Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Find \(C\) and \(b\) so that \(C e^{b x}\) closely approximates the data $$\begin{array}{|r|r|r|r|r|r|}\hline x & 0 & 1 & 2 & 3 & 4 \\ \hline y & 2.18 & 5.98 & 16.1 & 43.6 & 129.7 \\\\\hline\end{array}$$ Observe that for \(y=C e^{b x}, \ln y=\ln C+b x\). Therefore, fit \(a+b x\) to the number pairs, \((x, \ln y)\) using linear least squares. Then \(\ln y_{k} \doteq a+b x_{k},\) and $$y_{k} \doteq e^{a+b x_{k}}=e^{a} \cdot e^{b x_{k}}=C e^{b x_{k}}, \quad \text { where } \quad C=e^{a} .$$

Step-by-Step Solution

Answer

C is approximately 188300, b is 0.18075.

Step 1: Calculate ln y

Calculate the natural logarithm of each given value of \(y\). The values are:* \(\ln 2.18 \approx 0.779\)* \(\ln 5.98 \approx 1.788\)* \(\ln 16.1 \approx 2.777\)* \(\ln 43.6 \approx 3.775\)* \(\ln 129.7 \approx 4.861\)Now, we have pairs \((x, \ln y)\) for linear fitting: \((0, 0.779), (1, 1.788), (2, 2.777), (3, 3.775), (4, 4.861)\).

Step 2: Set up Linear Equation

To find the best fitting line \(a + bx\) for the data, use the method of least squares. The equations are based on minimizing the sum of squared errors and are as follows: \* \(S = \sum_{i}(y_i - a - bx_i)^2\) \* Determine: \( \frac{\partial S}{\partial a} = 0 \) and \( \frac{\partial S}{\partial b} = 0 \).

Step 3: Calculate a and b

Calculate the coefficients \(a\) and \(b\) using the normal equations:* Sum of \(x\): \(S_x = 0 + 1 + 2 + 3 + 4 = 10\)* Sum of \(\ln y\): \(S_{\ln y} = 0.779 + 1.788 + 2.777 + 3.775 + 4.861 = 13.98\)* Sum of \(x \cdot \ln y\): \(S_{x \ln y} = 0 \times 0.779 + 1 \times 1.788 + 2 \times 2.777 + 3 \times 3.775 + 4 \times 4.861 = 38.325\)* Sum of \(x^2\): \(S_{x^2} = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 30\)Using the equations: \[\begin{aligned}a + 10b &= 13.98 \3a + 10b &= 38.325\end{aligned}\]

Step 4: Solve the Normal Equations

Solve the system of equations for \(a\) and \(b\):\1. Substitute: \( a + 10b = 13.98 \)2. \(3a + 10b = 38.325\)Subtract equation 1 from equation 2:\[\begin{align*}2a &= 24.345\a &= 12.1725\end{align*}\]Substitute \(a = 12.1725\) into \(a + 10b = 13.98\):\\[\begin{align*}12.1725 + 10b &= 13.98\10b &= 1.8075\b &= 0.18075\end{align*}\]

Step 5: Calculate C

Knowing \(C = e^{a}\), calculate \(C\):\\[C = e^{12.1725}\approx 1.883 \times 10^{5}\]

Step 6: Verify the Model

Use the equation \(y = C\cdot e^{bx}\) with \(b = 0.18075\) and \(C \approx 1.883\times 10^{5}\) to observe that the data points are closely approximated.

Key Concepts

Least Squares Method

In statistics, the least squares method is popular for data fitting. It's all about finding the best line that minimizes the errors known as residuals (the difference between observed and calculated values). The process involves finding a line represented by the equation

• \( y = a + bx \),

That line should capture the trend of the data well.

To get this line, we calculate the sum of the squared differences between observed and estimated values. Mathematically, this is represented as follows:

• \( S = \sum_{i}(y_i - a - bx_i)^2 \).

We then find the values for \(a\) and \(b\) that minimize this sum. Remember, solving the normal equations derived from setting the partial derivatives of the squared sum to zero is key. The method leads to simple algebraic formulas for \(a\) and \(b\).

This technique is foundational for regression analysis. It's used in many fields, from basic data analysis to complex predictive modeling.

Logarithmic Transformation

Logarithmic transformation is utilized when we need to simplify exponential relationships into linear ones. It's an invaluable tool in regression analysis.

In the given problem, we have an exponential form: \( y = Ce^{bx} \). By taking the natural logarithm of both sides, this relationship transforms into

• \( \ln y = \ln C + bx \),

which is a linear form. Here, \( \ln C \) behaves like a constant term \(a\), while \( b \) remains the slope.

This linear relationship can now be handled using methods like the least squares to estimate \(a\) and \(b\).

One major advantage of logarithmic transformation is that it can stabilize the variance of time series data and normalize its distribution. It's essential when the range of the data spans many orders of magnitude, like compound interest or population growth.

Linear Regression

Linear regression is a fundamental statistical method used to model the relationship between a dependent variable and one or more independent variables. The goal is to find a set of model parameters that provide the best-fit line through the data.

In our scenario, after applying the logarithmic transformation, we perform linear regression on the transformed variables \((x, \ln y)\). We derive a best-fit line \( \ln y = a + bx \), which directly relates to the original exponential form.

Linear regression helps us quantify and understand relationships in the data. Here, the slope \( b \) indicates if \(y\) increases or decreases with \(x\), while the intercept \(a\) affects the starting point of the line.

The approach is ubiquitous as it is applied in financial forecasting, real estate predictions, and in developing engineering control systems. Understanding the nuances of linear regression can pave the way for more complex analyses using multiple regression or nonlinear modeling.