Programming with Python

Analyzing Temperature Data

While a lot of powerful tools are built into languages like Python, even more live in the libraries.

In order to load our temperature data, we need to import a library called NumPy. In general you should use this library if you want to do fancy things with numbers, especially if you have matrices or arrays. We can load NumPy using:

import numpy

Importing a library is like getting a box of lab equipment out of a storage locker, inventorying the contents, and setting the box on the bench. Libraries provide additional functionality to Python just like a new piece of equipment adds functionality to a lab space. Once we’ve loaded the library, we can use NumPy to read the data file:

numpy.loadtxt('data/temperature.csv', delimiter=',')

array([[ 264.,  264.,  264., ...,  263.,  263.,  264.],
       [ 263.,  264.,  264., ...,  262.,  262.,  263.],
       [ 301.,  302.,  302., ...,  301.,  301.,  301.],
       [ 292.,  293.,  293., ...,  291.,  292.,  292.]])

The expression numpy.loadtxt(...) is a function call that tells Python to run the function loadtxt from the numpy library. This dotted notation, with the syntax thing.component, is used everywhere in Python to refer to the parts of things.

The function call to numpy.loadtxt has two parameters: the name of the file we want to read and the delimiter that separates values on a line. Both need to be character strings (or strings for short) so we write them in quotes.

Within the Jupyter iPython notebook, pressing Shift+Enter runs the commands in the selected cell. Because we haven’t told iPython what to do with the output of numpy.loadtxt, the notebook just displays the output to the screen. In this case, the output is the data we loaded from the temperature file. By default, only a few rows and columns are shown (with ... to omit elements when displaying big arrays). To save space, Python displays numbers as 1. instead of 1.0 when there’s nothing interesting after the decimal point.

The function call numpy.loadtxt read the temperature file but didn’t save it somewhere we can access it. In order to use the data within Python, we need to assign assign the array to a variable. A variable is just a name that refers to an object. Python’s variables must begin with a letter and are case sensitive. We can assign a variable name to an object using =.

To fully understand what variables and objects are related, let’s consider the simplest “collection” of data: a single value. The line below assigns the value 55 to a variable weight_kg:

weight_kg = 55

Once a variable is assigned to an object, we can print the value in the object to the screen:

print weight_kg

55

and do arithmetic with it:

print 'weight in pounds:', 2.2 * weight_kg

weight in pounds: 121.0

We can also change a variable’s value by assigning it a new one:

weight_kg = 57
print 'weight in kilograms is now:', weight_kg

weight in kilograms is now: 57

As the example above shows, we can print several things at once by separating them with commas.

If we imagine the variable as a sticky note with a name written on it, ]variable assignment](reference.html#assignment) is like putting the sticky note on a particular box containing some information:

Variables as Sticky Notes

This means that assigning a value to any one variable does not change the values of objects associated with other variables. For example, let’s store the subject’s weight in pounds in a variable:

weight_lb = 2.2 * weight_kg
print 'weight in kilograms:', weight_kg, 'and in pounds:', weight_lb

weight in kilograms: 57 and in pounds: 125.4

Creating Another Variable

and then change weight_kg:

weight_kg = 100.0
print 'weight in kilograms is now:', weight_kg, 'and weight in pounds is still:', weight_lb

weight in kilograms is now: 100.0 and weight in pounds is still: 125.4

Updating a Variable

Since the variable name weight_lb doesn’t “remember” how it came to be associated with “125.4”, its value isn’t automatically updated when weight_kg changes. This is different from the way spreadsheets work, where changing the value of a cell cascades to any values that are calculated from it.

Just as we can assign a single value to a variable, we can also assign an array of values to a variable using the same syntax. Let’s re-run numpy.loadtxt and assign the output of the function call to a variable:

data = numpy.loadtxt('data/temperature.csv',delimiter=',')

This statement doesn’t produce any visible output but we can look at the object using print:

print data

[[ 264.  264.  264. ...,  263.  263.  264.]
 [ 263.  264.  264. ...,  262.  262.  263.]
 [ 301.  302.  302. ...,  301.  301.  301.]
 [ 292.  293.  293. ...,  291.  292.  292.]]

By default, the temperature data has units of 10ths of a degree Fahrenheit. Let’s load it again and convert it to degrees Fahrenheit by dividing every item in the array by 10. To avoid accidentally introducing errors through integer division, we should divide the data by a float just in case the data imports as integers. These small details make your code more robust and reusable!

data = numpy.loadtxt('data/temperature.csv',delimiter=',') / 10.
print data

[[ 26.4  26.4  26.4 ...,  26.3  26.3  26.4]
 [ 26.3  26.4  26.4 ...,  26.2  26.2  26.3]
 [ 30.1  30.2  30.2 ...,  30.1  30.1  30.1]
 [ 29.2  29.3  29.3 ...,  29.1  29.2  29.2]]

mass = 47.5
age = 122
mass = mass * 2.0
age = age - 20

first, second = 'Grace', 'Hopper'
third, fourth = second, first
print third, fourth

Let’s see what type of object the variable data refers to:

print type(data)

<class 'numpy.ndarray'>

The output tells us that data currently refers to an N-dimensional array created by the NumPy library. We can see what its shape is like this:

print data.shape

(4, 365)

This tells us that data has 4 rows and 365 columns. This array contains daily average temperature normals (data averaged over 30 years) for four weather stations around Flagstaff, AZ. The rows are the individual stations and the columns are the average temperatures for each day of the year.

The object associated with the variable data contains not just the values in the array but also information about the array. These are the members or attributes. This extra information describes data in the same way an adjective describes a noun: data.shape is an attribute of data that described the dimensions of data. We use the same dotted notation for the attributes of variables that we use for the functions in libraries because they have the same part-and-whole relationship.

If we want to pull a single number from the array, we use the index of that entry in square brackets:

print 'first value in data:', data[0, 0]

first value in data: 26.4

print 'middle value in data:', data[2, 180]

middle value in data: 69.6

An index like [2, 180] selects a single element of an array, but we can select whole sections as well. A section of an array is called a slice. For example, we can select the first ten days (columns) of values for the first two stations (rows) like this:

print data[0:2, 0:10]

[[ 26.4  26.4  26.4  26.5  26.5  26.6  26.6  26.6  26.7  26.7]
 [ 26.3  26.4  26.4  26.5  26.5  26.6  26.6  26.7  26.7  26.8]]

The slice 0:2 means, “Start at index 0 and go up to, but not including, index 2.” Again, the up-to-but-not-including takes a bit of getting used to, but the rule is that the difference between the upper and lower bounds is the number of values in the slice.

We don’t have to start slices at 0:

print data[2:4, 0:10]

[[ 30.1  30.2  30.2  30.2  30.3  30.3  30.4  30.4  30.5  30.6]
 [ 29.2  29.3  29.3  29.4  29.4  29.5  29.5  29.6  29.6  29.7]]

We don’t always have to include the upper and lower bound on the slice. If we don’t include the lower bound, Python uses 0 by default; if we don’t include the upper, the slice runs to the end of the axis, and if we don’t include either (i.e., if we just use ‘:’ on its own), the slice includes everything:

small = data[:2,360:]
print 'small is:'
print small

small is:
[[ 26.3  26.3  26.3  26.3  26.4]
 [ 26.1  26.1  26.2  26.2  26.3]]

element = 'oxygen'
print 'first three characters:', element[0:3]
print 'last three characters:', element[3:6]

first three characters: oxy
last three characters: gen

We can perform common mathematical operations on arrays and create new ones with the results. The simplest operations with data are arithmetic: add, subtract, multiply, and divide. When you do such operations on arrays, the operation is done on each individual element of the array. Thus:

doubledata = data * 2.0

will create a new array doubledata whose elements have the value of two times the value of the corresponding elements in data:

print 'original:'
print data[:2,360:]
print 'doubledata:'
print doubledata[:2,360:]

original:
[[ 26.3  26.3  26.3  26.3  26.4]
 [ 26.1  26.1  26.2  26.2  26.3]]
doubledata:
[[ 52.6  52.6  52.6  52.6  52.8]
 [ 52.2  52.2  52.4  52.4  52.6]]

We can also perform mathematical operations using two arrays of the same size. In that case, The operation will use the corresponding elements of each of the two arrays. Thus:

tripledata = doubledata + data

will give you an array where tripledata[0,0] = doubledata[0,0] + data[0,0] and so on for all other elements of the arrays.

print 'tripledata:'
print tripledata[:2,360:]

tripledata:
[[ 78.9  78.9  78.9  78.9  79.2]
 [ 78.3  78.3  78.6  78.6  78.9]]

We can also perform statistical operations on arrays. If we want to find the average temperature on all days across all stations, for example, we can just ask the array for its mean value

print data.mean()

45.783767123

mean is a method of the array, i.e., a function that belongs to it in the same way that the member shape does. If variables are nouns, methods are verbs: they are operations that the object can perform. We write data.mean() with empty parenthesis because it is a request to the object data for an action. data.shape doesn’t need them because it is just a “characteristic” of data.

NumPy arrays have lots of useful methods:

print 'maximum temperature:', data.max()
print 'minimum temperature:', data.min()
print 'standard deviation:', data.std()

maximum temperature: 71.4
minimum temperature: 26.1
standard deviation: 13.379753803

When analyzing data, we often want to look at partial statistics such as the maximum value per station or the average value per day. One way to do this is to create a new temporary array that contains just the data we want and then look at the statistics of the full sub-array:

station_0 = data[0, :] # 0 on the first axis, everything on the second
print 'maximum temperature for station 0:', station_0.max()

maximum temperature for station 0: 64.1

We don’t actually need to store the row in a variable of its own. Instead, we can combine the selection and the method call:

print 'maximum temperature for station 2:', data[2, :].max()

maximum temperature for station 2: 71.4

If we need the maximum temperature for every stations or the average for each day across all four stations, we can perform the operation across an axis. Most array methods allow us to specify the axis we want to work on. If we ask for the average across axis 0 (rows in our 2-D example), we get:

print data.mean(axis=0)

[ 28.     28.075  28.075  28.15   28.175  28.25   28.275  28.325  28.375
  28.45   28.475  28.55   28.6    28.675  28.7    28.75   28.825  28.875
  28.925  29.     29.05   29.075  29.175  29.2    29.275  29.35   29.4
  29.5    29.575  29.625  29.7    29.825  29.875  29.975  30.075  30.175
  30.25   30.375  30.5    30.6    30.75   30.9    31.05   31.175  31.325
  31.475  31.65   31.8    31.975  32.125  32.3    32.5    32.65   32.825
  33.025  33.2    33.4    33.575  33.75   33.95   34.125  34.325  34.525
  34.725  34.875  35.075  35.275  35.45   35.625  35.8    35.975  36.15
  36.325  36.5    36.65   36.825  37.025  37.15   37.325  37.5    37.65
  37.825  38.     38.2    38.35   38.525  38.725  38.9    39.075  39.275
  39.45   39.65   39.85   40.05   40.275  40.5    40.7    40.9    41.2
  41.4    41.65   41.9    42.175  42.425  42.675  42.95   43.2    43.5
  43.75   44.05   44.325  44.6    44.9    45.2    45.475  45.775  46.05
  46.35   46.6    46.925  47.2    47.5    47.775  48.075  48.325  48.6
  48.9    49.125  49.4    49.65   49.925  50.175  50.425  50.675  50.95
  51.15   51.4    51.65   51.9    52.15   52.375  52.625  52.85   53.1
  53.325  53.575  53.825  54.1    54.325  54.575  54.85   55.1    55.375
  55.65   55.925  56.2    56.475  56.775  57.075  57.35   57.65   57.95
  58.25   58.55   58.875  59.175  59.5    59.8    60.15   60.45   60.75
  61.05   61.35   61.675  61.975  62.25   62.525  62.825  63.075  63.35
  63.6    63.85   64.05   64.275  64.475  64.7    64.9    65.05   65.2
  65.35   65.475  65.6    65.7    65.8    65.875  65.925  65.975  66.025
  66.05   66.05   66.05   66.05   66.025  66.     65.95   65.925  65.825
  65.8    65.725  65.625  65.575  65.45   65.375  65.275  65.175  65.075
  64.95   64.85   64.75   64.65   64.55   64.425  64.3    64.2    64.075
  63.975  63.825  63.725  63.575  63.475  63.325  63.2    63.075  62.95
  62.8    62.675  62.525  62.35   62.175  62.025  61.85   61.675  61.475
  61.3    61.05   60.85   60.65   60.375  60.175  59.9    59.65   59.35
  59.1    58.825  58.5    58.2    57.875  57.575  57.2    56.875  56.5
  56.15   55.775  55.425  55.05   54.675  54.3    53.925  53.5    53.175
  52.75   52.35   52.     51.575  51.225  50.825  50.45   50.075  49.675
  49.325  48.975  48.6    48.275  47.875  47.55   47.2    46.875  46.55
  46.2    45.9    45.575  45.25   44.95   44.625  44.325  44.     43.7
  43.4    43.15   42.85   42.55   42.25   41.925  41.625  41.325  41.025
  40.725  40.375  40.075  39.775  39.425  39.125  38.775  38.45   38.125
  37.775  37.45   37.1    36.75   36.425  36.05   35.725  35.375  35.025
  34.7    34.35   34.     33.7    33.325  33.     32.675  32.35   32.05
  31.775  31.475  31.175  30.925  30.675  30.4    30.175  29.925  29.75
  29.525  29.325  29.125  28.975  28.85   28.7    28.575  28.45   28.325
  28.25   28.15   28.1    28.05   28.025  27.975  27.95   27.95   27.9
  27.9    27.925  27.925  27.95   28.   ]

As a quick check, we can ask this array what its shape is:

print data.mean(axis=0).shape

(365,)

The expression (365,) tells us we have an N×1 vector, so this is the average temperature per day for all stations. If we average across axis 1 (columns in our 2-D example), we get:

print data.mean(axis=1)

[ 43.8430137   43.00684932  49.98767123  46.29753425]

which is the average temperature per station across all days.

Plotting

The mathematician Richard Hamming once said, “The purpose of computing is insight, not numbers,” and the best way to develop insight is often to visualize data. Visualization deserves an entire lecture (or course) of its own, but we can explore a few features of Python’s matplotlib library here. While there is no “official” plotting library in Python, this package is the de facto standard.

First, we will import the pyplot module from matplotlib and use two of its functions to create and display a heat map of our data:

import matplotlib.pyplot
image  = matplotlib.pyplot.imshow(data)
matplotlib.pyplot.show(image)

Heatmap of the Data

It’s very hard to see what the image shows when it’s that small. Let’s change the aspect ratio:

image  = matplotlib.pyplot.imshow(data)
matplotlib.pyplot.axes().set_aspect('auto')
matplotlib.pyplot.show(image)

Heatmap of the Data with different aspect ratio

Blue regions in this heat map are low values, while red shows high values. As we can see, temperature rises and falls over the year.

Let’s take a look at the average temperature between all the stations over time by making a line plot:

ave_temp = data.mean(axis=0)
ave_plot = matplotlib.pyplot.plot(ave_temp)
matplotlib.pyplot.show(ave_plot)

Average Temperature Over Time

It’s interesting to examine how the temperature trends vary between stations throughout the year. We can make four different line plots that show the data from each of the stations separately. To summarize what we’ve learned so far, let’s write every command we need to import the data and create the plots:

import numpy as np
import matplotlib.pyplot as plt

data = np.loadtxt('data/temperature.csv', delimiter=',') / 10.

plt.plot(data[0,:])
plt.show()

plt.plot(data[1,:])
plt.show()

plt.plot(data[2,:])
plt.show()

plt.plot(data[3,:])
plt.show()

It is still difficult to compare the temperature between stations with these plots. Instead, let’s plot the data for all each of the stations separately in the same plot. We can include the argument hold=True the first time we call plt.plot to force all subsequent calls to plt.plot to use the same axes (until it reaches plt.show()). We can add labels to the axes using the xlabel() and ylabel() functions and the title with title().

import numpy as np
import matplotlib.pyplot as plt

data = np.loadtxt('data/temperature.csv', delimiter=',') / 10.

plt.plot(data[0,:], hold=True)
plt.plot(data[1,:])
plt.plot(data[2,:])
plt.plot(data[3,:])

plt.xlabel('Day')
plt.ylabel('Avg temperature, degrees F')
plt.title('Temperature normals near Flagstaff, AZ')

plt.show()

Temperature for each of the four stations

Subplots

To better visualize data, we often want to arrange separate plots in layouts with multiple rows and columns. The script below plots the data from each of the stations separately in a single figure using subplots. Type the code for yourself so you get a sense of what it does.

This script uses a number of new commands. The function matplotlib.pyplot.figure() creates a space into which we will place all of our plots. The parameter figsize tells Python how big to make this space. Each subplot is placed into the figure using the subplot command. The subplot command takes 3 parameters. The first denotes how many total rows of subplots there are, the second parameter refers to the total number of subplot columns, and the final parameters denotes which subplot your variable is referencing. Each subplot is stored in a different variable (axes1, axes2, axes3, axes4). Once a subplot is created, the axes are can be titled using the set_xlabel() (or set_ylabel()) method for the axes. plt.show() is called only when the entire figure is set up:

import numpy as np
import matplotlib.pyplot as plt

data = np.loadtxt('data/temperature.csv', delimiter=',')

fig = plt.figure(figsize=(5.0, 9.0))

axes1 = fig.add_subplot(4, 1, 1)
axes2 = fig.add_subplot(4, 1, 2)
axes3 = fig.add_subplot(4, 1, 3)
axes4 = fig.add_subplot(4, 1, 4)

axes1.set_ylabel('station 0')
axes1.plot(data[0,:])

axes2.set_ylabel('station 1')
axes2.plot(data[1,:])

axes3.set_ylabel('station 2')
axes3.plot(data[2,:])

axes4.set_ylabel('station 3')
axes4.plot(data[3,:])

plt.show(fig)

Temperature at each Station as Subplots