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Introduction to Statistics

III - Time Series

Definition: data for a variable at different points in time.

Simple Example of relating two time series:

  Time 1 Time 2 Time 3 Time 4 Time 5
Time Series (TS) I: 7 8 5 4 6
Time Series (TS) II: 2 3 0 -1 1

Time Series Plot

TS I: Mean = 6
TS II: Mean = 1

TS I Standard Deviation:

S = Square root of ( 1/(N-1) ) [Sum of (each data set - mean)2]

S = Square root of 1/4 (7-6)2 + (8-6)2 + (5-6)2 + (4-6)2 + (6-6)2

S = Square root of 1/4 (1 + 4 + 1 + 4 + 0)

S = Square root of 1/4 (10)

S = Square root of 10/4

S = Square root of 2.5

S ~ 1.58

TS II Standard Deviation:

S = Square root of 1/4 (1 + 4 + 1 + 4 + 0)

S = Square root of 10/4

S = Square root of 2.5

S ~ 1.58

Visually, it's obvious that these two time series are related. We can make this even more obvious by plotting one time series versus the other in a "scatter plot":

Scatter Plot

The two time series are linearly related. For example all the points in the scatter plot lie in a nice straight line. How do you express this relationship with numbers?

IV - Linear Correlation Coefficient

The statistical definition of "relatedness" of two time series is called correlation. We can calculate a "correlation coefficient" r that is a measure of how two time series are related. If r = 1, the two series are perfectly positively correlated, which means that as one variable gets larger, the other one does too. If r = -1, the two time series are perfectly negatively correlated, which means that as one variable gets larger the other one gets smaller. If r = 0, then the two variables are not related.

How do you calculate a correlation coefficient?

r = sum ( each time period of (It - Imean) (IIt - IImean) ) / ( (N-1)*(SI)*(SII) )

Where It is the value of Time Series I at time equals t and Imean is the mean of Time Series I.

Example using TS I and TS II:

r = [((7-6)(2-1) + (8-6)(3-1) + (5-6)(0-1) + (4-6)(-1- 1) + 0) ] / [ (5-1)(1.58)(1.58) ]

r = (1 + 4 + 1 + 4 + 0) / (4 * 1.58 * 1.58)

r = 10/10

r = 1

Therefore, the two time series are perfectly positively correlated. But in real life, r is almost never 1, -1, or 0. In the next section we learn how to interpret the significance of r in a real-life situation.

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