Python Lesson 2

Lesson outline

1. Working with pics.

2. More about NumPy.

3. Introduction to data representation with matplotlib.

4. Exercises

Working with pics

Import NumPy as in the previous lesson and pyplot and image libraries from matplotlib.

import numpy as np
from matplotlib import pyplot as plt # import pyplot function from matplotlib library
from matplotlib import image # import image function from matplotlib library

Read a jpg figure into an array and show it

imgarray=image.imread("iberian-lynx.png") 
imgplot=plt.imshow(imgarray)

Let's examine the array shape

imgarray.shape 

This is an RGB array with three values assigned per pixel. Let's now perform some basic manipulation of this array. We select the three RGB channels, the red, green and blue ones, transforming them to 2D arrays and showing them as heat maps

red_imgarray, gr_imgarray, bl_imgarray = imgarray[:,:,0], imgarray[:,:,1], imgarray[:,:,2] 
plt.imshow(red_imgarray, cmap='gray')
plt.show()
plt.imshow(gr_imgarray, cmap='gray')
plt.show()
plt.imshow(bl_imgarray, cmap='gray')

Change to a color map

imgplot=plt.imshow(red_imgarray,cmap="hot")

More about NumPy

Apart from reading data from files or, as we will see in the next lesson, transforming native Python structures into NumPy ndarrays using np.array NumPy provides a set of commands for the creation of arrays

• ones: Given array dimensions, it outputs an array with the given shape filled with the value 1.

• ones_like: Given an array, it outputs an array with the same dimensions and filled with the value 1.

• zeros: Given array dimensions, it outputs an array with the given shape filled with the value 0.

• zeros_like: Given an array, it outputs an array with the same dimensions and filled with the value 0.

• empty: Given array dimensions, it outputs an array with the same dimensions and with empty values (unitialized, be careful, getting into the wild side…).

• empty_like: Given an array, it outputs an array with the same dimensions and with unitialized values.

• full: Given array dimensions, it outputs an array with the same dimensions and with all elements equal to a given value.

• full_like: Given an array, it outputs an array with the same dimensions and with all elements equal to a given value.

• eye, identitiy: Given a square array dimension, it outputs a unit array (diagonal array) with the given shape.

• arange: Given start, stop [and step/] values, creates a 1D ndarray of evenly spaced values with /start as its first element, start + step the second, start + 2 step the third, and so on.

• linspace: Given start, stop [and N/] values, creates a 1D ndarray of exactly /N evenly spaced values with start as its first element and stop as the last one.

NumPy offers many types of data, with different dtype, for its storage in arrays. We are mainly interested in numerical data types, that are indicated by the prefix float (floating point numbers) or int (exact integer numbers) followed by a number indicating the number of bits per element. The standard double-precision floating point value is float64 (requires storage in 8 bytes) and the standard integer is int64. NumPy accepts complex values.

One of the main advantages of NumPy is vectorization, the possibility of performing simultaneously batches of operations in arrays without explicit loops. For example, we define a couple of arrays of random numbers and perform some operations with them

array_a = np.random.randn(3,3)
array_b = np.random.randn(3,3)

print(array_a, "\n\n", array_b, "\n\n", 10.0/(array_a + array_b))
#
print("\n\n")
#
print(array_a, "\n\n",array_b,"\n\n", np.sqrt(array_a**2 + array_b**2))

The function np.sqrt is an example of what is called an universal function (ufunc) that performs element-wise operations in data arrays. You can find a list of such NumPy functions in https://docs.scipy.org/doc/numpy-1.14.0/reference/ufuncs.html. Among them you can find the mathematical constants np.pi and np.e and the imaginary unit denoted as 1j..

One needs to be very aware that when working with NumPy arrays -and other data structures- Python uses the so called pass by reference and not the pass by value strategy of other programming languages. This means that an assignment implies a reference to data in the righthand side. This is completely different of what happens when we work with scalar data. If we execute

scalar_c = 8.5
scalar_c_2 = scalar_c
array_c = array_b[:2,:2]
#
print("array_b = ", array_b, "\n\n","array_c = ", array_c)
print("scalar_c = ", scalar_c, "\n\n","scalar_c_2 = ", scalar_c_2)
#
print("\n\n")
#
array_c[:] = 100.0
scalar_c_2 = 100.0
#
print("array_b = ", array_b, "\n\n","array_c = ", array_c)
print("scalar_c = ", scalar_c, "\n\n","scalar_c_2 = ", scalar_c_2)

Therefore array_b and array_c are bound to the same ndarray object. This is due to the need of optimizing the work with large matrices. A side effect of this is that you cannot assign values to elements of an array that has not been previously created (the function np.zeros is often used for this purpose). If you want a copy of the original matrix you can either use the copy method

array_d = array_a[:2,:2].copy()
#
print(array_a,  "\n\n", array_d)
#
print("\n\n")
#
array_d[:] = 1000.0
#
print(array_a,  "\n\n", array_d)

NumPy also allows to index using integer arrays, something called fancy indexing. In this case the resulting array is copied and it is not a reference to the original array. This can be seen in the following example

array_e = np.empty((10,10))
for value in range(-10,10):
    array_e[:, value] = value

print(array_e) 

array_f = array_e[2:5,[-1,5,2,3,2]] # Selecting a subset of columns and slicing the rows 

print(array_f)

print(array_e) 

NumPy arrays can be transposed using the transpose method or the special T attribute

print(array_a)
print(array_a.transpose())
print()
print(array_f)
print(array_f.T)

This is useful for example when computing the inner matrix product using np.dot

print(np.dot(array_a.T, array_a))
print("")
print(np.dot(array_f.T, array_f))

However, to perform matrix multiplication it is preferred using np.matmul or the a @ b notation.

Two or more NumPy arrays can also be concatenated, building up a large array from smaller ones. This can be done with the hstack and vstack methods.

arr_a = np.random.randn(2,4)
arr_b = np.random.randn(2,4)
arr_horizontal=np.hstack((arr_a,arr_b))
print(arr_horizontal)
arr_vertical=np.vstack([arr_a,arr_b])
print(arr_vertical)

Notice that in the hstack~(~vstack) case the number of rows(columns) in the arrays combined should be the same. These two are convenience functions, wrappers to the more general function concatenate

arr_v = np.concatenate([arr_a,arr_b])
arr_h = np.concatenate([arr_a,arr_b],axis=1)
print(arr_h)
print(arr_v)

Data in an array can also be flattened, tranforming the array into a vector (a one-dimensional array). This can be done with the NumPy ravel or flatten functions, both can act as a function or an array method.

arr_c = np.random.randn(4,4)
vec_c_0 = arr_c.ravel() # Equivalent to np.ravel(arr_c)
vec_c_1 = arr_c.flatten()  # Equivalent to np.flatten(arr_c)
if (np.array_equal(vec_c_0, vec_c_1)): # Comparing two arrays. 
    print(vec_c_0)

Note how we check if the two vectors created are equal. The NumPy function np.array_equal check if two arrays have identical shape and elements. You cannot check if two arrays are equal using the usual == conditional operator (try it). Both methods leave arr_c unchanged, but the ravel method provides an ndarray vector with access to the original data, while flatten copy the data and creates an independent object.

print(arr_c)
vec_c_0[0] = 1000.0
print(arr_c)
vec_c_1[0] = 10.0
print(arr_c)

The comparison between arrays yields Boolean arrays

print(array_a,  "\n\n",array_b,  "\n\n",array_a > array_b)

And you can use this Boolean arrays for indexing. In the example that follows we define a new matrix that only has negative non-zero elements, replacing the positive elements by zero.

boolean = array_a > 0
print(boolean)
array_e  = array_a.copy()
array_e[boolean] = 0

print(array_a,  "\n\n",array_e)

This is called vectorized computation, one of the greatest advantages of NumPy. We can, for example, select the positive elements of an array If you want to create a new array with the same shape of arr_c and with 0 in negative elements and 1 in positive elements you can easily do this in vectorized form, without loops (see Lesson 3)

arr_e = np.copy(arr_c)
arr_e[arr_c>0]=1
arr_e[arr_c<0]=0
print(arr_e)

Be aware that Boolean selection will NOT fail if the Boolean array has not the correct shape and this can be error prone. We will learn a better way for doing this in Lesson 5, using the np.where function.

Working with array you can construct complex conditionals combining simpler expressions with the logical operators & (and) and | (or) (the keywords and and or do not work in this context. For example

arr_f = np.copy(arr_c)
bool_mask = (arr_c > 1) | (arr_c < -1)
arr_f[bool_mask] = 2.0
print(arr_c,"\n", arr_f)

Selecting data with Booleans arrays always creates a copy of the original date, even if the data are unchanged.

Basic Data Plotting

We repeat what we did in the first lesson, reading one of the files with monthly temperatures and stripping the year from the array.

metdata_orig = np.loadtxt(fname='files/TData/T_Alicante_EM.csv', delimiter=',', skiprows=1)
metdata = metdata_orig[:,1:]

We can plot the array directly as a heat map

plt.imshow(metdata)

Using a different color map

plt.imshow(metdata, cmap="BrBG")

This is of limited utility. Let's compute and plot the mean monthly temperatures

ave_monthly = np.mean(metdata, axis=0)
ave_monthly_plot = plt.plot(ave_monthly)

and the average annual temperatures

ave_annual  = np.mean(metdata, axis=1)
ave_annual_plot = plt.plot(ave_annual)

In the same fashion we can also plot the maximum and minimum monthly temperatures

max_monthly = np.max(metdata, axis=0)
min_monthly = np.min(metdata, axis=0)
max_monthly_plot = plt.plot(max_monthly)
min_monthly_plot = plt.plot(min_monthly)

And the annual maximum and minimum temperatures

max_annual = np.max(metdata, axis=1)
min_annual = np.min(metdata, axis=1)
max_annual_plot = plt.plot(max_annual)
min_annual_plot = plt.plot(min_annual)

This is the most basic plotting in pyplot. You can improve the figure appearence as follows

fig, ax = plt.subplots()
ax.plot(max_monthly)
ax.plot(min_monthly)
ax.set_title("Cyprus Temperature Dataset")
ax.set_xlabel("Month (0-11)")
ax.set_ylabel("Max and Min average T (ºC)")

You can now solve exercises 2.1 and 2.2

We can combine several plots in a multi-panel figure

fig, ax = plt.subplots(nrows=2, ncols=2)
fig.tight_layout(pad=3.0)
ax[0,0].plot(max_monthly)
ax[0,1].plot(min_monthly)
ax[1,0].plot(max_annual)
ax[1,1].plot(min_annual)

You can now solve exercise 2.3.

Exercises

• Exercise 2.1: Plot the monthly and annual difference between max and min temperatures as a function of the month (1-12) and the year (1961-2096), respectively. In this case try to combine the plt.plot and plt.scatter functions. Hint: the plot function accept the syntax plt.plot(x,y).

• Exercise 2.2: Plot the standard deviation of the monthly and annual temperatures as a function of the month (1-12) and the year (1961-2096), respectively. Hint: check the std function in NumPy.

• Exercise 2.3: Prepare a plot with two panels (arranged as you wish) which depicts the annual dependence of the average Spring and Fall temperatures for meteorological seasons: Spring (Mar, Apr, May) and Fall (Sep, Oct, Nov).