Regression I: K-nearest neighbors {#regression1}¶

library(knitr)
library(plotly)
library(stringr)

knitr::opts_chunk$set(fig.align = "center")
reticulate::use_miniconda('r-reticulate')

print_tidymodels <- function(tidymodels_object) {
  if(!is_latex_output()) {
    tidymodels_object
  } else {
    output <- capture.output(tidymodels_object)
    
    for (i in seq_along(output)) {
      if (nchar(output[i]) <= 80) {
        cat(output[i], sep = "\n")
      } else {
        cat(str_sub(output[i], start = 1, end = 80), sep = "\n")
        cat(str_sub(output[i], start = 81, end = nchar(output[i])), sep = "\n")
      }
    }
  }
}
theme_update(axis.title = element_text(size = 12)) # modify axis label size in plots 

Overview¶

This chapter continues our foray into answering predictive questions. Here we will focus on predicting numerical variables and will use regression to perform this task. This is unlike the past two chapters, which focused on predicting categorical variables via classification. However, regression does have many similarities to classification: for example, just as in the case of classification, we will split our data into training, validation, and test sets, we will use tidymodels workflows, we will use a K-nearest neighbors (KNN) approach to make predictions, and we will use cross-validation to choose K. Because of how similar these procedures are, make sure to read Chapters @ref(classification) and @ref(classification2) before reading this one—we will move a little bit faster here with the concepts that have already been covered. This chapter will primarily focus on the case where there is a single predictor, but the end of the chapter shows how to perform regression with more than one predictor variable, i.e., multivariable regression. It is important to note that regression can also be used to answer inferential and causal questions, however that is beyond the scope of this book.

Chapter learning objectives¶

By the end of the chapter, readers will be able to do the following:

Recognize situations where a simple regression analysis would be appropriate for making predictions.
Explain the K-nearest neighbor (KNN) regression algorithm and describe how it differs from KNN classification.
Interpret the output of a KNN regression.
In a dataset with two or more variables, perform K-nearest neighbor regression in R using a tidymodels workflow.
Execute cross-validation in R to choose the number of neighbors.
Evaluate KNN regression prediction accuracy in R using a test data set and the root mean squared prediction error (RMSPE).
In the context of KNN regression, compare and contrast goodness of fit and prediction properties (namely RMSE vs RMSPE).
Describe the advantages and disadvantages of K-nearest neighbors regression.

The regression problem¶

Regression, like classification, is a predictive \index{predictive question} problem setting where we want to use past information to predict future observations. But in the case of regression, the goal is to predict numerical values instead of categorical values. The variable that you want to predict is often called the response variable. \index{response variable} For example, we could try to use the number of hours a person spends on exercise each week to predict their race time in the annual Boston marathon. As another example, we could try to use the size of a house to predict its sale price. Both of these response variables—race time and sale price—are numerical, and so predicting them given past data is considered a regression problem.

Just like in the \index{classification!comparison to regression} classification setting, there are many possible methods that we can use to predict numerical response variables. In this chapter we will focus on the K-nearest neighbors algorithm [@knnfix; @knncover], and in the next chapter we will study linear regression. In your future studies, you might encounter regression trees, splines, and general local regression methods; see the additional resources section at the end of the next chapter for where to begin learning more about these other methods.

Many of the concepts from classification map over to the setting of regression. For example, a regression model predicts a new observation’s response variable based on the response variables for similar observations in the data set of past observations. When building a regression model, we first split the data into training and test sets, in order to ensure that we assess the performance of our method on observations not seen during training. And finally, we can use cross-validation to evaluate different choices of model parameters (e.g., K in a K-nearest neighbors model). The major difference is that we are now predicting numerical variables instead of categorical variables.

\newpage

Note: You can usually tell whether a\index{categorical variable}\index{numerical variable} variable is numerical or categorical—and therefore whether you need to perform regression or classification—by taking two response variables X and Y from your data, and asking the question, “is response variable X more than response variable Y?” If the variable is categorical, the question will make no sense. (Is blue more than red? Is benign more than malignant?) If the variable is numerical, it will make sense. (Is 1.5 hours more than 2.25 hours? Is $500,000 more than $400,000?) Be careful when applying this heuristic, though: sometimes categorical variables will be encoded as numbers in your data (e.g., “1” represents “benign”, and “0” represents “malignant”). In these cases you have to ask the question about the meaning of the labels (“benign” and “malignant”), not their values (“1” and “0”).

Exploring a data set¶

In this chapter and the next, we will study a data set \index{Sacramento real estate} of 932 real estate transactions in Sacramento, California originally reported in the Sacramento Bee newspaper. We first need to formulate a precise question that we want to answer. In this example, our question is again predictive: \index{question!regression} Can we use the size of a house in the Sacramento, CA area to predict its sale price? A rigorous, quantitative answer to this question might help a realtor advise a client as to whether the price of a particular listing is fair, or perhaps how to set the price of a new listing. We begin the analysis by loading and examining the data, and setting the seed value.

\index{seed!set.seed}

library(tidyverse)
library(tidymodels)
library(gridExtra)

set.seed(5)

sacramento <- read_csv("data/sacramento.csv")
sacramento

The scientific question guides our initial exploration: the columns in the data that we are interested in are sqft (house size, in livable square feet) and price (house sale price, in US dollars (USD)). The first step is to visualize the data as a scatter plot where we place the predictor variable (house size) on the x-axis, and we place the target/response variable that we want to predict (sale price) on the y-axis. \index{ggplot!geom_point} \index{visualization!scatter}

Note: Given that the y-axis unit is dollars in Figure @ref(fig:07-edaRegr), we format the axis labels to put dollar signs in front of the house prices, as well as commas to increase the readability of the larger numbers. We can do this in R by passing the dollar_format function (from the scales package) to the labels argument of the scale_y_continuous function.

eda <- ggplot(sacramento, aes(x = sqft, y = price)) +
  geom_point(alpha = 0.4) +
  xlab("House size (square feet)") +
  ylab("Price (USD)") +
  scale_y_continuous(labels = dollar_format()) + 
  theme(text = element_text(size = 12))

eda

The plot is shown in Figure @ref(fig:07-edaRegr). We can see that in Sacramento, CA, as the size of a house increases, so does its sale price. Thus, we can reason that we may be able to use the size of a not-yet-sold house (for which we don’t know the sale price) to predict its final sale price. Note that we do not suggest here that a larger house size causes a higher sale price; just that house price tends to increase with house size, and that we may be able to use the latter to predict the former.

K-nearest neighbors regression¶

Much like in the case of classification, we can use a K-nearest neighbors-based \index{K-nearest neighbors!regression} approach in regression to make predictions. Let’s take a small sample of the data in Figure @ref(fig:07-edaRegr) and walk through how K-nearest neighbors (KNN) works in a regression context before we dive in to creating our model and assessing how well it predicts house sale price. This subsample is taken to allow us to illustrate the mechanics of KNN regression with a few data points; later in this chapter we will use all the data.

To take a small random sample of size 30, we’ll use the function slice_sample, and input the data frame to sample from and the number of rows to randomly select. \index{slice_sample}

# hidden seed
set.seed(10)

small_sacramento <- slice_sample(sacramento, n = 30)

Next let’s say we come across a 2,000 square-foot house in Sacramento we are interested in purchasing, with an advertised list price of $350,000. Should we offer to pay the asking price for this house, or is it overpriced and we should offer less? Absent any other information, we can get a sense for a good answer to this question by using the data we have to predict the sale price given the sale prices we have already observed. But in Figure @ref(fig:07-small-eda-regr), you can see that we have no observations of a house of size exactly 2,000 square feet. How can we predict the sale price?

small_plot <- ggplot(small_sacramento, aes(x = sqft, y = price)) +
  geom_point() +
  xlab("House size (square feet)") +
  ylab("Price (USD)") +
  scale_y_continuous(labels = dollar_format()) +
  geom_vline(xintercept = 2000, linetype = "dotted") + 
  theme(text = element_text(size = 12))

small_plot

We will employ the same intuition from the classification chapter, and use the neighboring points to the new point of interest to suggest/predict what its sale price might be. For the example shown in Figure @ref(fig:07-small-eda-regr), we find and label the 5 nearest neighbors to our observation of a house that is 2,000 square feet. \index{mutate}\index{slice}\index{arrange}\index{abs}

nearest_neighbors <- small_sacramento |>
  mutate(diff = abs(2000 - sqft)) |>
  arrange(diff) |>
  slice(1:5) #subset the first 5 rows

nearest_neighbors

nearest_neighbors <- mutate(nearest_neighbors, twothou = rep(2000, 5))

nn_plot <- small_plot +
  geom_segment(
    data = nearest_neighbors,
    aes(x = twothou, xend = sqft, y = price, yend = price), color = "orange"
  )

nn_plot

Figure @ref(fig:07-knn3-example) illustrates the difference between the house sizes of the 5 nearest neighbors (in terms of house size) to our new 2,000 square-foot house of interest. Now that we have obtained these nearest neighbors, we can use their values to predict the sale price for the new home. Specifically, we can take the mean (or average) of these 5 values as our predicted value, as illustrated by the red point in Figure @ref(fig:07-predictedViz-knn).

prediction <- nearest_neighbors |>
  summarise(predicted = mean(price))

prediction

nn_plot +
  geom_point(aes(x = 2000, y = prediction[[1]]), color = "red", size = 2.5)

Our predicted price is $r format(round(prediction[[1]]), big.mark=",", nsmall=0, scientific = FALSE) (shown as a red point in Figure @ref(fig:07-predictedViz-knn)), which is much less than $350,000; perhaps we might want to offer less than the list price at which the house is advertised. But this is only the very beginning of the story. We still have all the same unanswered questions here with KNN regression that we had with KNN classification: which $K$ do we choose, and is our model any good at making predictions? In the next few sections, we will address these questions in the context of KNN regression.

One strength of the KNN regression algorithm that we would like to draw attention to at this point is its ability to work well with non-linear relationships (i.e., if the relationship is not a straight line). This stems from the use of nearest neighbors to predict values. The algorithm really has very few assumptions about what the data must look like for it to work.

Training, evaluating, and tuning the model¶

As usual, we must start by putting some test data away in a lock box that we will come back to only after we choose our final model. Let’s take care of that now. Note that for the remainder of the chapter we’ll be working with the entire Sacramento data set, as opposed to the smaller sample of 30 points that we used earlier in the chapter (Figure @ref(fig:07-small-eda-regr)). \index{training data} \index{test data}

sacramento_split <- initial_split(sacramento, prop = 0.75, strata = price)
sacramento_train <- training(sacramento_split)
sacramento_test <- testing(sacramento_split)

Next, we’ll use cross-validation \index{cross-validation} to choose $K$. In KNN classification, we used accuracy to see how well our predictions matched the true labels. We cannot use the same metric in the regression setting, since our predictions will almost never exactly match the true response variable values. Therefore in the context of KNN regression we will use root mean square prediction error \index{root mean square prediction error|see{RMSPE}}\index{RMSPE} (RMSPE) instead. The mathematical formula for calculating RMSPE is:

\[\text{RMSPE} = \sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}(y_i - \hat{y}_i)^2}\]

where:

$n$ is the number of observations,
$y_i$ is the observed value for the $i^\text{th}$ observation, and
$\hat{y}_i$ is the forecasted/predicted value for the $i^\text{th}$ observation.

In other words, we compute the squared difference between the predicted and true response value for each observation in our test (or validation) set, compute the average, and then finally take the square root. The reason we use the squared difference (and not just the difference) is that the differences can be positive or negative, i.e., we can overshoot or undershoot the true response value. Figure @ref(fig:07-verticalerrors) illustrates both positive and negative differences between predicted and true response values. So if we want to measure error—a notion of distance between our predicted and true response values—we want to make sure that we are only adding up positive values, with larger positive values representing larger mistakes. If the predictions are very close to the true values, then RMSPE will be small. If, on the other-hand, the predictions are very different from the true values, then RMSPE will be quite large. When we use cross-validation, we will choose the $K$ that gives us the smallest RMSPE.

# save the seed
seedval <- .Random.seed

# (synthetic) new prediction points
pts <- tibble(sqft = c(1250, 1850, 2250), price = c(250000, 200000, 500000))
finegrid <- tibble(sqft = seq(from = 900, to = 3900, by = 10))

# fit the model
sacr_recipe_hid <- recipe(price ~ sqft, data = small_sacramento) |>
  step_scale(all_predictors()) |>
  step_center(all_predictors())

sacr_spec_hid <- nearest_neighbor(weight_func = "rectangular", 
                                  neighbors = 4) |>
  set_engine("kknn") |>
  set_mode("regression")

sacr_fit_hid <- workflow() |>
  add_recipe(sacr_recipe_hid) |>
  add_model(sacr_spec_hid) |>
  fit(data = small_sacramento)

sacr_full_preds_hid <- sacr_fit_hid |> 
                           predict(finegrid) |>
                           bind_cols(finegrid)

sacr_new_preds_hid <- sacr_fit_hid |> 
                           predict(pts) |>
                           bind_cols(pts)

# plot the vertical prediction errors
errors_plot <- ggplot(small_sacramento, aes(x = sqft, y = price)) +
  geom_point() +
  xlab("House size (square feet)") +
  ylab("Price (USD)") +
  scale_y_continuous(labels = dollar_format()) +
  geom_line(data = sacr_full_preds_hid, 
            aes(x = sqft, y = .pred), 
            color = "blue") + 
  geom_segment(
    data = sacr_new_preds_hid,
    aes(x = sqft, xend = sqft, y = price, yend = .pred), 
    color = "red") + 
  geom_point(data = sacr_new_preds_hid, 
             aes(x = sqft, y = price), 
             color = "black")

# restore the seed
.Random.seed <- seedval

errors_plot

Note: When using many code packages (tidymodels included), the evaluation output we will get to assess the prediction quality of our KNN regression models is labeled “RMSE”, or “root mean squared error”. Why is this so, and why not RMSPE? \index{RMSPE!comparison with RMSE} In statistics, we try to be very precise with our language to indicate whether we are calculating the prediction error on the training data (in-sample prediction) versus on the testing data (out-of-sample prediction). When predicting and evaluating prediction quality on the training data, we say RMSE. By contrast, when predicting and evaluating prediction quality on the testing or validation data, we say RMSPE. The equation for calculating RMSE and RMSPE is exactly the same; all that changes is whether the $y$s are training or testing data. But many people just use RMSE for both, and rely on context to denote which data the root mean squared error is being calculated on.

Now that we know how we can assess how well our model predicts a numerical value, let’s use R to perform cross-validation and to choose the optimal $K$. First, we will create a recipe for preprocessing our data. Note that we include standardization in our preprocessing to build good habits, but since we only have one predictor, it is technically not necessary; there is no risk of comparing two predictors of different scales. Next we create a model specification for K-nearest neighbors regression. Note that we use set_mode("regression") now in the model specification to denote a regression problem, as opposed to the classification problems from the previous chapters. The use of set_mode("regression") essentially tells tidymodels that we need to use different metrics (RMSPE, not accuracy) for tuning and evaluation. Then we create a 5-fold cross-validation object, and put the recipe and model specification together in a workflow. \index{tidymodels}\index{recipe}\index{workflow}

sacr_recipe <- recipe(price ~ sqft, data = sacramento_train) |>
  step_scale(all_predictors()) |>
  step_center(all_predictors())

sacr_spec <- nearest_neighbor(weight_func = "rectangular", 
                              neighbors = tune()) |>
  set_engine("kknn") |>
  set_mode("regression")

sacr_vfold <- vfold_cv(sacramento_train, v = 5, strata = price)

sacr_wkflw <- workflow() |>
  add_recipe(sacr_recipe) |>
  add_model(sacr_spec)

sacr_wkflw

print_tidymodels(sacr_wkflw)

Next we run cross-validation for a grid of numbers of neighbors ranging from 1 to 200. The following code tunes the model and returns the RMSPE for each number of neighbors. In the output of the sacr_results results data frame, we see that the neighbors variable contains the value of $K$, the mean (mean) contains the value of the RMSPE estimated via cross-validation, and the standard error (std_err) contains a value corresponding to a measure of how uncertain we are in the mean value. A detailed treatment of this is beyond the scope of this chapter; but roughly, if your estimated mean is 100,000 and standard error is 1,000, you can expect the true RMSPE to be somewhere roughly between 99,000 and 101,000 (although it may fall outside this range). You may ignore the other columns in the metrics data frame, as they do not provide any additional insight. \index{cross-validation!collect_metrics}

gridvals <- tibble(neighbors = seq(from = 1, to = 200, by = 3))

sacr_results <- sacr_wkflw |>
  tune_grid(resamples = sacr_vfold, grid = gridvals) |>
  collect_metrics() |>
  filter(.metric == "rmse")

# show the results
sacr_results

sacr_tunek_plot <- ggplot(sacr_results, aes(x = neighbors, y = mean)) +
  geom_point() +
  geom_line() +
  labs(x = "Neighbors", y = "RMSPE")

sacr_tunek_plot

Figure @ref(fig:07-choose-k-knn-plot) visualizes how the RMSPE varies with the number of neighbors $K$. We take the minimum RMSPE to find the best setting for the number of neighbors:

# show only the row of minimum RMSPE
sacr_min <- sacr_results |>
  filter(mean == min(mean))

sacr_min

The smallest RMSPE occurs when $K =$ r sacr_min |> pull(neighbors).

kmin <- sacr_min |> pull(neighbors)

Underfitting and overfitting¶

Similar to the setting of classification, by setting the number of neighbors to be too small or too large, we cause the RMSPE to increase, as shown in Figure @ref(fig:07-choose-k-knn-plot). What is happening here?

Figure @ref(fig:07-howK) visualizes the effect of different settings of $K$ on the regression model. Each plot shows the predicted values for house sale price from our KNN regression model for 6 different values for $K$: 1, 3, r kmin, 41, 250, and 932 (i.e., the entire dataset). For each model, we predict prices for the range of possible home sizes we observed in the data set (here 500 to 5,000 square feet) and we plot the predicted prices as a blue line.

gridvals <- c(1, 3, kmin, 41, 250, 932)

plots <- list()

for (i in 1:6) {
  if (i < 6){
      sacr_spec <- nearest_neighbor(weight_func = "rectangular", 
                                    neighbors = gridvals[[i]]) |>
        set_engine("kknn") |>
        set_mode("regression")

      sacr_wkflw <- workflow() |>
        add_recipe(sacr_recipe) |>
        add_model(sacr_spec)

      sacr_preds <- sacr_wkflw |>
        fit(data = sacramento_train) |>
        predict(sacramento_train) |>
        bind_cols(sacramento_train)

      plots[[i]] <- ggplot(sacr_preds, aes(x = sqft, y = price)) +
        geom_point(alpha = 0.4) +
        xlab("House size (square feet)") +
        ylab("Price (USD)") +
        scale_y_continuous(labels = dollar_format()) +
        geom_line(data = sacr_preds, aes(x = sqft, y = .pred), color = "blue") +
        ggtitle(paste0("K = ", gridvals[[i]])) +
        theme(text = element_text(size = 20), axis.title=element_text(size=20))
      } else {
      plots[[i]] <- ggplot(sacr_preds, aes(x = sqft, y = price)) +
        geom_point(alpha = 0.4) +
        xlab("House size (square feet)") +
        ylab("Price (USD)") +
        scale_y_continuous(labels = dollar_format()) +
        geom_hline(data = sacr_preds, 
                   mapping = aes(x = sqft), 
                   yintercept = mean(sacr_preds$price), 
                   color = "blue") +
        ggtitle(paste0("K = ", gridvals[[i]])) +
  theme(text = element_text(size = 20), axis.title=element_text(size=20)) 
  }
}

grid.arrange(grobs = plots, ncol = 2)

Figure @ref(fig:07-howK) shows that when $K$ = 1, the blue line runs perfectly through (almost) all of our training observations. This happens because our predicted values for a given region (typically) depend on just a single observation. In general, when $K$ is too small, the line follows the training data quite closely, even if it does not match it perfectly. If we used a different training data set of house prices and sizes from the Sacramento real estate market, we would end up with completely different predictions. In other words, the model is influenced too much by the data. Because the model follows the training data so closely, it will not make accurate predictions on new observations which, generally, will not have the same fluctuations as the original training data. Recall from the classification chapters that this behavior—where the model is influenced too much by the noisy data—is called overfitting; we use this same term in the context of regression. \index{overfitting!regression}

What about the plots in Figure @ref(fig:07-howK) where $K$ is quite large, say, $K$ = 250 or 932? In this case the blue line becomes extremely smooth, and actually becomes flat once $K$ is equal to the number of datapoints in the entire data set. This happens because our predicted values for a given x value (here, home size), depend on many neighboring observations; in the case where $K$ is equal to the size of the dataset, the prediction is just the mean of the house prices in the dataset (completely ignoring the house size). In contrast to the $K=1$ example, the smooth, inflexible blue line does not follow the training observations very closely. In other words, the model is not influenced enough by the training data. Recall from the classification chapters that this behavior is called underfitting; we again use this same term in the context of regression. \index{underfitting!regression}

Ideally, what we want is neither of the two situations discussed above. Instead, we would like a model that (1) follows the overall “trend” in the training data, so the model actually uses the training data to learn something useful, and (2) does not follow the noisy fluctuations, so that we can be confident that our model will transfer/generalize well to other new data. If we explore the other values for $K$, in particular $K$ = r sacr_min |> pull(neighbors) (as suggested by cross-validation), we can see it achieves this goal: it follows the increasing trend of house price versus house size, but is not influenced too much by the idiosyncratic variations in price. All of this is similar to how the choice of $K$ affects K-nearest neighbors classification, as discussed in the previous chapter.

Evaluating on the test set¶

To assess how well our model might do at predicting on unseen data, we will assess its RMSPE on the test data. To do this, we will first re-train our KNN regression model on the entire training data set, using $K =$ r sacr_min |> pull(neighbors) neighbors. Then we will use predict to make predictions on the test data, and use the metrics function again to compute the summary of regression quality. Because we specify that we are performing regression in set_mode, the metrics function knows to output a quality summary related to regression, and not, say, classification.

kmin <- sacr_min |> pull(neighbors)

sacr_spec <- nearest_neighbor(weight_func = "rectangular", neighbors = kmin) |>
  set_engine("kknn") |>
  set_mode("regression")

sacr_fit <- workflow() |>
  add_recipe(sacr_recipe) |>
  add_model(sacr_spec) |>
  fit(data = sacramento_train)

sacr_summary <- sacr_fit |>
  predict(sacramento_test) |>
  bind_cols(sacramento_test) |>
  metrics(truth = price, estimate = .pred) |>
  filter(.metric == 'rmse')

sacr_summary

Our final model’s test error as assessed by RMSPE is $`r format(round(sacr_summary |> pull(.estimate)), big.mark=",", nsmall=0, scientific=FALSE)`. Note that RMSPE is measured in the same units as the response variable. In other words, on new observations, we expect the error in our prediction to be *roughly* $r format(round(sacr_summary |> pull(.estimate)), big.mark=",", nsmall=0, scientific=FALSE). From one perspective, this is good news: this is about the same as the cross-validation RMSPE estimate of our tuned model (which was $`r format(round(sacr_min |> pull(mean)), big.mark=",", nsmall=0, scientific=FALSE)`), so we can say that the model appears to generalize well to new data that it has never seen before. However, much like in the case of KNN classification, whether this value for RMSPE is *good*—i.e., whether an error of around $r format(round(sacr_summary |> pull(.estimate)), big.mark=",", nsmall=0, scientific=FALSE) is acceptable—depends entirely on the application. In this application, this error is not prohibitively large, but it is not negligible either; $r format(round(sacr_summary |> pull(.estimate)), big.mark=",", nsmall=0, scientific=FALSE) might represent a substantial fraction of a home buyer’s budget, and could make or break whether or not they could afford put an offer on a house.

Finally, Figure @ref(fig:07-predict-all) shows the predictions that our final model makes across the range of house sizes we might encounter in the Sacramento area—from 500 to 5000 square feet. You have already seen a few plots like this in this chapter, but here we also provide the code that generated it as a learning challenge.

sacr_preds <- tibble(sqft = seq(from = 500, to = 5000, by = 10))

sacr_preds <- sacr_fit |>
  predict(sacr_preds) |>
  bind_cols(sacr_preds)

plot_final <- ggplot(sacramento_train, aes(x = sqft, y = price)) +
  geom_point(alpha = 0.4) +
  geom_line(data = sacr_preds, 
            mapping = aes(x = sqft, y = .pred), 
            color = "blue") +
  xlab("House size (square feet)") +
  ylab("Price (USD)") +
  scale_y_continuous(labels = dollar_format()) +
  ggtitle(paste0("K = ", kmin)) + 
  theme(text = element_text(size = 12))

plot_final

Multivariable KNN regression¶

As in KNN classification, we can use multiple predictors in KNN regression. In this setting, we have the same concerns regarding the scale of the predictors. Once again, predictions are made by identifying the $K$ observations that are nearest to the new point we want to predict; any variables that are on a large scale will have a much larger effect than variables on a small scale. But since the recipe we built above scales and centers all predictor variables, this is handled for us.

Note that we also have the same concern regarding the selection of predictors in KNN regression as in KNN classification: having more predictors is not always better, and the choice of which predictors to use has a potentially large influence on the quality of predictions. Fortunately, we can use the predictor selection algorithm from the classification chapter in KNN regression as well. As the algorithm is the same, we will not cover it again in this chapter.

We will now demonstrate a multivariable KNN regression \index{K-nearest neighbors!multivariable regression} analysis of the Sacramento real estate \index{Sacramento real estate} data using tidymodels. This time we will use house size (measured in square feet) as well as number of bedrooms as our predictors, and continue to use house sale price as our outcome/target variable that we are trying to predict. It is always a good practice to do exploratory data analysis, such as visualizing the data, before we start modeling the data. Figure @ref(fig:07-bedscatter) shows that the number of bedrooms might provide useful information to help predict the sale price of a house.

plot_beds <- sacramento |>
  ggplot(aes(x = beds, y = price)) +
  geom_point(alpha = 0.4) + 
  labs(x = 'Number of Bedrooms', y = 'Price (USD)') + 
  theme(text = element_text(size = 12))

plot_beds

Figure @ref(fig:07-bedscatter) shows that as the number of bedrooms increases, the house sale price tends to increase as well, but that the relationship is quite weak. Does adding the number of bedrooms to our model improve our ability to predict price? To answer that question, we will have to create a new KNN regression model using house size and number of bedrooms, and then we can compare it to the model we previously came up with that only used house size. Let’s do that now!

First we’ll build a new model specification and recipe for the analysis. Note that we use the formula price ~ sqft + beds to denote that we have two predictors, and set neighbors = tune() to tell tidymodels to tune the number of neighbors for us.

sacr_recipe <- recipe(price ~ sqft + beds, data = sacramento_train) |>
  step_scale(all_predictors()) |>
  step_center(all_predictors())

sacr_spec <- nearest_neighbor(weight_func = "rectangular", 
                              neighbors = tune()) |>
  set_engine("kknn") |>
  set_mode("regression")

Next, we’ll use 5-fold cross-validation to choose the number of neighbors via the minimum RMSPE:

gridvals <- tibble(neighbors = seq(1, 200))

sacr_multi <- workflow() |>
  add_recipe(sacr_recipe) |>
  add_model(sacr_spec) |>
  tune_grid(sacr_vfold, grid = gridvals) |>
  collect_metrics() |>
  filter(.metric == "rmse") |>
  filter(mean == min(mean))

sacr_k <- sacr_multi |>
              pull(neighbors)

sacr_multi

Here we see that the smallest estimated RMSPE from cross-validation occurs when $K =$ r sacr_k. If we want to compare this multivariable KNN regression model to the model with only a single predictor as part of the model tuning process (e.g., if we are running forward selection as described in the chapter on evaluating and tuning classification models), then we must compare the accuracy estimated using only the training data via cross-validation. Looking back, the estimated cross-validation accuracy for the single-predictor model was r format(round(sacr_min$mean), big.mark=",", nsmall=0, scientific = FALSE). The estimated cross-validation accuracy for the multivariable model is r format(round(sacr_multi$mean), big.mark=",", nsmall=0, scientific = FALSE). Thus in this case, we did not improve the model by a large amount by adding this additional predictor.

Regardless, let’s continue the analysis to see how we can make predictions with a multivariable KNN regression model and evaluate its performance on test data. We first need to re-train the model on the entire training data set with $K =$ r sacr_k, and then use that model to make predictions on the test data.

sacr_spec <- nearest_neighbor(weight_func = "rectangular", 
                              neighbors = sacr_k) |>
  set_engine("kknn") |>
  set_mode("regression")

knn_mult_fit <- workflow() |>
  add_recipe(sacr_recipe) |>
  add_model(sacr_spec) |>
  fit(data = sacramento_train)

knn_mult_preds <- knn_mult_fit |>
  predict(sacramento_test) |>
  bind_cols(sacramento_test)

knn_mult_mets <- metrics(knn_mult_preds, truth = price, estimate = .pred) |>
                     filter(.metric == 'rmse')
knn_mult_mets

This time, when we performed KNN regression on the same data set, but also included number of bedrooms as a predictor, we obtained a RMSPE test error of r format(round(knn_mult_mets |> pull(.estimate)), big.mark=",", nsmall=0, scientific=FALSE). Figure @ref(fig:07-knn-mult-viz) visualizes the model’s predictions overlaid on top of the data. This time the predictions are a surface in 3D space, instead of a line in 2D space, as we have 2 predictors instead of 1.

xvals <- seq(from = min(sacramento_train$sqft), 
             to = max(sacramento_train$sqft), 
             length = 50)
yvals <- seq(from = min(sacramento_train$beds), 
             to = max(sacramento_train$beds), 
             length = 50)

zvals <- knn_mult_fit |>
  predict(crossing(xvals, yvals) |> 
            mutate(sqft = xvals, beds = yvals)) |>
  pull(.pred)

zvalsm <- matrix(zvals, nrow = length(xvals))

plot_3d <- plot_ly() |>
  add_markers(
    data = sacramento_train,
    x = ~sqft,
    y = ~beds,
    z = ~price,
    marker = list(size = 2, opacity = 0.4, color = "red")
  ) |>
  layout(scene = list(
    xaxis = list(title = "Size (sq ft)"),
    zaxis = list(title = "Price (USD)"),
    yaxis = list(title = "Bedrooms")
  )) |>
  add_surface(
    x = ~xvals,
    y = ~yvals,
    z = ~zvalsm,
    colorbar = list(title = "Price (USD)")
  )

if(!is_latex_output()){  
  plot_3d
} else {
  scene = list(camera = list(eye = list(x = -2.1, y = -2.2, z = 0.75)))
  plot_3d <- plot_3d  |> layout(scene = scene)
  save_image(plot_3d, "img/plot3d_knn_regression.png", scale = 10)
  knitr::include_graphics("img/plot3d_knn_regression.png")
}

We can see that the predictions in this case, where we have 2 predictors, form a surface instead of a line. Because the newly added predictor (number of bedrooms) is related to price (as price changes, so does number of bedrooms) and is not totally determined by house size (our other predictor), we get additional and useful information for making our predictions. For example, in this model we would predict that the cost of a house with a size of 2,500 square feet generally increases slightly as the number of bedrooms increases. Without having the additional predictor of number of bedrooms, we would predict the same price for these two houses.

Strengths and limitations of KNN regression¶

As with KNN classification (or any prediction algorithm for that matter), KNN regression has both strengths and weaknesses. Some are listed here:

Strengths: K-nearest neighbors regression

is a simple, intuitive algorithm,
requires few assumptions about what the data must look like, and
works well with non-linear relationships (i.e., if the relationship is not a straight line).

Weaknesses: K-nearest neighbors regression

becomes very slow as the training data gets larger,
may not perform well with a large number of predictors, and
may not predict well beyond the range of values input in your training data.

Exercises¶

Practice exercises for the material covered in this chapter can be found in the accompanying worksheets repository in the “Regression I: K-nearest neighbors” row. You can launch an interactive version of the worksheet in your browser by clicking the “launch binder” button. You can also preview a non-interactive version of the worksheet by clicking “view worksheet.” If you instead decide to download the worksheet and run it on your own machine, make sure to follow the instructions for computer setup found in Chapter @ref(move-to-your-own-machine). This will ensure that the automated feedback and guidance that the worksheets provide will function as intended.

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Regression I: K-nearest neighbors {#regression1}

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