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Correlation and Regression



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Chapter 9 : Correlation and Regression



Correlation arrow_upward


  • A correlation is a single number that describes the degree of relationship between two variables.
  • It is often measured with Pearson’s Correlation Coefficient and it is represented with the letter r.

  • Pearson’s Correlation Coefficient Formula arrow_upward


  • Where, n equals the number of values.
  • In this formula:

  • Multiply  by  and sum

    Sum all the  scores

    Sum all the  scores

    Square all  and sum

    Square all  and sum

    Square the sum of

    Square the sum of



    Types of Correlation arrow_upward


  • The different types of correlation are:
    • Positive Correlation
    • Negative Correlation
    • Linear Correlation
    • Non Linear Correlation
    • Simple Correlation
    • Multiple Correlation
    • Partial Correlation

    Positive Correlation arrow_upward


  • When the values of two variables x and y move in the same direction, the correlation is said to be positive.
  • That is in positive correlation, when there is an increase in x, there will be an increase in y also. Similarly when there is a decrease in x, there will be a decrease in y also.

  • Negative Correlation arrow_upward


  • When the values of two variables x and y move in opposite direction, we say correlation is negative.
  • That is in negative correlation, when there is an increase in x, there will be a decrease in y. Similarly when there is a decrease in x, there will be an increase in y.

  • Linear Correlation arrow_upward


  • When the change in one variable results in the constant change in the other variable, we say the correlation is linear.
  • When there is a linear correlation, the points plotted will be in a straight line.


  • Non-linear Correlation arrow_upward


  • When the amount of change in one variable is not in a constant ratio to the change in the other variable, we say that the correlation is non-linear.

  • Simple Correlation arrow_upward


  • If there are only two variable under study, the correlation is said to be simple.

  • Multiple Correlations arrow_upward


  • When one variable is related to a number of other variables, the correlation is not simple. It is multiple if there is one variable on one side and a set of variables on the other side.
  • Example:
  • Relationship between yield with both rainfall and fertilizer together is multiple correlations.

  • Partial Correlation arrow_upward


  • The correlation is partial if we study the relationship between two variables keeping all other variables constant.
  • Example:
  • The relationship between yield and rainfall at a constant temperature is partial correlation.

  • Regression arrow_upward


  • A statistical measure that attempts to determine the strength of the relationship between one dependent variable (usually denoted by Y) and a series of other changing variables (known as independent variables).

  • Types of Regression arrow_upward



    Linear Regression:

  • It analyzes the relationship between two variables: x and y.
    • y = a + bx


    Exponential Regression:

  • It takes the input signal and fits an exponential function to it
  • where, t is the variable along the x-axis.

  • Quadratic Regression:

  • A quadratic function is a function f (x) of the form   for fixed constants a, b, and c.

  • Simple Linear Regression arrow_upward


  • Simple Linear Regression of one dependent variable (Y) and one independent variable (X).
  • The model is:
  •     Where,

        y= Values of the dependent variable

        x = Values of the independent variable

       a, b= “Regression coefficients” (what we want to find)

         = Residual or error


    Coefficient of Determination arrow_upward


  • The Coefficient of Determination  is the proportion of the total variation in the dependent variable  that is explained or accounted for by the variation in the independent variable
    • It is the square of the coefficient of correlation.
    • It ranges from 0 to 1.
    • It does not give any information about the direction of the relationship between the variables.


    Regression: How good is the Fit? arrow_upward


  • A line of best fit is a straight line that best represents the data on a scatter plot.
  • We measure the fit with the coefficient of determination, r2
  • r2 is the proportion of variation in Y explained by the regression.
    • Values range from 0 to 1
    • 0 indicates no relationship, 1 indicates perfect relationship.
  • Find individual residuals or errors
  • Then, the sum of all the residual is
  •  Observed value of the dependent variable for the ith observation.

     Estimated value of the dependent variable for the ith observation.


    Slope for the Estimated Regression Equation arrow_upward


  • The slope of the regression line is calculated by this formula:
  • Where,

    x = Value of independent variables

    y = Value of dependent variables


    Y- Intercept for the Estimated Regression Equation arrow_upward


    Where,

     = Mean value for independent variable

    = Mean value for dependent variable


    Good Regression Fit arrow_upward


  • Most of the points lie on the line:


  • Thank You from Kimavi arrow_upward


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