Problem I. Daily Minimum Temperature Continued (rstudio)

Question

Problem I. Daily Minimum Temperature Continued We will again use the daily minimum temperatures.csv file. The Daily Minimum Temperatures dataset describes the minimum daily temperatures over 10 years (1981-1990) in the city Melbourne, Australia. The units are in degrees Celsius and there are 3650 observations. The data were recorded by the Australian Bureau of Meteorology. Note on leap years, the day December 31st has been removed so that they still have 365 days. a. Recall in HW7 you fit an AR(1) model to this time series data. Did an AR(1) seem like a truly suitable model for the data? b. Choose a suitable order and lag of differencing for this time series. Make a time series plot of the differenced time series. Is the differenced time series stationary? Explain both your choice of order and lag, as well as your answer for whether it is stationary. c. Look at the acf and pacf of the differenced time series from part b. Does it seem like any particular AR or MA model will be suitable? Pick the model you think is most likely to be suitable and explain your answer. [Note don’t use both AR and MA terms, just do one or the other.] d. Fit the model you selected in part c, and run diagnostics. Write down the fitted model. e. Now instead fit the model ARMA(2,2). Write down the fitted model. f. Forecast out the temperature for January 1st 1991 using the model selected in part c and the model from part e. Comment on if the two predictions are similar.

Answer 1

Hello Lily, here is a suggested solution:

Part a: Suitability of AR(1) Model

In Homework 7, you fit an AR(1) model to the daily minimum temperature data. To assess whether an AR(1) model was truly suitable for this data, you should have looked at the residuals of the model and the autocorrelation function (ACF) of the residuals. A suitable model would have residuals that appear to be white noise (i.e., no significant autocorrelation at lags other than lag 0).

If the residuals exhibited significant autocorrelation at higher lags or did not appear to be white noise, then the AR(1) model was likely not the best fit. Additionally, model selection criteria such as AIC (Akaike Information Criterion) or BIC (Bayesian Information Criterion) should also have been considered.

Part b: Differencing the Time Series

To determine a suitable order and lag of differencing for the time series, you should plot the original time series and its ACF and PACF. If the time series shows a clear trend or seasonal patterns, differencing can help achieve stationarity.

1. Order of Differencing (d): Usually, start with d=1. This means taking the difference between consecutive observations.
2. Lag of Differencing: Typically, the lag is set to 1 if there's no clear seasonality. If seasonality is suspected, use a lag that corresponds to the period of seasonality (e.g., 12 for monthly data).

Plot the Differenced Time Series:

import pandas as pd

import matplotlib.pyplot as plt

# Load the data

data = pd.read_csv('daily-minimum-temperatures.csv', parse_dates=['Date'], index_col='Date')

# Difference the data

differenced_data = data.diff().dropna()

# Plot the differenced data

plt.figure(figsize=(10, 6))

plt.plot(differenced_data)

plt.title('Differenced Daily Minimum Temperatures')

plt.xlabel('Date')

plt.ylabel('Differenced Temperature (Celsius)')

plt.show()

Stationarity Check: To determine if the differenced time series is stationary, you can visually inspect the plot and perform statistical tests such as the Augmented Dickey-Fuller (ADF) test. A stationary series has a constant mean and variance over time.