WFM 5201: Data Management and Statistical Analysis

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WFM 5201 Data Management and Statistical Analysis
Lecture-6 Correlation and Regression Analysis

• Akm Saiful Islam

Institute of Water and Flood Management
(IWFM) Bangladesh University of Engineering and
Technology (BUET)
June, 2008
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Correlation

• Correlation is concerned with describing the
  direction (positive or negative) and strength of
  a relationship between two variables.
• Correlation makes no distinction between the two
  variables (it is a measure of how they vary
  jointly), whereas regression theory depends on a
  dependent variable being affected by an
  error-free independent variable.

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Correlation coefficient

• The direction and strength of the relationship
  can be expressed by means of a correlation
  coefficient r, which is mathematically defined
  as
• The sum of cross products of deviations

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Correlation coefficient

• The sum of squared deviations for X
• The sum of squared deviations for Y

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Pearsons r
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Correlation coefficient

• A correlation coefficient varies from -1 to 1
• -1 indicating a perfect negative relationship
  (one increase while other decrease),
• 0 indicating no relationship
• 1 indicating a perfect positive relationship.
• The size of the correlation indicates the
  strength of the relationship for example, the
  correlation coefficient -0.89 indicates a
  stronger relationship than a coefficient of 0.60.

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Linear Regression

• Regression is primarily concerned with using the
  relationship for the purpose of predicting one
  variable from knowledge of the other
• Correlation, on the other hand, is primarily
  concerned with discovering whether or not a
  relationship exists in the first place, and then
  specifying the strength and direction of this
  relationship.

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Linear Regression

• The simple linear regression equation is given
  as
• X given data
• b0 intercept of regression line
• b1 slope of regression line

It is also known as least squares method
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Regression line
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Coefficient of Regression
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Coefficient of Determination

• The decomposition of the sample variation of
  leads to a measure of the "goodness of fit",
  which is known as the coefficient of
  determination and denoted by R2.

Note
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Coefficient of determination

• is a measure commonly used to describe how well
  the sample regression line fits the observed
  data.
• Range
• 0 means poorest , 1 best fit of regression model

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Exercise-1 Fit regression equation between Boro
production and rainfall and find R2
Year Boro Production Rainfall
1975-76 424536 216
1976-77 152273 319
1977-78 437007 164
1978-79 278287 141
1979-80 417225 237
1980-81 500207 197
1981-82 395940 255
1982-83 418170 221
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Deviations or Errors

• The sum of squares of these deviations from the
  fitted line is

Total Explained unexplained deviation
deviation deviation
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Total, explained, and unexplained deviation
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Regression diagnostics

• Patterns for residual plots (a) satisfactory (b)
  funnel, (c) double bow (d) non-linear