Data Visualization

# Data Visualization
## Introduction to Time Series
### Dr Thiyanga Talagala
### 2020-03-12

---

## Time series

- A time series is a sequence of observations taken sequentially in time.

## Time series data vs Cross sectional data

- Time series data

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Year </th>
   <th style="text-align:left;"> Values </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 2012 </td>
   <td style="text-align:left;"> 120 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2013 </td>
   <td style="text-align:left;"> 122 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2014 </td>
   <td style="text-align:left;"> 140 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2015 </td>
   <td style="text-align:left;"> 150 </td>
  </tr>
</tbody>
</table>

- a set of observations, along with some information about what times those observations were recorded.

- usually discrete and equally spaced time intervals.

]

- Cross-sectional data

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> ID </th>
   <th style="text-align:left;"> Values </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> 200 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> 350 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:left;"> 480 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:left;"> 250 </td>
  </tr>
</tbody>
</table>

- observations that come from different individuals or groups at a single point in time.

]
---

## Deterministic  vs Non-deterministic time series

.pull-left[
- **Deterministic time series:** future values can be exactly determined by using some mathematical function.

![](lecture5ts_files/figure-html/unnamed-chunk-3-1.png)

`$$y_t = cos(2\pi t)$$`

]

.pull-right[
- **Non-deterministic time series:** future values can be determined only in terms of a probability distribution.

![](lecture5ts_files/figure-html/unnamed-chunk-4-1.png)

]
---

## Frequency of a time series: Seasonal periods

- Frequency: number of observation per natural time interval of measurement (usually year,  but sometimes a week, a day or an hour)

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Data </th>
   <th style="text-align:left;"> Frequency </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Annual </td>
   <td style="text-align:left;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Quarterly </td>
   <td style="text-align:left;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Monthly </td>
   <td style="text-align:left;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Weekly </td>
   <td style="text-align:left;"> 52 or 52.18 </td>
  </tr>
</tbody>
</table>

- Multiple frequency setting

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Data </th>
   <th style="text-align:left;"> Minute </th>
   <th style="text-align:left;"> Hour </th>
   <th style="text-align:left;"> Day </th>
   <th style="text-align:left;"> Week </th>
   <th style="text-align:left;"> Year </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Daily </td>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;"> 7 </td>
   <td style="text-align:left;"> 365.25 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Hourly </td>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;"> 24 </td>
   <td style="text-align:left;"> 168 </td>
   <td style="text-align:left;"> 8766 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Half-Hourly </td>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;"> 48 </td>
   <td style="text-align:left;"> 336 </td>
   <td style="text-align:left;"> 17532 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Minutes </td>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;"> 60 </td>
   <td style="text-align:left;"> 1440 </td>
   <td style="text-align:left;"> 10080 </td>
   <td style="text-align:left;"> 525960 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Seconds </td>
   <td style="text-align:left;"> 60 </td>
   <td style="text-align:left;"> 3600 </td>
   <td style="text-align:left;"> 86400 </td>
   <td style="text-align:left;"> 604800 </td>
   <td style="text-align:left;"> 31557600 </td>
  </tr>
</tbody>
</table>

---
.pull-left[

## Monthly time series

![](lecture5ts_files/figure-html/unnamed-chunk-7-1.png)

- Length of the series: 72

- Monthly seasonality

]

## Half-hourly Time Series

![](lecture5ts_files/figure-html/unnamed-chunk-8-1.png)

- Length of the series: 4032

- Daily seasonality and weekly seasonality

]

--
Note: Monthly seasonality with high-frequency data (daily, hourly, etc.) is tricky due to variable month lengths. You can't specify that using seasonal periods. It could be possibly handled using a dummy variable.

---

# Your turn

- What are the frequencies for a monthly time series with semi-annual and annual pattern?

---

# `ts` object in R

- Annual time series

```r
y <- ts(c(120, 122, 140, 150), start=2012)
y
```

```
Time Series:
Start = 2012 
End = 2015 
Frequency = 1 
[1] 120 122 140 150
```

---

# `ts` object in R

- Quarterly time series

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Quarter </th>
   <th style="text-align:left;"> Values </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 2012-Q1 </td>
   <td style="text-align:left;"> 120 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-Q2 </td>
   <td style="text-align:left;"> 122 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-Q3 </td>
   <td style="text-align:left;"> 140 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-Q4 </td>
   <td style="text-align:left;"> 150 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2013-Q1 </td>
   <td style="text-align:left;"> 200 </td>
  </tr>
</tbody>
</table>

```r
y <- ts(c(120, 122, 140, 150, 200), start=c(2012, 1), frequency = 4)
y
```

```
     Qtr1 Qtr2 Qtr3 Qtr4
2012  120  122  140  150
2013  200               
```

---
# `ts` object in R

- Monthly time series

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Month </th>
   <th style="text-align:left;"> Values </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 2012-Jan </td>
   <td style="text-align:left;"> 120 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-Feb </td>
   <td style="text-align:left;"> 122 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-March </td>
   <td style="text-align:left;"> 140 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-April </td>
   <td style="text-align:left;"> 150 </td>
  </tr>
</tbody>
</table>

```r
y <- ts(c(120, 122, 140, 150), start=c(2012, 1), frequency = 12)
y
```

```
     Jan Feb Mar Apr
2012 120 122 140 150
```

---
# `ts` object in R

- Weekly time series

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Week </th>
   <th style="text-align:left;"> Values </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 2012-W1 </td>
   <td style="text-align:left;"> 120 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-W2 </td>
   <td style="text-align:left;"> 122 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-W3 </td>
   <td style="text-align:left;"> 140 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2012-W4 </td>
   <td style="text-align:left;"> 150 </td>
  </tr>
</tbody>
</table>

```r
y <- ts(c(120, 122, 140, 150), start=c(2012, 1), frequency = 52)
y
```

```
Time Series:
Start = c(2012, 1) 
End = c(2012, 4) 
Frequency = 52 
[1] 120 122 140 150
```

---

## Time series plots

```r
y <- ts(c(120, 122, 140, 150, 200, 250), start=2012)
y
```

```
Time Series:
Start = 2012 
End = 2017 
Frequency = 1 
[1] 120 122 140 150 200 250
```

```r
class(y)
```

```
[1] "ts"
```

]

```r
autoplot(y)
```

![](lecture5ts_files/figure-html/unnamed-chunk-19-1.png)

]

---

## Add title and labels

```r
autoplot(y)
```

![](lecture5ts_files/figure-html/unnamed-chunk-20-1.png)

]

```r
autoplot(y)+ylab("Number of sales")+
  xlab("Year")+
  ggtitle("Time series plot of sales from 2012 to 2017")
```

![](lecture5ts_files/figure-html/unnamed-chunk-21-1.png)

]

---

## Your turn

Create plots of the following time series: dengue counts in Gampaha (Use `mozzie` package), a10 series (`fpp2` package).

Use help() to find out about the data in each series.

Modify the axes labels and title.

---

#### Time series patterns

**Trend**

- Long-term increase or decrease in the data.

**Seasonal**

- A seasonal pattern exists when a series is influenced by seasonal factors (e.g.,
the quarter of the year, the month, or day of the week). Seasonality is always
of a **fixed** and **known period**. Hence, seasonal time series are sometimes called
periodic time series.

- Period is unchanging and associated with some aspect of the
calendar.

**Cyclic**

- A cyclic pattern exists when data exhibit rises and falls that are not of fixed
period. The duration of these fluctuations is usually of at least 2 years.

In general,

- the average length of cycles is longer than the length of a seasonal pattern.

- the magnitude of cycles tends to be more variable than the magnitude of seasonal patterns.
  
---

## Cyclic pattern

![](lecture5ts_files/figure-html/unnamed-chunk-22-1.png)
  
---

## Cyclic and seasonal pattern

![](lecture5ts_files/figure-html/unnamed-chunk-23-1.png)

---

## Multiple seasonal pattern

![](lecture5ts_files/figure-html/unnamed-chunk-24-1.png)
  
---

## Seasonal pattern

![](lecture5ts_files/figure-html/unnamed-chunk-25-1.png)
---

## Trend

![](lecture5ts_files/figure-html/unnamed-chunk-26-1.png)

---

## Trend and seasonal

![](lecture5ts_files/figure-html/unnamed-chunk-27-1.png)