Time series graphics
MATH840 | Time Series
Ihor Miroshnychenko
Kyiv School of Economics
The seasonal period
| Quarters |
|
|
|
|
4 |
| Months |
|
|
|
|
12 |
| Weeks |
|
|
|
|
52 |
| Days |
|
|
|
7 |
365.25 |
| Hours |
|
|
24 |
168 |
8766 |
| Minutes |
|
60 |
1440 |
10080 |
525960 |
| Seconds |
60 |
3600 |
86400 |
604800 |
31557600 |
Graphics
Plots allow us to identify:
- Patterns
- Unusual observations
- Changes over time
- Relationships between variables
Time plots
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Time plots
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Time plots
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Time series patterns
- Trend: Long-term increase or decrease in data (not necessarily linear)
- Seasonal: Regular patterns at fixed, known intervals (yearly, monthly, weekly, daily)
- Cyclic: Rises and falls without fixed frequency, typically lasting 2+ years (often tied to economic conditions)
Time series patterns
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Seasonal plots
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Seasonal plots
- Data plotted against time within each seasonal period
- Useful for identifying seasonal patterns and anomalies
- Can be used for multiple seasonal periods
Multiple seasonal periods
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Multiple seasonal periods
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Multiple seasonal periods
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Seasonal subseries plots
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Seasonal subseries plots
- Data for each season collected together in time plot as separate time series
- Enables the underlying seasonal pattern to be seen clearly, and changes in seasonality over time to be visualized
Example: Australian holiday tourism
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Example: Australian holiday tourism
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Example: Australian holiday tourism
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Scatterplots
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Correlation
Measures the extent of linear relationship between two variables.
\[
r=\frac{\sum\left(x_t-\bar{x}\right)\left(y_t-\bar{y}\right)}{\sqrt{\sum\left(x_t-\bar{x}\right)^2} \sqrt{\sum\left(y_t-\bar{y}\right)^2}}
\]
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Scatterplot matrices
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Scatterplot matrices
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Lag plots
- Each graph shows \(y_t\) against \(y_{t-k}\) for lag (k).
- The autocorrelations are the correlations associated with these lag plots.
- \(r_1\) = correlation between \(y_t\) and \(y_{t-1}\).
- \(r_2\) = correlation between \(y_t\) and \(y_{t-2}\).
- And so on…
Autocorrelation
\[
r_k=\frac{\sum_{t=k+1}^T\left(y_t-\bar{y}\right)\left(y_{t-k}-\bar{y}\right)}{\sum_{t=1}^T\left(y_t-\bar{y}\right)^2},
\]
| Lag |
|
| 1 |
-0.052981 |
| 2 |
-0.758175 |
| 3 |
-0.026234 |
| 4 |
0.802205 |
| 5 |
-0.077471 |
| 6 |
-0.657451 |
| 7 |
0.001195 |
| 8 |
0.707254 |
| 9 |
-0.088756 |
Autocorrelation plots
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Trend and seasonality in ACF plots
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Trend and seasonality in ACF plots
- When the data have trend, ACF for small lags tend to be large and positive.
- When data are seasonal, ACF tends to be large and positive at seasonal lags (e.g., 12, 24 for monthly data with yearly seasonality).
- When data are trended and seasonal, ACF tends to be large and positive at small lags and seasonal lags.
White noise
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White noise ACF
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CI for ACF
- Sampling distribution of \(r_k\) for white noise data is asymptotically \(N(0, 1/T)\).
- Approximate 95% confidence interval for \(r_k\) is
\[
\pm \frac{1.96}{\sqrt{T}}
\]
where \(T\) is the length of the series.
