Estatstica e Modelos Probabilsticos - COE241 Aula passada Aula de - - PowerPoint PPT Presentation
Estatstica e Modelos Probabilsticos - COE241 Aula passada Aula de hoje Goodness of fit: v.a. Regresso linear discreta Goodness of fit: v.a. contnua Kolmogorov- Smirnov Rosa Leo 2013 Scatter Diagrams A scatter plot scatter
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Aula de hoje Regressão linear Aula passada Goodness of fit: v.a. discreta Goodness of fit: v.a. contínua Kolmogorov- Smirnov
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X n−1
2
=n−1S
2
0
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A scatter plot scatter plot is a graph that may be used to represent the relationship between two variables. Also referred to as a scatter scatter diagram diagram.
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A dependent variable dependent variable is the variable to be predicted or explained in a regression model. This variable is assumed to be functionally related to the independent variable.
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An independent variable independent variable is the variable related to the dependent variable in a regression equation. The independent variable is used in a regression model to estimate the value
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X Y
(a) Linear (a) Linear
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X Y
(b) Linear (b) Linear
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X Y
(c) Curvilinear (c) Curvilinear
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X Y
(d) Curvilinear (d) Curvilinear
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(e) No (e) No Relationship Relationship
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The correlation coefficient correlation coefficient is a quantitative measure of the strength
from + 1.0 to - 1.0. A correlation of ± 1.0 indicates a perfect linear relationship, whereas a correlation of 0 indicates no linear relationship.
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CORRELATION COEFFICIENT CORRELATION COEFFICIENT
where:
ρ = correlation coefficient σx = standard deviation of X σy = standard deviation of Y
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SAMPLE CORRELATION COEFFICIENT SAMPLE CORRELATION COEFFICIENT
where:
r = Sample correlation coefficient n = Sample size x = Value of the independent variable y = Value of the dependent variable
2 2
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Simple linear regression analysis Simple linear regression analysis analyzes the linear relationship that exists between a dependent variable and a single independent variable.
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SIMPLE LINEAR REGRESSION MODEL SIMPLE LINEAR REGRESSION MODEL
where: y = Value of the dependent variable x = Value of the independent variable a = Population’s y-intercept b = Slope of the population regression line = Error term, or residual
ε
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REGRESSION COEFFICIENTS REGRESSION COEFFICIENTS In the simple regression model, there are two coefficients: the intercept and the slope.
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The interpretation of the regression regression slope coefficient slope coefficient is that it gives the average change in the dependent variable for a unit increase in the independent variable. The slope coefficient may be positive or negative, depending on the relationship between the two variables.
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The least squares criterion least squares criterion is used for determining a regression line that minimizes the sum of squared residuals.
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A residual
residual is the difference between the actual value of the dependent variable and the value predicted by the regression model.
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X Y
4
300 20 100 400
x y 60 150 ˆ + =
Years with Company Sales in Thousands 390 390 312 312
Residual = 312 - 390 = -78