kcmeans

kcmeans: Conditional Expectation Function Estimation with K-Conditional-Means

https://github.com/thomaswiemann/kcmeans

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kcmeans: Conditional Expectation Function Estimation with K-Conditional-Means

README.Rmd

---
output: github_document
---



```{r, echo = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

# kcmeans


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``kcmeans`` is an implementation of the K-Conditional-Means (KCMeans) regression estimator analyzed by Wiemann (2023; [arxiv:2311.17021](https://arxiv.org/abs/2311.17021)) for conditional expectation function estimation using categorical features. The implementation leverages the unconditional KMeans implementation in one dimension using dynamic programming of the [``Ckmeans.1d.dp``](https://CRAN.R-project.org/package=Ckmeans.1d.dp) package.

See the working paper [Optimal Categorical Instrumental Variables](https://arxiv.org/abs/2311.17021) for further discussion of the KCMeans estimator.

## Installation

Install the latest development version from GitHub (requires [devtools](https://github.com/r-lib/devtools) package):

```{r, eval = FALSE}
if (!require("devtools")) {
  install.packages("devtools")
}
devtools::install_github("thomaswiemann/kcmeans", dependencies = TRUE)
```

Install the latest public release from CRAN:
```{r, eval = FALSE}
install.packages("kcmeans")
```

## Usage

To illustrate ``kcmeans``, consider simulating a small dataset with a continuous outcome variable ``y``, two observed predictors -- a categorical variable ``Z`` and a continuous variable ``X`` -- and an (unobserved) Gaussian error. As in Wiemann (2023), the reduced form has an unobserved lower-dimensional representation dependent on the latent categorical variable ``Z0``.
```{r}
# Load package
library(kcmeans)
# Set seed
set.seed(51944)
# Sample parameters
nobs = 800 # sample size
# Sample data
X <- rnorm(nobs)
Z <- sample(1:20, nobs, replace = T)
Z0 <- Z %% 4 # lower-dimensional latent categorical variable
y <- Z0 + X + rnorm(nobs)
```

``kcmeans`` is then computed by combining the categorical feature with the continuous feature. By default, the categorical feature is the first column. Alternatively, the column corresponding to the categorical feature can be set via the ``which_is_cat`` argument. Computation is _very_ quick -- indeed the dynamic programming algorithm of the leveraged ``Ckmeans.1d.dp`` package is polynomial in the number of values taken by the categorical feature ``Z``. See also ``?kcmeans`` for details.

```{r}
system.time({
kcmeans_fit <- kcmeans(y = y, X = cbind(Z, X), K = 4)
})
```

We may now use the ``predict.kcmeans`` method to construct fitted values and/or compute predictions of the lower-dimensional latent categorical feature ``Z0``. See also ``?predict.kcmeans`` for details.
```{r}
# Predicted values for the outcome + R^2
y_hat <- predict(kcmeans_fit, cbind(Z, X))
round(1 - mean((y - y_hat)^2) / mean((y - mean(y))^2), 3)

# Predicted values for the latent categorical feature + missclassification rate
Z0_hat <- predict(kcmeans_fit, cbind(Z, X), clusters = T) - 1
mean((Z0 - Z0_hat)!=0)


```

Finally, it is also straightforward to compute standard errors for the final coefficients, e.g., using ``summary.lm``:

```{r}
# Compute the linear regression object and call summary.lm
lm_fit <- lm(y ~ as.factor(Z0_hat) + X)
summary(lm_fit)
```


## Choice of K via Cross-Validation

Since the cardinality of the support of the underlying low-dimensional latent categorical variable is often unknown, it is useful to consider multiple KCMeans estimators with varying values for K. The below code snippet uses the [``ddml``](https://thomaswiemann.com/ddml/) package to compute the cross-validation mean-square prediction error (MSPE) of three KCMeans estimators (see also ``?ddml::crossval`` for details). 

In addition, the KCMeans MSPEs are compared to the MSPE of three alternative conditional expectation function estimators:

1. Ordinary least squares (see also ``?ddml::ols``)
2. Lasso with cross-validated penalty parameter (see also ``?ddml::mdl_glmnet``)
3. Ridge with cross-validated penalty parameter 


```{r}
# load the ddml package
library(ddml)

# one-hot encoding for ols, lasso, and ridge
Z_indicators <- model.matrix(~ as.factor(Z)) 

# Combine features and create indices
X_all <- cbind(Z, X, Z_indicators)
indx_factor <- 1:2
indx_indicators <- 2:(2 + ncol(Z_indicators))

# Create the learners, assign indicators to ols, lasso, and ridge
learner_list <- list(list(fun = kcmeans,
                          args = list(K = 2),
                          assign_X = indx_factor),
                     list(fun = kcmeans,
                          args = list(K = 4),
                          assign_X = indx_factor),
                     list(fun = kcmeans,
                          args = list(K = 6),
                          assign_X = indx_factor),
                     list(fun = ols,
                          assign_X = indx_indicators),
                     list(fun = mdl_glmnet,
                          assign_X = indx_indicators),
                     list(fun = mdl_glmnet,
                          args = list(alpha = 0),
                          assign_X = indx_indicators))

# Compute the cross-valdiation MSPE
cv_res <- crossval(y = y, X = X_all, 
                   learners = learner_list, 
                   cv_folds = 20, silent = T)

```

The results show that KCMeans with K=4 and K=6 achieve the smallest MSPE among the considered estimators.

```{r}
# Print the results
names(cv_res$mspe) <- c("KCMeans (K=2)", "KCMeans (K=4)", "KCMeans (K=6)",
                        "OLS", "Lasso", "Ridge")
round(cv_res$mspe, 4)

# Which learner is the best?
names(which.min(cv_res$mspe))
```

# References
Wiemann T (2023). "Optimal Categorical Instruments." https://arxiv.org/abs/2311.17021

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