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How can you ensure that your data is sorted for an accurate plot?
How can you ensure that your data is sorted for an accurate plot?
Sort the data by date from oldest to newest
What type of chart should you choose for time series data to connect data points with a line?
What type of chart should you choose for time series data to connect data points with a line?
Line with Markers
How can you edit the chart title in Excel?
How can you edit the chart title in Excel?
Click on the default chart title to edit
What is the purpose of adding axis titles to a chart?
What is the purpose of adding axis titles to a chart?
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Which option allows you to adjust the date format in Excel charts?
Which option allows you to adjust the date format in Excel charts?
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What are the formulas for calculating autocovariance and autocorrelation when k = 2?
What are the formulas for calculating autocovariance and autocorrelation when k = 2?
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What do the autocorrelations r1, r2, ..., rK form together?
What do the autocorrelations r1, r2, ..., rK form together?
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What is the purpose of constructing ACF plots in time series analysis?
What is the purpose of constructing ACF plots in time series analysis?
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What is the most basic forecasting method in time series analysis?
What is the most basic forecasting method in time series analysis?
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What does Theil's U -statistic measure in time series analysis?
What does Theil's U -statistic measure in time series analysis?
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What is the value of Theil's U-statistic for the beer data in Table 8?
What is the value of Theil's U-statistic for the beer data in Table 8?
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What is the purpose of using seasonal plots in time series analysis?
What is the purpose of using seasonal plots in time series analysis?
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The autocorrelation statistic r1 is equal to 0.868 for the given data. The autocovariance statistic c1 is equal to ____.
The autocorrelation statistic r1 is equal to 0.868 for the given data. The autocovariance statistic c1 is equal to ____.
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The autocovariance c2 is greater than the autocovariance c1 for the data in Example 3.
The autocovariance c2 is greater than the autocovariance c1 for the data in Example 3.
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What is time series analysis?
What is time series analysis?
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Which fields commonly use time series data?
Which fields commonly use time series data?
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Forecasting has become irrelevant to decision makers.
Forecasting has become irrelevant to decision makers.
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A time series is a sequence of data points measured at successive points in __________.
A time series is a sequence of data points measured at successive points in __________.
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What are examples of time series data?
What are examples of time series data?
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What are the two main purposes of time series analysis?
What are the two main purposes of time series analysis?
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Match the following time series patterns with their descriptions:
Match the following time series patterns with their descriptions:
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What are the two major categories of forecasting methods according to Makridakis et al.?
What are the two major categories of forecasting methods according to Makridakis et al.?
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The autocorrelation r1 tells us whether there is a correlation between yt and yt________
The autocorrelation r1 tells us whether there is a correlation between yt and yt________
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When k = 2, the sample autocovariance is given by c2 = ________ (yt − ȳ)(yt−2 − ȳ)
When k = 2, the sample autocovariance is given by c2 = ________ (yt − ȳ)(yt−2 − ȳ)
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The autocorrelations r1, r2, ..., rK form the ________ function (ACF)
The autocorrelations r1, r2, ..., rK form the ________ function (ACF)
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Plotting the autocorrelations is referred to as a ________ or an ACF plot
Plotting the autocorrelations is referred to as a ________ or an ACF plot
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When k = 0, the autocovariance is just equal to the ________ and the autocorrelation is equal to 1
When k = 0, the autocovariance is just equal to the ________ and the autocorrelation is equal to 1
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In R, we can use the ________ function to calculate the autocovariance and autocorrelation
In R, we can use the ________ function to calculate the autocovariance and autocorrelation
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The ACF plot is shown in ________ for 20 lags
The ACF plot is shown in ________ for 20 lags
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The autocovariance statistic is denoted by ________
The autocovariance statistic is denoted by ________
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The autocorrelation statistic r1 is equal to ______ for the given data.
The autocorrelation statistic r1 is equal to ______ for the given data.
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The autocovariance ______ up to lag 2 for the given data are 36.6, 31.7, and 25.3.
The autocovariance ______ up to lag 2 for the given data are 36.6, 31.7, and 25.3.
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The ______ statistic r2 is equal to 0.692 for the given data.
The ______ statistic r2 is equal to 0.692 for the given data.
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The ______ type = 'covariance' and type = 'correlation' methods were used to calculate the autocovariance and autocorrelation.
The ______ type = 'covariance' and type = 'correlation' methods were used to calculate the autocovariance and autocorrelation.
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The ______ method produced U-statistics that are more than 1 for both MA(3) and MA(5).
The ______ method produced U-statistics that are more than 1 for both MA(3) and MA(5).
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According to Makridakis et al., the two main categories of forecasting methods are ______.
According to Makridakis et al., the two main categories of forecasting methods are ______.
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The sample autocovariance for the kth lag is given by ck = 1/n * Σ(yt - ȳ)(yt-k - ȳ), where yt is the observation of the time series at time t and ȳ is the ______.
The sample autocovariance for the kth lag is given by ck = 1/n * Σ(yt - ȳ)(yt-k - ȳ), where yt is the observation of the time series at time t and ȳ is the ______.
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The sample autocorrelation for the kth lag is given by rk = Σ(yt - ȳ)(yt-k - ȳ) / Σ(yt - ȳ)^2, which can be interpreted as the ______ between yt and yt-k.
The sample autocorrelation for the kth lag is given by rk = Σ(yt - ȳ)(yt-k - ȳ) / Σ(yt - ȳ)^2, which can be interpreted as the ______ between yt and yt-k.
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The autocovariance and autocorrelation are two important ______ that are often used to summarise time series data.
The autocovariance and autocorrelation are two important ______ that are often used to summarise time series data.
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Yt-1 is described as “lagged” by one ______, yt-2 as “lagged” by two ______, and so on.
Yt-1 is described as “lagged” by one ______, yt-2 as “lagged” by two ______, and so on.
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The observations yt-1, yt-2, etc. are described as “lagged” by one, two, etc. ______.
The observations yt-1, yt-2, etc. are described as “lagged” by one, two, etc. ______.
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The sample autocovariance and autocorrelation for one lag are respectively given by c1 and ______.
The sample autocovariance and autocorrelation for one lag are respectively given by c1 and ______.
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The sample autocorrelation for the kth lag is given by rk = Σ(yt - ȳ)(yt-k - ȳ) / Σ(yt - ȳ)^2, where the sum is taken from t = ______ to n.
The sample autocorrelation for the kth lag is given by rk = Σ(yt - ȳ)(yt-k - ȳ) / Σ(yt - ȳ)^2, where the sum is taken from t = ______ to n.
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The formula for calculating the sample autocovariance ck is similar to the formula for calculating the ______ and correlation coefficient for cross-sectional data.
The formula for calculating the sample autocovariance ck is similar to the formula for calculating the ______ and correlation coefficient for cross-sectional data.
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Study Notes
Introduction to Time Series Analysis
- Time series analysis is a statistical methodology that deals with time series data, which are data points indexed in time order.
- Time series data are prevalent in fields such as economics, finance, and medicine.
- Forecasting is essential to decision makers in various fields, and many forecasting techniques have been developed over the centuries.
Applications of Time Series Data
- Examples of time series data:
- Business: weekly interest rates, daily closing stock prices, monthly price indices, yearly sales figures
- Meteorology: daily high and low temperatures, annual precipitation and drought indices, hourly wind speeds
- Agriculture: annual figures for crop and livestock production, soil erosion, and export sales
- Biology: electrical activity of the heart at millisecond intervals
- Ecology: abundance of an animal species
Purposes of Time Series Analysis
- Two main purposes of time series analysis:
- Understanding or modeling the stochastic mechanism that gives rise to an observed series
- Forecasting the future values of a series based on the history of that series and possibly other related series or factors
Types of Time Series Data
- Four types of time series data:
- Time series with increasing trend and seasonal behavior (e.g., Australian electricity data)
- Time series with decreasing trend (e.g., US Treasury bill contracts)
- Time series with constant trend and random spikes (e.g., Sales of product C)
- Time series with increasing trend, seasonal behavior, and cyclical pattern (e.g., Australian clay brick data)
Steps in a Forecasting Task
- Five basic steps in a forecasting task:
- Problem definition: Define the problem and determine what needs to be forecasted
- Gathering information: Obtain past data and other relevant information
- Preliminary analysis: Analyze the data to identify patterns and trends
- Choosing and fitting models: Choose an appropriate forecasting model and fit it to the data
- Using and evaluating a forecasting model: Use the model to make forecasts and evaluate its performance
Basic Forecasting Tools
- Four example data sets used to illustrate time series analysis methodology:
- Cross-sectional customer survey data
- Monthly global oil prices (in USD)
- Daily stock prices (in ZAR)
- Australian beer production (in megalitres)
Graphical Summaries
- Types of graphical summaries:
- Time plots and time series patterns
- Seasonal plots
- Visualizations for cross-sectional data (scatter plots, histograms, bar plots)
Typical Time Series Patterns
- Five typical time series patterns:
- Horizontal (H) pattern: Fluctuates horizontally around a constant mean
- Seasonal (S) pattern: Influenced by seasonal factors (e.g., quarterly, monthly, or daily)
- Cyclical (C) pattern: Exhibits rises and falls that are not of a fixed period
- Trend (T) pattern: Exhibits a long-term increase or decrease
- Irregular (E) component: No identifiable pattern### Time Series Analysis
- Time series data is a sequence of data points measured at regular time intervals.
- Characteristics of time series data:
- Horizontal pattern: a consistent pattern over time.
- Seasonal pattern: a pattern that repeats at fixed intervals (e.g., daily, weekly, monthly).
- Cyclical pattern: a long-term pattern that is not seasonal.
- Trend pattern: a persistent upward or downward movement over time.
- Random pattern: a pattern that is unpredictable.
Autocovariance and Autocorrelation
- Autocovariance (ck) measures the covariance between a time series and its lagged values:
- ck = Σ(yt - ȳ)(yt-k - ȳ) / n, where ȳ is the mean of the time series.
- Autocorrelation (rk) measures the correlation between a time series and its lagged values:
- rk = ck / Σ(yt - ȳ)² / n.
Forecasting Methods
- Naïve Forecast 1 (NF1) method: uses the most recent observation as a forecast.
- Moving Average (MA) method: uses the average of past k observations as a forecast:
- ŷt+1 = (yt + yt-1 + ... + yt-k+1) / k.
Measuring Forecasting Accuracy
- Mean Error (ME): measures the average difference between forecasts and actual values.
- Mean Absolute Error (MAE): measures the average absolute difference between forecasts and actual values.
- Mean Square Error (MSE): measures the average squared difference between forecasts and actual values.
- Mean Percentage Error (MPE): measures the average percentage difference between forecasts and actual values.
- Mean Absolute Percentage Error (MAPE): measures the average absolute percentage difference between forecasts and actual values.
Theil's U-Statistic
- Measures the goodness-of-fit of a forecasting model compared to the NF1 method:
- U < 1: the model is better than the NF1 method.
- U = 1: the model is as good as the NF1 method.
- U > 1: the NF1 method is better than the model.
Key Formulas
- Autocovariance: ck = Σ(yt - ȳ)(yt-k - ȳ) / n
- Autocorrelation: rk = ck / Σ(yt - ȳ)² / n
- Moving Average: ŷt+1 = (yt + yt-1 + ... + yt-k+1) / k
- Mean Error: ME = Σ(yt - ŷt) / n
- Mean Absolute Error: MAE = Σ|yt - ŷt| / n
- Mean Square Error: MSE = Σ(yt - ŷt)² / n
- Mean Percentage Error: MPE = Σ(yt - ŷt) / yt / n
- Mean Absolute Percentage Error: MAPE = Σ|yt - ŷt| / yt / n
- Theil's U-Statistic: U = Σ(ŷt+1 - yt+1)² / Σ(yt+1 - yt)² / n
Autocorrelation and Autocovariance
- Autocorrelation measures the correlation between a time series
ytand its lagged valuesyt-k. - The sample autocovariance for the kth lag is calculated as
ck = (1/n) * Σ[(yt - ȳ) * (yt-k - ȳ)], whereytis the observation at timet,ȳis the mean, andnis the number of observations. - The sample autocorrelation for the kth lag is calculated as
rk = (Σ[(yt - ȳ) * (yt-k - ȳ)]) / (Σ[(yt - ȳ)2]), whererkcan be interpreted as the correlation betweenytandyt-k.
Autocorrelation Function (ACF)
- The autocorrelations
r1,r2, ...,rKform the autocorrelation function (ACF). - The ACF can be plotted to visualize the autocorrelations, known as a correlogram or ACF plot.
- In R, the ACF can be easily calculated using the
acf()function.
Example Calculations
- For the beer data, the autocovariances and autocorrelations were calculated using R, resulting in
c1 = 158.81,c2 = 21.36,r1 = 0.421, andr2 = 0.057. - The ACF plot for the beer data is shown in Figure 8.
Interpretation of Autocorrelation
- When
k = 0, the autocovariance is equal to the mean squared deviation (MSD), and the autocorrelation is equal to 1. - The autocorrelation
rkcan be interpreted as the correlation betweenytandyt-k. - Autocorrelation is used to summarize time series data and plays a similar role to covariance and correlation coefficient for cross-sectional data.
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Learn about time series analysis, a statistical methodology for data indexed in time order, and its applications in fields like economics, finance, and medicine.
