基于R语言在特定点进行边界内外的插值与外推

Alright, let's work through how to do interpolation and extrapolation for your dataset in R. I'll walk you through practical, adaptable code examples since you mentioned handling points both inside and outside your boundary values.

First, let's formalize your x vector and add example y values (replace these with your actual response data—you'll need paired x/y values for interpolation):

# Your provided x sequence
x <- c(1600L, 1650L, 1675L, 1700L, 1725L, 1775L, 1800L, 1825L, 1850L, 1875L, 1880L, 1885L, 1900L, 1920L, 1925L, 1930L, 1935L, 1940L, 1945L, 1950L, 1955L, 1960L, 1965L, 1975L, 1980L, 1985L, 1990L, 1995L, 2000L, 2005L, 2010L, 2015L, 2020L, 2025L, 2030L, 2035L, 2040L, 2045L, 2050L, 2055L, 2060L, 2065L, 2070L, 2075L, 2080L, 2085L, 2090L, 2095L, 2100L, 2105L, 2110L, 2115L, 2120L, 2125L, 2130L, 2135L, 2140L, 2145L, 2150L, 2155L, 2160L, 2165L, 2170L, 2175L, 2180L, 2185L, 2190L, 2195L)

# Example y values (replace with your real data!)
set.seed(123) # For reproducibility
y <- rnorm(length(x), mean = 50 + 0.02*x, sd = 2)

First, clarify which points count as your "boundary"—for this example, let's say we want to:

• Interpolate values inside the range 1950 to 2100
• Extrapolate values outside that range (e.g., 1500 or 2200)

We'll use these test points to demonstrate:

test_x <- c(1500, 1975, 2050, 2200)

Linear Interpolation & Extrapolation #

This is the simplest approach. Use approxfun for interpolation inside the data range, and either extend a linear fit or use boundary values for extrapolation:

Option 1: Use Nearest Boundary Values for Extrapolation

# Create linear interpolation function; rule=2 uses nearest edge for out-of-range points
linear_fun <- approxfun(x, y, method = "linear", rule = 2)

# Generate predictions
linear_pred <- linear_fun(test_x)

# View results
data.frame(Test_X = test_x, Linear_Prediction = linear_pred)

Option 2: True Linear Extrapolation

If you want to extend the linear trend from your data instead of clamping to boundary values, fit a linear model:

# Fit linear regression to your data
lm_model <- lm(y ~ x)

# Predict for test points
lm_pred <- predict(lm_model, newdata = data.frame(x = test_x))

# View results
data.frame(Test_X = test_x, Linear_Extrapolation = lm_pred)

Smoothing Spline (For Curved Interpolation/Extrapolation) #

If your data has a non-linear trend, splines will give smoother interpolations and can extrapolate beyond your boundaries:

# Create natural spline function (extrapolates smoothly)
spline_fun <- splinefun(x, y, method = "natural")

# Generate predictions
spline_pred <- spline_fun(test_x)

# View results
data.frame(Test_X = test_x, Spline_Prediction = spline_pred)

If you need fine-grained control (e.g., only interpolate between specific boundaries and use different extrapolation rules for low/high ends), build a custom function:

# Define your custom boundaries
lower_bound <- 1950
upper_bound <- 2100

custom_interp_extrap <- function(new_x) {
  pred <- numeric(length(new_x))
  
  # Interpolate inside boundary using splines
  inside_idx <- new_x >= lower_bound & new_x <= upper_bound
  pred[inside_idx] <- splinefun(
    x[x >= lower_bound & x <= upper_bound],
    y[x >= lower_bound & x <= upper_bound],
    method = "natural"
  )(new_x[inside_idx])
  
  # Extrapolate below lower bound using linear fit on nearby edge points
  low_edge_idx <- x >= lower_bound - 50 & x <= lower_bound
  low_lm <- lm(y[low_edge_idx] ~ x[low_edge_idx])
  pred[new_x < lower_bound] <- predict(low_lm, newdata = data.frame(x = new_x[new_x < lower_bound]))
  
  # Extrapolate above upper bound using linear fit on nearby edge points
  high_edge_idx <- x >= upper_bound & x <= upper_bound + 50
  high_lm <- lm(y[high_edge_idx] ~ x[high_edge_idx])
  pred[new_x > upper_bound] <- predict(high_lm, newdata = data.frame(x = new_x[new_x > upper_bound]))
  
  return(pred)
}

# Test the custom function
custom_pred <- custom_interp_extrap(test_x)
data.frame(Test_X = test_x, Custom_Prediction = custom_pred)

Pick the method that best matches your data's trend—linear is great for simple relationships, splines work for curved data, and the custom function lets you tailor behavior exactly to your boundaries.

内容的提问来源于stack exchange，提问作者Novic