4.8 Fitting Exponential Models to Data

Exponential and Logarithmic Models

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Construction of exponential models

Exponential models describe situations where a quantity grows or decays at a constant percentage rate over time. They show up everywhere: bacteria doubling in a petri dish, money compounding in a savings account, or a car losing value year after year.

The general form is $y = ab^x$, where:

• $a$ is the initial value (the y-intercept, or the output when $x = 0$)
• $b$ is the growth or decay factor
  • If $b > 1$, you have growth. A value of $b = 1.05$ means a 5% increase per unit time.
  • If $0 < b < 1$, you have decay. A value of $b = 0.85$ means a 15% decrease per unit time.
• $x$ is the independent variable, usually representing time

To construct an exponential model from data:

1. Identify the initial value $a$ from the y-intercept or the first data point (the value when $x = 0$).
2. Calculate the growth/decay factor $b$ by finding the ratio of consecutive y-values: $b = \frac{y_{n+1}}{y_n}$. If the data is truly exponential, this ratio should be roughly constant across your data points.
3. Write the model by substituting $a$ and $b$ into $y = ab^x$.

For example, if a bacterial colony starts at 200 cells and triples every hour, then $a = 200$ and $b = 3$, giving you $y = 200 \cdot 3^x$.

The correlation coefficient (often $r$ or $r^2$ from regression) tells you how well the exponential model fits the data. Values closer to 1 (or -1 for decay) indicate a stronger fit.

Development of logarithmic models

Logarithmic models are the inverse of exponential models. While an exponential model answers "what's the quantity after this much time?", a logarithmic model answers "how much time until the quantity reaches this value?"

The general form is $y = \log_b(x)$, where $b$ is the base (the same as the growth or decay factor from the corresponding exponential model).

To convert an exponential model into a logarithmic one:

1. Start with the exponential equation, say $y = b^x$.
2. Swap $x$ and $y$ to get $x = b^y$.
3. Apply $\log_b$ to both sides: $y = \log_b(x)$.

Logarithmic models are especially useful for:

• Finding doubling time or half-life (how long until a quantity doubles or halves)
• Scales that compress huge ranges of values into manageable numbers, like the Richter scale (earthquake magnitude) and the decibel scale (sound intensity). These use logarithmic scales because the raw quantities span many orders of magnitude.

Creation of logistic models

Not everything grows without bound. When there's a cap on growth (limited food, limited customers, limited resources), the logistic model is more realistic than a pure exponential. Growth starts fast, then slows as it approaches a maximum.

The logistic model takes the form:

$P(t) = \frac{K}{1 + Ae^{-rt}}$

• $P(t)$ is the population (or quantity) at time $t$
• $K$ is the carrying capacity, the maximum sustainable value
• $A$ is a constant determined by the initial condition
• $r$ is the growth rate, controlling how quickly $P(t)$ approaches $K$

To build a logistic model:

1. Identify the carrying capacity $K$ from context. This might be the total market size, the maximum population an environment supports, etc.

2. Find $A$ using the initial value: $A = \frac{K}{P(0)} - 1$. For instance, if $K = 10{,}000$ and $P(0) = 500$, then $A = \frac{10{,}000}{500} - 1 = 19$.

3. Estimate $r$ by plugging in another known data point and solving.

Logistic growth has three distinct phases:

1. Early phase: The population is small relative to $K$, so growth looks nearly exponential.
2. Middle phase: Growth slows as resources become scarcer. The inflection point (fastest growth rate) occurs at $P = \frac{K}{2}$.
3. Late phase: The population levels off, approaching $K$ asymptotically but never exceeding it.

Real-world examples include the spread of diseases through a population, adoption of new technology (like smartphones), and growth of invasive species in a new ecosystem.

Model Fitting and Analysis

Once you have data, you need to determine which model fits best and how to find it.

Regression analysis uses your calculator or software to find the equation that best matches a dataset. For exponential data, you'd use exponential regression (often labeled ExpReg on a graphing calculator). The least squares method works behind the scenes, minimizing the sum of squared differences between the actual data points and the values the model predicts.

Residuals are the differences between observed values and predicted values: $\text{residual} = y_{\text{observed}} - y_{\text{predicted}}$. If your model is a good fit, residuals should be small and scattered randomly (no clear pattern). A pattern in the residuals suggests the wrong type of model was chosen.

Goodness of fit is measured by $r^2$, which tells you what proportion of the variability in the data your model explains. An $r^2$ of 0.97 means the model accounts for 97% of the variation.

Data transformation is a useful technique for checking whether data is exponential. If you take the logarithm of the y-values and the resulting plot looks linear, the original data follows an exponential pattern. This works because taking $\log$ of both sides of $y = ab^x$ gives $\log(y) = \log(a) + x\log(b)$, which is a linear equation in $x$ and $\log(y)$.