olsRegressionhwk3.html

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<div class="highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
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<div class="highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
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<p><strong>Question 1:</strong> Use the following data set to and Numpy code to find the best fitting polynomial of the form
$$
    y = \beta_3x^3 + \beta_2 x^2 + \beta_1 x + \beta_0
$$</p>
<p>That is, solve for $\beta_3,\,\beta_2,\,\beta_1$ and $\beta_0$.</p>
<p>Represent your answer as an array variable named $beta$.</p>
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<div class="highlight hl-ipython3"><pre><span></span><span class="n">data</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s2">"./data/hw_data.csv"</span><span class="p">)</span>
<span class="n">data</span>
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<div class="highlight hl-ipython3"><pre><span></span><span class="c1"># Data for x and y defined outside the function</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="s1">'x'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="s1">'y'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>

<span class="k">def</span> <span class="nf">calculate_matrix</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
    <span class="c1"># Create the matrix M using vstack and transpose it </span>
    <span class="n">M</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">T</span>
    <span class="k">return</span> <span class="n">M</span>

<span class="c1"># Call the function to create M</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">calculate_matrix</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>

<span class="c1"># Solve for the coefficients using least squares</span>
<span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">lstsq</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">rcond</span><span class="o">=</span><span class="kc">None</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
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<div class="highlight hl-ipython3"><pre><span></span><span class="c1"># HIDDEN TEST CELL</span>
<span>Hidden Tests Redacted</span>
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<pre><span style="color: red; font-weight: bold;">One or more test cases in this cell did not pass.</span>
Instructor hints: 
	1. The variable 'data' is not defined.
	2. The variable 'data' must be a Pandas DataFrame.
	3. The dataframe 'data' must contain 'x' and 'y' columns.
	4. The function 'calculate_matrix' is not defined.
	5. The output of calculate_matrix should be a numpy array.
	6. The matrix 'M' is incorrect.
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<p>Since we don't have a square matrix, we must represent it as follows:
$$
(M^T \cdot M)\beta = M^Ty
$$</p>
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<div class="highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">calculate_beta</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
    <span class="c1"># Use M.T to compute the normal equations and solve for beta</span>
    <span class="n">MtM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span>  <span class="c1"># Compute M^T * M</span>
    <span class="n">MtY</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>  <span class="c1"># Compute M^T * y</span>
    <span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">MtM</span><span class="p">,</span> <span class="n">MtY</span><span class="p">)</span>  <span class="c1"># Solve (M^T * M) beta = M^T * y</span>
    <span class="k">return</span> <span class="n">beta</span>

<span class="n">beta</span> <span class="o">=</span> <span class="n">calculate_beta</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
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<div class="highlight hl-ipython3"><pre><span></span><span class="c1"># HIDDEN TEST CELL</span>
<span>Hidden Tests Redacted</span>
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<pre><span style="color: red; font-weight: bold;">One or more test cases in this cell did not pass.</span>
Instructor hints: 
	1. The variable 'data' is not defined.
	2. The variable 'data' must be a Pandas DataFrame.
	3. The dataframe 'data' must contain 'x' and 'y' columns.
	4. The function 'calculate_beta' is not defined.
	5. The output of calculate_beta should be a numpy array.
	6. The beta values calculated by 'calculate_beta' are incorrect.
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<p>Plot your results against the true <code>y</code> values.
Create a new variable <code>yhat</code> to represent your estimate.
Compare the results using <a href="https://numpy.org/doc/stable/reference/generated/numpy.linalg.norm.html"><code>np.linalg.norm</code></a>.
Plot <code>yhat</code> and <code>y</code> with <code>^</code> and <code>o</code> respectively. Finally, provide a legend.</p>
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<div class="highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">plot_estimates</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
    <span class="c1"># Calculate predicted values (yhat)</span>
    <span class="n">yhat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
    
    <span class="c1"># Print the norm of the residuals (y - yhat)</span>
    <span class="n">residual_norm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">yhat</span><span class="p">)</span>
    <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Norm of residuals: </span><span class="si">{</span><span class="n">residual_norm</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
    
    <span class="c1"># Plot the original data points and the estimated values</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s1">'o'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'True y'</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">yhat</span><span class="p">,</span> <span class="s1">'^'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'Estimated yhat'</span><span class="p">)</span>
    
    <span class="c1"># Do not edit or remove the code presented below. </span>
    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Student Plot'</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'y'</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>  <span class="c1"># Add legend</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="s1">'student_plot.png'</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>

<span class="n">plot_estimates</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
</pre></div>
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<div class="highlight hl-ipython3"><pre><span></span><span class="c1"># HIDDEN TEST CELL</span>
<span>Hidden Tests Redacted</span>
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<pre><span style="color: green; font-weight: bold;">Congratulations! All test cases in this cell passed.</span>
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<p><strong>Question 2:</strong> Consider the data below. This data comes from the US Bureau of Labor Statistics. It includes the importance of a skills to all US occupations. There is also an "Annual Wage" column detailing the average annual wage for US workers of each occupation nationwide.</p>
<p>Use OLS regression to estimate the impact of each skill on an occupation's annual wage.</p>
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<div class="highlight hl-ipython3"><pre><span></span><span class="n">data</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s2">"./data/skillsData.csv"</span><span class="p">)</span>
<span class="n">data</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
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<div class="highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>

<span class="k">def</span> <span class="nf">calculate_sorted_beta</span><span class="p">(</span><span class="n">data</span><span class="p">):</span>
    <span class="c1"># Target variable (annual wage)</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="s2">"Annual Wage"</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>  

    <span class="c1"># Features (todas las columnas excepto Annual Wage)</span>
    <span class="n">M</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">"Annual Wage"</span><span class="p">])</span><span class="o">.</span><span class="n">values</span>  
    
    <span class="c1"># Añadimos columna de 1s para el intercepto</span>
    <span class="n">M</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">column_stack</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">M</span><span class="p">])</span>  
    
    <span class="c1"># Calculamos G = M^T M</span>
    <span class="n">G</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">M</span>  
    
    <span class="c1"># Calculamos y2 = M^T y</span>
    <span class="n">y2</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span>  
    
    <span class="c1"># Resolvemos betas: (M^T M)^(-1) M^T y</span>
    <span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">@</span> <span class="n">y2</span>  
    
    <span class="c1"># Construimos índice: intercepto + nombres de features</span>
    <span class="n">feature_names</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"Intercept"</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">"Annual Wage"</span><span class="p">])</span><span class="o">.</span><span class="n">columns</span><span class="p">)</span>
    
    <span class="c1"># Convertimos en Pandas Series y ordenamos</span>
    <span class="n">beta_series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">feature_names</span><span class="p">)</span><span class="o">.</span><span class="n">sort_values</span><span class="p">(</span><span class="n">ascending</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
    
    <span class="k">return</span> <span class="n">beta_series</span>

<span class="n">calculate_sorted_beta</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
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<a name="cell-a8cdbec3db0b72e7"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-a8cdbec3db0b72e7</code></span>
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<div class="highlight hl-ipython3"><pre><span></span><span class="c1"># HIDDEN TEST CELL</span>
<span>Hidden Tests Redacted</span>
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<pre><span style="color: green; font-weight: bold;">Congratulations! All test cases in this cell passed.</span>
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<div class="highlight hl-ipython3"><pre><span></span><span class="c1"># HIDDEN TEST CELL</span>
<span>Hidden Tests Redacted</span>
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<pre><span style="color: red; font-weight: bold;">One or more test cases in this cell did not pass.</span>
Instructor hints: 
	1. The function 'calculate_sorted_beta' is not defined.
	2. The dataset 'data' is not defined.
	3. Data should be a Pandas DataFrame.
	4. f
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<div class="highlight hl-ipython3"><pre><span></span><span class="c1"># HIDDEN TEST CELL</span>
<span>Hidden Tests Redacted</span>
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<pre><span style="color: green; font-weight: bold;">Congratulations! All test cases in this cell passed.</span>
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<a name="cell-ed03ce7327210dc7"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-ed03ce7327210dc7</code></span>
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<div class="highlight hl-ipython3"><pre><span></span><span class="c1"># HIDDEN TEST CELL</span>
<span>Hidden Tests Redacted</span>
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<pre><span style="color: red; font-weight: bold;">One or more test cases in this cell did not pass.</span>
Instructor hints: 
	1. The function 'calculate_sorted_beta' is not defined.
	2. The dataset 'data' is not defined.
	3. Data should be a Pandas DataFrame.
	4. Computers and Electronics
	5. Depth Perception
	6. Food Production
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