In this lab, you'll practice your knowledge of correlation, autocorrelation, and partial autocorrelation by working on three different datasets.

In this lab you will:

• Plot and discuss the autocorrelation function (ACF) for a time series
• Plot and discuss the partial autocorrelation function (PACF) for a time series

We'll be looking at the exchange rates dataset again.

• First, run the following cell to import all the libraries and the functions required for this lab
• Then import the data in 'exch_rates.csv'
• Change the data type of the 'Frequency' column
• Set the 'Frequency' column as the index of the DataFrame

# Import all packages and functions
import pandas as pd
import numpy as np
import matplotlib.pylab as plt
%matplotlib inline
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
from matplotlib.pylab import rcParams

# Import data
xr = None

# Change the data type of the 'Frequency' column 


# Set the 'Frequency' column as the index

Plot all three exchange rates in one graph:

# Plot here

You can see that the EUR/USD and AUD/USD exchange rates are somewhere between 0.5 and 2, whereas the Danish Krone is somewhere between 4.5 and 9. Now let's look at the correlations between these time series.

# Correlation

Next, look at the plots of the differenced (1-lag) series. Use subplots to plot them rather than creating just one plot.

# 1-lag differenced series 
xr_diff = None

# Plot

Calculate the correlation of this differenced time series.

# Correlation 

Next, let's look at the "lag-1 autocorrelation" for the EUR/USD exchange rate.

• Create a "lag-1 autocorrelation" series
• Combine both the original and the shifted ("lag-1 autocorrelation") series into a DataFrame
• Plot these time series, and look at the correlation coefficient

# Isolate the EUR/USD exchange rate
eur = xr[['Euro']]

# "Shift" the time series by one period
eur_shift_1 = None

# Combine the original and shifted time series
lag_1 = None

# Plot 

# Correlation

Repeat this for a "lag-50 autocorrelation".

# "Shift" the time series by 50 periods
eur_shift_50 = None

# Combine the original and shifted time series
lag_50 = None

# Plot

# Correlation

Knowing this, let's plot the ACF now.

# Plot ACF

The series is heavily autocorrelated at first, and then there is a decay. This is a typical result for a series that is a random walk, generally you'll see heavy autocorrelations first, slowly tailing off until there is no autocorrelation anymore.

Next, let's look at the partial autocorrelation function plot.

# Plot PACF

This is interesting! Remember that Partial Autocorrelation Function gives the partial correlation of a time series with its own lagged values, controlling for the values of the time series at all shorter lags. When controlling for 1 period, the PACF is only very high for one-period lags, and basically 0 for shorter lags. This is again a typical result for random walk series!

Let's work with the air passenger dataset you have seen before. Plot the ACF and PACF for both the differenced and regular series.

Note: When plotting the PACF, make sure you specify method='ywm' in order to avoid any warnings.

# Import and process the air passenger data
air = pd.read_csv('passengers.csv')
air['Month'] = pd.to_datetime(air['Month'])
air.set_index('Month', inplace=True)
air.head()

# Plot ACF (regular)

# Plot PACF (regular)

# Generate a differenced series
air_diff = None

# Plot ACF (differenced)

# Plot PACF (differenced)

Are you getting the hang of interpreting ACF and PACF plots? For one final time, plot the ACF and PACF for both the NYSE time series.

Note: When plotting the PACF, make sure you specify method='ywm' in order to avoid any warnings.

# Import and process the NYSE data
nyse = pd.read_csv('NYSE_monthly.csv') 
nyse['Month'] = pd.to_datetime(nyse['Month'])
nyse.set_index('Month', inplace=True)
nyse.head()

# Plot ACF

# Plot PACF

Great, you've now been introduced to ACF and PACF. Let's move into more serious modeling with autoregressive and moving average models!