Effective Programming Practices for Economists. 10. Some scientific tools for Python

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1 Effective Programming Practices for Economists 10. Some scientific tools for Python Hans-Martin von Gaudecker Department of Economics, Universität Bonn

2 A NumPy primer The main NumPy object is the homogeneous multidimensional array Array: Table of elements, indexed by a tuple of integers Homogenous: Elements are of the same type Multidimensional: Can have several dimensions (called axes in NumPy-speak, since dimensions is ambiguous. Number of axes is often called rank ) For creating an ndarray object, you can use numpy.array Be sure to do the import correctly, Python s standard library also provides an array object... Licensed under the Creative Commons Attribution License 2/27

3 Basic usage of ndarray objects Create an array using numpy.array... >>> import numpy as np >>> a = np.array([[1.0, 2.2], [-1, 3.0]]) >>> a array([[ 1., 2.2], [-1., 3. ]]) Sequence: 1d, sequence of sequences: 2d, seq. of seq. s of seq. s: 3d, and so on Be careful to use a sequence and not np.array(1, 2, 3, 4) or np.array([1, 2], [3, 4]))... or fancier constructs: >>> b = np.arange(10).reshape(2, 5) >>> b array([[0, 1, 2, 3, 4], [5, 6, 7, 8, 9]]) Licensed under the Creative Commons Attribution License 3/27

4 Basic usage of ndarray objects See basic properties via attributes shape, ndim, size, and dtype: >>> for i in a, b:... print(i.shape)... print(i.ndim)... print(i.size)... print(i.dtype) (2, 2) 2 4 float64 (2, 5) 2 10 int64 Licensed under the Creative Commons Attribution License 4/27

5 Basic usage of ndarray objects Convenience functions for array creation: zeros([shape]), ones([shape]), diag([sequence on diagonal]): >>> c = np.zeros([1, 2, 3]) >>> c.ndim >>> c 3 array([[[ 0., 0., 0.], [ 0., 0., 0.]]]) >>> d = np.diag([1] * 3) >>> d.ndim >>> d 2 array([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) Licensed under the Creative Commons Attribution License 5/27

6 Basic usage of ndarray objects And one more convenience function: linspace(lower, upper, n_grid_points): >>> np.linspace(-.5,.5, 9) array([-0.5, , -0.25, , 0., 0.125, 0.25, 0.375, 0.5 ]) Useful to evaluate or plot functions on a grid... Licensed under the Creative Commons Attribution License 6/27

7 Basic operations with ndarray objects All basic operations apply element-wise >>> a = np.array([20,30,40,50]) >>> b = np.arange(4) >>> c = a - b >>> c array([20, 29, 38, 47]) >>> b ** 2 array([0, 1, 4, 9]) >>> 10 * sin(a) array([ , , , ]) >>> a < 35 array([true, True, False, False], dtype=bool) Licensed under the Creative Commons Attribution License 7/27

8 Basic operations with ndarray objects The operator performs element-by-element multiplication >>> A = np.array([[1, 1],... [0, 1]]) >>> B = np.array([[2, 0],... [3, 4]]) >>> A * B array([[2, 0], [0, 4]]) Use NumPy s dot() function for the matrix product... >>> np.dot(a, B) array([[5, 4], [3, 4]]) Licensed under the Creative Commons Attribution License 8/27

9 Indexing, slicing and iterating One dimensional arrays can be indexed, sliced and iterated over like lists and other Python sequences Multidimensional arrays can be indexed / sliced with one index per axis. These indices are given in a tuple separated by commas: >>> B[1, 1] 4 >>> B[0, :] array([2, 0]) >>> B[0] array([2, 0]) Licensed under the Creative Commons Attribution License 9/27

10 Indexing, slicing and iterating Slices also return views (references to the slice of the original array in memory) >>> c = B[0, :] >>> c[:] = [1, 1] >>> B array([[1, 1], [3, 4]]) Use the copy() method of an ndarray to get a copy >>> d = B[0, :].copy() >>> d[:] = [3, 3] >>> B array([[1, 1], [3, 4]]) Licensed under the Creative Commons Attribution License 10/27

11 Some useful references Tutorial (very clear examples): User guide (getting more complete): Reference (detailed explanations, but no easy read): Nice, slightly technical exposition: Walt, Colbert, and Varoquaux (2011) If you re used to Matlab... numpy-for-matlab-users.html... continued, but also R included: Licensed under the Creative Commons Attribution License 11/27

12 Matplotlib Publication-quality 2-dimensional plots >>> import matplotlib.pyplot as plt >>> plt.plot([-500, 0, 550], [-1000, 0, 550]) >>> plt.show() Licensed under the Creative Commons Attribution License 12/27

13 Change the axes, add title >>> plt.axis((-600.0, 600.0, , )) (-600.0, 600.0, , ) >>> plt.title('piecewise-linear utility function') <matplotlib.text.text object at 0x1027c3110> >>> plt.xlabel('gamble outcome') <matplotlib.text.text object at 0x1027b7cd0> >>> plt.ylabel('utility evaluation') <matplotlib.text.text object at 0x1027b9c50> Licensed under the Creative Commons Attribution License 13/27

14 Saving the figure >>> plt.savefig('linear_utility_3.png') 1000 Piecewise-linear utility function 500 Utility evaluation Gamble outcome Licensed under the Creative Commons Attribution License 14/27

15 Example: Comparing utility functions import os import numpy as np from matplotlib import pyplot from src.examples.scientific_tools.utility_functions import ce # Set the number of gridpoints for evaluation of the certainty equivalents. N_GRIDPOINTS = 101 # Define the outcomes of the lottery. z_low = z_high = # Define the parameters of the utility function. preference_specifications = [{'func': 'pow', 'gamma': 0.3, 'lambda': 2.0}] preference_specifications.append({'func': 'exp', 'gamma': 0.001, 'lambda': 2.0}) # Labels for the utility functions to be used in output. preference_label = {'exp': 'Exponential', 'pow': 'Power'} legend = [] # Define grids for probabilities and certainty equivalent evaluations. p_grid = np.linspace(0, 1, N_GRIDPOINTS) ce_grid = np.empty((len(preference_specifications), N_GRIDPOINTS)) Licensed under the Creative Commons Attribution License 15/27

16 # Define the basic characteristics of the figure. pyplot.plot() pyplot.title('certainty equivalent of [{}, {}]'.format(z_low, z_high)) pyplot.xlabel('probability of {} to occur'.format(z_high)) pyplot.ylabel('certainty equivalent') pyplot.axis([0, 1, -750, 600]) pyplot.grid(true) # Calculate and plot the utility values. for i, preference_specification in enumerate(preference_specifications): for j, p_high in enumerate(p_grid): lottery = {z_low: p_high, z_high: p_high} ce_grid[i, j] = ce(lottery, preference_specification) # Add plot to the figure, define legend. pyplot.plot(p_grid, ce_grid[i]) legend.append(r'{utl} utility, $\gamma$={gam}, $\lambda$={lam}'.format( utl=preference_label[preference_specification['func']], gam=preference_specification['gamma'], lam=preference_specification['lambda'] )) pyplot.legend(legend, loc='lower center') pyplot.savefig(os.path.join(out_path, 'utility_exponential_vs_power.pdf')) Licensed under the Creative Commons Attribution License 16/27

17 Example: Comparing utility functions 600 Certainty equivalent of [-500.0, 550.0] 400 Certainty equivalent Power utility, γ=0.3, λ=2.0 Exponential utility, γ=0.001, λ= Probability of to occur Licensed under the Creative Commons Attribution License 17/27

18 What else you can do with Matplotlib... Licensed under the Creative Commons Attribution License 18/27

19 What else you can do with Matplotlib... All the examples from the previous slide and many more you find along with the relevant source code at: Great way to get started Another good reference read is Tosi (2009) Licensed under the Creative Commons Attribution License 19/27

20 A very quick introduction to SciPy SciPy contains all kinds of useful tools for scientific applications Builds on top of NumPy, many things require to be passed ndarrays Often routines are just interfaces to classic (and fast) C or Fortran libraries Licensed under the Creative Commons Attribution License 20/27

21 A very quick introduction to SciPy Build in modular form import scipy not very useful Rather use statements like: from scipy.optimize import minimize Most relevant components to typical economics application: scipy.integrate scipy.optimize scipy.stats scipy.linalg Licensed under the Creative Commons Attribution License 21/27

22 Typical usage of scipy.optimize >>> from scipy.optimize import minimize >>> def g(x):... return x ** 2... >>> minimize(g, 200) success: True status: 0 jac: array([ e-11]) nfev: 15 njev: 5 hess: array([[ ]]) message: 'Optimization terminated successfully.' x: array([ e-09]) fun: e-17 Licensed under the Creative Commons Attribution License 22/27

23 Typical usage of scipy.optimize scipy.minimize provides a unified interface to many optimizers Full syntax of underlying functions is often scary... fmin_bfgs( f, x0, fprime, args, gtol, norm, epsilon, maxiter, full_output, disp, retall, callback )... but very seldomly needed Look through it if you use things seriously Licensed under the Creative Commons Attribution License 23/27

24 Further extensions... pandas very useful data structures + descriptives: StatsModels statistics & econometrics OLS, ML, time series, much more stuff Seaborn statistical visualisation on top of MPL: So called scikits provide specialised extensions to SciPy, see Index of extra tools loosely related to SciPy (careful not everything continues to be maintained): Licensed under the Creative Commons Attribution License 24/27

25 References I Tosi, Sandro (2009). Matplotlib for Python Developers. Birmingham, UK: Packt Publishing. Walt, Stéfan van der, S. Chris Colbert, and Gaël Varoquaux (2011). The NumPy Array: A Structure for Efficient Numerical Computation. In: Computing in Science & Engineering 13.2, pp Licensed under the Creative Commons Attribution License 25/27

26 Acknowledgements This course is designed after and borrows a lot from the Software Carpentry course designed by Greg Wilson for scientists and engineers. The Software Carpentry course material is made available under a Creative Commons Attribution License, as is this course s material. Licensed under the Creative Commons Attribution License 26/27

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