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Modeling Nonlinear Data

Modeling Nonlinear Data. Section 4.1. Linear, Exponential, Power. y = a + bx Linear model: y changes by common difference for equal increments of change in x. y = ab x Exponential model: y changes by a common ratio for equal increments of change in x. y = ax b

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Modeling Nonlinear Data

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  1. Modeling Nonlinear Data Section 4.1

  2. Linear, Exponential, Power • y = a + bx • Linear model: y changes by common difference for equal increments of change in x. • y = abx • Exponential model: y changes by a common ratio for equal increments of change in x. • y = axb • Power Model: Power relationship between x and y

  3. Properties of Logs/Exponents • logbx = y if by = x • log(AB) = logA + logB • log(A/B) = logA – logB • logXp = p logX • 10log x = x • (xa)(xb) = xa+b

  4. Exponential Regression y = abx • Plot data points and check scatterplot, check correlation and residuals, not linear? • Check if common ratio. • Transform data into linear by using log y. • Check scatterplot, linear? • LSRL: log ŷ = a + bx • Undo log (remember ŷ = log ŷ ). • Use equation to make predictions.

  5. Linear or Exponential?

  6. Ex: Year vs Crude Oil Production

  7. Do Ratio Test

  8. Transform to Linear (log y)

  9. Year vs. log(Oil Production)

  10. LSRL

  11. Perform Inverse Transformation

  12. Check Residuals

  13. Example: Exponential Growth of a Savings Account; Principle - $2, rate – 6% per yr compounded monthly, time = 3 years. x = time; y = account balance x months 0, 12, 24, …, 360 (30 years) y = 2(1.005)x • Data in lists: L1 (x), L2 (y). • Check scatterplot, correlation, residuals, common difference or ratio, linear? • Make L3 (log y). • Scatterplot of L1 andL3,linear? Do LSRL. • Undo Log (remember ŷ = log y). • Use exponential model for predictions.

  14. Power Regression y = axb • One quantity is proportional to the second quantity raised to a power. • All pass through the origin. • A relationship between height and weight seem to follow this model. • Log(x) and log(y) • LSRL on log(x), log(y) • Undo transformation

  15. Example:

  16. 1. Plot length vs weight, linear? Power regression make sense?

  17. 2. Take log of both x and y and plot, linear? 3. Find the LSRL: log(weight) = -1.8994 + 3.0494 log(length)

  18. 4. Residuals show that a linear model is good for the logs of x and y?

  19. 5. Undo logs of LSRL: log ŷ = -1.8994 + 3.0494 log x Then plot power regression over scatterplot of original data, good fit?

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