some update of pca

doc/pub/DimRed/html/._DimRed-bs018.html

@@ -258,7 +258,7 @@ $$
 When you are done with these steps, print out \( \mu_n \) to verify it is
 close to \( \mu \) and plot your mean centered data to verify it is
 centered at the origin! Compare your code with the functionality from <b>Scikit-Learn</b> discussed above.
-The following code elements perform these operations using <b>pandas</b> or your own functionality for doing so. 
+The following code elements perform these operations using <b>pandas</b> or using our own functionality for doing so. The latter, using <b>numpy</b> is rather simply through the <b>mean()</b> function. 
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
@@ -277,7 +277,7 @@ Alternatively, you could also have used the functions we discussed earlier for s
 That is, we could have used the <b>StandardScaler</b> function in <b>Scikit-Learn</b>, a function which 
 ensures that for each feature/predictor we study the mean value is
 zero and the variance is one (every column in the design/feature
-matrix).
+matrix).  You would then not get the same results, since we divide by the variance. For diagonal covariance matrix elements will then be one, while the non-diagonal ones will be divided by \( 2\sqrt{2} \) for our specific case.
 
 <h3 id="___sec19" class="anchor">Compute the sample covariance </h3>
 
@@ -319,11 +319,11 @@ following tasks:
 
 $$
 \begin{equation*}
-x_i \approx \tilde{x}_i := \mu_n + \langle x_i, v_0 \rangle v_0
+x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
 \end{equation*}
 $$
 
-where \( v_0 \) is the first principal component. What do you observe?
+where \( v_0 \) is the first principal component.
 
 <p>
 Collecting all these steps we can write our own PCA function and

doc/pub/DimRed/html/DimRed-reveal.html

@@ -1068,7 +1068,7 @@ $$
 When you are done with these steps, print out \( \mu_n \) to verify it is
 close to \( \mu \) and plot your mean centered data to verify it is
 centered at the origin! Compare your code with the functionality from <b>Scikit-Learn</b> discussed above.
-The following code elements perform these operations using <b>pandas</b> or your own functionality for doing so. 
+The following code elements perform these operations using <b>pandas</b> or using our own functionality for doing so. The latter, using <b>numpy</b> is rather simply through the <b>mean()</b> function. 
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -1087,7 +1087,7 @@ Alternatively, you could also have used the functions we discussed earlier for s
 That is, we could have used the <b>StandardScaler</b> function in <b>Scikit-Learn</b>, a function which 
 ensures that for each feature/predictor we study the mean value is
 zero and the variance is one (every column in the design/feature
-matrix).
+matrix).  You would then not get the same results, since we divide by the variance. For diagonal covariance matrix elements will then be one, while the non-diagonal ones will be divided by \( 2\sqrt{2} \) for our specific case.
 
 <h3 id="___sec19">Compute the sample covariance </h3>
 
@@ -1131,12 +1131,12 @@ following tasks:
 <p>&nbsp;<br>
 $$
 \begin{equation*}
-x_i \approx \tilde{x}_i := \mu_n + \langle x_i, v_0 \rangle v_0
+x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
 \end{equation*}
 $$
 <p>&nbsp;<br>
 
-where \( v_0 \) is the first principal component. What do you observe?
+where \( v_0 \) is the first principal component.
 
 <p>
 Collecting all these steps we can write our own PCA function and

doc/pub/DimRed/html/DimRed-solarized.html

@@ -1044,7 +1044,7 @@ $$
 When you are done with these steps, print out \( \mu_n \) to verify it is
 close to \( \mu \) and plot your mean centered data to verify it is
 centered at the origin! Compare your code with the functionality from <b>Scikit-Learn</b> discussed above.
-The following code elements perform these operations using <b>pandas</b> or your own functionality for doing so. 
+The following code elements perform these operations using <b>pandas</b> or using our own functionality for doing so. The latter, using <b>numpy</b> is rather simply through the <b>mean()</b> function. 
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -1063,7 +1063,7 @@ Alternatively, you could also have used the functions we discussed earlier for s
 That is, we could have used the <b>StandardScaler</b> function in <b>Scikit-Learn</b>, a function which 
 ensures that for each feature/predictor we study the mean value is
 zero and the variance is one (every column in the design/feature
-matrix).
+matrix).  You would then not get the same results, since we divide by the variance. For diagonal covariance matrix elements will then be one, while the non-diagonal ones will be divided by \( 2\sqrt{2} \) for our specific case.
 
 <h3 id="___sec19">Compute the sample covariance </h3>
 
@@ -1105,11 +1105,11 @@ following tasks:
 
 $$
 \begin{equation*}
-x_i \approx \tilde{x}_i := \mu_n + \langle x_i, v_0 \rangle v_0
+x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
 \end{equation*}
 $$
 
-where \( v_0 \) is the first principal component. What do you observe?
+where \( v_0 \) is the first principal component.
 
 <p>
 Collecting all these steps we can write our own PCA function and

doc/pub/DimRed/html/DimRed.html

@@ -1049,7 +1049,7 @@ $$
 When you are done with these steps, print out \( \mu_n \) to verify it is
 close to \( \mu \) and plot your mean centered data to verify it is
 centered at the origin! Compare your code with the functionality from <b>Scikit-Learn</b> discussed above.
-The following code elements perform these operations using <b>pandas</b> or your own functionality for doing so. 
+The following code elements perform these operations using <b>pandas</b> or using our own functionality for doing so. The latter, using <b>numpy</b> is rather simply through the <b>mean()</b> function. 
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
@@ -1068,7 +1068,7 @@ Alternatively, you could also have used the functions we discussed earlier for s
 That is, we could have used the <b>StandardScaler</b> function in <b>Scikit-Learn</b>, a function which 
 ensures that for each feature/predictor we study the mean value is
 zero and the variance is one (every column in the design/feature
-matrix).
+matrix).  You would then not get the same results, since we divide by the variance. For diagonal covariance matrix elements will then be one, while the non-diagonal ones will be divided by \( 2\sqrt{2} \) for our specific case.
 
 <h3 id="___sec19">Compute the sample covariance </h3>
 
@@ -1110,11 +1110,11 @@ following tasks:
 
 $$
 \begin{equation*}
-x_i \approx \tilde{x}_i := \mu_n + \langle x_i, v_0 \rangle v_0
+x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
 \end{equation*}
 $$
 
-where \( v_0 \) is the first principal component. What do you observe?
+where \( v_0 \) is the first principal component.
 
 <p>
 Collecting all these steps we can write our own PCA function and

doc/pub/DimRed/ipynb/DimRed.ipynb

@@ -1161,7 +1161,7 @@
     "When you are done with these steps, print out $\\mu_n$ to verify it is\n",
     "close to $\\mu$ and plot your mean centered data to verify it is\n",
     "centered at the origin! Compare your code with the functionality from **Scikit-Learn** discussed above.\n",
-    "The following code elements perform these operations using **pandas** or your own functionality for doing so."
+    "The following code elements perform these operations using **pandas** or using our own functionality for doing so. The latter, using **numpy** is rather simply through the **mean()** function."
    ]
   },
   {
@@ -1191,7 +1191,7 @@
     "That is, we could have used the **StandardScaler** function in **Scikit-Learn**, a function which \n",
     "ensures that for each feature/predictor we study the mean value is\n",
     "zero and the variance is one (every column in the design/feature\n",
-    "matrix).\n",
+    "matrix).  You would then not get the same results, since we divide by the variance. For diagonal covariance matrix elements will then be one, while the non-diagonal ones will be divided by $2\\sqrt{2}$ for our specific case.\n",
     "\n",
     "### Compute the sample covariance\n",
     "\n",
@@ -1257,7 +1257,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "x_i \\approx \\tilde{x}_i := \\mu_n + \\langle x_i, v_0 \\rangle v_0\n",
+    "x_i \\approx \\tilde{x}_i = \\mu_n + \\langle x_i, v_0 \\rangle v_0\n",
     "$$"
    ]
   },
@@ -1265,7 +1265,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "where $v_0$ is the first principal component. What do you observe?\n",
+    "where $v_0$ is the first principal component. \n",
     "\n",
     "Collecting all these steps we can write our own PCA function and\n",
     "compare this with the functionality included in **Scikit-Learn**.  \n",

doc/src/DimRed/DimRed.do.txt

@@ -812,7 +812,7 @@ and the mean-centered data $\bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \}$ takes
 When you are done with these steps, print out $\mu_n$ to verify it is
 close to $\mu$ and plot your mean centered data to verify it is
 centered at the origin! Compare your code with the functionality from _Scikit-Learn_ discussed above.
-The following code elements perform these operations using _pandas_ or your own functionality for doing so. 
+The following code elements perform these operations using _pandas_ or using our own functionality for doing so. The latter, using _numpy_ is rather simply through the _mean()_ function. 
 !bc pycod
 df = pd.DataFrame(X)
 # Pandas does the centering for us
@@ -828,7 +828,7 @@ Alternatively, you could also have used the functions we discussed earlier for s
 That is, we could have used the _StandardScaler_ function in _Scikit-Learn_, a function which 
 ensures that for each feature/predictor we study the mean value is
 zero and the variance is one (every column in the design/feature
-matrix).
+matrix).  You would then not get the same results, since we divide by the variance. For diagonal covariance matrix elements will then be one, while the non-diagonal ones will be divided by $2\sqrt{2}$ for our specific case.
 
 === Compute the sample covariance ===
 
@@ -860,12 +860,13 @@ following tasks:
 * We plot the mean centered data and lines along the first and second principal components
 * Then we project the mean centered data onto the first and second principal components, and plot the projected data. 
 * Finally, we approximate the data as
+
 !bt
 \begin{equation*}
-x_i \approx \tilde{x}_i := \mu_n + \langle x_i, v_0 \rangle v_0
+x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
 \end{equation*}
 !et
-where $v_0$ is the first principal component. What do you observe?
+where $v_0$ is the first principal component. 
 
 Collecting all these steps we can write our own PCA function and
 compare this with the functionality included in _Scikit-Learn_.