Time Series Analysis and Forecasting

Time Series Analysis and Forecasting

The Method of Free Hand Curve (2020)

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Introduction to Free Hand Curve

The Method of Free Hand Curve is a simple and non-mathematical technique used to fit a curve to a set of data points by visually estimating the trend. It provides a quick, approximate visualization of the relationship between variables.

It is used to

  • Plot Data Points: Scatter plot the data points on a graph.
  • Draw Curve: Using a smooth, freehand line, draw a curve that best represents the overall trend of the data points.
  • No Rigid Rules: The curve is drawn based on the analyst’s judgment, without strict mathematical formulas.

Exploring Data Patterns

This method is subjective and less precise than mathematical curve-fitting techniques but is useful for initial data exploration or when data patterns are irregular.

The secular trend is measured by the method of the free hand curve in the following steps:

  • Take the time periods along the $x$-axis by taking appropriate scaling
  • Plot the points for observed values of the $Y$ variable as the dependent variable against the given time periods
  • Join these plotted points by line segments to get a historigram
  • Draw a free-hand smooth curve (or a straight line) through the histogram

In this method we draw the given times series data on graph paper, then we draw a free-hand trend line through the plotted graph according to the trend of the graph. Then we read trend values from this free-hand trend line.

It is generally preferred to use a curve instead of a straight line to show the secular trend.

Merits (Free Hand Curve)

  • The free-hand curve method is simple, easy, and quick for measuring secular trends.
  • A well-fitted trend line (or curve) approximates the trend closely based on a mathematical model.

Demerits (Free Hand Curve)

  • It is a rough and crude method.
  • It is greatly affected by personal bias as different persons may fit different trends to the same data.
    The estimates are not reliable due to personal bias.

Practical Example of Curve

Question: The following time series shows the number of road accidents in Punjab from 1972 to 1978.

Year1972197319741975197619771978
No. of Accidents2493263826993038374540794688
  • Obtain the historigram showing the number of road accidents and a free-hand trend line by drawing a straight line
  • Find the trend values for this time series

Solution:

Method of Free Hand Curve
YearValueTotalMeanTrend value
19722493  2200
19732638  2550
19742699  2950
1975303823338033403340
19763745  3650
19774079  4050
19784688  4499

The usefulness of the Method of Free Hand Curve

The method of freehand curve is useful for:

  1. Exploratory Data Analysis (EDA): As a preliminary step free free-hand curve method helps us to understand the basic characteristics of the data and identify potential relationships between variables.
  2. Visual Communication: It also helps to present trends in the data in a clear and easily understandable way for non-statistical audiences.
  3. Limited Data: When you have a relatively small dataset, a free hand curve might be sufficient to get a basic idea of the central tendency.

Real-Life Examples of Free-Hand Curve

Some real-life examples where the Method of Free Hand Curve can be applied:

  • Economics:
    • Plotting and estimating trends in stock market prices over time.
    • Visualizing the relationship between inflation and unemployment rates.
  • Meteorology:
    • Drawing temperature trends over days or months based on scattered weather data.
    • Estimating rainfall patterns over a year.
  • Education:
    • Plotting student performance trends over semesters to identify improvements or declines.
    • Visualizing the relationship between study hours and exam scores.
  • Healthcare:
    • Tracking a patient’s blood pressure or glucose levels over time to observe trends.
    • Estimating the progression of a disease based on irregular data points.
  • Business:
    • Visualizing sales trends over months or years to identify seasonal patterns.
    • Estimating the relationship between advertising expenditure and revenue growth.
  • Sports:
    • Plotting an athlete’s performance metrics (e.g., running speed, scores) over time to observe trends.
    • Visualizing the relationship between training hours and performance improvement.

Summary

By understanding the method of free hand curves and its limitations, one can use it as a valuable tool for initial data exploration and visualization alongside other statistical techniques for a more robust analysis. The freehand curve provides a quick, intuitive way to understand trends or relationships without relying on complex mathematical models.

MCQs Intermediate Mathematics Part-I Quadratic Equations

The Secular Trend Example and Uses (2020)

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For the estimation of the secular trend of a time series, the most commonly used method is to fit a straight line $\hat{y} = a+bx$, an exponential curve $\hat{y}=ab^x$, and a second-degree parabola $\hat{y}=a +bx+ cx^2$, etc, where $y$ is the value of a time series variable, $x$ representing the time and all others are constants (the intercept $a$, and the slope $b$). The method of least squares is a widely used method to determine the values of the constants appearing in such an equation.

The Secular Trend is used

  • For prediction (or projection) into the future
  • The detrending process (removal of trend) in a time series for studying other non-trend fluctuations.
  • It is used for historical description

The secular trend can be represented either by a straight line or by some type of smooth curve. It is measured by the following methods:

Least Squares Method (secular trend)

The secular trend may be used in determining how a time series has grown in the past or in making a forecast. The trend line is used to adjust a series to eliminate the effect of the secular trend to isolate non-trend fluctuations.

Note that

  • These trends can be positive or negative. For example, the advancement of technology offers new opportunities but also raises concerns about job displacement and privacy.
  • These trends can be interrelated. For instance, urbanization might be fueled by technological advancements that allow people to work remotely.
  • Identifying secular trends can be challenging, as they unfold over a long period. However, by analyzing historical data, monitoring current developments, and considering expert opinions, one can gain valuable insights into the long-term direction of change.

By understanding and utilizing secular trends, individuals, businesses, and policymakers can make informed decisions, prepare for future challenges, and capitalize on emerging opportunities in a constantly evolving world.

Time Series Data Analysis

R Programming Language

Coding Time Variable (2020)

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Coding Time Variable by Taking Origin at the Beginning

Suppose we have time-series data for the years 1990, 1991, 1992, and 1994.

We can take the origin of a time series at the beginning and assign $x = 0$ to the first period and $1, 2, 3, …$ to other periods. The code for the year will be

Coding Time Variable

Coding Time Variable by Taking Middle Years as Zero

To simplify the trend calculations, the time variable $t$ (year variable) is coded by taking deviations $t-\overline{t}$, where $\overline{t}$ is the average number computed as $\overline{t}=\frac{First\, Period + Last\, Period}{2}$. Taking $x=t-\overline{t}$ we get
$$\sum x = 0 = \sum x^3 = \sum x^5 = \cdots$$

There are two cases when coding a Time Variable (when taking zero in the Middle):

  • When there are an odd number of Years:
    For an odd number of years (as in the period 1990 to 1994) the $\overline{t}$ is the middle point. The $\overline{t}$ is $\overline{t} = (1990+1994)/2=1992$ the code for the year $t$ is $x=t-\overline{t}$. For t=1990, we have $x=1990-1992 =0$. Thus the coded year is zero at $\overline{t}$. Now after taking x=0 at the middle of an odd number of years, we assign $-1, -2, …$ for the years before the middle of the year and $1,2,…$ for the years after the middle year.
    Year (t) $x=t-\overline{t}$
    1990 -2
    1991 -1
    1992 0
    1993 1
    1994 2
  • When there are even numbers of years
    Suppose we have time-series data for the years 1990, 1991, 1992, 1993, 1994, and 1995. The value of middle point is $\overline{t} = (1990+1995)/2 = 1992.5$. So $x=0$ halfway between the years 1992 and 1993 (in the middle of 1992 and 1993). For $t=1992$, we have $x=t-\overline{t}=1992-1992.5=-0.5$. Thus coding the middle of an even number of years as $x=0$, we assign $-0.5, -1.5, -2.5, …$ for the years before the middle year and $0.5, 1.5, 2.5, …$ for the years after the middle year as shown below
Year(t)$x=t-\overline{t}$$x=\frac{t-\overline{t}}{1/2}$
1990-2.5-5
1991-1.5-3
1992-0.51
19930.51
19941.53
19952.55
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To avoid decimals in the coded year, we can take the unit of measurement as $\frac{1}{2}$ year. Therefore, after coding $x=0$ in the middle of an even number of years, we assign $-1,-3, -5,…$ for the year before the middle year and $1,3,5,…$ for the years after the middle year as shown above.

Read more about Coding Time Variables in R

R Programming Language

Computer MCQs