.. for doctests >>> import matplotlib.pyplot as plt >>> plt.switch_backend("Agg") >>> import numpy as np >>> np.random.seed(0) .. _scipy: Scipy : high-level scientific computing ======================================= **Authors**: *GaĆ«l Varoquaux, Adrien Chauve, Andre Espaze, Emmanuelle Gouillart, Ralf Gommers* .. topic:: Scipy The :mod:scipy package contains various toolboxes dedicated to common issues in scientific computing. Its different submodules correspond to different applications, such as interpolation, integration, optimization, image processing, statistics, special functions, etc. .. tip:: :mod:scipy can be compared to other standard scientific-computing libraries, such as the GSL (GNU Scientific Library for C and C++), or Matlab's toolboxes. scipy is the core package for scientific routines in Python; it is meant to operate efficiently on numpy arrays, so that numpy and scipy work hand in hand. Before implementing a routine, it is worth checking if the desired data processing is not already implemented in Scipy. As non-professional programmers, scientists often tend to **re-invent the wheel**, which leads to buggy, non-optimal, difficult-to-share and unmaintainable code. By contrast, Scipy's routines are optimized and tested, and should therefore be used when possible. .. contents:: Chapters contents :local: :depth: 1 .. warning:: This tutorial is far from an introduction to numerical computing. As enumerating the different submodules and functions in scipy would be very boring, we concentrate instead on a few examples to give a general idea of how to use scipy for scientific computing. :mod:scipy is composed of task-specific sub-modules: =========================== ========================================== :mod:scipy.cluster Vector quantization / Kmeans :mod:scipy.constants Physical and mathematical constants :mod:scipy.fftpack Fourier transform :mod:scipy.integrate Integration routines :mod:scipy.interpolate Interpolation :mod:scipy.io Data input and output :mod:scipy.linalg Linear algebra routines :mod:scipy.ndimage n-dimensional image package :mod:scipy.odr Orthogonal distance regression :mod:scipy.optimize Optimization :mod:scipy.signal Signal processing :mod:scipy.sparse Sparse matrices :mod:scipy.spatial Spatial data structures and algorithms :mod:scipy.special Any special mathematical functions :mod:scipy.stats Statistics =========================== ========================================== .. tip:: They all depend on :mod:numpy, but are mostly independent of each other. The standard way of importing Numpy and these Scipy modules is:: >>> import numpy as np >>> from scipy import stats # same for other sub-modules The main :mod:scipy namespace mostly contains functions that are really numpy functions (try scipy.cos is np.cos). Those are exposed for historical reasons; there's no reason to use import scipy in your code. File input/output: :mod:scipy.io ---------------------------------- **Matlab files**: Loading and saving:: >>> from scipy import io as spio >>> a = np.ones((3, 3)) >>> spio.savemat('file.mat', {'a': a}) # savemat expects a dictionary >>> data = spio.loadmat('file.mat') >>> data['a'] array([[1., 1., 1.], [1., 1., 1.], [1., 1., 1.]]) .. warning:: **Python / Matlab mismatches**, *eg* matlab does not represent 1D arrays :: >>> a = np.ones(3) >>> a array([1., 1., 1.]) >>> spio.savemat('file.mat', {'a': a}) >>> spio.loadmat('file.mat')['a'] array([[1., 1., 1.]]) Notice the difference? | .. Comments to make doctests pass which require an image >>> from matplotlib import pyplot as plt >>> plt.imsave('fname.png', np.array([[0]])) **Image files**: Reading images:: >>> import imageio >>> imageio.imread('fname.png') # doctest: +ELLIPSIS Array(...) >>> # Matplotlib also has a similar function >>> import matplotlib.pyplot as plt >>> plt.imread('fname.png') # doctest: +ELLIPSIS array(...) .. seealso:: * Load text files: :func:numpy.loadtxt/:func:numpy.savetxt * Clever loading of text/csv files: :func:numpy.genfromtxt/:func:numpy.recfromcsv * Fast and efficient, but numpy-specific, binary format: :func:numpy.save/:func:numpy.load * More advanced input/output of images in scikit-image: :mod:skimage.io Special functions: :mod:scipy.special --------------------------------------- Special functions are transcendental functions. The docstring of the :mod:scipy.special module is well-written, so we won't list all functions here. Frequently used ones are: * Bessel function, such as :func:scipy.special.jn (nth integer order Bessel function) * Elliptic function (:func:scipy.special.ellipj for the Jacobian elliptic function, ...) * Gamma function: :func:scipy.special.gamma, also note :func:scipy.special.gammaln which will give the log of Gamma to a higher numerical precision. * Erf, the area under a Gaussian curve: :func:scipy.special.erf .. _scipy_linalg: Linear algebra operations: :mod:scipy.linalg ---------------------------------------------- .. tip:: The :mod:scipy.linalg module provides standard linear algebra operations, relying on an underlying efficient implementation (BLAS, LAPACK). * The :func:scipy.linalg.det function computes the determinant of a square matrix:: >>> from scipy import linalg >>> arr = np.array([[1, 2], ... [3, 4]]) >>> linalg.det(arr) -2.0 >>> arr = np.array([[3, 2], ... [6, 4]]) >>> linalg.det(arr) # doctest: +SKIP 0.0 >>> linalg.det(np.ones((3, 4))) Traceback (most recent call last): ... ValueError: expected square matrix * The :func:scipy.linalg.inv function computes the inverse of a square matrix:: >>> arr = np.array([[1, 2], ... [3, 4]]) >>> iarr = linalg.inv(arr) >>> iarr array([[-2. , 1. ], [ 1.5, -0.5]]) >>> np.allclose(np.dot(arr, iarr), np.eye(2)) True Finally computing the inverse of a singular matrix (its determinant is zero) will raise LinAlgError:: >>> arr = np.array([[3, 2], ... [6, 4]]) >>> linalg.inv(arr) # doctest: +SKIP Traceback (most recent call last): ... ...LinAlgError: singular matrix * More advanced operations are available, for example singular-value decomposition (SVD):: >>> arr = np.arange(9).reshape((3, 3)) + np.diag([1, 0, 1]) >>> uarr, spec, vharr = linalg.svd(arr) The resulting array spectrum is:: >>> spec # doctest: +ELLIPSIS array([14.88982544, 0.45294236, 0.29654967]) The original matrix can be re-composed by matrix multiplication of the outputs of svd with np.dot:: >>> sarr = np.diag(spec) >>> svd_mat = uarr.dot(sarr).dot(vharr) >>> np.allclose(svd_mat, arr) True SVD is commonly used in statistics and signal processing. Many other standard decompositions (QR, LU, Cholesky, Schur), as well as solvers for linear systems, are available in :mod:scipy.linalg. .. _intro_scipy_interpolate: Interpolation: :mod:scipy.interpolate --------------------------------------- :mod:scipy.interpolate is useful for fitting a function from experimental data and thus evaluating points where no measure exists. The module is based on the FITPACK Fortran subroutines_. .. _FITPACK Fortran subroutines : http://www.netlib.org/dierckx/index.html .. _netlib : http://www.netlib.org By imagining experimental data close to a sine function:: >>> measured_time = np.linspace(0, 1, 10) >>> noise = (np.random.random(10)*2 - 1) * 1e-1 >>> measures = np.sin(2 * np.pi * measured_time) + noise :class:scipy.interpolate.interp1d can build a linear interpolation function:: >>> from scipy.interpolate import interp1d >>> linear_interp = interp1d(measured_time, measures) .. image:: scipy/auto_examples/images/sphx_glr_plot_interpolation_001.png :target: scipy/auto_examples/plot_interpolation.html :scale: 60 :align: right Then the result can be evaluated at the time of interest:: >>> interpolation_time = np.linspace(0, 1, 50) >>> linear_results = linear_interp(interpolation_time) A cubic interpolation can also be selected by providing the kind optional keyword argument:: >>> cubic_interp = interp1d(measured_time, measures, kind='cubic') >>> cubic_results = cubic_interp(interpolation_time) :class:scipy.interpolate.interp2d is similar to :class:scipy.interpolate.interp1d, but for 2-D arrays. Note that for the interp family, the interpolation points must stay within the range of given data points. See the summary exercise on :ref:summary_exercise_stat_interp for a more advanced spline interpolation example. Optimization and fit: :mod:scipy.optimize ------------------------------------------- Optimization is the problem of finding a numerical solution to a minimization or equality. .. tip:: The :mod:scipy.optimize module provides algorithms for function minimization (scalar or multi-dimensional), curve fitting and root finding. :: >>> from scipy import optimize Curve fitting .............. .. Comment to make doctest pass >>> np.random.seed(0) .. image:: scipy/auto_examples/images/sphx_glr_plot_curve_fit_001.png :target: scipy/auto_examples/plot_curve_fit.html :align: right :scale: 50 Suppose we have data on a sine wave, with some noise: :: >>> x_data = np.linspace(-5, 5, num=50) >>> y_data = 2.9 * np.sin(1.5 * x_data) + np.random.normal(size=50) If we know that the data lies on a sine wave, but not the amplitudes or the period, we can find those by least squares curve fitting. First we have to define the test function to fit, here a sine with unknown amplitude and period:: >>> def test_func(x, a, b): ... return a * np.sin(b * x) .. image:: scipy/auto_examples/images/sphx_glr_plot_curve_fit_002.png :target: scipy/auto_examples/plot_curve_fit.html :align: right :scale: 50 We then use :func:scipy.optimize.curve_fit to find :math:a and :math:b:: >>> params, params_covariance = optimize.curve_fit(test_func, x_data, y_data, p0=[2, 2]) >>> print(params) [3.05931973 1.45754553] .. raw:: html
.. topic:: Exercise: Curve fitting of temperature data :class: green The temperature extremes in Alaska for each month, starting in January, are given by (in degrees Celcius):: max: 17, 19, 21, 28, 33, 38, 37, 37, 31, 23, 19, 18 min: -62, -59, -56, -46, -32, -18, -9, -13, -25, -46, -52, -58 1. Plot these temperature extremes. 2. Define a function that can describe min and max temperatures. Hint: this function has to have a period of 1 year. Hint: include a time offset. 3. Fit this function to the data with :func:scipy.optimize.curve_fit. 4. Plot the result. Is the fit reasonable? If not, why? 5. Is the time offset for min and max temperatures the same within the fit accuracy? :ref:solution  Finding the minimum of a scalar function ........................................ .. Comment to make doctest pass >>> np.random.seed(0) .. image:: scipy/auto_examples/images/sphx_glr_plot_optimize_example1_001.png :target: scipy/auto_examples/plot_optimize_example1.html :align: right :scale: 50 Let's define the following function: :: >>> def f(x): ... return x**2 + 10*np.sin(x) and plot it: .. doctest:: >>> x = np.arange(-10, 10, 0.1) >>> plt.plot(x, f(x)) # doctest:+SKIP >>> plt.show() # doctest:+SKIP This function has a global minimum around -1.3 and a local minimum around 3.8. Searching for minimum can be done with :func:scipy.optimize.minimize, given a starting point x0, it returns the location of the minimum that it has found: .. sidebar:: result type The result of :func:scipy.optimize.minimize is a compound object comprising all information on the convergence :: >>> result = optimize.minimize(f, x0=0) >>> result # doctest: +ELLIPSIS fun: -7.9458233756... hess_inv: array([[0.0858...]]) jac: array([-1.19209...e-06]) message: 'Optimization terminated successfully.' nfev: 18 nit: 5 njev: 6 status: 0 success: True x: array([-1.30644...]) >>> result.x # The coordinate of the minimum # doctest: +ELLIPSIS array([-1.30644...]) | **Methods**: As the function is a smooth function, gradient-descent based methods are good options. The lBFGS algorithm __ is a good choice in general:: >>> optimize.minimize(f, x0=0, method="L-BFGS-B") # doctest: +ELLIPSIS fun: array([-7.94582338]) hess_inv: <1x1 LbfgsInvHessProduct with dtype=float64> jac: array([-1.42108547e-06]) message: ...'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL' nfev: 12 nit: 5 status: 0 success: True x: array([-1.30644013]) Note how it cost only 12 functions evaluation above to find a good value for the minimum. | **Global minimum**: A possible issue with this approach is that, if the function has local minima, the algorithm may find these local minima instead of the global minimum depending on the initial point x0:: >>> res = optimize.minimize(f, x0=3, method="L-BFGS-B") >>> res.x array([3.83746709]) .. Comment to make doctest pass >>> np.random.seed(42) If we don't know the neighborhood of the global minimum to choose the initial point, we need to resort to costlier global optimization. To find the global minimum, we use :func:scipy.optimize.basinhopping (added in version 0.12.0 of Scipy). It combines a local optimizer with sampling of starting points:: >>> optimize.basinhopping(f, 0) # doctest: +SKIP nfev: 1725 minimization_failures: 0 fun: -7.9458233756152845 x: array([-1.30644001]) message: ['requested number of basinhopping iterations completed successfully'] njev: 575 nit: 100 .. seealso: Another available (but much less efficient) global optimizer is :func:scipy.optimize.brute (brute force optimization on a grid). More algorithms for different classes of global optimization problems exist, but this is out of the scope of :mod:scipy. Some useful packages for global optimization are OpenOpt, IPOPT_, PyGMO_ and PyEvolve_. .. note:: :mod:scipy used to contain the routine anneal, it has been removed in SciPy 0.16.0. .. _IPOPT: https://github.com/xuy/pyipopt .. _PyGMO: http://esa.github.io/pygmo/ .. _PyEvolve: http://pyevolve.sourceforge.net/ | **Constraints**: We can constrain the variable to the interval (0, 10) using the "bounds" argument: .. sidebar:: A list of bounds As :func:~scipy.optimize.minimize works in general with x multidimensionsal, the "bounds" argument is a list of bound on each dimension. :: >>> res = optimize.minimize(f, x0=1, ... bounds=((0, 10), )) >>> res.x # doctest: +ELLIPSIS array([0.]) .. tip:: What has happened? Why are we finding 0, which is not a mimimum of our function. .. topic:: **Minimizing functions of several variables** To minimize over several variables, the trick is to turn them into a function of a multi-dimensional variable (a vector). See for instance the exercise on 2D minimization below. .. note:: :func:scipy.optimize.minimize_scalar is a function with dedicated methods to minimize functions of only one variable. .. seealso:: Finding minima of function is discussed in more details in the advanced chapter: :ref:mathematical_optimization. .. topic:: Exercise: 2-D minimization :class: green .. image:: scipy/auto_examples/images/sphx_glr_plot_2d_minimization_002.png :target: scipy/auto_examples/plot_2d_minimization.html :align: right :scale: 50 The six-hump camelback function .. math:: f(x, y) = (4 - 2.1x^2 + \frac{x^4}{3})x^2 + xy + (4y^2 - 4)y^2 has multiple global and local minima. Find the global minima of this function. Hints: - Variables can be restricted to :math:-2 < x < 2 and :math:-1 < y < 1. - Use :func:numpy.meshgrid and :func:matplotlib.pyplot.imshow to find visually the regions. - Use :func:scipy.optimize.minimize, optionally trying out several of its methods. How many global minima are there, and what is the function value at those points? What happens for an initial guess of :math:(x, y) = (0, 0) ? :ref:solution  Finding the roots of a scalar function ........................................ To find a root, i.e. a point where :math:f(x) = 0, of the function :math:f above we can use :func:scipy.optimize.root: :: >>> root = optimize.root(f, x0=1) # our initial guess is 1 >>> root # The full result fjac: array([[-1.]]) fun: array([0.]) message: 'The solution converged.' nfev: 10 qtf: array([1.33310463e-32]) r: array([-10.]) status: 1 success: True x: array([0.]) >>> root.x # Only the root found array([0.]) Note that only one root is found. Inspecting the plot of :math:f reveals that there is a second root around -2.5. We find the exact value of it by adjusting our initial guess: :: >>> root2 = optimize.root(f, x0=-2.5) >>> root2.x array([-2.47948183]) .. note:: :func:scipy.optimize.root also comes with a variety of algorithms, set via the "method" argument. .. image:: scipy/auto_examples/images/sphx_glr_plot_optimize_example2_001.png :target: scipy/auto_examples/plot_optimize_example2.html :align: right :scale: 70 | Now that we have found the minima and roots of f and used curve fitting on it, we put all those results together in a single plot: .. raw:: html
.. seealso:: You can find all algorithms and functions with similar functionalities in the documentation of :mod:scipy.optimize. See the summary exercise on :ref:summary_exercise_optimize for another, more advanced example. Statistics and random numbers: :mod:scipy.stats ------------------------------------------------- .. Comment to make doctest pass >>> np.random.seed(0) The module :mod:scipy.stats contains statistical tools and probabilistic descriptions of random processes. Random number generators for various random process can be found in :mod:numpy.random. Distributions: histogram and probability density function .......................................................... Given observations of a random process, their histogram is an estimator of the random process's PDF (probability density function): :: >>> samples = np.random.normal(size=1000) >>> bins = np.arange(-4, 5) >>> bins array([-4, -3, -2, -1, 0, 1, 2, 3, 4]) >>> histogram = np.histogram(samples, bins=bins, density=True)[0] >>> bins = 0.5*(bins[1:] + bins[:-1]) >>> bins array([-3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5]) >>> from scipy import stats >>> pdf = stats.norm.pdf(bins) # norm is a distribution object >>> plt.plot(bins, histogram) # doctest: +ELLIPSIS [] >>> plt.plot(bins, pdf) # doctest: +ELLIPSIS [] .. image:: scipy/auto_examples/images/sphx_glr_plot_normal_distribution_001.png :target: scipy/auto_examples/plot_normal_distribution.html :scale: 70 .. sidebar:: **The distribution objects** :class:scipy.stats.norm is a distribution object: each distribution in :mod:scipy.stats is represented as an object. Here it's the normal distribution, and it comes with a PDF, a CDF, and much more. If we know that the random process belongs to a given family of random processes, such as normal processes, we can do a maximum-likelihood fit of the observations to estimate the parameters of the underlying distribution. Here we fit a normal process to the observed data:: >>> loc, std = stats.norm.fit(samples) >>> loc # doctest: +ELLIPSIS -0.045256707... >>> std # doctest: +ELLIPSIS 0.9870331586... .. topic:: Exercise: Probability distributions :class: green Generate 1000 random variates from a gamma distribution with a shape parameter of 1, then plot a histogram from those samples. Can you plot the pdf on top (it should match)? Extra: the distributions have many useful methods. Explore them by reading the docstring or by using tab completion. Can you recover the shape parameter 1 by using the fit method on your random variates? Mean, median and percentiles ............................. The mean is an estimator of the center of the distribution:: >>> np.mean(samples) # doctest: +ELLIPSIS -0.0452567074... The median another estimator of the center. It is the value with half of the observations below, and half above:: >>> np.median(samples) # doctest: +ELLIPSIS -0.0580280347... .. tip:: Unlike the mean, the median is not sensitive to the tails of the distribution. It is "robust" _. .. topic:: Exercise: Compare mean and median on samples of a Gamma distribution :class: green Which one seems to be the best estimator of the center for the Gamma distribution? The median is also the percentile 50, because 50% of the observation are below it:: >>> stats.scoreatpercentile(samples, 50) # doctest: +ELLIPSIS -0.0580280347... Similarly, we can calculate the percentile 90:: >>> stats.scoreatpercentile(samples, 90) # doctest: +ELLIPSIS 1.2315935511... .. tip:: The percentile is an estimator of the CDF: cumulative distribution function. Statistical tests ................. .. image:: scipy/auto_examples/images/sphx_glr_plot_t_test_001.png :target: scipy/auto_examples/plot_t_test.html :scale: 60 :align: right A statistical test is a decision indicator. For instance, if we have two sets of observations, that we assume are generated from Gaussian processes, we can use a T-test __ to decide whether the means of two sets of observations are significantly different:: >>> a = np.random.normal(0, 1, size=100) >>> b = np.random.normal(1, 1, size=10) >>> stats.ttest_ind(a, b) # doctest: +SKIP (array(-3.177574054...), 0.0019370639...) .. tip:: The resulting output is composed of: * The T statistic value: it is a number the sign of which is proportional to the difference between the two random processes and the magnitude is related to the significance of this difference. * the *p value*: the probability of both processes being identical. If it is close to 1, the two process are almost certainly identical. The closer it is to zero, the more likely it is that the processes have different means. .. seealso:: The chapter on :ref:statistics  introduces much more elaborate tools for statistical testing and statistical data loading and visualization outside of scipy. Numerical integration: :mod:scipy.integrate --------------------------------------------- Function integrals ................... The most generic integration routine is :func:scipy.integrate.quad. To compute :math:\int_0^{\pi / 2} sin(t) dt:: >>> from scipy.integrate import quad >>> res, err = quad(np.sin, 0, np.pi/2) >>> np.allclose(res, 1) # res is the result, is should be close to 1 True >>> np.allclose(err, 1 - res) # err is an estimate of the err True Other integration schemes are available: :func:scipy.integrate.fixed_quad, :func:scipy.integrate.quadrature, :func:scipy.integrate.romberg... Integrating differential equations ................................... :mod:scipy.integrate also features routines for integrating Ordinary Differential Equations (ODE) __. In particular, :func:scipy.integrate.odeint solves ODE of the form:: dy/dt = rhs(y1, y2, .., t0,...) As an introduction, let us solve the ODE :math:\frac{dy}{dt} = -2 y between :math:t = 0 \dots 4, with the initial condition :math:y(t=0) = 1. First the function computing the derivative of the position needs to be defined:: >>> def calc_derivative(ypos, time): ... return -2 * ypos .. image:: scipy/auto_examples/images/sphx_glr_plot_odeint_simple_001.png :target: scipy/auto_examples/plot_odeint_simple.html :scale: 70 :align: right Then, to compute y as a function of time:: >>> from scipy.integrate import odeint >>> time_vec = np.linspace(0, 4, 40) >>> y = odeint(calc_derivative, y0=1, t=time_vec) .. raw:: html
Let us integrate a more complex ODE: a damped spring-mass oscillator __. The position of a mass attached to a spring obeys the 2nd order *ODE* :math:y'' + 2 \varepsilon \omega_0 y' + \omega_0^2 y = 0 with :math:\omega_0^2 = k/m with :math:k the spring constant, :math:m the mass and :math:\varepsilon = c/(2 m \omega_0) with :math:c the damping coefficient. We set:: >>> mass = 0.5 # kg >>> kspring = 4 # N/m >>> cviscous = 0.4 # N s/m Hence:: >>> eps = cviscous / (2 * mass * np.sqrt(kspring/mass)) >>> omega = np.sqrt(kspring / mass) The system is underdamped, as:: >>> eps < 1 True For :func:~scipy.integrate.odeint, the 2nd order equation needs to be transformed in a system of two first-order equations for the vector :math:Y = (y, y'): the function computes the velocity and acceleration:: >>> def calc_deri(yvec, time, eps, omega): ... return (yvec[1], -2.0 * eps * omega * yvec[1] - omega **2 * yvec[0]) .. image:: scipy/auto_examples/images/sphx_glr_plot_odeint_damped_spring_mass_001.png :target: scipy/auto_examples/plot_odeint_damped_spring_mass.html :scale: 70 :align: right Integration of the system follows:: >>> time_vec = np.linspace(0, 10, 100) >>> yinit = (1, 0) >>> yarr = odeint(calc_deri, yinit, time_vec, args=(eps, omega)) .. raw:: html
.. tip:: :func:scipy.integrate.odeint uses the LSODA (Livermore Solver for Ordinary Differential equations with Automatic method switching for stiff and non-stiff problems), see the ODEPACK Fortran library_ for more details. .. _ODEPACK Fortran library : http://people.sc.fsu.edu/~jburkardt/f77_src/odepack/odepack.html .. seealso:: **Partial Differental Equations** There is no Partial Differential Equations (PDE) solver in Scipy. Some Python packages for solving PDE's are available, such as fipy_ or SfePy_. .. _fipy: http://www.ctcms.nist.gov/fipy/ .. _SfePy: http://sfepy.org/doc/ Fast Fourier transforms: :mod:scipy.fftpack --------------------------------------------- The :mod:scipy.fftpack module computes fast Fourier transforms (FFTs) and offers utilities to handle them. The main functions are: * :func:scipy.fftpack.fft to compute the FFT * :func:scipy.fftpack.fftfreq to generate the sampling frequencies * :func:scipy.fftpack.ifft computes the inverse FFT, from frequency space to signal space | As an illustration, a (noisy) input signal (sig), and its FFT:: >>> from scipy import fftpack >>> sig_fft = fftpack.fft(sig) # doctest:+SKIP >>> freqs = fftpack.fftfreq(sig.size, d=time_step) # doctest:+SKIP .. |signal_fig| image:: scipy/auto_examples/images/sphx_glr_plot_fftpack_001.png :target: scipy/auto_examples/plot_fftpack.html :scale: 60 .. |fft_fig| image:: scipy/auto_examples/images/sphx_glr_plot_fftpack_002.png :target: scipy/auto_examples/plot_fftpack.html :scale: 60 ===================== ===================== |signal_fig| |fft_fig| ===================== ===================== **Signal** **FFT** ===================== ===================== As the signal comes from a real function, the Fourier transform is symmetric. The peak signal frequency can be found with freqs[power.argmax()] .. image:: scipy/auto_examples/images/sphx_glr_plot_fftpack_003.png :target: scipy/auto_examples/plot_fftpack.html :scale: 60 :align: right Setting the Fourrier component above this frequency to zero and inverting the FFT with :func:scipy.fftpack.ifft, gives a filtered signal. .. note:: The code of this example can be found :ref:here  .. topic:: numpy.fft Numpy also has an implementation of FFT (:mod:numpy.fft). However, the scipy one should be preferred, as it uses more efficient underlying implementations. | **Fully worked examples:** .. |periodicity_finding| image:: scipy/auto_examples/solutions/images/sphx_glr_plot_periodicity_finder_001.png :scale: 50 :target: scipy/auto_examples/solutions/plot_periodicity_finder.html .. |image_blur| image:: scipy/auto_examples/solutions/images/sphx_glr_plot_image_blur_002.png :scale: 50 :target: scipy/auto_examples/solutions/plot_image_blur.html =================================================================================================================== =================================================================================================================== Crude periodicity finding (:ref:link ) Gaussian image blur (:ref:link ) =================================================================================================================== =================================================================================================================== |periodicity_finding| |image_blur| =================================================================================================================== =================================================================================================================== | .. topic:: Exercise: Denoise moon landing image :class: green .. image:: ../data/moonlanding.png :scale: 70 1. Examine the provided image :download:moonlanding.png <../data/moonlanding.png>, which is heavily contaminated with periodic noise. In this exercise, we aim to clean up the noise using the Fast Fourier Transform. 2. Load the image using :func:matplotlib.pyplot.imread. 3. Find and use the 2-D FFT function in :mod:scipy.fftpack, and plot the spectrum (Fourier transform of) the image. Do you have any trouble visualising the spectrum? If so, why? 4. The spectrum consists of high and low frequency components. The noise is contained in the high-frequency part of the spectrum, so set some of those components to zero (use array slicing). 5. Apply the inverse Fourier transform to see the resulting image. :ref:Solution  | Signal processing: :mod:scipy.signal -------------------------------------- .. tip:: :mod:scipy.signal is for typical signal processing: 1D, regularly-sampled signals. .. image:: scipy/auto_examples/images/sphx_glr_plot_resample_001.png :target: scipy/auto_examples/plot_resample.html :scale: 65 :align: right **Resampling** :func:scipy.signal.resample: resample a signal to n points using FFT. :: >>> t = np.linspace(0, 5, 100) >>> x = np.sin(t) >>> from scipy import signal >>> x_resampled = signal.resample(x, 25) >>> plt.plot(t, x) # doctest: +ELLIPSIS [] >>> plt.plot(t[::4], x_resampled, 'ko') # doctest: +ELLIPSIS [] .. tip:: Notice how on the side of the window the resampling is less accurate and has a rippling effect. This resampling is different from the :ref:interpolation  provided by :mod:scipy.interpolate as it only applies to regularly sampled data. .. image:: scipy/auto_examples/images/sphx_glr_plot_detrend_001.png :target: scipy/auto_examples/plot_detrend.html :scale: 65 :align: right **Detrending** :func:scipy.signal.detrend: remove linear trend from signal:: >>> t = np.linspace(0, 5, 100) >>> x = t + np.random.normal(size=100) >>> from scipy import signal >>> x_detrended = signal.detrend(x) >>> plt.plot(t, x) # doctest: +ELLIPSIS [] >>> plt.plot(t, x_detrended) # doctest: +ELLIPSIS [] .. raw:: html
**Filtering**: For non-linear filtering, :mod:scipy.signal has filtering (median filter :func:scipy.signal.medfilt, Wiener :func:scipy.signal.wiener), but we will discuss this in the image section. .. tip:: :mod:scipy.signal also has a full-blown set of tools for the design of linear filter (finite and infinite response filters), but this is out of the scope of this tutorial. **Spectral analysis**: :func:scipy.signal.spectrogram compute a spectrogram --frequency spectrums over consecutive time windows--, while :func:scipy.signal.welch comptes a power spectrum density (PSD). .. |chirp_fig| image:: scipy/auto_examples/images/sphx_glr_plot_spectrogram_001.png :target: scipy/auto_examples/plot_spectrogram.html :scale: 45 .. |spectrogram_fig| image:: scipy/auto_examples/images/sphx_glr_plot_spectrogram_002.png :target: scipy/auto_examples/plot_spectrogram.html :scale: 45 .. |psd_fig| image:: scipy/auto_examples/images/sphx_glr_plot_spectrogram_003.png :target: scipy/auto_examples/plot_spectrogram.html :scale: 45 |chirp_fig| |spectrogram_fig| |psd_fig| Image manipulation: :mod:scipy.ndimage ----------------------------------------- .. include:: image_processing/image_processing.rst :start-line: 1 Summary exercises on scientific computing ----------------------------------------- The summary exercises use mainly Numpy, Scipy and Matplotlib. They provide some real-life examples of scientific computing with Python. Now that the basics of working with Numpy and Scipy have been introduced, the interested user is invited to try these exercises. .. only:: latex .. toctree:: :maxdepth: 1 summary-exercises/stats-interpolate.rst summary-exercises/optimize-fit.rst summary-exercises/image-processing.rst summary-exercises/answers_image_processing.rst .. only:: html **Exercises:** .. toctree:: :maxdepth: 1 summary-exercises/stats-interpolate.rst summary-exercises/optimize-fit.rst summary-exercises/image-processing.rst **Proposed solutions:** .. toctree:: :maxdepth: 1 summary-exercises/answers_image_processing.rst .. include the gallery. Skip the first line to avoid the "orphan" declaration .. include:: scipy/auto_examples/index.rst :start-line: 1 .. seealso:: **References to go further** * Some chapters of the advanced __ and the packages and applications __ parts of the scipy lectures * The scipy cookbook __