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3.2 Non-Linear Parameters - Exponential - notes

3.2 Non-Linear Parameters - Exponential - notes -...

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Objective [3.2] Notes Part 2 Exponential Be familiar with and recognize the algebraic form(s) for all three models, including their corresponding parameters and if those parameters have any meaning in the context of the functional relationship (e.g. does a parameter control the initial value, rate of change (shape), decreasing or increasing (flip), max/min, limiting values, etc.) In a linear model both parameters (linear and constant) have meaning in the “real” business world. The linear parameter was the constant rate of change of a relationship and the constant parameter is the initial value. This will NOT be that case for other models. Some parameters may not have any “real” business meaning. Some will be derived from other values that do hav e meaning and yet some may just be “fitting” parameters with no specific meaning. So what about the parameters in the three models we have left to study, quadratic, exponential and logistic. Do their parameters all have business meanings? How do we find their parameter values given some information (data)? What is the minimum information (data) we need to find values for all of them? Exponential Model Parameters There are two “forms” of the exponential model. Sometimes they are used interchangeably but technically one is to be used in “finite compounding “or “discrete” cases and the other in “continuous” cases. What I mean by discrete and continuous I will say later. For now it’s just ok to identify the two. You were already introduced to them b efore in objective 1.8. In objective 1.11 you learned that one of the parameters controls whether the model is strictly increasing or decreasing. Before you continue you may want to go back and review those two objectives. ( ) - finite or discrete ( ) - continuous t kt D t ab D t ae So can you recognize the parameters in each version? If not then you should review objective 1.10. They are a and b in the finite case and a and k in the continuous case? So like the linear model there are ONLY two parameters in a basic exponential model So is one of them an initial value??? ..... Yes. Is one of the a rate???…. almost and Yes! But not rate in the same sense as a linear model . In the linear model the rate of change is “absolute”. In other words it likely has units. But the rate in the exponential is a “relative” rate of change. When we use the word “ relative it refers to a percentage change. Relative rate of change is a % change . Relative (exponential) rate vs Absolute (linear) rate Say I have a relationship between time and price. So price is the output in say $ and time the input in months. If I say the price is increasing at a rate of $10/month and the current price is $15. In a month the price will be $25, but two months the price will be $35. Do you recognize that as a linear relationship!
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