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diff --git a/lecture-notes/advanced-python/slides/arrays.tex b/lecture-notes/advanced-python/slides/arrays.tex new file mode 100644 index 0000000..e4501c8 --- /dev/null +++ b/lecture-notes/advanced-python/slides/arrays.tex @@ -0,0 +1,341 @@ +\section{Arrays} + +\begin{frame}[fragile] + \frametitle{Arrays: Introduction} + \begin{itemize} + \item Similar to lists, but homogeneous + \item Much faster than arrays + \end{itemize} + \begin{lstlisting} + In[]: a1 = array([1,2,3,4]) + In[]: a1 # 1-D + In[]: a2 = array([[1,2,3,4],[5,6,7,8]]) + In[]: a2 # 2-D + \end{lstlisting} +\end{frame} + +\begin{frame}[fragile] + \frametitle{\texttt{arange} and \texttt{shape}} + \begin{lstlisting} + In[]: ar1 = arange(1, 5) + In[]: ar2 = arange(1, 9) + In[]: print ar2 + In[]: ar2.shape = 2, 4 + In[]: print ar2 + \end{lstlisting} + \begin{itemize} + \item \texttt{linspace} and \texttt{loadtxt} also returned arrays + \end{itemize} + \begin{lstlisting} + In[]: ar1.shape + In[]: ar2.shape + \end{lstlisting} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Special methods} + \begin{lstlisting} + In[]: identity(3) + \end{lstlisting} + \begin{itemize} + \item array of shape (3, 3) with diagonals as 1s, rest 0s + \end{itemize} + \begin{lstlisting} + In[]: zeros((4,5)) + \end{lstlisting} + \begin{itemize} + \item array of shape (4, 5) with all 0s + \end{itemize} + \begin{lstlisting} + In[]: a = zeros_like([1.5, 1, 2, 3]) + In[]: print a, a.dtype + \end{lstlisting} + \begin{itemize} + \item An array with all 0s, with similar shape and dtype as argument + \item Homogeneity makes the dtype of a to be float + \item \texttt{ones, ones\_like, empty, empty\_like} + \end{itemize} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Operations on arrays} + \begin{lstlisting} + In[]: a1 + In[]: a1 * 2 + In[]: a1 + \end{lstlisting} + \begin{itemize} + \item The array is not changed; New array is returned + \end{itemize} + \begin{lstlisting} + In[]: a1 + 3 + In[]: a1 - 7 + In[]: a1 / 2.0 + \end{lstlisting} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Operations on arrays \ldots} + \begin{itemize} + \item Like lists, we can assign the new array, the old name + \end{itemize} + \begin{lstlisting} + In[]: a1 = a1 + 2 + In[]: a1 + \end{lstlisting} + \begin{itemize} + \item \alert{Beware of Augmented assignment!} + \end{itemize} + \begin{lstlisting} + In[]: a, b = arange(1, 5), arange(1, 5) + In[]: print a, a.dtype, b, b.dtype + In[]: a = a/2.0 + In[]: b /= 2.0 + In[]: print a, a.dtype, b, b.dtype + \end{lstlisting} + \begin{itemize} + \item Operations on two arrays; element-wise + \end{itemize} + \begin{lstlisting} + In[]: a1 + a1 + In[]: a1 * a2 + \end{lstlisting} +\end{frame} + +\section{Accessing pieces of arrays} + +\begin{frame}[fragile] + \frametitle{Accessing \& changing elements} + \begin{lstlisting} + In[]: A = array([12, 23, 34, 45, 56]) + + In[]: C = array([[11, 12, 13, 14, 15], + [21, 22, 23, 24, 25], + [31, 32, 33, 34, 35], + [41, 42, 43, 44, 45], + [51, 52, 53, 54, 55]]) + + In[]: A[2] + In[]: C[2, 3] + \end{lstlisting} + \begin{itemize} + \item Indexing starts from 0 + \item Assign new values, to change elements + \end{itemize} + \begin{lstlisting} + In[]: A[2] = -34 + In[]: C[2, 3] = -34 + \end{lstlisting} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Accessing rows} + \begin{itemize} + \item Indexing works just like with lists + \end{itemize} + \begin{lstlisting} + In[]: C[2] + In[]: C[4] + In[]: C[-1] + \end{lstlisting} + \begin{itemize} + \item Change the last row into all zeros + \end{itemize} + \begin{lstlisting} + In[]: C[-1] = [0, 0, 0, 0, 0] + \end{lstlisting} + OR + \begin{lstlisting} + In[]: C[-1] = 0 + \end{lstlisting} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Accessing columns} + \begin{lstlisting} + In[]: C[:, 2] + In[]: C[:, 4] + In[]: C[:, -1] + \end{lstlisting} + \begin{itemize} + \item The first parameter is replaced by a \texttt{:} to specify we + require all elements of that dimension + \end{itemize} + \begin{lstlisting} + In[]: C[:, -1] = 0 + \end{lstlisting} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Slicing} + \begin{lstlisting} + In[]: I = imread('squares.png') + In[]: imshow(I) + \end{lstlisting} + \begin{itemize} + \item The image is just an array + \end{itemize} + \begin{lstlisting} + In[]: print I, I.shape + \end{lstlisting} + \begin{enumerate} + \item Get the top left quadrant of the image + \item Obtain the square in the center of the image + \end{enumerate} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Slicing \ldots} + \begin{itemize} + \item Slicing works just like with lists + \end{itemize} + \begin{lstlisting} + In[]: C[0:3, 2] + In[]: C[2, 0:3] + In[]: C[2, :3] + \end{lstlisting} + \begin{lstlisting} + In[]: imshow(I[:150, :150]) + + In[]: imshow(I[75:225, 75:225]) + \end{lstlisting} +\end{frame} + +\begin{frame} +\frametitle{Image after slicing} +\includegraphics[scale=0.45]{../advanced_python/images/slice.png}\\ +\end{frame} + + + +\begin{frame}[fragile] + \frametitle{Striding} + \begin{itemize} + \item Compress the image to a fourth, by dropping alternate rows and + columns + \item We shall use striding + \item The idea is similar to striding in lists + \end{itemize} + \begin{lstlisting} + In[]: C[0:5:2, 0:5:2] + In[]: C[::2, ::2] + In[]: C[1::2, ::2] + \end{lstlisting} + \begin{itemize} + \item Now, the image can be shrunk by + \end{itemize} + \begin{lstlisting} + In[]: imshow(I[::2, ::2]) + \end{lstlisting} +\end{frame} + +\section{Matrix Operations} + +\begin{frame}[fragile] + \frametitle{Matrix Operations using \texttt{arrays}} + We can perform various matrix operations on \texttt{arrays}\\ + A few are listed below. + + \begin{center} + \begin{tabular}{lll} + Operation & How? & Example \\ + \hline + Transpose & \texttt{.T} & \texttt{A.T} \\ + Product & \texttt{dot} & \texttt{dot(A, B)} \\ + Inverse & \texttt{inv} & \texttt{inv(A)} \\ + Determinant & \texttt{det} & \texttt{det(A)} \\ + Sum of all elements & \texttt{sum} & \texttt{sum(A)} \\ + Eigenvalues & \texttt{eigvals} & \texttt{eigvals(A)} \\ + Eigenvalues \& Eigenvectors & \texttt{eig} & \texttt{eig(A)} \\ + Norms & \texttt{norm} & \texttt{norm(A)} \\ + SVD & \texttt{svd} & \texttt{svd(A)} \\ + \end{tabular} + \end{center} +\end{frame} + +\section{Least square fit} + +\begin{frame}[fragile] + \frametitle{Least Square Fit} + \begin{lstlisting} + In[]: L, t = loadtxt("pendulum.txt", + unpack=True) + In[]: L + In[]: t + In[]: tsq = t * t + In[]: plot(L, tsq, 'bo') + In[]: plot(L, tsq, 'r') + \end{lstlisting} + \begin{itemize} + \item Both the plots, aren't what we expect -- linear plot + \item Enter Least square fit! + \end{itemize} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Matrix Formulation} + \begin{itemize} + \item We need to fit a line through points for the equation $T^2 = m \cdot L+c$ + \item In matrix form, the equation can be represented as $T_{sq} = A \cdot p$, where $T_{sq}$ is + $\begin{bmatrix} + T^2_1 \\ + T^2_2 \\ + \vdots\\ + T^2_N \\ + \end{bmatrix}$ + , A is + $\begin{bmatrix} + L_1 & 1 \\ + L_2 & 1 \\ + \vdots & \vdots\\ + L_N & 1 \\ + \end{bmatrix}$ + and p is + $\begin{bmatrix} + m\\ + c\\ + \end{bmatrix}$ + \item We need to find $p$ to plot the line + \end{itemize} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Least Square Fit Line} + \begin{lstlisting} + In[]: A = array((L, ones_like(L))) + In[]: A.T + In[]: A + \end{lstlisting} + \begin{itemize} + \item We now have \texttt{A} and \texttt{tsq} + \end{itemize} + \begin{lstlisting} + In[]: result = lstsq(A, tsq) + \end{lstlisting} + \begin{itemize} + \item Result has a lot of values along with m and c, that we need + \end{itemize} + \begin{lstlisting} + In[]: m, c = result[0] + In[]: print m, c + \end{lstlisting} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Least Square Fit Line} + \begin{itemize} + \item Now that we have m and c, we use them to generate line and plot + \end{itemize} + \begin{lstlisting} + In[]: tsq_fit = m * L + c + In[]: plot(L, tsq, 'bo') + In[]: plot(L, tsq_fit, 'r') + \end{lstlisting} +\end{frame} + +\begin{frame} +\frametitle{Least Square Fit Line} +\includegraphics[scale=0.45]{../advanced_python/images/lst-sq-fit.png}\\ +\end{frame} + + |