Math Problem Statement

Values of x and y are as follows, where x is the input and y is the output for the entire table. Tabulate the information: x: 1, 2, 3, 4, 5 and let the corresponding values for the given x values be y: 4, 18, 4, -24, 3

A function f(x) which is given as the best fit for the given data is recorded as f(x)= -(x-1)^2(x-3)(x-5)(x-7)+c

Find the value of c such that Sum Squared Error is as minimum as possible.

Solution

To find the value of cc such that the Sum Squared Error (SSE) is minimized, we'll follow these steps:

1. Define the Sum Squared Error (SSE)

The Sum Squared Error is calculated as:

SSE=i=1n(yif(xi))2SSE = \sum_{i=1}^{n} (y_i - f(x_i))^2

Where:

  • yiy_i are the given output values (from the table).
  • xix_i are the given input values (from the table).
  • f(x)f(x) is the function that models the data.

2. Given Function f(x)f(x)

The given function is:

f(x)=(x1)2(x3)(x5)(x7)+cf(x) = -(x-1)^2(x-3)(x-5)(x-7) + c

3. Known Data Points

From the table:

  • x:1,2,3,4,5x: 1, 2, 3, 4, 5
  • y:4,18,4,24,3y: 4, 18, 4, -24, 3

4. Set Up the SSE Equation

We will calculate the SSE for each of the given data points and then find the value of cc that minimizes this error.

SSE=(y1f(1))2+(y2f(2))2+(y3f(3))2+(y4f(4))2+(y5f(5))2SSE = (y_1 - f(1))^2 + (y_2 - f(2))^2 + (y_3 - f(3))^2 + (y_4 - f(4))^2 + (y_5 - f(5))^2

Substitute f(x)=(x1)2(x3)(x5)(x7)+cf(x) = -(x-1)^2(x-3)(x-5)(x-7) + c into the equation:

SSE=(4f(1))2+(18f(2))2+(4f(3))2+(24f(4))2+(3f(5))2SSE = (4 - f(1))^2 + (18 - f(2))^2 + (4 - f(3))^2 + (-24 - f(4))^2 + (3 - f(5))^2

Now we need to compute f(x)f(x) for each xx value (i.e., x=1,2,3,4,5x = 1, 2, 3, 4, 5) to complete this sum. We will simplify this process by evaluating f(x)f(x) at each point and determining the value of cc that minimizes SSE.

Let me calculate the function values at each point for a range of cc values and determine the optimal one. numpy as np from scipy.optimize import minimize

Given data

x_values = np.array([1, 2, 3, 4, 5]) y_values = np.array([4, 18, 4, -24, 3])

Define the function f(x) = -(x-1)^2(x-3)(x-5)(x-7) + c

def f(x, c): return -(x-1)**2 * (x-3) * (x-5) * (x-7) + c

Define the Sum Squared Error (SSE) function to minimize

def sse(c): # Calculate the sum of squared errors for all data points error = y_values - f(x_values, c) return np.sum(error**2)

Minimize the SSE function to find the optimal c

result = minimize(sse, x0=0) # Initial guess for c is 0

The optimal value of c

optimal_c = result.x[0] optimal_cThe value of cc that minimizes the Sum Squared Error (SSE) is approximately 3.43.4.

Would you like further details on the calculation or have any additional questions?

Here are some related questions you might find useful:

  1. How can I interpret the significance of the function f(x)f(x) in relation to the data points?
  2. How does changing the value of cc impact the SSE and the curve fit?
  3. How can we plot the function with the optimized cc value to visualize the fit?
  4. What are some other methods for minimizing the SSE besides using the optimization function?
  5. How does polynomial regression relate to this method for finding the best fit?

Tip: The Sum Squared Error is a common method for fitting models to data, but it can be sensitive to outliers. Always consider visualizing your data alongside the fitted model to assess the quality of the fit.

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Math Problem Analysis

Mathematical Concepts

Optimization
Sum Squared Error (SSE)
Polynomial Functions
Least Squares Method

Formulas

SSE = Σ(y_i - f(x_i))^2
f(x) = -(x-1)^2(x-3)(x-5)(x-7) + c

Theorems

Sum of Squared Errors Minimization
Optimization Algorithms (e.g., gradient descent or scipy.optimize)

Suitable Grade Level

Grades 11-12