A New(?) Regression Clustering Algorithm

13 Aug, 2017 · by Justin Silverman · Read in about 94 min · (19842 Words)
R Machine Learning

Motivation
My Solution - Hybrid K-Means/Linear-Regression with Transformation
Starting on Boring Simulated Data
Now a more interesting simulated dataset
More realistic, presense of observational noise
Conclusions and future directions

Motivation

I am a fan of the Stack Exchange forums. In particular, I like Cross Validated and Stack Overflow. An interesting question regarding clustering was posted recently. Essentially someone had the following dataset.

plot(Retirees)

Essentially the poster wanted a way of clustering the observations into the “lines” that are fairly easy to observe in the data. I am going to ignore the fact that these lines are actually the result of artifact (e.g., conversion of discrete values to percentages and then plotting the percentages vs. a variable used to calculate the percentages) and just pretend they are real as I think its still an interesting problem. I am actually going to simulate some non-artifactual data and use this as well.

Not too surprisingly, the poster stated that K-means clustering had not preformed too well.

# Function to enable easier plotting of the results.
plot_groupings <- function(d, g, col=rainbow(length(unique(g))), 
                           plot.lines=TRUE, point.color=FALSE, ...){
  K <- length(unique(g))
  if (point.color) {
    plot(d, col=g, ...)}
  else {
    plot(d, ...)
  }
  if (plot.lines){
    for (i in 1:K){
      d.local <- d[g == i,]
      d.local <- d.local[order(d.local[,1]), ]
      lines(d.local, col = col[i], lwd=2)
    }
  }
}

kmean.clusters <- kmeans(Retirees, 10, nstart=20)
plot_groupings(Retirees, kmean.clusters$cluster)

We see that this is fairly obviously not capturing the functional pattern that is readily apparent by eye.

My Solution - Hybrid K-Means/Linear-Regression with Transformation

My Solution ended up being an approach to regression clustering that is a hybrid between regression and k-means clustering. I would be shocked if this is a new method, I bet it is a hacky-approximation to a better method. That said, I am not aware of anything exactly like it and I was quite happy with how well it seemed to work.

UPDATE: Not surprisingly, this is not a new method. A very smart professor at Duke has recently informed me that this is basically a mixture of regressions model (or a mixture of experts mod). So, don’t I feel silly with the title for this post. Still, I left the title in to grip the readers attention! (Is it working?) Anyways, A brief Google search led me to the FlexMix package. It turns out this package is awesome, the authors of this package really did a great job and its super powerful. It’s faster than my implementation, probably more robust, and is a fully probabilistic model. So here’s the deal: You can keep reading this post (and I hope you do), hopefully see some pretty pictures and see how my method (and the FlexMix package) both school K-Means clustering and DBSCAN (which really is not a fair comparison because those methods were not even designed for this problem), or you can just head over and read the Vignettes for the FlexMix package.

Back to a description of my algorithm: Like K-Means the algorithm is based off of iterative calculation of group parameters and then assignment of observations to the “nearest” group. Unlike K-means instead of just calculating the mean, I am actually calculating a regression for each group. And instead of calculating distance to the nearest group mean, I am turning the squared residuals into a probability and then using a sampling step to assign the observation to groups based on that probability. Importantly, this sampling step is one component of the algorithm that helps prevent it from getting stuck in local-optima. The other component that helps avoid local optima is that I run the algorithm from multiple starting points (multiple “units”) in parallel, for each iteration I take the “unit” that is doing the worst (based on sum of squared residuals) and replace it with the “unit” that is doing the best.

I have also made heavy use of the purrr::map function and the lm.fit (rather than normal lm) function for speed. I find that the algorithm works pretty well and surprisingly quickly.

Starting on Boring Simulated Data

To start off I will simulate some data similar to the data provided by the user.

library(tidyverse)
set.seed(4)

lambda <- seq(0.01, .1, by=0.01)
x <- 5:100
pars <- expand.grid("lambda" = lambda, "x" = x)
dat <- cbind(pars$x, exp(-pars$lambda*pars$x))

In answering the user’s question I also suggested a log-transformation (if the patterns were not purely artifactual).

dat.log <- cbind(dat[,1], log(dat[,2]))

par(mfrow=c(1,2))
plot(dat, main="Original")
plot(dat.log, main="Log-transformed")

Next I implement the actual Regression-K-Means Algorithm.

# Function to convert vector/matrix to probabilities (e.g., normalize to 1)
miniclo <- function (c){
  if (is.vector(c)) 
    c <- matrix(c, nrow = 1)
  (c/rowSums(c)) 
}

# The Main Function
# K: Number of Clusters
# n.unit: Number of parallel optimizations to run
# n.iter: Number of iterations to run the optimization for
# warmup.frac: what portion of iterations could be warmup (added stocastic varition to
#   escape local optima)
func_cluster <- function(dat, K, n.unit=4, n.iter=100, warmup.frac=0.9){
  # Create Design Matrix and Response
  X <- as.matrix(cbind(1, dat[,1])) # Design matrix, add intercept
  Y <- dat[,2] # Responses
  n <- nrow(X) # Number of observations
  
  # Random Initialize the group assignments. 
  gs <- as.list(1:n.unit) %>% 
    map(~sample(1:K, size = nrow(dat), replace = TRUE)) 
  
  # Create a safe version that deals with degeneration that can occur in some units 
  # code will allow this to be fixed. 
  safe_lm <- safely(function(i, X, Y) lm.fit(X[i,], Y[i]))
  
  for (i in 1:n.iter) {
    # Start out by fitting linear regressons to each group
    fits <- gs %>% 
      map(~split(1:n, .x)) %>% 
      #at_depth(2, ~lm.fit(X[.x,], Y[.x])) # Fit models
      at_depth(2, safe_lm, X, Y) %>% 
      at_depth(2, "result")
    
    # Replace Nulls with Best non-null from last iteration
    nulls <- fits %>% 
      at_depth(2, is.null) %>% 
      map(unlist) %>% 
      map(any) %>% 
      unlist()
    
    if (any(nulls)){
      d <- rank(best.worst.store, ties.method="random")
      d <- which(d==min(d[!nulls]))
      fits[nulls] <- fits[d[rep(1, sum(nulls))]]
    }
    
    # Calculate the squared residuals of each data-point to each
    # groups fitted linear model. Note I also add a small value (0.0001)
    # to avoid zero values that can show up later when converting to probabilities
    # and inverting. 
    sq.resids <- fits %>% 
      at_depth(2, "coefficients") %>% # Extract Coefficients
      at_depth(2, ~((Y-X%*%.x)^2)+0.0001) %>%  # Predict and Compute Squared Residuals
      map(bind_cols)
    
    ncolumns <- map(sq.resids, ncol)
    nulls <- ncolumns < K
    if (any(nulls)){
      d <- rank(best.worst.store, ties.method="random")
      d <- which(d==min(d[!nulls]))
      sq.resids[nulls] <- sq.resids[d[rep(1, sum(nulls))]]
    }    

    # Store which "unit" which of the n.unit optimiztions did the 
    # best and which did the worst
    best.worst.store <- sq.resids %>% 
      map(sum) %>% 
      unlist()
    best.worst <- c(which.min(best.worst.store), which.max(best.worst.store))
  
    # Compute new group assignements, notice I convert the relative
    # squared residual of each model into a probability and then use the 
    # inverse of this probability as the probability that a data-point
    # belongs to that group
     new.gs <- sq.resids  %>% 
      map(as.matrix) %>% 
      map(miniclo)  %>% # Add small value to fix zeros
      map(~.x^(-1)) %>% 
      map(miniclo)
    if (i < round(n.iter*warmup.frac)){ # Warmup with extra stochastic variation to get out of local-optima
      new.gs <- new.gs %>% 
      map(~apply(.x, 1, function(x) sample(1:K, 1, replace = TRUE, prob = x))) # Add sampling to get over some minima
    } else { # just take maxima
      new.gs <- new.gs %>% 
      map(max.col, ties.method="random") # Add sampling to get over some minima
    }
    
  
    # Replace the worst unit with the best
    new.gs[[best.worst[2]]] <- new.gs[[best.worst[1]]]
  
    # Update the cluster assignemnts for each unit
    gs <- new.gs
  }
  return(gs[[best.worst[1]]])
}



# Now investigate results of best unit and plot 
best.g <- func_cluster(dat.log, K=10, n.iter=200)

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## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

par(mfrow=c(1,2))
plot_groupings(dat.log, best.g, main="Log Transformed Results")
plot_groupings(dat, best.g, main="Results on Original Scale")

As you can see, the algorithm does quite well and in this case nearly all points are classified correctly (it actually improves if n.iter and n.unit are increased).

Finally I applied it to the data provided by the poster. I transformed both the X and Y axis to linearize the data. I could have also used a non-linear function rather than a linear one but this seemed easier.

dat <- Retirees 
dat.log <- cbind(log(dat[,1]), log(dat[,2]/100))

best.g <- func_cluster(dat.log, K=10, n.iter=200)

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

## Warning: at_depth() is deprecated, please use `modify_depth()` instead

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par(mfrow=c(1,2))
plot_groupings(dat.log, best.g, main="Log Transformed Results")
plot_groupings(dat, best.g, main="Results on Original Scale")

Its not perfect (again, it actually improves with more iterations) but it is definitely doing a ton better than regular K-Means we tried originally.

Now a more interesting simulated dataset

About 7 years ago I did some work with a very smart friend on some image analysis algorithms for detecting linear segments in images. This recent problem of functional clustering actually made me want to simulate this problem again and try it out.

limits <- as.list(1:10) %>%
  map(~c(rnorm(1, -75, 25), rnorm(1, 75, 25))) %>% 
  map(~.x[order(.x)]) %>% 
  map(~seq(.x[1], .x[2], length.out=25))

dat <- rnorm(10, 0, 5) %>% 
  as.list() %>% 
  map2(limits, ~.x*.y+rnorm(1, 0, 500)) %>% 
  map2(limits, ~cbind("y" = .x, "x" = .y)) %>% # Sorry about the swapping of x and y here (can't change purrr)
  map(as.data.frame) %>% 
  bind_rows(.id="group") %>% 
  mutate(group = factor(group, as.character(1:10))) 

# Now try to cluster (without knowing true groupings)
dat.run <- as.matrix(dat[,c(3,2)])
best.g <- func_cluster(dat.run, K=10, n.iter=250, n.unit=6)


par(mfrow=c(1,2))
plot_groupings(dat.run, best.g, plot.lines = F, point.color = F, main="True")
plot_groupings(dat[,c("x", "y")], dat$group, main="My Algorithms Results")

More realistic, presense of observational noise

Lets also see how well it works in the presence of some added noise.

limits <- as.list(1:6) %>%
  map(~c(rnorm(1, -75, 25), rnorm(1, 75, 25))) %>% 
  map(~.x[order(.x)]) %>% 
  map(~seq(.x[1], .x[2], length.out=25))

dat <- rnorm(6, 0, 5) %>% 
  as.list() %>% 
  map2(limits, ~.x*.y+rnorm(1, 0, 500)) %>% 
  map2(limits, ~cbind("y" = .x, "x" = .y)) %>% # Sorry about the swapping of x and y here (can't change purrr)
  map(as.data.frame) %>% 
  bind_rows(.id="group") %>% 
  mutate(group = factor(group, as.character(1:10))) %>% 
  mutate(y = y + rnorm(nrow(.), 0, 40)) # Add gaussian observation noise to dependent variable

# Now to cluster without knowing true groupings. 
dat.run <- as.matrix(dat[,c(3,2)])

best.g <- func_cluster(dat.run, K=6, n.iter=500, n.unit=6)

# Finally K-Means and DBSCAN as a point of comparison
library(dbscan)
library(flexmix)

## Loading required package: lattice

kmean.clusters <- kmeans(dat.run, 10, nstart=20)
db <- dbscan(dat.run, eps = 30, minPts = 2)
mod <- flexmix(y ~ x, data = dat, k=6)

par(mfrow=c(3,2))
plot_groupings(dat.run, dat$group, plot.lines=F, point.color=F, main="Data Seen by Classifiers")
plot_groupings(dat.run, dat$group, main="True")
plot_groupings(dat.run, best.g, plot.lines = T, point.color = F, main="My Algorithms Results")
plot_groupings(dat.run, clusters(mod), main="FlexMix")
plot_groupings(dat.run, kmean.clusters$cluster, main="K-Means")
plot_groupings(dat.run, db$cluster, main="DBSCAN")

Conclusions and future directions

So it looks to me like it is actually doing pretty darn well (note that differences in the the color of the lines between the True and Computed results doesn’t matter, only that points in the same cluster are given the same color). It’s beating K-Means and DBSCAN at least. I am particularly happy because for a number of these lines I couldn’t visually pick out those groups if I hadn’t colored the true values in the prior plot.

The approach and method I describe here are very general. I have used a mixture of univariate linear regression models for unsupervised learning. But there is no reason why it must stop there, multivariate regression with crazy types of covariates/features, with non-gaussian errors or even non-parametric regression models could be used. So keep this in mind, its a really cool method. Also remember the FlexMix package is really cool and can already do a lot of this and more.