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Modeling Non-linear Data

Topic Review on "Title":
  • Linear growth: occurs when a variable is added by a fixed number in each equal time period.
  • Exponential growth: occurs when a variable is multiplied by a fixed number greater than 1 in each equal time period.
  • Exponential growth works like compound interest. 
  • Graph of exponential function:
  •  

  • This exponential function represents future value,
  • FV = P(1+r)n, where p is the principal deposit, r is the rate of return and n is the number of months/years.
  • Common Ratio: The ratio of yn/yn-1 for equal-interval values of x confirms the existence of exponential growth. 
  • Remember that exponential and logarithmic functions are inverses of each other.
  • The more data points you have, the more accurate your resulting Least Squares Regression Line.
  • Algebraic Properties of Logarithms:
  • Logbx = y if and only if by = x
  • Log (AB)  = Log A+ Log B
  • Log (A/B) = Log A–Log B
  • Log xp = p Log x
  • For Studentized Residual
  • Studentized residual is a residual adjusted by dividing it by an estimate of its standard deviation.
  • The reason why we need studentization is because the variances of the residuals differ, even though the variances of the true errors are all equal to each other.
  • variance of the ith residual \mbox{var}(\widehat{\varepsilon}_i)=\sigma^2(1-h_{ii}).
  • Studentized residuals  {\widehat{\varepsilon}_i\over \widehat{\sigma} \sqrt{1-h_{ii}\  }}
  • appropriate estimate of σ is \widehat{\sigma}

Line Callout 1: Design Matrix  is used in certain statistical models, such as glm model  X=\left[\begin{matrix}1 & x_1 \\  \vdots & \vdots \\ 1 & x_n \end{matrix}\right]

 

Line Callout 1: Hat Matrix  orthogonal projection onto the column space of the design matrix

H = X(XTX) 1XT

  

Line Callout 1: Leverage   is the ith diagonal entry in the hat matrix.    hii   


Rapid Study Kit for "Title":
Flash Movie Flash Game Flash Card
Core Concept Tutorial Problem Solving Drill Review Cheat Sheet

"Title" Tutorial Summary :

This tutorial describes the way to model non-linear data.  We will identify exponential and power functions. Through which we can transform exponential and power functions into linear functions. And then by using properties of logarithms, we can create a model for non-linear data.

We will also discuss the relationship between errors and residuals. Then we will also introduce studentized residual for further applications.


Tutorial Features:

Specific Tutorial Features:

  • Studentized residuals are used because the variances of the residuals differ, even though the variances of the true errors are all equal to each other.
  • Least-squares regression techinques is used to minimize residual between the data points and the curve.

Series Features:

  • Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
  • Definition slides introduce terms as they are needed.
  • Visual representation of concepts
  • Animated examples—worked out step by step
  • A concise summary is given at the conclusion of the tutorial.

"Title" Topic List:
  • Linear growth
  • Exponential growth
  • Common Ratio
  • Algebraic Properties of Logarithms
  • Studentized Residual


See all 24 lessons in Introductory Statistics, including concept tutorials, problem drills and cheat sheets:
Teach Yourself Introductory Statistics Visually in 24 Hours

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