scipy - peter beerli · scipy • is a collection of mathematical algorithms and convenience...

Scipy

MingYe(2015)andPeterBeerli(2017update)

SciPy• Is a collection of mathematical algorithms andconveniencefunctionsbuiltontheNumpyextensionofPython.

• It adds significant power to the interactive Pythonsession by providing the user with high-levelcommands and classes for manipulating andvisualizingdata.

• IsthecorepackageforscientificroutinesinPython

• Operate efficiently on numpy arrays, so that numpyandscipyworkhandinhand.

• Contains various toolboxes dedicated to commonissuesinscientificcomputingsuchas:

interpolation, integration, optimization, imageprocessing,statistics,specialfunctions,etc.

TheadditionalbenefitofbasingSciPy:

• Making a powerful programming language availablefor use in developing sophisticated programs andspecializedapplications.

• Everything from parallel programming to web anddata-base subroutines and classes have been madeavailabletothePythonprogrammer.

• scipy.cluster:Vectorquantization/Kmeans• scipy.constants:Physicalandmathematicalconstants• scipy.fftpack:Fouriertransform• scipy.integrate:Integrationroutines• scipy.interpolate:Interpolation• scipy.io:Datainputandoutput• scipy.linalg:Linearalgebraroutines• scipy.ndimage:n-dimensionalimagepackage• scipy.odr:Orthogonaldistanceregression• scipy.optimize:Optimization• scipy.signal:Signalprocessing• scipy.sparse:Sparsematrices• scipy.spatial:Spatialdatastructuresandalgorithms• scipy.special:Anyspecialmathematicalfunctions• scipy.stats:Statistics

Fileinput/output:scipy.io matlabfiles:sio.loadmat sio.savemat sio.whosmat

Savemat:SaveadictionaryofnamesandarraysintoaMATLAB-style.matfile.

• ReadingImage:

LinearAlgebra:scipy.linalg• Thescipy.linalg.det()functioncomputesthedeterminantofasquarematrix

Thescipy.linalg.inv()functioncomputestheinverseofasquarematrix:

SVD:SingularValueDecomposition

• Theoriginalmatrixcanbere-composedbymatrixmultiplicationoftheoutputsofsvdwithnp.dot:

Optimizationandfit:scipy.optimize

Optimizationistheproblemoffindinganumericalsolutiontoaminimizationorequality.Thescipy.optimizemoduleprovidesusefulalgorithmsforfunctionminimization(scalarormultidimensional),curvefittingandrootfinding.

Thisfunctionhasaglobalminimumaround-1.3andalocalminimumaround3.8

• Thegeneralandefficientwaytofindaminimumforthisfunctionistoconductagradientdescentstartingfromagiveninitialpoint.TheBFGSalgorithmisagoodwayofdoingthis:

Apossibleissuewiththisapproachisthat,ifthefunctionhaslocalminimathealgorithmmayfindtheselocalminimainsteadoftheglobalminimumdependingontheinitialpoint:

If we don’t know the neighborhood of the globalminimumtochoosetheinitialpoint,weneedtoresortto costlier global optimization. To find the globalminimum, the simplest algorithm is the brute forcealgorithm, inwhich the function isevaluatedoneachpointofagivengrid:

BruteForcealgorithmbecomesquiteslowforlargergridsizes.Simulatedannealingcanbeagoodalternative:

Scipy.optimize.anneal()

Forlocalminimum,wecanconstraintthevariabletotheintervalanduse:

• Let s = s0

• For k = 0 through kmax (exclusive): • T ← temperature(k ⁄ kmax)

• Pick a random neighbour, snew ← neighbour(s)• If P(E(s), E(snew), T) ≥ random(0, 1), move to the new state:

• s ← snew

• Output: the final state s

Rootfinding

Tofindaroot,apointwheref(x)=0

Onlyonerootisfound.Butthereisasecondrootaround-2.5.Wefindtheexactvalueofitbyadjustingourinitialguess:

scipy.optimize.fsolve()

Othermodules:

Curvefitting:Supposewehavedatasampledfromfwithsomenoises.ifweknowthefunctionalformofthefunctionfromwhichthesample,werebutnottheamplitudesoftheterms,wecanfindthosebyleastsquarescurvefitting.1. wehavetodefinethefunctiontofit:2. Wehavedatasampledfromfwithsomenoise:3. canusescipy.optimize.curve_fit()tofindaandb

Plotthecurveandfittedpoints:

HistogramandprobabilitydensityfunctionGiven observations of a random process, their histogram is anestimatoroftherandomprocess’sPDF(probabilitydensityfunction):

Scipystatistic

cdf:CumulativeDistributionFunction

Interpolation:scipy.interpolate Thescipy.interpolateisusefulforfittingafunctionfromexperimentaldataandthusevaluatingpointswherenomeasureexists.

Thescipy.interpolate.interp1dclasscanbuildalinearinterpolationfunction:

Thenthescipy.interpolate.linear_interpinstanceneedstobeevaluatedatthetimeofinterest:

Acubicinterpolationcanalsobeselectedbyprovidingthekindoptionalkeywordargument:

Geometricaltransformationsonimages