如何用R求解方程By=Ax+C的系数A、B并计算均值和标准差

Hey there! Let's walk through exactly how to implement this in R. We'll break it down into clear, actionable steps:

1. Define Your Input Data #

First, let's start by defining the vectors you provided:

# Input data vectors
y <- c(-9216.656, -9134.369, -9186.813, -8780.633, -9006.787)
x <- c(908.4656, 932.6687, 1015.0424, 816.9116, 703.2510)
C <- c(-8.965961, -8.883207, -8.935910, -8.546677, -8.766842)

2. Generate All Pairwise Index Combinations #

We need every possible pair of data points to build our 10 systems of equations. The combn() function makes this easy—it generates all unique 2-element combinations of indices from our 5 data points:

# Get all 10 pairwise index combinations (5 choose 2)
pair_indices <- combn(length(x), 2)

3. Solve Each Linear System #

For each pair of indices, we'll rewrite the equation By = Ax + C into the standard linear form Ax - By = -C, then construct the coefficient matrix and constant vector to solve for A and B using R's built-in solve() function. We'll store all solutions in a data frame:

# Initialize an empty data frame to store results
results <- data.frame(A = numeric(), B = numeric())

# Loop through each pair of indices
for (i in 1:ncol(pair_indices)) {
  # Extract the two indices for the current pair
  idx1 <- pair_indices[1, i]
  idx2 <- pair_indices[2, i]
  
  # Build the coefficient matrix for the system:
  # [x1  -y1]   [A]   = [-C1]
  # [x2  -y2]   [B]     [-C2]
  coeff_matrix <- matrix(
    c(x[idx1], -y[idx1], x[idx2], -y[idx2]),
    nrow = 2, byrow = TRUE
  )
  
  # Build the constant term vector
  const_vector <- c(-C[idx1], -C[idx2])
  
  # Solve the linear system and store the results
  solution <- solve(coeff_matrix, const_vector)
  results <- rbind(results, data.frame(A = solution[1], B = solution[2]))
}

# Print all 10 A/B pairs
cat("All 10 sets of A and B:\n")
print(results)

4. Calculate Mean & Standard Deviation #

Finally, we'll compute the mean and standard deviation for both A and B using base R functions:

# Compute summary statistics
summary_stats <- data.frame(
  Metric = c("Mean", "Standard Deviation"),
  A = c(mean(results$A), sd(results$A)),
  B = c(mean(results$B), sd(results$B))
)

# Print the summary stats
cat("\nSummary statistics for A and B:\n")
print(summary_stats)

Quick Notes: #

• The solve() function works reliably here because each pairwise system is a 2x2 matrix (two equations, two unknowns) with no linear dependence between the equations.
• The combn() function ensures we only get unique pairs (no duplicates like (1,2) and (2,1)), which gives us exactly the 10 distinct systems we need.

内容的提问来源于stack exchange，提问作者imtaiky