192 lines
7.2 KiB
TeX
192 lines
7.2 KiB
TeX
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\chapter{Design pattern}
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Many code fragments are variations of some basic pattern. These
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pattern are used in many different variations in many different
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contexts. In this chapter we summarize a few of these \enterm[design
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pattern]{design pattern}.
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\section{Looping over vector elements}
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Iterating over vector elements by means of a \mcode{for}-loop is a very commen pattern:
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\begin{pagelisting}[caption={\varcode{for}-loop for accessing vector elements by indices}]
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x = [2:3:20]; % Some vector.
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for i=1:length(x) % For loop over the indices of the vector.
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i % This is the index (an integer number)
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% to be used for accessing vector elements.
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x(i) % This is the value of the i-th element of the vector.
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a = x(i); % The value of the i-th vector element
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% is assigned to the variable a.
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% Use the value of the i-th vector element by passing it
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% as an argument to a function:
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do_something(x(i));
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end
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\end{pagelisting}
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\noindent
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If the result of the computation within the loop are single numbers
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that are to be stored in a vector, you should create this vector with
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the right size before the loop:
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\begin{pagelisting}[caption={\varcode{for}-loop for writing a vector}]
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x = [1.2 2.3 2.6 3.1]; % Some vector.
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% Create a vector for the results, as long as the number of loops:
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y = zeros(length(x),1);
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for i=1:length(x)
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% Write the result of the computation at
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% the i-th position in the y vector:
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y(i) = get_something(x(i));
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end
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% Now the result vector can be further processed:
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mean(y);
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\end{pagelisting}
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\noindent
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The computation within the loop could also result in a vector of some
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length and not just a single number. If the length of this vector
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(here 10) is known beforehand, then you should create a matrix of
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appropriate size for storing the results:
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\begin{pagelisting}[caption={\varcode{for}-loop for writing rows of a matrix}]
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x = [2:3:20]; % Some vector.
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% Create space for results -
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% as many rows as loops, as many columns as needed:
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y = zeros(length(x), 10);
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for i=1:length(x)
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% Write the return value of the function get_something - now a
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% column vector with 10 elements - into the i-th row of the matrix y:
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y(i, :) = get_some_more(x(i));
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end
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% Process the results stored in matrix y:
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mean(y, 1)
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\end{pagelisting}
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\noindent
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Another possibility is that the result vectors (here of unknown size)
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need to be combined into a single large vector:
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\begin{pagelisting}[caption={\varcode{for}-loop for appending vectors}]
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x = [2:3:20]; % Some vector.
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y = []; % Empty vector for storing the results.
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for i=1:length(x)
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% The function get_something() returns a vector of unspecified size:
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z = get_somehow_more(x(i));
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% The content of z is appended to the result vector y:
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y = [y; z(:)];
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% The z(:) syntax ensures that we append column-vectors.
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end
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% Process the results stored in the vector z:
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mean(y)
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\end{pagelisting}
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\section{Scaling and shifting random numbers, zeros, and ones}
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Random number generators usually return random numbers of a given mean
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and standard deviation. Multiply those numbers by a factor to change their standard deviation and add a number to shift the mean.
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\begin{pagelisting}[caption={Scaling and shifting of random numbers}]
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% 100 random numbers drawn from a normal distribution
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% with mean 0 and standard deviation 1:
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x = randn(100, 1);
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% 100 random numbers drawn from a normal distribution
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% with mean 4.8 and standard deviation 2.3:
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mu = 4.8;
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sigma = 2.3;
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y = randn(100, 1)*sigma + mu;
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\end{pagelisting}
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\noindent
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The same principle can be useful for in the context of the functions
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\mcode{zeros()} or \mcode{ones()}:
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\begin{pagelisting}[caption={Scaling and shifting of \varcode{zeros()} and \varcode{ones()}}]
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x = -1:0.01:2; % Vector of x-values for plotting
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plot(x, exp(-x.*x));
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% Plot for the same x-values a horizontal line with y=0.8:
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plot(x, zeros(size(x))+0.8);
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% ... or a line with y=0.5:
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plot(x, ones(size(x))*0.5);
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\end{pagelisting}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Plotting a mathematical function}
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A mathematical function $y=f(x)$ assigns to each value of $x$ a single
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$y$-value. For drawing a function with the computer we need compute a
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table of values with many $x$-values and the corresponding function
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values $y=f(x)$.
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We first create a vector with useful $x$-values. They range from the
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smallest to the largest value we want to plot. We also need to set a
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step size that is small enough to result in a smooth plot of the
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function. We then compute for each value $x_i$ of this vector the
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corresponding function value $y_i$ and store them in a vector. We then
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can plot the values of the $y$ vector against the ones of the $x$
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vector.
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The following scripts compute and plot the function $f(x)=e^{-x^2}$:
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\begin{pagelisting}[caption={Plotting a mathematical function --- very detailed}]
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xmin = -1.0;
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xmax = 2.0;
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dx = 0.01; % Step size
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x = xmin:dx:xmax; % Vector with x-values.
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y = exp(-x.*x); % No for loop! '.*' for multiplying the vector elements.
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plot(x, y);
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\end{pagelisting}
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\begin{pagelisting}[caption={Plotting a mathematical function --- shorter}]
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x = -1:0.01:2;
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y = exp(-x.*x);
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plot(x, y);
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\end{pagelisting}
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\begin{pagelisting}[caption={Plotting a mathematical function --- compact}]
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x = -1:0.01:2;
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plot(x, exp(-x.*x));
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\end{pagelisting}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Normalizing histograms}
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For estimating probabilities or probability densities from histograms
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we need to normalize them appropriately.
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The \mcode{histogram()} function does this automatically with the appropriate arguments:
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\begin{pagelisting}[caption={Probability density with the \varcode{histogram()}-function}]
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x = randn(100, 1); % Some real-valued data.
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histogram(x, 'Normalization', 'pdf');
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\end{pagelisting}
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\begin{pagelisting}[caption={Probability with the \varcode{histogram()}-function}]
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x = randi(6, 100, 1); % Some integer-valued data.
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histogram(x, 'Normalization', 'probability');
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\end{pagelisting}
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\noindent
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Alternatively one can normalize the histogram data as returned by the
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\code{hist()}-function manually:
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\begin{pagelisting}[caption={Probability density with the \varcode{hist()}- and \varcode{bar()}-function}]
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x = randn(100, 1); % Some real-valued data.
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[h, b] = hist(x); % Compute histogram.
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h = h/sum(h)/(b(2)-b(1)); % Normalization to a probability density.
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bar(b, h); % Plot the probability density.
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\end{pagelisting}
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\begin{pagelisting}[caption={Probability with the \varcode{hist()}- and \varcode{bar()}-function}]
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x = randi(6, 100, 1); % Some integer-valued data.
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[h, b] = hist(x); % Compute histogram.
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h = h/sum(h); % Normalize to probability.
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bar(b, h); % Plot the probabilities.
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\end{pagelisting}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\printsolutions
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