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<body>
<h1 id="homework-5">Homework #5</h1>
<h2 id="due-414-by-1159pm">due 4/14 by 11:59pm</h2>
<p><strong>1.</strong> Create a new github repository called ‘05_curve_fitting’.</p>
<ol style="list-style-type: lower-alpha">
<li><p>Add rcc02007 and zhs15101 as collaborators.</p></li>
<li><p>Clone the repository to your computer.</p></li>
</ol>
<p><strong>2.</strong> Create a least-squares function called <code>least_squares.m</code> that accepts a Z-matrix and dependent variable y as input and returns the vector of best-fit constants, a, the best-fit function evaluated at each point <span class="math inline">\(f(x_{i})\)</span>, and the coefficient of determination, r2.</p>
<div class="sourceCode"><pre class="sourceCode matlab"><code class="sourceCode matlab">[a,fx,r2]=least_squares(Z,y);</code></pre></div>
<p>Test your function on the sets of data in script <code>problem_2_data.m</code> and show that the following functions are the best fit lines. Report the coefficient of determination in your README.</p>
<ol style="list-style-type: lower-alpha">
<li><p>y=0.3745+0.98644x+0.84564/x</p></li>
<li><p>y= 22.47-1.36x+0.28x^2</p></li>
<li><p>y=4.0046e<sup>(-1.5x)+2.9213e</sup>(-0.3x)+1.5647e^(-0.05x)</p></li>
<li><p>y=0.99sin(t)+0.5sin(3t)</p></li>
</ol>
<p><strong>3.</strong> Load the data from the <a href="https://github.com/cooperrc/stats_and_linear_regression/blob/master/compiled_data.csv">class dart experiment</a>. Show your work in a script with filename <code>dart_statistics.m</code> in your repository.</p>
<ol style="list-style-type: lower-alpha">
<li><p>Which user (0-32) was the most accurate dart thrower e.g. mean(x)+mean(y) was closest to 0 cm?</p></li>
<li><p>Which user (0-32) was the most precise dart thrower e.g. std(x)+std(y) was closest to 0 cm?</p></li>
</ol>
<p><strong>4.</strong> 100 steel rods are going to be used to support a 1000 kg structure. The rods will buckle when the load in any rod exceeds the <a href="https://en.wikipedia.org/wiki/Euler%27s_critical_load">critical buckling load</a></p>
<p><span class="math inline">\(P_{cr}=\frac{\pi^3 Er^4}{16L^2}\)</span> <img src="./equations/buckle.png" alt="buckle" /></p>
<p>where E=200e9 Pa, r=0.01 m +/-0.001 m, and L is the length of the rod which can only be controlled to 1% tolerance. Create a Monte Carlo model <code>buckle_monte_carlo.m</code> that predicts the mean and standard deviation of the buckling load for 100 samples with normally distributed dimensions r and L.</p>
<p><code>[mean_buckle_load,std_buckle_load]=buckle_monte_carlo(E,r_mean,r_std,L_mean,L_std)</code></p>
<ol style="list-style-type: lower-alpha">
<li><p>What is the mean_buckle_load and std_buckle_load for L=5 m?</p></li>
<li><p>What length, L, should the beams be so that only 2.5% will reach the critical buckling load?</p></li>
</ol>
<p><strong>5.</strong> The drag coefficient for spheres such as sporting balls is known to vary as a function of the Reynolds number Re, a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces:</p>
<p><span class="math inline">\(Re = \frac{\rho V D}{\mu}\)</span> <img src="./equations/reynolds.png" alt="Reynolds" /></p>
<p>where ρ= the fluid’s density (kg/m3), V= its velocity (m/s), D= diameter (m), and μ= dynamic viscosity (N⋅s/m2). The following table provides values for a smooth spherical ball:</p>
<table>
<thead>
<tr class="header">
<th align="left">Re (x1e-4)</th>
<th align="left">C_D</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">2</td>
<td align="left">0.52</td>
</tr>
<tr class="even">
<td align="left">5.8</td>
<td align="left">0.52</td>
</tr>
<tr class="odd">
<td align="left">16.8</td>
<td align="left">0.52</td>
</tr>
<tr class="even">
<td align="left">27.2</td>
<td align="left">0.5</td>
</tr>
<tr class="odd">
<td align="left">29.9</td>
<td align="left">0.49</td>
</tr>
<tr class="even">
<td align="left">33.9</td>
<td align="left">0.44</td>
</tr>
<tr class="odd">
<td align="left">36.3</td>
<td align="left">0.18</td>
</tr>
<tr class="even">
<td align="left">40</td>
<td align="left">0.074</td>
</tr>
<tr class="odd">
<td align="left">46</td>
<td align="left">0.067</td>
</tr>
<tr class="even">
<td align="left">60</td>
<td align="left">0.08</td>
</tr>
<tr class="odd">
<td align="left">100</td>
<td align="left">0.12</td>
</tr>
<tr class="even">
<td align="left">200</td>
<td align="left">0.16</td>
</tr>
<tr class="odd">
<td align="left">400</td>
<td align="left">0.19</td>
</tr>
</tbody>
</table>
<ol style="list-style-type: lower-alpha">
<li>Create a function <code>sphere_drag.m</code> that outputs the drag coefficient based on the given table and an input Reynolds number using a spline interpolation of either linear (‘linear’), piecewise cubic (‘pchip’), or cubic (‘cubic’):</li>
</ol>
<p><code>[Cd_out]=sphere_drag(Re_in,spline_type)</code></p>
<ol start="2" style="list-style-type: lower-alpha">
<li>Use the following physical constants to plot the drag force vs velocity for a baseball: ρ= 1.3 kg/m3, V= 4 - 40 (m/s), D= 23.5 cm, and μ=1.78e-5 Pa-s. Plot all three interpolation methods on a single plot. Show the plot in your README.</li>
</ol>
<p><strong>6.</strong> Evaluate the integral of the following function:</p>
<p><span class="math inline">\(\int_{2}^{3}f(x) = \int_{2}^{3} 1/6x^3 + 1/2x^2+x dx\)</span></p>
<ol style="list-style-type: lower-alpha">
<li><p>analytically</p></li>
<li><p>with 1-point Gauss quadrature</p></li>
<li><p>with 2-point Gauss quadrature</p></li>
<li><p>with 3-point Gauss quadrature</p></li>
<li><p>include the results in a table in your README</p></li>
</ol>
<table>
<thead>
<tr class="header">
<th align="left">method</th>
<th align="left">value</th>
<th align="left">error</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">analytical</td>
<td align="left">…</td>
<td align="left">0%</td>
</tr>
<tr class="even">
<td align="left">1 Gauss point</td>
<td align="left">…</td>
<td align="left">..%</td>
</tr>
<tr class="odd">
<td align="left">…</td>
<td align="left">…</td>
<td align="left">…</td>
</tr>
</tbody>
</table>
</body>
</html>
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# Homework #5
## due 11/17 by 11:59pm


### Problem 1 due 11/9

**1\.** Create a new github repository called '05_curve_fitting'.

a. Add rcc02007 and zhs15101 as collaborators.

b. Clone the repository to your computer.


**2\.** Create a least-squares function called `least_squares.m` that accepts a Z-matrix and
dependent variable y as input and returns the vector of best-fit constants, a, the
best-fit function evaluated at each point $f(x_{i})$, and the coefficient of
determination, r2.

```matlab
[a,fx,r2]=least_squares(Z,y);
```

Test your function on the sets of data in script `problem_2_data.m` and show that the
following functions are the best fit lines. Report the coefficient of determination in your README.

a. y=0.3745+0.98644x+0.84564/x

b. y= 22.47-1.36x+0.28x^2

c. y=4.0046e^(-1.5x)+2.9213e^(-0.3x)+1.5647e^(-0.05x)

d. y=0.99sin(t)+0.5sin(3t)

**3\.** Load the data from the [class dart
experiment](https://github.com/cooperrc/stats_and_linear_regression/blob/master/compiled_data.csv).
Show your work in a script with filename `dart_statistics.m` in your repository.

a. Which user (0-32) was the most accurate dart thrower e.g. mean(x)+mean(y) was closest to 0 cm?

b. Which user (0-32) was the most precise dart thrower e.g. std(x)+std(y) was closest to 0 cm?

**4\.** 100 steel rods are going to be used to support a 1000 kg structure. The
rods will buckle when the load in any rod exceeds the [critical buckling
load](https://en.wikipedia.org/wiki/Euler%27s_critical_load)

$P_{cr}=\frac{\pi^3 Er^4}{16L^2}$
![buckle](./equations/buckle.png)

where E=200e9 Pa, r=0.01 m +/-0.001 m, and L is the length of the rod which can only be
controlled to 1% tolerance. Create a Monte
Carlo model `buckle_monte_carlo.m` that predicts the mean and standard deviation of the buckling load for 100
samples with normally distributed dimensions r and L.

`[mean_buckle_load,std_buckle_load]=buckle_monte_carlo(E,r_mean,r_std,L_mean,L_std)`

a. What is the mean_buckle_load and std_buckle_load for L=5 m?

b. What length, L, should the beams be so that only 2.5% will reach the critical buckling
load?

**5\.** The drag coefficient for spheres such as sporting balls is known to vary as a
function of the Reynolds number Re, a dimensionless number that gives a measure of the
ratio of inertial forces to viscous forces:

$Re = \frac{\rho V D}{\mu}$
![Reynolds](./equations/reynolds.png)

where ρ= the fluid’s density (kg/m3), V= its velocity (m/s), D= diameter (m), and μ=
dynamic viscosity (N⋅s/m2). The
following table provides values for a smooth spherical ball:

|Re (x1e-4) | C_D |
|---|---|
|2 | 0.52|
|5.8| 0.52|
|16.8| 0.52 |
|27.2|0.5|
|29.9|0.49 |
|33.9|0.44|
|36.3 | 0.18|
|40| 0.074 |
|46|0.067 |
|60|0.08 |
|100| 0.12|
|200| 0.16|
|400| 0.19|

a. Create a function `sphere_drag.m` that outputs the drag coefficient based on the given
table and an input Reynolds number using a spline interpolation of either linear
('linear'),
piecewise cubic ('pchip'), or cubic ('cubic'):

`[Cd_out]=sphere_drag(Re_in,spline_type)`

b. Use the following physical constants to plot the drag force vs velocity for a baseball:
ρ= 1.3 kg/m3, V= 4 - 40 (m/s), D= 23.5 cm, and μ=1.78e-5 Pa-s. Plot all three
interpolation methods on a single plot. Show the plot in your README.

**6\.** Evaluate the integral of the following function:

$\int_{2}^{3}f(x) = \int_{2}^{3} 1/6x^3 + 1/2x^2+x dx$

a. analytically

b. with 1-point Gauss quadrature

c. with 2-point Gauss quadrature

d. with 3-point Gauss quadrature

e. include the results in a table in your README

|method|value|error|
|---|---|---|
|analytical| ... | 0%|
|1 Gauss point | ... | ..% |
|... | ... | ... |


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P_{cr}=\frac{\pi^2 EI}{4L^2}

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\int_{2}^{3}f(x) = \int_{2}^{3} 1/6x^3 + 1/2x^2+x dx

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Re = \frac{\rho V D}{\mu}

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% part a
xa=[1 2 3 4 5]';
ya=[2.2 2.8 3.6 4.5 5.5]';

% part b

xb=[0 2 4 6 8 10 12 14 16 18]'
yb=[21.5 20.84 23.19 22.69 30.27 40.11 43.31 54.79 70.88 89.48]';

% part c

xc=[0.5 1 2 3 4 5 6 7 9]';
yc=[6 4.4 3.2 2.7 2.2 1.9 1.7 1.4 1.1]';

% part d

xd=[0.00000000e+00 1.26933037e-01 2.53866073e-01 3.80799110e-01 5.07732146e-01 6.34665183e-01 7.61598219e-01 8.88531256e-01 1.01546429e+00 1.14239733e+00 1.26933037e+00 1.39626340e+00 1.52319644e+00 1.65012947e+00 1.77706251e+00 1.90399555e+00 2.03092858e+00 2.15786162e+00 2.28479466e+00 2.41172769e+00 2.53866073e+00 2.66559377e+00 2.79252680e+00 2.91945984e+00 3.04639288e+00 3.17332591e+00 3.30025895e+00 3.42719199e+00 3.55412502e+00 3.68105806e+00 3.80799110e+00 3.93492413e+00 4.06185717e+00 4.18879020e+00 4.31572324e+00 4.44265628e+00 4.56958931e+00 4.69652235e+00 4.82345539e+00 4.95038842e+00 5.07732146e+00 5.20425450e+00 5.33118753e+00 5.45812057e+00 5.58505361e+00 5.71198664e+00 5.83891968e+00 5.96585272e+00 6.09278575e+00 6.21971879e+00 6.34665183e+00 6.47358486e+00 6.60051790e+00 6.72745093e+00 6.85438397e+00 6.98131701e+00 7.10825004e+00 7.23518308e+00 7.36211612e+00 7.48904915e+00 7.61598219e+00 7.74291523e+00 7.86984826e+00 7.99678130e+00 8.12371434e+00 8.25064737e+00 8.37758041e+00 8.50451345e+00 8.63144648e+00 8.75837952e+00 8.88531256e+00 9.01224559e+00 9.13917863e+00 9.26611167e+00 9.39304470e+00 9.51997774e+00 9.64691077e+00 9.77384381e+00 9.90077685e+00 1.00277099e+01 1.01546429e+01 1.02815760e+01 1.04085090e+01 1.05354420e+01 1.06623751e+01 1.07893081e+01 1.09162411e+01 1.10431742e+01 1.11701072e+01 1.12970402e+01 1.14239733e+01 1.15509063e+01 1.16778394e+01 1.18047724e+01 1.19317054e+01 1.20586385e+01 1.21855715e+01 1.23125045e+01 1.24394376e+01 1.25663706e+01]';
yd=[9.15756288e-02 3.39393873e-01 6.28875306e-01 7.67713096e-01 1.05094584e+00 9.70887288e-01 9.84265740e-01 1.02589034e+00 8.53218113e-01 6.90197665e-01 5.51277193e-01 5.01564914e-01 5.25455797e-01 5.87052838e-01 5.41394658e-01 7.12365594e-01 8.14839678e-01 9.80181855e-01 9.44430709e-01 1.06728057e+00 1.15166322e+00 8.99464065e-01 7.77225453e-01 5.92618124e-01 3.08822183e-01 -1.07884730e-03 -3.46563271e-01 -5.64836023e-01 -8.11931510e-01 -1.05925186e+00 -1.13323611e+00 -1.11986890e+00 -8.88336727e-01 -9.54113139e-01 -6.81378679e-01 -6.02369117e-01 -4.78684439e-01 -5.88160325e-01 -4.93580777e-01 -5.68747320e-01 -7.51641934e-01 -8.14672884e-01 -9.53191554e-01 -9.55337518e-01 -9.85995556e-01 -9.63373597e-01 -1.01511061e+00 -7.56467517e-01 -4.17379564e-01 -1.22340361e-01 2.16273929e-01 5.16909714e-01 7.77031694e-01 1.00653798e+00 9.35718089e-01 1.00660116e+00 1.11177057e+00 9.85485116e-01 8.54344900e-01 6.26444042e-01 6.28124048e-01 4.27764254e-01 5.93991751e-01 4.79248018e-01 7.17522492e-01 7.35927848e-01 9.08802925e-01 9.38646871e-01 1.13125860e+00 1.07247935e+00 1.05198782e+00 9.41647332e-01 6.98801244e-01 4.03193543e-01 1.37009682e-01 -1.43203880e-01 -4.64369445e-01 -6.94978252e-01 -1.03483196e+00 -1.10261288e+00 -1.12892727e+00 -1.03902484e+00 -8.53573083e-01 -7.01815315e-01 -6.84745997e-01 -6.14189417e-01 -4.70090797e-01 -5.95052432e-01 -5.96497000e-01 -5.66861911e-01 -7.18239679e-01 -9.52873043e-01 -9.37512847e-01 -1.15782985e+00 -1.03858206e+00 -1.03182712e+00 -8.45121554e-01 -5.61821980e-01 -2.83427014e-01 -8.27056140e-02]';
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Expand Up @@ -21,3 +21,45 @@ columns with your netid on each row as such,
|rcc02007 | 1 | 30 |
|rcc02007 | ...| ... |
|rcc02007 | ...| ... |

# Extra Credit Assignment \#3
## Due 11/17 by 11:59pm

**Nonlinear Regression - Logistic Regression**

![logistic regression of Challenger O-ring failure](http://www.stat.ufl.edu/~winner/cases/challenger.ppt)

Use the Temperature and failure data from the Challenger O-rings
[challenger_oring.csv](./challenger_oring.csv). Your independent variable is temperature and your dependent
variable is failure (1=fail, 0=pass). Create a function called `cost_logistic.m` that
takes the vector `a`, and independent variable `x` and dependent variable `y`. Use the
function, $\sigma(t)=\frac{1}{1+e^{-t}}$ where $t=a_{0}+a_{1}x$. Use the cost function,

$J(a_{0},a_{1})=1/m\sum_{i=1}^{n}\left[-y_{i}\log(\sigma(t_{i}))-(1-y_{i})\log((1-\sigma(t_{i})))\right]$

and gradient

$\frac{\partial J}{\partial a_{i}}=
1/m\sum_{k=1}^{N}\left(\sigma(t_{k})-y_{k}\right)x_{k}^{i}$

where $x_{k}^{i} is the k-th value of temperature raised to the i-th power (0, and 1)

a. edit `cost_logistic.m` so that the output is `[J,grad]` or [cost, gradient]

b. use the following code to solve for a0 and a1

```matlab
% Set options for fminunc
options = optimset('GradObj', 'on', 'MaxIter', 400);
% Run fminunc to obtain the optimal theta
% This function will return theta and the cost
[theta, cost] = ...
fminunc(@(a)(costFunction(a, x, y)), initial_a, options);
```

c. plot the data and the best-fit logistic regression model

```matlab
plot(x,y, x, sigma(a(1)+a(2)*x))
```

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