]</span></p>
<p>Thus, we can use the logistic function to simplify the eqution,</p>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
&=\left[ \frac{1}{1+e^{-\theta ^Tx}}\cdot \left(
1-\frac{1}{1+e^{-\theta ^Tx}} \right) \right] \cdot x
\\
&=g\left( \theta ^Tx \right) \cdot \left( 1-g\left( \theta ^Tx
\right) \right) \cdot x
\end{aligned}
\end{equation}\]</span></p>
<p>We plug back this portion to the gradient of cost function (1),</p>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
\frac{\partial}{\partial \theta _j}J\left( \theta \right)
&=-\frac{1}{m}\left[ \sum_{i=1}^m{\left( y^{\left( i \right)}\cdot
\frac{1}{g\left( \theta ^Tx \right)}-\left( 1-y^{\left( i \right)}
\right) \cdot \frac{1}{1-g\left( \theta ^Tx \right)} \right)}\cdot
g\left( \theta ^Tx \right) \cdot \left( 1-g\left( \theta ^Tx \right)
\right) \cdot x^{\left( i \right)} \right]
\\
&=-\frac{1}{m}\left[ \sum_{i=1}^m{\left[ y^{\left( i \right)}\cdot
\left( 1-g\left( \theta ^Tx \right) \right) \cdot x^{\left( i
\right)}-\left( 1-y^{\left( i \right)} \right) \cdot g\left( \theta ^Tx
\right) \cdot x^{\left( i \right)} \right]} \right]
\\
&=-\frac{1}{m}\sum_{i=1}^m{\left[ y^{\left( i \right)}\cdot
x^{\left( i \right)}-y^{\left( i \right)}\cdot g\left( \theta ^Tx
\right) \cdot x^{\left( i \right)}-g\left( \theta ^Tx \right) \cdot
x^{\left( i \right)}+y^{\left( i \right)}\cdot g\left( \theta ^Tx
\right) \cdot x^{\left( i \right)} \right]}
\\
&\mathrm{because}-y^{\left( i \right)}\cdot g\left( \theta ^Tx
\right) \cdot x^{\left( i \right)}\,\,\mathrm{offsets} +y^{\left( i
\right)}\cdot g\left( \theta ^Tx \right) \cdot x^{\left( i \right)}
\\
&=-\frac{1}{m}\sum_{i=1}^m{\left[ y^{\left( i \right)}\cdot
x^{\left( i \right)}-g\left( \theta ^Tx \right) \cdot x^{\left( i
\right)} \right]}
\\
&=-\frac{1}{m}\sum_{i=1}^m{\left[ \left[ y^{\left( i
\right)}-g\left( \theta ^Tx \right) \right] \cdot x^{\left( i \right)}
\right]}
\\
&=\frac{1}{m}\sum_{i=1}^m{\left[ \left[ h_{\theta}\left( x^{\left( i
\right)} \right) -y^{\left( i \right)} \right] \cdot x^{\left( i
\right)} \right]}
\end{aligned}
\end{equation}\]</span></p>
<p>which is the same as linear regression's</p>
<p>Q.E.D.</p>
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<h3 id="introduction">Introduction</h3>
<p>In my second semester of High School Junior year, when we were
studying vector calculus <span class="citation"
data-cites="Calculus">@Calculus</span> 3, our teacher gave us an
assignment: proof of Stokes formula. CALCULUS EARLY TRANSCENDENTALS
EIGHTH EDITION JAMES STEWART (our Calculus3 textbook) only has a special
form of stokes' theorem proof. Thus, I looked up the prood at old
Chinese Text Book. It is a pity that our high school calculus education
with American calculus education are quite different compared to
Chinese's. I searched on zhihu, Bilibili and other public platforms, but
failed to find any suitable results. Luckily, I finally figure it out
when I went back home during the epidemic.</p>
<h3 id="stokes-theorem">Stoke's Theorem</h3>
<p>For surface S in <span class="math inline">\(R^3\)</span> space is an
oriented surface in <span class="math inline">\(x,y,z\)</span>
coordinate with boundary <span class="math inline">\(\partial
S\)</span>. Let <span class="math inline">\(R\)</span> be abounded, open
region in <span class="math inline">\(s, t\)</span> plane with smooth
boundary <span class="math inline">\(\partial R\)</span> . Suppose that
<span class="math inline">\(F\)</span> is a continuously differentiable
vector field. <span class="math inline">\(\vec{r}\)</span> is a smooth
parametrization that maps <span class="math inline">\(R\)</span> to
<span class="math inline">\(S\)</span>, and <span
class="math inline">\(\partial R\)</span> to <span
class="math inline">\(\partial S\)</span>. Then</p>
<p><span class="math display">\[
\iint_S{curl\,\,\vec{F}\,\,·\,\,d\vec{S}}=\iint_S{\left( \nabla \times
\vec{F} \right) \,\,·\,\,\vec{n}\,\,dS}=\int_{\partial
S}{\vec{F}\,\,·\,\,\vec{T}\,\,ds}=\,\,\int_{\partial
S}{\vec{F}\,\,·\,\,d\vec{r}}
\]</span></p>
<figure>
<img src="/img/stks.png" alt="stks" />
<figcaption aria-hidden="true">stks</figcaption>
</figure>
<h3 id="proof">Proof</h3>
<p>Let</p>
<p><span class="math display">\[
\vec{r}\left( s,t \right) =\left[ \begin{array}{c}
x\left( s,t \right)\\
y\left( s,t \right)\\
z\left( s,t \right)\\
\end{array} \right] \,\,: R\rightarrow S.
\]</span></p>
<p>Then.</p>
<p><span class="math display">\[
d\vec{r}\mid_{\partial R}^{}\,\,=\left[ \begin{matrix}
\frac{\partial x}{\partial s}& \frac{\partial x}{\partial
t}\\
\frac{\partial y}{\partial s}& \frac{\partial y}{\partial
t}\\
\frac{\partial z}{\partial s}& \frac{\partial z}{\partial
t}\\
\end{matrix} \right] \left[ \begin{array}{c}
ds\\
dt\\
\end{array} \right] \,\,=\left[ \begin{array}{c}
\frac{\partial x}{\partial s}ds+\frac{\partial x}{\partial t}dt\\
\frac{\partial y}{\partial s}ds+\frac{\partial y}{\partial t}dt\\
\frac{\partial z}{\partial s}ds+\frac{\partial z}{\partial t}dt\\
\end{array} \right] \,\,=\left[ \begin{array}{c}
\frac{\partial x}{\partial s}\\
\frac{\partial y}{\partial s}\\
\frac{\partial z}{\partial s}\\
\end{array} \right] ds+\left[ \begin{array}{c}
\frac{\partial x}{\partial t}\\
\frac{\partial y}{\partial t}\\
\frac{\partial z}{\partial t}\\
\end{array} \right] dt\,\,=\frac{\partial \vec{r}}{\partial
s}ds+\frac{\partial \vec{r}}{\partial t}dt
\]</span></p>
<p>Hence.</p>
<p><span class="math display">\[
\int_{\partial S}{\vec{F}·}d\vec{r}=\int_{\partial S}{\left(
\vec{F}·\frac{\partial \vec{r}}{\partial s}ds+\vec{F}·\frac{\partial
\vec{r}}{\partial s} \right)}
\]</span></p>
<p>We define a 2-dimensional vector field <span
class="math inline">\(G=(G1,G2)\)</span> on the s,t, plane by</p>
<p><span class="math display">\[
G_1=\vec{F}\,\,·\,\,\frac{\partial \vec{r}}{\partial
s}\,\, and\,\, G_2=\vec{F}\,\,·\,\,\frac{\partial \vec{r}}{\partial t}
\]</span></p>
<p>Therefore, we put G into the original line integral</p>
<p><span class="math display">\[
\int_{\partial S}{\vec{F}\,\,·\,\,d\vec{r}=}\int_{\partial R}{\left(
G_1ds+G_2dt \right) =\int_R{\begin{array}{c}
\left( \frac{\partial G_2}{\partial s}-\frac{\partial G_1}{\partial
t} \right) dsdt\,\,,\\
\end{array}}\,\,}
\]</span></p>
<p><span class="math display">\[
\int_{\partial S}{\vec{F}\,\,·\,\,d\vec{r}=}\int_{\partial R}{\left(
G_1ds+G_2dt \right) \,\,}
\]</span></p>
<p>On the other hand.</p>
<p><span class="math display">\[
\iint_S{curl\,\,\vec{F}\,\,·\,\,d\vec{S}\,\,=\,\,\iint_R{curl\,\,\vec{F}\mid_{\vec{r}}^{}·\frac{\partial
\vec{r}}{\partial s}\times \frac{\partial \vec{r}}{\partial t}dsdt}}
\]</span></p>
<p>We expand it, get</p>
<p><span class="math display">\[
curl\,\,\vec{F}\mid_{\vec{r}}^{}\cdot \frac{\partial \vec{r}}{\partial
s}\times \frac{\partial \vec{r}}{\partial t}\,\,=\,\,\left|
\begin{matrix}
\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial
z}& \frac{\partial F_1}{\partial z}-\frac{\partial
F_3}{\partial x}& \frac{\partial F_2}{\partial
x}-\frac{\partial F_1}{\partial y}\\
\frac{\partial x}{\partial s}& \frac{\partial y}{\partial
s}& \frac{\partial z}{\partial s}\\
\frac{\partial x}{\partial t}& \frac{\partial y}{\partial
t}& \frac{\partial z}{\partial t}\\
\end{matrix} \right|
\]</span></p>
<p><span class="math display">\[
=\frac{\partial \vec{F}}{\partial s}·\frac{\partial \vec{r}}{\partial
t}-\frac{\partial \vec{F}}{\partial t}·\frac{\partial \vec{r}}{\partial
s}\,\,=\frac{\partial G_2}{\partial s}-\frac{\partial G_1}{\partial t}
\]</span></p>
<p>Hence.</p>
<p><span class="math display">\[
\int_R{curl\,\,F\mid_{\vec{r}}^{}\cdot \frac{\partial \vec{r}}{\partial
s}}\times \frac{\partial \vec{r}}{\partial
t}\,\,dsdt\,\,=\,\,\int_R{\begin{array}{c}
\left( \frac{\partial G_2}{\partial s}-\frac{\partial G_1}{\partial
t} \right) dsdt\,\,\\
\end{array}}
\]</span></p>
<p>For the line integral part,we have</p>
<p><span class="math display">\[
\int_{\partial S}{\vec{F}\,\,·\,\,d\vec{r}=}\int_{\partial
R}{\begin{array}{c}
\left( G_1ds+G_2dt \right) \,\,\\
\end{array}}
\]</span></p>
<p>Use Green's Theorem, we know</p>
<p><span class="math display">\[
\int_{\partial R}{\begin{array}{c}
\left( G_1ds+G_2dt \right)\\
\end{array}}=\int_R{\begin{array}{c}
\left( \frac{\partial G_2}{\partial s}-\frac{\partial G_1}{\partial
t} \right) dsdt\,\,,\\
\end{array}}
\]</span></p>
<p>So, we can conclude that</p>
<p><span class="math display">\[
\int_{\partial S}{\vec{F}·}d\vec{r}=\int_R{\begin{array}{c}
\left( \frac{\partial G_2}{\partial s}-\frac{\partial G_1}{\partial
t} \right) dsdt=\int_R{curl\,\,F\mid_{\vec{r}}^{}\cdot \frac{\partial
\vec{r}}{\partial s}}\times \frac{\partial \vec{r}}{\partial t}\\
\end{array}}dsdt=\iint_S{curl\,\,F\,\,·\,\,d\vec{S}}
\]</span></p>
<p>Thus, we finished our proof of Stokes' Theorem.</p>
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<h3 id="machine-learning-by-andrew-ng">Machine Learning By Andrew
Ng</h3>
<h4 id="content">Content</h4>
<ul>
<li>Introduction</li>
<li>Potential Submit Problem</li>
<li>ex1.m</li>
<li>computerCost.m</li>
<li>gradientDescent.m</li>
<li>ex1_multi.m</li>
<li>featureNormalize.m</li>
<li>computeCostMulti.m</li>
<li>gradientDescentMulti.m</li>
<li>normalEqn.m</li>
</ul>
<h4 id="introduction">Introduction</h4>
<p>I have listened for so many times that the course: Machine Learning
by Andrew Ng is one of the best introductory course for machine
learning. Now, I have finished the first two week's classed and pass the
first programming project. I truly fathom out the deliberate design of
this course. The passion and meticulous designed project impress me as
well. It is so particular that I feel like someone is teaching me
hand-by-hand. Coursera uses a kind of judging method which is not like
the common OJ that you upload your program and they run it and test it
with their test data. Coursera uses an aggregate local submit module
that can do all the stuff. You just need to generate a token that linked
to your coursera account so that it can upload your progress. It is so
cooooool!!!!!! <img src="/img/ML/ex1/f1.png" alt="f1" /> <img
src="/img/ML/ex1/f2.png" alt="f2" /> ##### Potential Submit Problem Once
I tried to submit the program, it says <img src="/img/ML/ex1/d1.png"
alt="d1" /> According to
https://blog.csdn.net/qq_44498043/article/details/105904715 and
https://blog.csdn.net/weixin_45923568/article/details/104193579 , I
found out that the error is caused by the vulnerable grammar that it
used before. And in the latest version of matlab, matlab banned those
kinds of grammar in order to increase the stability. So all we need is
to go to the coursera website and download the new version of submit
module.</p>
<h4 id="ex1.m">ex1.m</h4>
<pre class="line-numbers language-matlab" data-language="matlab"><code class="language-matlab"><span class="token comment">%% Initialization</span>
clear <span class="token punctuation">;</span> close all<span class="token punctuation">;</span> clc
<span class="token comment">%% ==================== Part 1: Basic Function ====================</span>
<span class="token comment">% Complete warmUpExercise.m</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Running warmUpExercise ... \n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'5x5 Identity Matrix: \n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">warmUpExercise</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Program paused. Press enter to continue.\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">pause</span><span class="token punctuation">;</span>
<span class="token comment">%% ======================= Part 2: Plotting =======================</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Plotting Data ...\n'</span><span class="token punctuation">)</span>
data <span class="token operator">=</span> <span class="token function">load</span><span class="token punctuation">(</span><span class="token string">'ex1data1.txt'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
X <span class="token operator">=</span> <span class="token function">data</span><span class="token punctuation">(</span><span class="token operator">:</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">;</span> y <span class="token operator">=</span> <span class="token function">data</span><span class="token punctuation">(</span><span class="token operator">:</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
m <span class="token operator">=</span> <span class="token function">length</span><span class="token punctuation">(</span>y<span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token comment">% number of training examples</span>
<span class="token comment">% Plot Data</span>
<span class="token comment">% Note: You have to complete the code in plotData.m</span>
<span class="token function">plotData</span><span class="token punctuation">(</span>X<span class="token punctuation">,</span> y<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Program paused. Press enter to continue.\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">pause</span><span class="token punctuation">;</span>
<span class="token comment">%% =================== Part 3: Cost and Gradient descent ===================</span>
X <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token function">ones</span><span class="token punctuation">(</span>m<span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token function">data</span><span class="token punctuation">(</span><span class="token operator">:</span><span class="token punctuation">,</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">]</span><span class="token punctuation">;</span> <span class="token comment">% Add a column of ones to x</span>
theta <span class="token operator">=</span> <span class="token function">zeros</span><span class="token punctuation">(</span><span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token comment">% initialize fitting parameters</span>
<span class="token comment">% Some gradient descent settings</span>
iterations <span class="token operator">=</span> <span class="token number">1500</span><span class="token punctuation">;</span>
alpha <span class="token operator">=</span> <span class="token number">0.01</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'\nTesting the cost function ...\n'</span><span class="token punctuation">)</span>
<span class="token comment">% compute and display initial cost</span>
J <span class="token operator">=</span> <span class="token function">computeCost</span><span class="token punctuation">(</span>X<span class="token punctuation">,</span> y<span class="token punctuation">,</span> theta<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'With theta = [0 ; 0]\nCost computed = %f\n'</span><span class="token punctuation">,</span> J<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Expected cost value (approx) 32.07\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token comment">% further testing of the cost function</span>
J <span class="token operator">=</span> <span class="token function">computeCost</span><span class="token punctuation">(</span>X<span class="token punctuation">,</span> y<span class="token punctuation">,</span> <span class="token punctuation">[</span><span class="token operator">-</span><span class="token number">1</span> <span class="token punctuation">;</span> <span class="token number">2</span><span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'\nWith theta = [-1 ; 2]\nCost computed = %f\n'</span><span class="token punctuation">,</span> J<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Expected cost value (approx) 54.24\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Program paused. Press enter to continue.\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">pause</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'\nRunning Gradient Descent ...\n'</span><span class="token punctuation">)</span>
<span class="token comment">% run gradient descent</span>
theta <span class="token operator">=</span> <span class="token function">gradientDescent</span><span class="token punctuation">(</span>X<span class="token punctuation">,</span> y<span class="token punctuation">,</span> theta<span class="token punctuation">,</span> alpha<span class="token punctuation">,</span> iterations<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token comment">% print theta to screen</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Theta found by gradient descent:\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'%f\n'</span><span class="token punctuation">,</span> theta<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Expected theta values (approx)\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">' -3.6303\n 1.1664\n\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token comment">% Plot the linear fit</span>
hold on<span class="token punctuation">;</span> <span class="token comment">% keep previous plot visible</span>
<span class="token function">plot</span><span class="token punctuation">(</span><span class="token function">X</span><span class="token punctuation">(</span><span class="token operator">:</span><span class="token punctuation">,</span><span class="token number">2</span><span class="token punctuation">)</span><span class="token punctuation">,</span> X<span class="token operator">*</span>theta<span class="token punctuation">,</span> <span class="token string">'-'</span><span class="token punctuation">)</span>
<span class="token function">legend</span><span class="token punctuation">(</span><span class="token string">'Training data'</span><span class="token punctuation">,</span> <span class="token string">'Linear regression'</span><span class="token punctuation">)</span>
hold off <span class="token comment">% don't overlay any more plots on this figure</span>
<span class="token comment">% Predict values for population sizes of 35,000 and 70,000</span>
predict1 <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">3.5</span><span class="token punctuation">]</span> <span class="token operator">*</span>theta<span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'For population = 35,000, we predict a profit of %f\n'</span><span class="token punctuation">,</span><span class="token punctuation">...</span>
predict1<span class="token operator">*</span><span class="token number">10000</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
predict2 <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">7</span><span class="token punctuation">]</span> <span class="token operator">*</span> theta<span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'For population = 70,000, we predict a profit of %f\n'</span><span class="token punctuation">,</span><span class="token punctuation">...</span>
predict2<span class="token operator">*</span><span class="token number">10000</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Program paused. Press enter to continue.\n'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">pause</span><span class="token punctuation">;</span>
<span class="token comment">%% ============= Part 4: Visualizing J(theta_0, theta_1) =============</span>
<span class="token function">fprintf</span><span class="token punctuation">(</span><span class="token string">'Visualizing J(theta_0, theta_1) ...\n'</span><span class="token punctuation">)</span>
<span class="token comment">% Grid over which we will calculate J</span>
theta0_vals <span class="token operator">=</span> <span class="token function">linspace</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">10</span><span class="token punctuation">,</span> <span class="token number">10</span><span class="token punctuation">,</span> <span class="token number">100</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
theta1_vals <span class="token operator">=</span> <span class="token function">linspace</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">4</span><span class="token punctuation">,</span> <span class="token number">100</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token comment">% initialize J_vals to a matrix of 0's</span>
J_vals <span class="token operator">=</span> <span class="token function">zeros</span><span class="token punctuation">(</span><span class="token function">length</span><span class="token punctuation">(</span>theta0_vals<span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token function">length</span><span class="token punctuation">(</span>theta1_vals<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token comment">% Fill out J_vals</span>
<span class="token keyword">for</span> <span class="token number">i</span> <span class="token operator">=</span> <span class="token number">1</span><span class="token operator">:</span><span class="token function">length</span><span class="token punctuation">(</span>theta0_vals<span class="token punctuation">)</span>
<span class="token keyword">for</span> <span class="token number">j</span> <span class="token operator">=</span> <span class="token number">1</span><span class="token operator">:</span><span class="token function">length</span><span class="token punctuation">(</span>theta1_vals<span class="token punctuation">)</span>
t <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token function">theta0_vals</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token function">theta1_vals</span><span class="token punctuation">(</span><span class="token number">j</span><span class="token punctuation">)</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token function">J_vals</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">,</span><span class="token number">j</span><span class="token punctuation">)</span> <span class="token operator">=</span> <span class="token function">computeCost</span><span class="token punctuation">(</span>X<span class="token punctuation">,</span> y<span class="token punctuation">,</span> t<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">end</span>
<span class="token keyword">end</span>
<span class="token comment">% Because of the way meshgrids work in the surf command, we need to</span>
<span class="token comment">% transpose J_vals before calling surf, or else the axes will be flipped</span>
J_vals <span class="token operator">=</span> J_vals<span class="token operator">'</span><span class="token punctuation">;</span>
<span class="token comment">% Surface plot</span>
figure<span class="token punctuation">;</span>
<span class="token function">surf</span><span class="token punctuation">(</span>theta0_vals<span class="token punctuation">,</span> theta1_vals<span class="token punctuation">,</span> J_vals<span class="token punctuation">)</span>
<span class="token function">xlabel</span><span class="token punctuation">(</span><span class="token string">'\theta_0'</span><span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token function">ylabel</span><span class="token punctuation">(</span><span class="token string">'\theta_1'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token comment">% Contour plot</span>
figure<span class="token punctuation">;</span>
<span class="token comment">% Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100</span>
<span class="token function">contour</span><span class="token punctuation">(</span>theta0_vals<span class="token punctuation">,</span> theta1_vals<span class="token punctuation">,</span> J_vals<span class="token punctuation">,</span> <span class="token function">logspace</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">3</span><span class="token punctuation">,</span> <span class="token number">20</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token function">xlabel</span><span class="token punctuation">(</span><span class="token string">'\theta_0'</span><span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token function">ylabel</span><span class="token punctuation">(</span><span class="token string">'\theta_1'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
hold on<span class="token punctuation">;</span>
<span class="token function">plot</span><span class="token punctuation">(</span><span class="token function">theta</span><span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token function">theta</span><span class="token punctuation">(</span><span class="token number">2</span><span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token string">'rx'</span><span class="token punctuation">,</span> <span class="token string">'MarkerSize'</span><span class="token punctuation">,</span> <span class="token number">10</span><span class="token punctuation">,</span> <span class="token string">'LineWidth'</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<h5 id="computercost.m">computerCost.m</h5>
<pre class="line-numbers language-matlab" data-language="matlab"><code class="language-matlab"><span class="token keyword">function</span> J <span class="token operator">=</span> <span class="token function">computeCost</span><span class="token punctuation">(</span>X<span class="token punctuation">,</span> y<span class="token punctuation">,</span> theta<span class="token punctuation">)</span>
<span class="token comment">%COMPUTECOST Compute cost for linear regression</span>
<span class="token comment">% J = COMPUTECOST(X, y, theta) computes the cost of using theta as the</span>
<span class="token comment">% parameter for linear regression to fit the data points in X and y</span>
<span class="token comment">% Initialize some useful values</span>
m <span class="token operator">=</span> <span class="token function">length</span><span class="token punctuation">(</span>y<span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token comment">% number of training examples</span>
<span class="token comment">% You need to return the following variables correctly </span>