Given the following data set,
x 0 1 2 3 4 5 6
y 0.06 0.156 0.4056 1.05456 2.741856 7.1288256 18.53494656
Determine by regression the best exponential function that fits the data
Function =
To determine the best exponential function that fits the given data set, you can use the method of least squares regression. This involves finding the equation of the exponential function that minimizes the sum of the squared differences between the actual y-values and the predicted y-values from the equation.
Here are the steps to find the best exponential function:
Step 1: Plot the data points on a graph.
Step 2: Determine the form of the exponential function. In this case, it will be of the form y = a * exp(b*x).
Step 3: Take the natural logarithm of both sides of the equation to linearize the function:
ln(y) = ln(a) + b*x.
Step 4: Create a new data set where x remains the same, and y is replaced with its natural logarithm.
x 0 1 2 3 4 5 6
ln(y) -2.8134 -1.8609 -0.9016 -0.0484 0.9988 1.967 2.9221
Step 5: Calculate the sum of x, sum of ln(y), sum of x squared, and sum of x multiplied by ln(y).
∑x = 0 + 1 + 2 + 3 + 4 + 5 + 6 = 21
∑ln(y) = -2.8134 - 1.8609 - 0.9016 - 0.0484 + 0.9988 + 1.967 + 2.9221 = -1.735
∑x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91
∑x * ln(y) = (0 * -2.8134) + (1 * -1.8609) + (2 * -0.9016) + (3 * -0.0484) + (4 * 0.9988) + (5 * 1.967) + (6 * 2.9221) = 6.7179
Step 6: Use the following formulas to find the values of a and b:
b = (n * ∑x * ln(y) - ∑x * ∑ln(y)) / (n * ∑x^2 - (∑x)^2)
a = e^((∑ln(y) - b * ∑x)/n)
Where n is the number of data points (in this case, 7).
Step 7: Calculate the values of a and b using the formula from step 6.
b = (7 * 6.7179 - 21 * -1.735) / (7 * 91 - 21^2)
b ≈ 0.9997
a = e^((-1.735 - 0.9997 * 21) / 7)
a ≈ 0.0575
Step 8: The exponential function that fits the data is given by y ≈ 0.0575 * exp(0.9997 * x).
Therefore, the best exponential function that fits the given data set is:
y ≈ 0.0575 * exp(0.9997 * x).