Let and be the prediction errors of two statistical/machine learning algorithms. and have relatively low bias, and high variances and . They are also correlated, having a Pearson correlation coefficient equal to .

Aggregating models and might result in a predictive model with lower prediction error variance than and . But not all the times. For those who attended statistics/probability/portfolio optimization classes, this may probably sound obvious; you can directly jump to the illustrative R part, below.

Let , with , be the prediction error of the ensemble model built with and . We have :

And from the fact that :

We get :

Now, let’s see how changes with an increase of , the ensemble’s allocation for model :

When is close to , that is, when the ensemble contains almost only model , we have :

That’s the relative change in the variance of the ensemble prediction error, induced by introducing model in an ensemble containing almost only . Hence, if , increasing the allocation of model won’t increase or decrease the variance of ensemble prediction at all. The variance will decrease if we introduce in the ensemble a model , so that . If , it won’t decrease, no matter the combination of and .

For a simple illustrative example in R, I create simulated data observations :

  # number of observations
  n <- 100
  u <- 1:n

  # Simulated observed data
  intercept <- 5 
  slope <- 0.2
  set.seed(2)

  ## data
  y <-  intercept + slope*u + rnorm(n)
  plot(u, y, type = 'l', main = "Simulated data observations")
  points(u, y, pch = 19)

I fit a linear regression model to the data, as a benchmark model :

  # Defining training and test sets
  train <- 1:(0.7*n)
  test <- -train
  u.train <- u[(train)]
  y.train <- y[(train)]
  u.test <- u[(test)]
  y.test <- y[(test)]

  # Fitting the benchmark model to the training set 
  fit.lm <- lm(y.train ~ u.train)

  (slope.coeff <- fit.lm$coefficients[2])
  (intercept.coeff <- fit.lm$coefficients[1])

## u.train
## 0.1925
## (Intercept)
## 5.292

Obtaining the predicted values from the benchmark model, and prediction error

  # Predicted values from benchmark model on the test set
  y.pred <- intercept.coeff + slope.coeff*u.test   
  
  # prediction error from linear regression
  pred.err <- y.pred - y.test
  (mean.pred.err <- mean(pred.err))
  (var.pred.err <- var(pred.err))

## [1] -0.1802
## [1] 1.338

Now, I consider two other models, and :

and

with and being 2 correlated gaussians with zero mean and constant variances (well, not really “models” that i fit, but these help me to build fictitious various predictions with different correlations for the illustration). The slope and intercept are those obtained from the benchmark model.

 # Alternative model 1 (low bias, high variance, oscillating)
  m <- length(y.pred)
  eps1 <- rnorm(m, mean = 0, sd = 1.5)
  y.pred1 <- intercept.coeff + slope.coeff*u.test + sin(u.test) + eps1

 # prediction error for model 1
  pred.err1 <- y.pred1 - y.test

We can visualize the predictions of model , and the benchmark, with different prediction errors correlations between model and , to get an intuition of the possible ensemble predictions :

  # Different prediction errors correlations for 1 and 2
  rho.vec <- c(-1, -0.8, 0.6, 1)  

  # Independent random gaussian numbers defining model 2 errors
  eps <- rnorm(m, mean = 0, sd = 2)

  # Plotting the predictions with different correlations
  par(mfrow=c(2, 2))  
  for (i in 1:4)
  {
    rho <- rho.vec[i]

    # Correlated gaussian numbers (Cholesky decomposition)
    eps2 <- rho*eps1 + sqrt(1 - rho^2)*eps

    # Alternative  model 2 (low bias, higher variance than 1, oscillating)
    y.pred2 <- intercept.coeff + slope.coeff*u.test - 0.35*sin(u.test) + eps2    

    # prediction error for model 2
    pred.err2 <- y.pred2 - y.test    

    # predictions from 1 & 2 correlation
    corr.pred12 <- round(cor(pred.err1, pred.err2), 2)

    # Plot
    plot(u.test, y.test, type = "p", 
         xlab = "test set values", ylab = "predicted values",
         main = paste("models 1 & 2 pred. values n correlation :", 
                      corr.pred12))
    points(u.test, y.test, pch = 19)
    lines(u.test, y.pred, lwd = 2)
    lines(u.test, y.pred1, 
          col = "blue", lwd = 2)
    lines(u.test, y.pred2, 
          col = "red", lwd = 2)
  }

Now including allocations of models and , we can see how the ensemble variance evolves as a function of allocation and correlation :

  # Allocation for model 1 in the ensemble
  alpha.vec <- seq(from = 0, to = 1, by = 0.05)  

  # Correlations between model 1 and model 2
  rho.vec <- seq(from = -1, to = 1, by = 0.05)  

  # Results matrices
  nb.row <- length(alpha.vec)
  nb.col <- length(rho.vec)  

  ## Average prediction errors of the ensemble
  mean.pred.err.ens <- matrix(0, nrow = nb.row, ncol = nb.col)
  rownames(mean.pred.err.ens) <- paste0("pct. 1 : ", alpha.vec*100, "%")
  colnames(mean.pred.err.ens) <- paste0("corr(1, 2) : ", rho.vec)

  ## Variance of prediction error of the ensemble
  var.pred.err.ens <- matrix(0, nrow = nb.row, ncol = nb.col)
  rownames(var.pred.err.ens) <- paste0("pct. 1 : ", alpha.vec*100, "%")
  colnames(var.pred.err.ens) <- paste0("corr(1, 2) : ", rho.vec)

  # loop on correlations and allocations 
  for (i in 1:nb.row)
  {
    for (j in 1:nb.col)
    {
      alpha <- alpha.vec[i]
      rho <- rho.vec[j]

      # Alternative model 2 (low bias, higher variance, oscillating)
      eps2 <- rho*eps1 + sqrt(1 - rho^2)*eps
      y.pred2 <- intercept.coeff + slope.coeff*u.test - 0.35*sin(u.test) + eps2
      pred.err2 <- y.pred2 - y.test

      # Ensemble prediction error
      z <- alpha*pred.err1 + (1-alpha)*pred.err2
      mean.pred.err.ens[i, j] <- mean(z)      
      var.pred.err.ens[i, j] <-  var(z)
    }
  }
  res.var <- var.pred.err.ens
 
  # Heat map for the variance of the ensemble
  filled.contour(alpha.vec, rho.vec, res.var, color = terrain.colors, 
                 main = "prediction error variance for the ensemble", 
                 xlab = "allocation in x1", ylab = "correlation between 1 and 2")

Hence, the lower the correlation between and , the lower the variance of the ensemble model prediction. This, combined with an allocation of comprised between and seem to be the building blocks for the final model. The ensemble models biases helps in making a choice of allocation (this is actually an optimization problem that can be directly solved with portfolio theory).

Finding the final ensemble, with lower variance and lower bias :

# Final ensemble

## Allocation
alpha.vec[which.min(as.vector(res.var))]
## [1] 0.45

## Correlation
res.bias <- abs(mean.pred.err.ens)
which.min(res.bias[which.min(as.vector(res.var)), ])
 ## corr(1, 2) : 
-0.7
 ## 7

Creating the final model with these parameters :

    rho <- -0.7
    eps2 <- rho*eps1 + sqrt(1 - rho^2)*eps

    # Alternative model 2 (low bias, higher variance, oscillating)
    y.pred2 <- intercept.coeff + slope.coeff*u.test - 0.35*sin(u.test) + eps2        

    # Final ensemble prediction
    y.pred.ens <- 0.45*y.pred1 + 0.55*y.pred2
  
    # Plot
    plot(u.test, y.test, type = "p", 
         xlab = "test set", ylab = "predicted values",
         main = "Final ensemble model (green)")
    points(u.test, y.test, pch = 19)
    # benchmark 
    lines(u.test, y.pred, lwd = 2)
    # model 1
    lines(u.test, y.pred1, col = "blue", lwd = 2)
    # model 2
    lines(u.test, y.pred2, col = "red", lwd = 2)
    # ensemble model with 1 and 2
    points(u.test, y.pred.ens, col = "green", pch = 19)
    lines(u.test, y.pred.ens, col = "green", lwd = 2)

Performance of the final model :

    # Benchmark
    pred.err <- y.pred - y.test
    # Model 1 
    pred1.err <- y.pred1 - y.test
    # Model 2
    pred2.err <- y.pred2 - y.test
    # Ensemble model
    pred.ens.err <- y.pred.ens - y.test
    
    # Bias comparison 
    bias.ens <- mean(y.pred.ens - y.test)  
    bias.ens_vsbench <- (bias.ens/mean(y.pred - y.test) - 1)*100
    bias.ens_vs1 <- (bias.ens/mean(y.pred1 - y.test) - 1)*100
    bias.ens_vs2 <- (bias.ens/mean(y.pred2 - y.test) - 1)*100
  
    # Variance comparison
    var.ens <- var(y.pred.ens - y.test)
    var.ens_vsbench <- (var.ens/var(y.pred - y.test) - 1)*100
    var.ens_vs1 <- (var.ens/var(y.pred1 - y.test) - 1)*100
    var.ens_vs2 <- (var.ens/var(y.pred2 - y.test) - 1)*100
  
   cbind(c(bias.ens_vsbench, bias.ens_vs1, bias.ens_vs2), c(var.ens_vsbench, var.ens_vs1, var.ens_vs2))
##        [,1]  [,2]
## [1,] -95.31 46.27
## [2,] -112.12 -62.93
## [3,] -88.33 -45.53