14.1 Time Series Patterns

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected—time. Because each date is paired with the temperature reading for that day, we do not have to think of the data as being random. Instead, we can use the times given to impose a chronological order on the data.

A time series is a set of observations for a particular variable taken at regular intervals of time or over successive periods of time. The interval in a time series may be taken every second, every minute, every hour, every day, every week, every month, every year, or at any other regular time interval. The data in a time series must be indexed in chronological order in order to spot patterns or trends that occur in the data over time. If these trends or patterns continue into the future, we can use the past pattern to create a model and make predictions about what will happen in the future.

Time Series Graphs

A time series graph, also called a time series plot, is a useful tool to help us identify any underlying patterns in the time series data. A time series graph is a graphical presentation of the relationship between time (in chronological order) and the time series variable.   The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

Components of a Time Series

Time series forecasting models use historical time series data to develop a forecast of the future. Although most models can be applied to any time series data, the choice of model for a particular time series depends on the patterns or components present in the time series. Some models are better at capturing certain components than other models. To get the most accurate predictions, we want to select a model that is good at modelling the components present in the given time series. By analyzing the graph of a time series, we can often identify the components present in the time series, allowing us to break down a complex time series into more manageable parts in order to build better models and forecasts. Time series data typically exhibits one or more of four possible components that help us understand the underlying patterns present in a time series.

Trend Pattern

A trend pattern is the long-term movement or general direction of the data over a period of time. A trend could be upward, downward, or a flat trend and is generally the result of long-term factors or influences on the data. For example, in a time series for the price of a stock the overall increase or decrease in value of the stock over several years is considered a trend. Often, we look for linear trends in the data, and when a linear trend is present, we can use a trendline or simple linear regression model to model the data. But trends do not have to follow a linear model. Other examples of trends in a time series include quadratic trends or exponential trends.

Seasonality or Seasonal Patterns

Seasonal patterns are repeated patterns that occur over a one-year period due to seasonal influences. Repeated and predictable highs and lows in the time series data that occur at the same time each year indicate seasonality in the data. For example, retail sales experience predictable highs during the holiday season every year, which creates a seasonal pattern.

Cycles or Cyclic Patterns

Unlike seasonality, which occurs at regular intervals, cyclic patterns occur at irregular intervals and are often influenced by multi-year economic or business cycles. These fluctuations are much longer in duration, typically spanning multiple years. Because of the extended time frame required for cycles, identifying a cyclic pattern in a time series is challenging.

Irregular Variation or Random Fluctuations

Random variation or irregular components in a time series are unpredictable or unforeseen changes in the data that cannot be attributed to any of the other components. Random fluctuations are often unexplained, reflecting the inherent randomness and unpredictability in the system.

Exercises

1. For each of the following time series, identify the components of the time series present in the data.
   1. 
   2. 
   3. 
   Click to see Answer

   1. seasonal effects, random variation
   2. trend, random variation
   3. seasonal effects, trend, random variation

    

2. The number of pizzas sold each week at a local pizza restaurant is given in the table below.
   

   Week	Number of Pizzas Sold
   1	120
   2	135
   3	140
   4	115
   5	130
   6	145
   7	160
   8	170
   9	125
   10	130
   11	150
   12	140
   13	155
   14	180
   15	165
   16	170
   17	190

   1. Create a time series graph for this data.
   2. What components of a time series are present in the time series?
   Click to see Answer

   1. 
   2. trend, random variation

    

3. The number of coffees sold each day at a local coffee shop is given in the table below.
   

   Day	Number of Coffees Sold
   1	120
   2	100
   3	115
   4	130
   5	110
   6	150
   7	165
   8	110
   9	120
   10	125
   11	135
   12	130
   13	152
   14	160
   15	110
   16	120
   17	125
   18	127
   19	130
   20	155
   21	162
   22	120
   23	125
   24	130
   25	135
   26	137
   27	160
   28	155
   29	130
   30	140
   31	145
   32	139
   33	142
   34	165
   35	163
   36	130
   37	120
   38	125
   39	135
   40	145
   41	160
   42	170
   43	130
   44	137
   45	130
   46	135
   47	132
   48	163
   49	168
   50	125
   51	130
   52	120
   53	115
   54	125
   55	156
   56	161

   1. Create a time series graph for this data.
   2. What components of a time series are present in the time series?
   Click to see Answer

   1. 
   2. seasonal effects, random variation

    

4. The quarterly demand, in [latex]1000[/latex]s, for a particular product is recorded in the table below.
   

   Quarter	Demand (in [latex]1000[/latex]s)
   1	50
   2	75
   3	85
   4	70
   5	60
   6	95
   7	100
   8	85
   9	70
   10	115
   11	125
   12	110
   13	80
   14	140
   15	145
   16	130
   17	90
   18	165
   19	175
   20	155

   1. Create a time series graph for this data.
   2. What components of a time series are present in the time series?
   Click to see Answer

   1. 
   2. trend, seasonal effects, random variation

    

Attribution

“2.2 Histograms, Frequency Polygons, and Time Series Graphs“ in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.