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 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:
Where:
- are the given output values (from the table).
- are the given input values (from the table).
- is the function that models the data.
2. Given Function
The given function is:
3. Known Data Points
From the table:
4. Set Up the SSE Equation
We will calculate the SSE for each of the given data points and then find the value of that minimizes this error.
Substitute into the equation:
Now we need to compute for each value (i.e., ) to complete this sum. We will simplify this process by evaluating at each point and determining the value of that minimizes SSE.
Let me calculate the function values at each point for a range of 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 that minimizes the Sum Squared Error (SSE) is approximately .
Would you like further details on the calculation or have any additional questions?
Here are some related questions you might find useful:
- How can I interpret the significance of the function in relation to the data points?
- How does changing the value of impact the SSE and the curve fit?
- How can we plot the function with the optimized value to visualize the fit?
- What are some other methods for minimizing the SSE besides using the optimization function?
- 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