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  • Reordering matrix elements to reflect column and row clustering in naiive python

    - by bgbg
    Hello, I'm looking for a way to perform clustering separately on matrix rows and than on its columns, reorder the data in the matrix to reflect the clustering and putting it all together. The clustering problem is easily solvable, so is the dendrogram creation (for example in this blog or in "Programming collective intelligence"). However, how to reorder the data remains unclear for me. Eventually, I'm looking for a way of creating graphs similar to the one below using naive Python (with any "standard" library such as numpy, matplotlib etc, but without using R or other external tools).

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  • Efficient algorithm for finding largest eigenpair of small general complex matrix

    - by mklassen
    I am looking for an efficient algorithm to find the largest eigenpair of a small, general (non-square, non-sparse, non-symmetric), complex matrix, A, of size m x n. By small I mean m and n is typically between 4 and 64 and usually around 16, but with m not equal to n. This problem is straight forward to solve with the general LAPACK SVD algorithms, i.e. gesvd or gesdd. However, as I am solving millions of these problems and only require the largest eigenpair, I am looking for a more efficient algorithm. Additionally, in my application the eigenvectors will generally be similar for all cases. This lead me to investigate Arnoldi iteration based methods, but I have neither found a good library nor algorithm that applies to my small general complex matrix. Is there an appropriate algorithm and/or library?

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  • Clustering [assessment] algorithm with distance matrix as an input

    - by Max
    Can anyone suggest some clustering algorithm which can work with distance matrix as an input? Or the algorithm which can assess the "goodness" of the clustering also based on the distance matrix? At this moment I'm using a modification of Kruskal's algorithm (http://en.wikipedia.org/wiki/Kruskal%27s_algorithm) to split data into two clusters. It has a problem though. When the data has no distinct clusters the algorithm will still create two clusters with one cluster containing one element and the other containing all the rest. In this case I would rather have one cluster containing all the elements and another one which is empty. Are there any algorithms which are capable of doing this type of clustering? Are there any algorithms which can estimate how well the clustering was done or even better how many clusters are there in the data? The algorithms should work only with distance(similarity) matrices as an input.

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  • print matrix in dialog box

    - by Edan
    Hello, I'm having a little difficulty to print a matrix array on dialog box. The matrix is integer and as far as i understood i need to change it into string? anyway, here's the code: public void print_Matrix(int row, int column) { for (int i = 0; i <= row; i++) { for (int j = 0; j <= column; j++) { JOptionPane.showMessageDialog(null, matrix_Of_Life); } } what I need to do in order to print array into dialog box? thanks.

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  • Static Typing and Writing a Simple Matrix Library

    - by duckworthd
    Aye it's been done a million times before, but damnit I want to do it again. I'm writing a simple Matrix Library for C++ with the intention of doing it right. I've come across something that's fairly obvious in mathematics, but not so obvious to a strongly typed system -- the fact that a 1x1 matrix is just a number. To avoid this, I started walking down the hairy path of matrices as a composition of vectors, but also stumbled upon the fact that two vectors multiplied together could either be a number or a dyad, depending on the orientation of the two. My question is, what is the right way to deal with this situation in a strongly typed language like C++ or Java?

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  • Confusion Matrix with number of classified/misclassified instances on it (Python/Matplotlib)

    - by Pinkie
    I am plotting a confusion matrix with matplotlib with the following code: from numpy import * import matplotlib.pyplot as plt from pylab import * conf_arr = [[33,2,0,0,0,0,0,0,0,1,3], [3,31,0,0,0,0,0,0,0,0,0], [0,4,41,0,0,0,0,0,0,0,1], [0,1,0,30,0,6,0,0,0,0,1], [0,0,0,0,38,10,0,0,0,0,0], [0,0,0,3,1,39,0,0,0,0,4], [0,2,2,0,4,1,31,0,0,0,2], [0,1,0,0,0,0,0,36,0,2,0], [0,0,0,0,0,0,1,5,37,5,1], [3,0,0,0,0,0,0,0,0,39,0], [0,0,0,0,0,0,0,0,0,0,38] ] norm_conf = [] for i in conf_arr: a = 0 tmp_arr = [] a = sum(i,0) for j in i: tmp_arr.append(float(j)/float(a)) norm_conf.append(tmp_arr) plt.clf() fig = plt.figure() ax = fig.add_subplot(111) res = ax.imshow(array(norm_conf), cmap=cm.jet, interpolation='nearest') cb = fig.colorbar(res) savefig("confmat.png", format="png") But I want to the confusion matrix to show the numbers on it like this graphic (the right one): http://i48.tinypic.com/2e30kup.jpg How can I plot the conf_arr on the graphic?

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  • Creating a Large Matrix in ff

    - by Ryan Rosario
    I am trying to create a huge matrix in ff, and I know that ff is good for this sort of thing. But, there is a major problem. The dimensions of the matrix exceed .Machine$max_integer! I am running on a 64 bit machine, using 64bit R and 64bit ff. Is there any way to get around this problem? It's been suggested that R is using the MAXINT value from stdint.h. Is there any way to fix this without changing that file and possibly breaking build? > ffMatrix <- ff(vmode="boolean", dim=c(1e10,1e10)) Error in if (length < 0 || length > .Machine$integer.max) stop("length must be between 1 and .Machine$integer.max") : missing value where TRUE/FALSE needed In addition: Warning message: In ff(vmode = "boolean", dim = c(1e+10, 1e+10)) : NAs introduced by coercion > 1e+10 > .Machine$integer.max [1] TRUE

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  • print web on dot matrix receipt printer

    - by nightingale2k1
    Hi, I need to print a receipt from my web based apps using dot matrix printer epson tm-u220d (pos printer). I need to know, should I generate the receipt in html or in plain text ? I ever saw some commands for dot matrix printer to change the font size, line feed etc .. but I don't remember that commands. if I have to use plain text I need to use that commands. anyone knows where i can get the references ? Thanks

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  • Projection matrix + world plane ~> Homography from image plane to world plane

    - by B3ret
    I think I have my wires crossed on this, it should be quite easy. I have a projection matrix from world coordinates to image coordinates (4D homogeneous to 3D homgeneous), and therefore I also have the inverse projection matrix from image coordinates to world "rays". I want to project points of the image back onto a plane within the world (which is given of course as 4D homogeneous vector). The needed homography should be uniquely identified, yet I can not figure out how to compute it. Of course I could also intersect the back-projected rays with the world plane, but this seems not a good way, knowing that there MUST be a homography doing this for me. Thanks in advance, Ben

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  • Get positions for NAs only in the "middle" of a matrix column

    - by Abiel
    I want to obtain an index that refers to the positions of NA values in a matrix where the index is true if a given cell is NA and there is at least one non-NA value before and after it in the column. For example, given the following matrix [,1] [,2] [,3] [,4] [1,] NA 1 NA 1 [2,] 1 NA NA 2 [3,] NA 2 NA 3 the only value of the index that comes back TRUE should be [2,2]. Is there a compact expression for what I want to do? If I had to I could loop through columns and use something like min(which(!is.na(x[,i]))) to find the first non-NA value in each column, and then set all values before that to FALSE (and the same for all values after the max). This way I would not select leading and trailing NA values. But this seems a bit messy, so I'm wondering if there is a cleaner expression that does this without loops.

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  • Tool to diagonalize large matrices

    - by Xodarap
    I want to compute a diffusion kernel, which involves taking exp(b*A) where A is a large matrix. In order to play with values of b, I'd like to diagonalize A (so that exp(A) runs quickly). My matrix is about 25k x 25k, but is very sparse - only about 60k values are non-zero. Matlab's "eigs" function runs of out memory, as does octave's "eig" and R's "eigen." Is there a tool to find the decomposition of large, sparse matrices? Dunno if this is relevant, but A is an adjacency matrix, so it's symmetric, and it is full rank.

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  • extract data from an array without using loop in R

    - by Manolo
    I have a vector v with row positions: v<-c(10,3,100,50,...) with those positions I want to extract elements of a matrix, having a column fixed, for example lets suppose my column number is 2, so I am doing: data<-c() data<-c(matrix[[v]][[2]]) matrix has the data in the following format: [[34]] [1] "200_s_at" "4853" "1910" "3554" "2658" So for example, I want to extract from the row 342 the value 1910 only, column 2, and do the same with the next rows but I got an error when I want to do that, is it possible to do it directly? or should I have a loop that read one by one the positions in v and fill the data vector like: #algorithm for i<-1 to length(v) pos<-v[i] data[[i]]<-c(matriz[[pos]][[2]]) next i Thanks

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  • Compute rolling window covariance matrix

    - by user1665355
    I am trying to compute a rolling window (shifting by 1 day) covariance matrix for a number of assets. Say my df looks like this: df <- data.frame(x = 0:4, y = 5:9,z=1:5,u=4:8) How would a possible for loop look like if I want to calculate a covariance matrix on a rolling basis by shifting the rolling window by 1 day? Or should I use some apply family function? What time series class would be preferrable if I want to create a time series object for the loop above? I simply can't get it... Best Regards

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  • Generate (in R) a matrix of all possible outcomes for throwing n dice (ignoring order)

    - by Brani
    In cases where order does matter, it's rather easy to generate the matrix of all possible outcomes. One way for doing this is using expand.grid as shown here. What if it doesn't? If I'm right, the number of possible combinations is (S+N-1)!/S!(N-1)!, where S is the number of dice, each with N sides numbered 1 through N. (It is different from the well known combinations formula because it is possible for the same number to appear on more than one dice). For example, when throwing four six-sided dice, N=6 and S=4, so the number of possible combinations is (4+6-1)!/4!(6-1)! = 9!/4!x5! = 126. How can I generate a matrix of these 126 possible outcomes? Thank you.

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  • Generate a matrix of all possible outcomes for throwing n dice (ignoring order)

    - by Brani
    In cases where order does matter, it's rather easy to generate the matrix of all possible outcomes. One way for doing this is using expand.grid as shown here. What if it doesn't? If I'm right, the number of possible combinations is (S+N-1)!/S!(N-1)!, where S is the number of dice, each with N sides numbered 1 through N. (It is different from the well known combinations formula because it is possible for the same number to appear on more than one dice). For example, when throwing four six-sided dice, N=6 and S=4, so the number of possible combinations is (4+6-1)!/4!(6-1)! = 9!/4!x5! = 126. How can I generate a matrix of these 126 possible outcomes? Thank you.

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  • Calculating the null space of a matrix

    - by Ainsworth
    I'm attempting to solve a set of equations of the form Ax = 0. A is known 6x6 matrix and I've written the below code using SVD to get the vector x which works to a certain extent. The answer is approximately correct but not good enough to be useful to me, how can I improve the precision of the calculation? Lowering eps below 1.e-4 causes the function to fail. from numpy.linalg import * from numpy import * A = matrix([[0.624010149127497 ,0.020915658603923 ,0.838082638087629 ,62.0778180312547 ,-0.336 ,0], [0.669649399820597 ,0.344105317421833 ,0.0543868015800246 ,49.0194290212841 ,-0.267 ,0], [0.473153758252885 ,0.366893577716959 ,0.924972565581684 ,186.071352614705 ,-1 ,0], [0.0759305208803158 ,0.356365401030535 ,0.126682113674883 ,175.292109352674 ,0 ,-5.201], [0.91160934274653 ,0.32447818779582 ,0.741382053883291 ,0.11536775372698 ,0 ,-0.034], [0.480860406786873 ,0.903499596111067 ,0.542581424762866 ,32.782593418975 ,0 ,-1]]) def null(A, eps=1e-3): u,s,vh = svd(A,full_matrices=1,compute_uv=1) null_space = compress(s <= eps, vh, axis=0) return null_space.T NS = null(A) print "Null space equals ",NS,"\n" print dot(A,NS)

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  • Transform LINQ Dataset into a Matrix for export

    - by Mad Halfling
    Hi folks, I've got a data table with columns in which include Item, Category and Value (and others, but those are the only relevant ones for this problem) that I access via LINQ in a C# ASP.Net MVC app. I want to transform these into a matrix and output that as a CSV file to pull into Excel as matrix with the items down the side, the categories across the top and the values in the row cells. However, I don't know how many, or what, categories there will be in this table, nor will there always be a record for each item/category combination. I've written this by looping round, getting my "master category" list, then looking again for each item, filling in either blank or Value, depending on whether the item/category record exists, but as there are currently 27000 records in the table, this isn't as fast as I'd like. Is there a slicker and faster way I can do this, maybe via LINQ (firing into a quicker SQL statement so the DB server can do the leg-work), or will any method essentially come back to what I am doing? Thx MH

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  • handling matrix data in python

    - by Ovisek
    I was trying to progressively subtract values of a 3D matrix. The matrix looks like: ATOM 1223 ZX SOD A 11 2.11 -1.33 12.33 ATOM 1224 ZY SOD A 11 -2.99 -2.92 20.22 ATOM 1225 XH HEL A 12 -3.67 9.55 21.54 ATOM 1226 SS ARG A 13 -6.55 -3.09 42.11 ... here the last three columns are representing values for axes x,y,z respectively. now I what I wanted to do is, take the values of x,y,z for 1st line and subtract with 2nd,3rd,4th line in a iterative way and print the values for each axes. I was using: for line in map(str.split,inp): x = line[-3] y = line[-2] z = line[-1] for separating the values, but how to do in iterative way. should I do it by using Counter.

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  • A Taxonomy of Numerical Methods v1

    - by JoshReuben
    Numerical Analysis – When, What, (but not how) Once you understand the Math & know C++, Numerical Methods are basically blocks of iterative & conditional math code. I found the real trick was seeing the forest for the trees – knowing which method to use for which situation. Its pretty easy to get lost in the details – so I’ve tried to organize these methods in a way that I can quickly look this up. I’ve included links to detailed explanations and to C++ code examples. I’ve tried to classify Numerical methods in the following broad categories: Solving Systems of Linear Equations Solving Non-Linear Equations Iteratively Interpolation Curve Fitting Optimization Numerical Differentiation & Integration Solving ODEs Boundary Problems Solving EigenValue problems Enjoy – I did ! Solving Systems of Linear Equations Overview Solve sets of algebraic equations with x unknowns The set is commonly in matrix form Gauss-Jordan Elimination http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination C++: http://www.codekeep.net/snippets/623f1923-e03c-4636-8c92-c9dc7aa0d3c0.aspx Produces solution of the equations & the coefficient matrix Efficient, stable 2 steps: · Forward Elimination – matrix decomposition: reduce set to triangular form (0s below the diagonal) or row echelon form. If degenerate, then there is no solution · Backward Elimination –write the original matrix as the product of ints inverse matrix & its reduced row-echelon matrix à reduce set to row canonical form & use back-substitution to find the solution to the set Elementary ops for matrix decomposition: · Row multiplication · Row switching · Add multiples of rows to other rows Use pivoting to ensure rows are ordered for achieving triangular form LU Decomposition http://en.wikipedia.org/wiki/LU_decomposition C++: http://ganeshtiwaridotcomdotnp.blogspot.co.il/2009/12/c-c-code-lu-decomposition-for-solving.html Represent the matrix as a product of lower & upper triangular matrices A modified version of GJ Elimination Advantage – can easily apply forward & backward elimination to solve triangular matrices Techniques: · Doolittle Method – sets the L matrix diagonal to unity · Crout Method - sets the U matrix diagonal to unity Note: both the L & U matrices share the same unity diagonal & can be stored compactly in the same matrix Gauss-Seidel Iteration http://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method C++: http://www.nr.com/forum/showthread.php?t=722 Transform the linear set of equations into a single equation & then use numerical integration (as integration formulas have Sums, it is implemented iteratively). an optimization of Gauss-Jacobi: 1.5 times faster, requires 0.25 iterations to achieve the same tolerance Solving Non-Linear Equations Iteratively find roots of polynomials – there may be 0, 1 or n solutions for an n order polynomial use iterative techniques Iterative methods · used when there are no known analytical techniques · Requires set functions to be continuous & differentiable · Requires an initial seed value – choice is critical to convergence à conduct multiple runs with different starting points & then select best result · Systematic - iterate until diminishing returns, tolerance or max iteration conditions are met · bracketing techniques will always yield convergent solutions, non-bracketing methods may fail to converge Incremental method if a nonlinear function has opposite signs at 2 ends of a small interval x1 & x2, then there is likely to be a solution in their interval – solutions are detected by evaluating a function over interval steps, for a change in sign, adjusting the step size dynamically. Limitations – can miss closely spaced solutions in large intervals, cannot detect degenerate (coinciding) solutions, limited to functions that cross the x-axis, gives false positives for singularities Fixed point method http://en.wikipedia.org/wiki/Fixed-point_iteration C++: http://books.google.co.il/books?id=weYj75E_t6MC&pg=PA79&lpg=PA79&dq=fixed+point+method++c%2B%2B&source=bl&ots=LQ-5P_taoC&sig=lENUUIYBK53tZtTwNfHLy5PEWDk&hl=en&sa=X&ei=wezDUPW1J5DptQaMsIHQCw&redir_esc=y#v=onepage&q=fixed%20point%20method%20%20c%2B%2B&f=false Algebraically rearrange a solution to isolate a variable then apply incremental method Bisection method http://en.wikipedia.org/wiki/Bisection_method C++: http://numericalcomputing.wordpress.com/category/algorithms/ Bracketed - Select an initial interval, keep bisecting it ad midpoint into sub-intervals and then apply incremental method on smaller & smaller intervals – zoom in Adv: unaffected by function gradient à reliable Disadv: slow convergence False Position Method http://en.wikipedia.org/wiki/False_position_method C++: http://www.dreamincode.net/forums/topic/126100-bisection-and-false-position-methods/ Bracketed - Select an initial interval , & use the relative value of function at interval end points to select next sub-intervals (estimate how far between the end points the solution might be & subdivide based on this) Newton-Raphson method http://en.wikipedia.org/wiki/Newton's_method C++: http://www-users.cselabs.umn.edu/classes/Summer-2012/csci1113/index.php?page=./newt3 Also known as Newton's method Convenient, efficient Not bracketed – only a single initial guess is required to start iteration – requires an analytical expression for the first derivative of the function as input. Evaluates the function & its derivative at each step. Can be extended to the Newton MutiRoot method for solving multiple roots Can be easily applied to an of n-coupled set of non-linear equations – conduct a Taylor Series expansion of a function, dropping terms of order n, rewrite as a Jacobian matrix of PDs & convert to simultaneous linear equations !!! Secant Method http://en.wikipedia.org/wiki/Secant_method C++: http://forum.vcoderz.com/showthread.php?p=205230 Unlike N-R, can estimate first derivative from an initial interval (does not require root to be bracketed) instead of inputting it Since derivative is approximated, may converge slower. Is fast in practice as it does not have to evaluate the derivative at each step. Similar implementation to False Positive method Birge-Vieta Method http://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/polynomial%20methods/bv%20method.html C++: http://books.google.co.il/books?id=cL1boM2uyQwC&pg=SA3-PA51&lpg=SA3-PA51&dq=Birge-Vieta+Method+c%2B%2B&source=bl&ots=QZmnDTK3rC&sig=BPNcHHbpR_DKVoZXrLi4nVXD-gg&hl=en&sa=X&ei=R-_DUK2iNIjzsgbE5ID4Dg&redir_esc=y#v=onepage&q=Birge-Vieta%20Method%20c%2B%2B&f=false combines Horner's method of polynomial evaluation (transforming into lesser degree polynomials that are more computationally efficient to process) with Newton-Raphson to provide a computational speed-up Interpolation Overview Construct new data points for as close as possible fit within range of a discrete set of known points (that were obtained via sampling, experimentation) Use Taylor Series Expansion of a function f(x) around a specific value for x Linear Interpolation http://en.wikipedia.org/wiki/Linear_interpolation C++: http://www.hamaluik.com/?p=289 Straight line between 2 points à concatenate interpolants between each pair of data points Bilinear Interpolation http://en.wikipedia.org/wiki/Bilinear_interpolation C++: http://supercomputingblog.com/graphics/coding-bilinear-interpolation/2/ Extension of the linear function for interpolating functions of 2 variables – perform linear interpolation first in 1 direction, then in another. Used in image processing – e.g. texture mapping filter. Uses 4 vertices to interpolate a value within a unit cell. Lagrange Interpolation http://en.wikipedia.org/wiki/Lagrange_polynomial C++: http://www.codecogs.com/code/maths/approximation/interpolation/lagrange.php For polynomials Requires recomputation for all terms for each distinct x value – can only be applied for small number of nodes Numerically unstable Barycentric Interpolation http://epubs.siam.org/doi/pdf/10.1137/S0036144502417715 C++: http://www.gamedev.net/topic/621445-barycentric-coordinates-c-code-check/ Rearrange the terms in the equation of the Legrange interpolation by defining weight functions that are independent of the interpolated value of x Newton Divided Difference Interpolation http://en.wikipedia.org/wiki/Newton_polynomial C++: http://jee-appy.blogspot.co.il/2011/12/newton-divided-difference-interpolation.html Hermite Divided Differences: Interpolation polynomial approximation for a given set of data points in the NR form - divided differences are used to approximately calculate the various differences. For a given set of 3 data points , fit a quadratic interpolant through the data Bracketed functions allow Newton divided differences to be calculated recursively Difference table Cubic Spline Interpolation http://en.wikipedia.org/wiki/Spline_interpolation C++: https://www.marcusbannerman.co.uk/index.php/home/latestarticles/42-articles/96-cubic-spline-class.html Spline is a piecewise polynomial Provides smoothness – for interpolations with significantly varying data Use weighted coefficients to bend the function to be smooth & its 1st & 2nd derivatives are continuous through the edge points in the interval Curve Fitting A generalization of interpolating whereby given data points may contain noise à the curve does not necessarily pass through all the points Least Squares Fit http://en.wikipedia.org/wiki/Least_squares C++: http://www.ccas.ru/mmes/educat/lab04k/02/least-squares.c Residual – difference between observed value & expected value Model function is often chosen as a linear combination of the specified functions Determines: A) The model instance in which the sum of squared residuals has the least value B) param values for which model best fits data Straight Line Fit Linear correlation between independent variable and dependent variable Linear Regression http://en.wikipedia.org/wiki/Linear_regression C++: http://www.oocities.org/david_swaim/cpp/linregc.htm Special case of statistically exact extrapolation Leverage least squares Given a basis function, the sum of the residuals is determined and the corresponding gradient equation is expressed as a set of normal linear equations in matrix form that can be solved (e.g. using LU Decomposition) Can be weighted - Drop the assumption that all errors have the same significance –-> confidence of accuracy is different for each data point. Fit the function closer to points with higher weights Polynomial Fit - use a polynomial basis function Moving Average http://en.wikipedia.org/wiki/Moving_average C++: http://www.codeproject.com/Articles/17860/A-Simple-Moving-Average-Algorithm Used for smoothing (cancel fluctuations to highlight longer-term trends & cycles), time series data analysis, signal processing filters Replace each data point with average of neighbors. Can be simple (SMA), weighted (WMA), exponential (EMA). Lags behind latest data points – extra weight can be given to more recent data points. Weights can decrease arithmetically or exponentially according to distance from point. Parameters: smoothing factor, period, weight basis Optimization Overview Given function with multiple variables, find Min (or max by minimizing –f(x)) Iterative approach Efficient, but not necessarily reliable Conditions: noisy data, constraints, non-linear models Detection via sign of first derivative - Derivative of saddle points will be 0 Local minima Bisection method Similar method for finding a root for a non-linear equation Start with an interval that contains a minimum Golden Search method http://en.wikipedia.org/wiki/Golden_section_search C++: http://www.codecogs.com/code/maths/optimization/golden.php Bisect intervals according to golden ratio 0.618.. Achieves reduction by evaluating a single function instead of 2 Newton-Raphson Method Brent method http://en.wikipedia.org/wiki/Brent's_method C++: http://people.sc.fsu.edu/~jburkardt/cpp_src/brent/brent.cpp Based on quadratic or parabolic interpolation – if the function is smooth & parabolic near to the minimum, then a parabola fitted through any 3 points should approximate the minima – fails when the 3 points are collinear , in which case the denominator is 0 Simplex Method http://en.wikipedia.org/wiki/Simplex_algorithm C++: http://www.codeguru.com/cpp/article.php/c17505/Simplex-Optimization-Algorithm-and-Implemetation-in-C-Programming.htm Find the global minima of any multi-variable function Direct search – no derivatives required At each step it maintains a non-degenerative simplex – a convex hull of n+1 vertices. Obtains the minimum for a function with n variables by evaluating the function at n-1 points, iteratively replacing the point of worst result with the point of best result, shrinking the multidimensional simplex around the best point. Point replacement involves expanding & contracting the simplex near the worst value point to determine a better replacement point Oscillation can be avoided by choosing the 2nd worst result Restart if it gets stuck Parameters: contraction & expansion factors Simulated Annealing http://en.wikipedia.org/wiki/Simulated_annealing C++: http://code.google.com/p/cppsimulatedannealing/ Analogy to heating & cooling metal to strengthen its structure Stochastic method – apply random permutation search for global minima - Avoid entrapment in local minima via hill climbing Heating schedule - Annealing schedule params: temperature, iterations at each temp, temperature delta Cooling schedule – can be linear, step-wise or exponential Differential Evolution http://en.wikipedia.org/wiki/Differential_evolution C++: http://www.amichel.com/de/doc/html/ More advanced stochastic methods analogous to biological processes: Genetic algorithms, evolution strategies Parallel direct search method against multiple discrete or continuous variables Initial population of variable vectors chosen randomly – if weighted difference vector of 2 vectors yields a lower objective function value then it replaces the comparison vector Many params: #parents, #variables, step size, crossover constant etc Convergence is slow – many more function evaluations than simulated annealing Numerical Differentiation Overview 2 approaches to finite difference methods: · A) approximate function via polynomial interpolation then differentiate · B) Taylor series approximation – additionally provides error estimate Finite Difference methods http://en.wikipedia.org/wiki/Finite_difference_method C++: http://www.wpi.edu/Pubs/ETD/Available/etd-051807-164436/unrestricted/EAMPADU.pdf Find differences between high order derivative values - Approximate differential equations by finite differences at evenly spaced data points Based on forward & backward Taylor series expansion of f(x) about x plus or minus multiples of delta h. Forward / backward difference - the sums of the series contains even derivatives and the difference of the series contains odd derivatives – coupled equations that can be solved. Provide an approximation of the derivative within a O(h^2) accuracy There is also central difference & extended central difference which has a O(h^4) accuracy Richardson Extrapolation http://en.wikipedia.org/wiki/Richardson_extrapolation C++: http://mathscoding.blogspot.co.il/2012/02/introduction-richardson-extrapolation.html A sequence acceleration method applied to finite differences Fast convergence, high accuracy O(h^4) Derivatives via Interpolation Cannot apply Finite Difference method to discrete data points at uneven intervals – so need to approximate the derivative of f(x) using the derivative of the interpolant via 3 point Lagrange Interpolation Note: the higher the order of the derivative, the lower the approximation precision Numerical Integration Estimate finite & infinite integrals of functions More accurate procedure than numerical differentiation Use when it is not possible to obtain an integral of a function analytically or when the function is not given, only the data points are Newton Cotes Methods http://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas C++: http://www.siafoo.net/snippet/324 For equally spaced data points Computationally easy – based on local interpolation of n rectangular strip areas that is piecewise fitted to a polynomial to get the sum total area Evaluate the integrand at n+1 evenly spaced points – approximate definite integral by Sum Weights are derived from Lagrange Basis polynomials Leverage Trapezoidal Rule for default 2nd formulas, Simpson 1/3 Rule for substituting 3 point formulas, Simpson 3/8 Rule for 4 point formulas. For 4 point formulas use Bodes Rule. Higher orders obtain more accurate results Trapezoidal Rule uses simple area, Simpsons Rule replaces the integrand f(x) with a quadratic polynomial p(x) that uses the same values as f(x) for its end points, but adds a midpoint Romberg Integration http://en.wikipedia.org/wiki/Romberg's_method C++: http://code.google.com/p/romberg-integration/downloads/detail?name=romberg.cpp&can=2&q= Combines trapezoidal rule with Richardson Extrapolation Evaluates the integrand at equally spaced points The integrand must have continuous derivatives Each R(n,m) extrapolation uses a higher order integrand polynomial replacement rule (zeroth starts with trapezoidal) à a lower triangular matrix set of equation coefficients where the bottom right term has the most accurate approximation. The process continues until the difference between 2 successive diagonal terms becomes sufficiently small. Gaussian Quadrature http://en.wikipedia.org/wiki/Gaussian_quadrature C++: http://www.alglib.net/integration/gaussianquadratures.php Data points are chosen to yield best possible accuracy – requires fewer evaluations Ability to handle singularities, functions that are difficult to evaluate The integrand can include a weighting function determined by a set of orthogonal polynomials. Points & weights are selected so that the integrand yields the exact integral if f(x) is a polynomial of degree <= 2n+1 Techniques (basically different weighting functions): · Gauss-Legendre Integration w(x)=1 · Gauss-Laguerre Integration w(x)=e^-x · Gauss-Hermite Integration w(x)=e^-x^2 · Gauss-Chebyshev Integration w(x)= 1 / Sqrt(1-x^2) Solving ODEs Use when high order differential equations cannot be solved analytically Evaluated under boundary conditions RK for systems – a high order differential equation can always be transformed into a coupled first order system of equations Euler method http://en.wikipedia.org/wiki/Euler_method C++: http://rosettacode.org/wiki/Euler_method First order Runge–Kutta method. Simple recursive method – given an initial value, calculate derivative deltas. Unstable & not very accurate (O(h) error) – not used in practice A first-order method - the local error (truncation error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size In evolving solution between data points xn & xn+1, only evaluates derivatives at beginning of interval xn à asymmetric at boundaries Higher order Runge Kutta http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods C++: http://www.dreamincode.net/code/snippet1441.htm 2nd & 4th order RK - Introduces parameterized midpoints for more symmetric solutions à accuracy at higher computational cost Adaptive RK – RK-Fehlberg – estimate the truncation at each integration step & automatically adjust the step size to keep error within prescribed limits. At each step 2 approximations are compared – if in disagreement to a specific accuracy, the step size is reduced Boundary Value Problems Where solution of differential equations are located at 2 different values of the independent variable x à more difficult, because cannot just start at point of initial value – there may not be enough starting conditions available at the end points to produce a unique solution An n-order equation will require n boundary conditions – need to determine the missing n-1 conditions which cause the given conditions at the other boundary to be satisfied Shooting Method http://en.wikipedia.org/wiki/Shooting_method C++: http://ganeshtiwaridotcomdotnp.blogspot.co.il/2009/12/c-c-code-shooting-method-for-solving.html Iteratively guess the missing values for one end & integrate, then inspect the discrepancy with the boundary values of the other end to adjust the estimate Given the starting boundary values u1 & u2 which contain the root u, solve u given the false position method (solving the differential equation as an initial value problem via 4th order RK), then use u to solve the differential equations. Finite Difference Method For linear & non-linear systems Higher order derivatives require more computational steps – some combinations for boundary conditions may not work though Improve the accuracy by increasing the number of mesh points Solving EigenValue Problems An eigenvalue can substitute a matrix when doing matrix multiplication à convert matrix multiplication into a polynomial EigenValue For a given set of equations in matrix form, determine what are the solution eigenvalue & eigenvectors Similar Matrices - have same eigenvalues. Use orthogonal similarity transforms to reduce a matrix to diagonal form from which eigenvalue(s) & eigenvectors can be computed iteratively Jacobi method http://en.wikipedia.org/wiki/Jacobi_method C++: http://people.sc.fsu.edu/~jburkardt/classes/acs2_2008/openmp/jacobi/jacobi.html Robust but Computationally intense – use for small matrices < 10x10 Power Iteration http://en.wikipedia.org/wiki/Power_iteration For any given real symmetric matrix, generate the largest single eigenvalue & its eigenvectors Simplest method – does not compute matrix decomposition à suitable for large, sparse matrices Inverse Iteration Variation of power iteration method – generates the smallest eigenvalue from the inverse matrix Rayleigh Method http://en.wikipedia.org/wiki/Rayleigh's_method_of_dimensional_analysis Variation of power iteration method Rayleigh Quotient Method Variation of inverse iteration method Matrix Tri-diagonalization Method Use householder algorithm to reduce an NxN symmetric matrix to a tridiagonal real symmetric matrix vua N-2 orthogonal transforms     Whats Next Outside of Numerical Methods there are lots of different types of algorithms that I’ve learned over the decades: Data Mining – (I covered this briefly in a previous post: http://geekswithblogs.net/JoshReuben/archive/2007/12/31/ssas-dm-algorithms.aspx ) Search & Sort Routing Problem Solving Logical Theorem Proving Planning Probabilistic Reasoning Machine Learning Solvers (eg MIP) Bioinformatics (Sequence Alignment, Protein Folding) Quant Finance (I read Wilmott’s books – interesting) Sooner or later, I’ll cover the above topics as well.

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  • Approximate string matching with a letter confusion matrix?

    - by zigglenaut
    I'm trying to model a phonetic recognizer that has to isolate instances of words (strings of phones) out of a long stream of phones that doesn't have gaps between each word. The stream of phones may have been poorly recognized, with letter substitutions/insertions/deletions, so I will have to do approximate string matching. However, I want the matching to be phonetically-motivated, e.g. "m" and "n" are phonetically similar, so the substitution cost of "m" for "n" should be small, compared to say, "m" and "k". So, if I'm searching for [mein] "main", it would match the letter sequence [meim] "maim" with, say, cost 0.1, whereas it would match the letter sequence [meik] "make" with, say, cost 0.7. Similarly, there are differing costs for inserting or deleting each letter. I can supply a confusion matrix that, for each letter pair (x,y), gives the cost of substituting x with y, where x and y are any letter or the empty string. I know that there are tools available that do approximate matching such as agrep, but as far as I can tell, they do not take a confusion matrix as input. That is, the cost of any insertion/substitution/deletion = 1. My question is, are there any open-source tools already available that can do approximate matching with confusion matrices, and if not, what is a good algorithm that I can implement to accomplish this?

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  • Dot Matrix printing in C# ?

    - by Dale
    I'm trying to print to Dot Matrix printers (various models) out of C#, currently I'm using Win32 API (you can find alot of examples online) calls to send escape codes directly to the printer out of my C# application. This works great, but... My problem is because I'm generating the escape codes and not relying on the windows print system the printouts can't be sent to any "normal" printers or to things like PDF print drivers. (This is now causing a problem as we're trying to use the application on a 2008 Terminal Server using Easy Print [Which is XPS based]) The question is: How can I print formatted documents (invoices on pre-printed stationary) to Dot Matrix printers (Epson, Oki and Panasonic... various models) out of C# not using direct printing, escape codes etc. **Just to clarify, I'm trying things like GDI+ (System.Drawing.Printing) but the problem is that its very hard, to get things to line up like the old code did. (The old code sent the characters direct to the printer bypassing the windows driver.) Any suggestions how things could be improved so that they could use GDI+ but still line up like the old code did?

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