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Chapter 6 Linearized Nonlinear Equations 100309

# Chapter 6 Linearized Nonlinear Equations 100309 - Chapter 6...

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1 Chapter 6 Least Squares Curve Fitting of Linearized Non-Linear Models Contents Introduction ......................................................................................................................... 1 6.1 The Importance of Non-Polynomial Models ................................................................ 2 6.2 Converting Nonlinear Models to Linear Least Squares Problems ................................ 3 6.3 Linear Least Squares for the Exponential Model ......................................................... 3 6.3.1 Linearizing the Relationship between the Variables ........................................... 3 Example 6.1 Fitting Data with an Exponential Model ................................................. 4 6.3.2 The Coefficient of Determination for Non-Linear Models .................................. 6 6.4 Linearizing the Power Law Model ............................................................................... 8 Example 6.2 Fitting Data with a Power Law Model .................................................... 8 6.5 Linearizing the Saturation-Growth Model .................................................................. 10 Example 6.3 Fitting Data with a Saturation-Growth Model ....................................... 10 6.6 Other Non-Linear Models ........................................................................................... 12 Example 6.4 Other Nonlinear Models ........................................................................ 12 Chapter 6 Summary .......................................................................................................... 15 Exercises ........................................................................................................................... 16 Introduction Some phenomena obey a nonlinear relationship that cannot be captured by any order polynomial. As we saw in Chapter 5, polynomials are powerful, flexible functions capable of representing many different relationships between dependent and independent variables as continuous functions. The Weierstrass approximation theorem also tells us that for any analytic function on a fixed interval a polynomial exists that will represent the function to any desired degree of accuracy. This does not mean, however, that a polynomial is always the best representation of the data. When the data obey an exponential or power law relationship, for example, then polynomial models will never represent the data as efficiently. Another advantage of polynomial models is that when we perform a least squares curve fit for a polynomial model, the set of equations we must solve is a set of linear equations. For exponential, power law and other types of non-polynomial models, the least squares equations are no longer linear and we find ourselves looking at a system of nonlinear equations that must be solved simultaneously. Many nonlinear relationships, however, including the important exponential and power law relationships, are easily converted

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2 into linear relationships by a simple change of variable. This allows us to fit the models to the data using the same linear least squares equations developed in Chapter 5. But what is so special about certain non-polynomial models? 6.1 The Importance of Non ­ Polynomial Models The usefulness of the exponential function in representing the behavior of natural phenomena comes from one of its fundamental properties. The exponential function is the function which is equal to its own derivative: ݂ሺݔሻ ൌ ݂݀ ݀ݔ ݁ ݀ݕ ݀ݔ ሺ݁ ሺ6.1ሻ When the rate of growth or decay of a variable is proportional to the size of the variable, then it can be represented with an exponential relationship. The range of phenomena that can be represented this way is astounding. Population growth is an example. When a population’s growth is not resource limited, then the larger the population becomes, the faster it grows. So the unconstrained growth rate of a population is proportional to the size of the population, and the growth rate is just the time derivative of the size. The
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Chapter 6 Linearized Nonlinear Equations 100309 - Chapter 6...

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