In this article, we’ll talk about what Density Estimation is and the role it plays in statistical analysis. We’ll analyze two popular density estimation methods, histograms and kernel density estimators, and analyze their theoretical properties as well as how they perform in practice. Finally, we’ll look at how density estimation may be used as a tool for classification tasks. Hopefully after reading this article, you leave with an appreciation of density estimation as a fundamental statistical tool, and a solid intuition behind the density estimation approaches we discuss here. Ideally, this article will also spark an interest in learning more about density estimation and point you towards additional resources to help you dive deeper than what is discussed here! Contents: Background Concepts What is density estimation? Histograms Overview Theoretical Properties Demonstration of Theoretical Properties Kernel Density Estimators (KDE) Naive Density Estimator KDE: Overview Kernel and Bandwidth Density Estimation for Classification Wrap-up Sources Background Concepts Learning/refreshing on the following concepts will be helpful to fully appreciate the rest of what is discussed in this article. Bias and variance: important concepts for discussing the accuracy of the density estimation approaches discussed. The cumulative distribution function and probability density function. Parametric vs. non-parametric statistics: knowing the distinction will help understand the relevance of the density estimation methods discussed. O notation: used to describe the asymptotic behavior of the bias/variance of the density estimators. Kernel: kind of important for the kernel density estimator. What is density estimation? Density estimation is concerned with reconstructing the probability density function of a random variable, X, given a sample of random variates X1, X2,…, Xn. Density estimation plays a crucial role in statistical analysis. It may be used as a standalone method for analyzing the properties of a random variable’s distribution, such as modality, spread, and skew. Alternatively, density estimation may be used as a means for further statistical analysis, such as classification tasks, goodness-of-fit tests, and anomaly detection, to name a few. Some of you may recall that the probability distribution of a random variable X can be completely characterized by its cumulative distribution function (CDF), F(⋅). If X is a discrete random variable, then we can derive its probability mass function (PMF), p(⋅), from its CDF via the following relationship: p(Xi) = F(Xi) − F(Xi-1), where Xi-1 denotes the largest value within the discrete distribution of X that is less than Xi. If X is continuous, then its probability density function (PDF), p(⋅), may be derived by differentiating its CDF i.e. F′(⋅) = p(⋅). Based on this, you may be wondering why we need methods to estimate the probability distribution of X, when we can just exploit the relationships stated above. Certainly, given a sample of data X1,…, Xn, we may always construct an estimate of its CDF. If X is discrete, then constructing its PMF is straightforward, as it simply requires counting the frequency of observations for each distinct value that appears in our sample. However, if X is continuous, estimating its PDF is not so trivial. Notice that our estimate of the CDF, F(⋅), will necessarily follow a discrete distribution, since we have a finite amount of empirical data. Since F(⋅) is discrete, we cannot simply differentiate it to obtain an estimate of the PDF. Thus, this motivates the need for other methods of estimating p(⋅). To provide some additional motivation behind density estimation, the CDF may be suboptimal to use for analyzing the properties of the probability distribution of X. For example, consider the following display. PDF vs. CDF of data following a bimodal distribution. Certain properties of the distribution of X, such as its bimodal nature, are immediately clear from analyzing its PDF. However, these properties are harder to notice from analyzing its CDF, due to the cumulative nature of the distribution. For many folks, the PDF likely provides a more intuitive display of the distribution of X — it is larger at values of X that are more likely to “occur” and smaller for values of X that are less likely. Broadly speaking, density estimation approaches may be categorized as parametric or non-parametric. Parametric density estimation assumes X follows some distribution that may be characterized by some parameters (ex: X ∼ N(μ,σ)). Density estimation in this case involves estimating the relevant parameters for the parametric distribution of X, and then plugging in these parameter estimates to the corresponding density function formula for X. Non-parametric density estimation makes less rigid assumptions about the distribution of X, and estimates the shape of the density function directly from the empirical data. As a result, non-parametric density estimates will typically have lower bias and higher variance compared to parametric density estimates. Non-parametric methods may be desired when the underlying distribution of X is unknown and we’re working with a large amount of empirical data. For the rest of this article, we’ll focus on analyzing two popular non-parametric methods for density estimation: Histograms and kernel density estimators (KDEs). We’ll dig into how they work, the benefits and drawbacks of each approach, and how accurately they estimate the true density function of a random variable. Finally, we’ll examine how density estimation can be applied to classification problems, and how the quality of the density estimator can impact classification performance. Histograms Overview Histograms are a simple non-parametric approach for constructing a density estimate from a sample of data. Intuitively, this approach involves partitioning the range of our data into distinct equal length bins. Then, for any given point, assign its density to be equal to the proportion of points that reside within the same bin, normalized by the bin length. Formally, given a sample of n observations partition the domain into M bins such that For a given point x ∈ βl, where βl denotes the lth bin, the density estimate produced by the histogram will be Pointwise density estimate of the histogram. Since the histogram density estimator assigns uniform density to all points within the same bin, the density estimate will be discontinuous at all of its breakpoints where the density estimates differ. Histogram density estimate for the standard Gaussian. Uniform densities are assigned to all points within the same bin. Above, we have the histogram density estimate of the standard Gaussian distribution generated from a sample of 1000 data points. We see that x = 0 and x = −0.5 lie within the same bin, and thus have identical density estimates. Theoretical Properties Histograms are a simple and intuitive method for density estimation. They make no assumptions about the underlying distribution of the random variable. Histogram estimation simply requires tuning the bin width, h, and the point where the histogram bins originate from, t0. However, we’ll see very soon that the accuracy of the histogram estimator is highly dependent on tuning these parameters appropriately. As desired, the histogram estimator is a true density function. It is non-negative over its entire domain. It integrates to 1. Integral of the histogram density estimator. We can evaluate the accuracy of the histogram estimator for estimating the true density, p(⋅), by decomposing its mean squared error into its bias and variance terms. First, lets examine its bias at a given point x ∈ (bk-1, bk]. Expected value of the pointwise histogram density estimate. Let’s take a bit of a leap here. Using the Taylor series expansion, the fact that the PDF is the derivative of the CDF, and |x − bk-1| ≤ h, we can derive the following. Thus, we have which implies Asymptotic bias of the histogram density estimator. Therefore, the histogram estimator is an unbiased estimator of the true density, p(⋅), as the bin width approaches 0. Now, let’s analyze the variance of the histogram estimator. Notice that as h → ∞, we have Therefore, Asymptotic variance of the histogram density estimator. Now, we’re at a bit of an impasse; we see that as h → ∞, the bias of the histogram density estimate decreases, while its variance increases.  We’re typically concerned with the accuracy of the density estimate at large sample sizes (i.e. as n → ∞). Therefore, to maximize the accuracy of the histogram density estimate, we’ll want to tune h to achieve the following behavior: Choose h to be small to minimize bias. As h → 0 and n → ∞, we must have nh → ∞ to minimize variance. In other words, the large sample size should overpower the small bin width, asymptotically. This bias-variance trade-off is not unexpected: Small bin widths may capture the density around a particular point with high precision. However, density estimates may change from small random variations across data sets as less points will fall within the same bin. Large bin widths include more data points when computing the density estimate at a given point, which means density estimates will be more robust to small random variations in the data. Let’s illustrate this trade-off with some examples. Demonstration of Theoretical Properties First, we’ll look at how small bin widths may lead to large variance in the histogram density estimator. For this example, we’ll draw four samples of 50 random variates, where each sample is drawn from a standard Gaussian distribution. We’ll set a relatively small bin width (h = 0.2). set.seed(25) # Standard Gaussian mu <- 0 sd <- 1 # Parameters for density estimate n <- 50 h <- 0.2 # Generate 4 samples of standard Gaussian samples <- replicate(4, rnorm(n, mean = mu, sd = sd), simplify = FALSE) # Setup 2x2 plot par(mfrow = c(2, 2), mar = c(4, 4, 3, 1)) # Plot histograms titles <- paste("Sample", 1:4) invisible(mapply(plot_histogram, samples, title = titles, MoreArgs = list(binwidth = h, origin = 0, line = 0))) Histogram density estimates (h = 0.2) generated from 4 different samples of the standard Gaussian. Notice the high variability in density estimates across samples. It’s clear that the histogram density estimates vary quite a bit. For instance, we see that the pointwise density estimate at x = 0 ranges from approximately 0.2 in Sample 4 to approximately 0.6 in Sample 2. Additionally, the distribution of the density estimate produced in Sample 1 appears almost bimodal, with peaks around −1 and a little above 0. Let’s repeat this exercise to demonstrate how large bin widths may result in a density estimate with lower variance, but higher bias. For this example, let’s draw four samples from a bimodal distribution consisting of a mixture of two Gaussian distributions, N(0, 1) and N(3, 1). We’ll set a relatively large bin width (h = 2). set.seed(25) # Bimodal distribution parameters - mixture of N(0, 1) and N(4, 1) mu_1 <- 0 sd_1 <- 1 mu_2 <- 3 sd_2 <- 1 # Density estimation parameters n <- 100 h <- 2 # Generate 4 samples from bimodal distribution samples <- replicate(4, c(rnorm(n/2, mean = mu_1, sd = sd_1), rnorm(n/2, mean = mu_2, sd = sd_2)), simplify = FALSE) # Set up 2x2 plotting grid par(mfrow = c(2, 2), mar = c(4, 4, 3, 1)) # Plot histograms titles <- paste("Sample", 1:4) invisible(mapply(plot_histogram, samples, title = titles, MoreArgs = list(binwidth = h, origin = 0, line = 0))) Histogram density estimates (h = 2) generated from 4 different samples of a bimodal distribution. These histograms fail to capture the bimodal nature of the data. There is still some variation in the density estimates across the four histograms, but they appear stable relative to the density estimates we saw above with smaller bin widths. For instance, it appears that the pointwise density estimate at x = 0 is approximately 0.15 across all the histograms. However, it’s clear that these histogram estimators introduce a large amount of bias, as the bimodal distribution of the true density function is masked by the large bin widths. Additionally, we mentioned previously that the histogram estimator requires tuning the origin point, t0. Let’s look at an example that illustrates the impact that the choice of t0 can have on the histogram density estimate. set.seed(123) # Distribution and density estimation parameters # Bimodal distribution: mixture of N(0, 1) and N(5, 1) n <- 50 data <- c(rnorm(n/2, mean = 0, sd = 1), rnorm(n/2, mean = 5, sd = 1)) h <- 3 # Set up plotting grid par(mfrow = c(1, 2), mar = c(4, 4, 3, 1)) # Same bin width, different origins plot_histogram(data, binwidth = h, origin = 0, title = paste("Bin width = ", h, ", Origin = 0")) plot_histogram(data, binwidth = h, origin = 1, title = paste("Bin width = ", h, ", Origin = 1")) Histogram density estimates of a bimodal distribution with different origin points. Notice the histogram on the right fails to capture the bimodal nature of the data. The histogram density estimates above differ in their origin point by a magnitude of 1. The impact of the different origin point on the resulting histogram density estimates is evident. The histogram on the left captures the fact that the distribution is bimodal with peaks around 0 and 5. In contrast, the histogram on the right gives the impression that the density of X follows a unimodal distribution with a single peak around 5. Histograms are a simple and intuitive approach to density estimation. However, histograms will always produce density estimates that follow a discrete distribution, and we’ve seen that the resulting density estimate may be highly dependent on an arbitrary choice of the origin point. Next, we’ll look at an alternative method for density estimation, Kernel Density Estimation, that addresses these shortcomings. Kernel Density Estimators (KDE) Naive Density Estimator We’ll first look at the most basic form of a kernel density estimator, the naive density estimator. This approach is also known as the “moving histogram”; it is an extension of the traditional histogram density estimator that computes the density at a given point by examining the number of observations that fall within an interval that is centered around that point. Formally, the pointwise density estimate at x produced by the naive density estimator can be written as follows. Pointwise density estimate of the Naive Density Estimator. Its corresponding kernel is defined as follows. Naive Density Estimator kernel function. Unlike the traditional histogram density estimate, the density estimate produced by the moving histogram does not vary based on the choice of origin point. In fact, there is no concept of “origin point” in the moving histogram, as the density estimate at x only depends on the points that lie within the neighborhood (x − (h/2), x + (h/2)). Let’s examine the density estimate produced by the naive density estimator for the same bimodal distribution as we used above for highlighting the histogram’s dependency on origin point. set.seed(123) # Bimodal distribution - mixture of N(0, 1) and N(5, 1) data <- c(rnorm(n/2, mean = 0, sd = 1), rnorm(n/2, mean = 5, sd = 1)) # Density estimate parameters n <- 50 h <- 1 # Naive Density Estimator: KDE with rectangular kernel using half the bin width # Rectangular kernel counts points within (x - h, x + h) pdf_est <- density(data, kernel = "rectangular", bw = h/2) # Plot PDF plot(pdf_est, main = "NDE: Bimodal Gaussian", xlab = "x", ylab = "Density", col = "blue", lwd = 2) rug(data) polygon(pdf_est, col = rgb(0, 0, 1, 0.2), border = NA) grid() Naive Density Estimate of a bimodal distribution containing a mixture of N(0, 1) and N(5, 1). Clearly, the density estimate produced by the naive density estimator captures the bimodal distribution much more accurately than the traditional histogram. Additionally, the density at each point is captured with much finer granularity. That being said, the density estimate produced by the NDE is still quite “rough” i.e. the density estimate does not have smooth curvature. This is because each observation is weighted as “all or nothing” when computing the pointwise density estimate, which is apprent from its kernel, K0. Specifically, all points within the neighborhood (x − (h/2), x + (h/2)) contribute equally to the density estimate, while points outside the interval contribute nothing. Ideally, when computing the density estimate for x, we would like to weigh points in proportion to their distance from x, such that the points closer/farther from x have a higher/lower impact on its density estimate, respectively. This is essentially what the KDE does: it generalizes the naive density estimator by replacing the uniform density function with an arbitrary density function, the kernel. Intuitively, you can think of the KDE as a smoothed histogram. KDE: Overview The kernel density estimator generated from a sample X1,…, Xn, can be defined as follows: Pointwise density estimate of the KDE. Below are some popular choices for kernels used in density estimation. These are just several of the more popular kernels that are typically used for density estimation. For more information about kernel functions, check out the Wikipedia. If you’re seeking for some intuition behind what exactly a kernel function is (as I was), check out this quora thread. We can see that the KDE is a genuine density function. It is always non-negative, since K(⋅) is a density function. It integrates to 1. Integral of the KDE. Kernel and Bandwidth In practice, K(⋅) is chosen to be symmetric and unimodal around 0 (∫u⋅K(u)du = 0). Additionally, K(⋅) is typically scaled to have unit variance when used for density estimation (∫u2⋅K(u)du = 1). This scaling essentially standardizes the impact that the choice of bandwidth, h, has on the KDE, regardless of the kernel being used. Since the KDE at a given point is the weighted sum of its neighboring points, where the weights are computed by K(⋅), the smoothness of the density estimate is inherited from the smoothness of the kernel function. Smooth kernel functions will produce smooth KDEs. We can see that the Gaussian kernel depicted above is infinitely differentiable, so KDEs with the Gaussian kernel will produce density estimates with smooth curvature. On the other hand, the other kernel functions (Epanechnikov, rectangular, triangular) are not differentiable everywhere (ex: ±1), and in the case of the rectangular and triangular kernels, do not have smooth curvature. Thus, KDEs using these kernels will produce rougher density estimates. However, in practice, we’ll see that as long as the kernel function is continuous, the choice of the kernel has relatively little impact on the KDE compared to the choice of bandwidth. set.seed(123) # sample from standard Gaussian x <- rnorm(50) # kernel/bandwidths for KDEs kernels <- c("gaussian", "epanechnikov", "rectangular", "triangular") bandwidths <- c(0.5, 1, 2) colors_k <- rainbow(length(kernels)) colors_b <- rainbow(length(bandwidths)) plot_kde_comparison <- function(values, label, type = c("kernel", "bandwidth")) { type <- match.arg(type) plot(NULL, xlim = range(x) + c(-1, 1), ylim = c(0, 0.5), xlab = "x", ylab = "Density", main = paste("KDE with Different", label)) for (i in seq_along(values)) { if (type == "kernel") { d <- density(x, kernel = values[i]) col <- colors_k[i] } else { d <- density(x, bw = values[i], kernel = "gaussian") col <- colors_b[i] } lines(d$x, d$y, col = col, lwd = 2) } curve(dnorm(x), add = TRUE, lty = 2, lwd = 2) legend("topright", legend = c(as.character(values), "True Density"), col = c(if (type == "kernel") colors_k else colors_b, "black"), lwd = 2, lty = c(rep(1, length(values)), 2), cex = 0.8) rug(x) } plot_kde_comparison(kernels, "Kernels", type = "kernel") plot_kde_comparison(bandwidths, "Bandwidths", type = "bandwidth") We see that the KDEs for the standard Gaussian with various kernels are relatively similar, compared to the KDEs produced with various bandwidths. Accuracy of the KDE Let’s examine how accurately the KDE estimates the true density, p(⋅). As we did with the histogram estimator, we can decompose its mean squared error into its bias and variance terms. For details behind how to derive these bias and variance terms, check out lecture 6 of these notes. The bias and variance of the KDE at x can be expressed as follows. Asymptotic bias and variance of the KDE. Intuitively, these results give us the following insights: The effect of K(⋅) on the accuracy of the KDE is primarily captured via the term σ2K = ∫K(u)2du. The Epanechnikov kernel minimizes this integral, so theoretically it should produce the optimal KDE. However, we’ve seen that the choice of kernel has little practical impact on the KDE relative to its bandwidth. Additionally, the Epanechnikov kernel has a bounded support interval ([−1, 1]). As a result, it may produce rougher density estimates relative to kernels that are nonzero across the entire real number space (ex: Gaussian). Thus, the Gaussian kernel is commonly used in practice. Recall that the asymptotic bias and variance of the histogram estimator as h → ∞ was O(h) and O(1/(nh)), respectively. Comparing these against KDE tells us that the KDE improves upon the histogram density estimator primarily through decreased asymptotic bias. This is expected: the kernel smoothly varies the weight of the neighboring points of x when computing the pointwise density at x, instead of assigning uniform density to arbitrary fixed intervals of the domain. In other words, the KDE imposes a less rigid structure on the density estimate compared to the histogram approach. For histograms and KDEs, we’ve seen that the bandwidth h can have a significant impact on the accuracy of the density estimate. Ideally, we would select the h such that the mean squared error of the density estimator is minimized. However, it turns out that this theoretically optimal h depends on the curvature of the true density p(⋅), which is unknown practice (otherwise we wouldn’t need density estimation)! Some popular approaches for bandwidth selection include: Assuming the true density resembles some reference distribution p0(⋅) (ex: Gaussian), then plugging in the curvature of p0(⋅) to derive the bandwidth. This approach is simple, but it assumes the distribution of the data, so it may be a poor choice if you’re looking to build density estimates to explore your data. Non-parametric approaches to bandwidth selection, such as cross-validation and plug-in methods. The unbiased cross-validation and Sheather-Jones methods are popular bandwidth selectors and typically produce fairly accurate density estimates. For more information on the impact of bandwidth selection on the KDE, check out this blog post. set.seed(42) # Simulate data: a bimodal distribution x <- c(rnorm(150, mean = -2), rnorm(150, mean = 2)) # Define true density true_density <- function(x) { 0.5 * dnorm(x, mean = -2, sd = 1) + 0.5 * dnorm(x, mean = 2, sd = 1) } # Create plotting range x_grid <- seq(min(x) - 1, max(x) + 1, length.out = 500) xlim <- range(x_grid) ylim <- c(0, max(true_density(x_grid)) * 1.2) # Base plot plot(NULL, xlim = xlim, ylim = ylim, main = "KDE: Various Bandwidth Selection Methods", xlab = "x", ylab = "Density") # KDE with different bandwidths lines(density(x), col = "red", lwd = 2, lty = 4) h_scott <- 1.06 * sd(x) * length(x)^(-1/5) lines(density(x, bw = h_scott), col = "blue", lwd = 2, lty = 2) lines(density(x, bw = bw.ucv(x)), col = "darkgreen", lwd = 2, lty = 3) lines(density(x, bw = bw.SJ(x)), col = "purple", lwd = 2, lty = 4) # True density lines(x_grid, true_density(x_grid), col = "black", lwd = 2) # Add legend legend("topright", legend = c("Silverman (Default))", "Scott's Rule", "Unbiased CV", "Sheather-Jones", "True Density"), col = c("red", "blue", "darkgreen", "purple", "black"), lty = 1:6, lwd = 2, cex = 0.8) KDEs using various bandwidth selection methods where the underlying data follows a bimodal distribution. Notice the KDEs using the Sheather-Jones and Unbiased Cross-Validation methods produce density estimates closest to the true density. Density Estimation for Classification We’ve discussed a great deal about the underlying theory of histograms and KDE, and we’ve demonstrated how they perform at modeling the true density of some sample data. Now, we’ll look at how we can apply what we learned about density estimation for a simple classification task. For instance, say we want to build a classifier from a sample of n observations (x1, y1),…, (xn, yn), where each xi comes from a p-dimensional feature space, X, and yi corresponds to the target labels drawn from Y = {1,…, m}. Intuitively, we want to build a classifier such that for each observation, our classifier assigns it the class label k such that the following is satisfied. The Bayes classifier does precisely that, and computes the conditional probability above using the following equation. The Bayes Classifier This classifier relies on the following: πk = P(Y = k): the prior probability that an observation (xi, yi) belongs to the kth class (i.e. yi = k). This can be estimated by simply counting the proportion of points in each class from our sample data. fk(x) ≡ P(X = x | Y = k): the p-dimensional density function of X for all observations in target class k. This is harder to estimate: for each of the m target classes, we must determine the shape of the distribution for each dimension of X, and also whether there are any associations between the different dimensions. The Bayes classifier is optimal if the quantities above can be computed precisely. However, this is impossible to achieve in practice when working with a finite sample of data. For more detail behind why the Bayes classifier is optimal, check out this site. So the question becomes, how can we approximate the Bayes classifier? One popular method is the Naive Bayes classifier. Naive Bayes assumes class-conditional independence, which means that for each target class, it reduces the p-dimensional density estimation problem into p separate univariate density estimation tasks. These univariate densities may be estimated parametrically or non-parametrically. A typical parametric approach would assume that each dimension of X follows a univariate Gaussian distribution with class-specific mean and a diagonal co-variance matrix, whereas a non-parametric approach may model each dimension of X using a histogram or KDE. The parametric approach to univariate density estimation in Naive Bayes may be useful when we have a small amount of data relative to the size of the feature space, as the bias introduced by the Gaussian assumption may help reduce the variance of the classifier. However, the Gaussian assumption may not always be appropriate depending on the distribution of data that you’re working with. Let’s examine how parametric vs. non-parametric density estimates can impact the decision boundary of the Naive Bayes classifier. We’ll build two classifiers on the Iris dataset: one of them will assume each feature follows a Gaussian distribution, and the other will build kernel density estimates for each feature. # Parametric Naive Bayes param_nb <- naive_bayes(Species ~ ., data = train) # Nonparametric Naive Bayes # KDE with Gaussian kernel and Sheather-Jones bandwidth nonparam_nb <- naive_bayes(Species ~ ., data = train, usekernel = TRUE, kernel="gaussian", bw="sj") # play with bandwidth to see how it affects the classification boundaries! # Create grid for plotting decision boundaries x_seq <- seq(min(iris2D$Sepal.Length), max(iris2D$Sepal.Length), length.out = 200) y_seq <- seq(min(iris2D$Petal.Length), max(iris2D$Petal.Length), length.out = 200) grid <- expand.grid(Sepal.Length = x_seq, Petal.Length = y_seq) # Predict class for each point on grid grid$param_pred <- predict(param_nb, grid) grid$nonparam_pred <- predict(nonparam_nb, grid) # Plot decision boundaries nb_parametric <- ggplot() + geom_tile(data = grid, aes(x = Sepal.Length, y = Petal.Length, fill = param_pred), alpha = 0.3) + geom_point(data = train, aes(x = Sepal.Length, y = Petal.Length, color = Species), size = 2) + ggtitle("Parametric Naive Bayes Decision Boundary") + theme_minimal() nb_nonparametric <- ggplot() + geom_tile(data = grid, aes(x = Sepal.Length, y = Petal.Length, fill = nonparam_pred), alpha = 0.3) + geom_point(data = train, aes(x = Sepal.Length, y = Petal.Length, color = Species), size = 2) + ggtitle("Nonparametric Naive Bayes Decision Boundary") + theme_minimal() nb_parametric nb_nonparametric Decision boundaries produced by the parametric Naive Bayes classifier. Decision boundaries produced by the non-parametric Naive Bayes classifier. Notice the rough decision boundaries relative to that of its parametric counterpart. # Parametric Naive Bayes prediction on test data param_pred <- predict(param_nb, newdata = test) # Non-parametric Naive Bayes prediction on test data nonparam_pred <- predict(nonparam_nb, newdata = test) # Create confusion matrices param_cm <- confusionMatrix(param_pred, test$Species) nonparam_cm <- confusionMatrix(nonparam_pred, test$Species) output <- capture.output({ # Print confusion matrices cat("\n=== Parametric Naive Bayes Metrics ===\n") print(param_cm$table) cat("Parametric Naive Bayes Accuracy: ", param_cm$overall['Accuracy'], "\n\n") cat("=== Non-parametric Naive Bayes Metrics ===\n") print(nonparam_cm$table) cat("Nonparametric Naive Bayes Accuracy: ", nonparam_cm$overall['Accuracy'], "\n") }) cat(paste(output, collapse = "\n")) Classification performance for both Naive Bayes models. Non-parametric Naive Bayes achieved slightly better performance on our data. We see that the non-parametric Naive Bayes classifier achieves slightly better accuracy than its parametric counterpart. This is because the non-parametric density estimates produce a classifier with a more flexible decision boundary. As a result, several of the “virginica” observations that were incorrectly classified as “versicolor” by the parametric classifier ended up being classified correctly by the non-parametric model. That being said, the decision boundaries produced by non-parametric Naive Bayes appear to be rough and disconnected. Thus, there are some regions of the feature space where the classification boundary may be questionable, and fail to generalize well to new data. In contrast, the parametric Naive Bayes classifier produces smooth, connected decision boundaries that appear to accurately capture the general pattern of the feature distributions for each species. This distinction brings up an important point that “more flexible density estimation” does not equate to “better density estimation”, especially when applied to classification. After all, there’s a reason why Naive Bayes classification is popular. Although making less assumptions about the distribution of your data may seem desirable to produce unbiased density estimates, simplifying assumptions may be effective when there is insufficient empirical data to produce high quality estimates, or if the parametric assumptions are believed to be mostly accurate. In the latter case, parametric estimation will introduce little to no bias to the estimator, whereas non-parametric approaches may introduce large amounts of variance. Indeed, looking at the feature distributions below, the Gaussian assumption of parametric Naive Bayes doesn’t seem inappropriate. For the most part, it appears the class distributions for petal and sepal length appear to be unimodal and symmetric. iris_long <- pivot_longer(iris, cols = c(Sepal.Length, Petal.Length), names_to = "Feature", values_to = "Value") ggplot(iris_long, aes(x = Value, fill = Species)) + geom_density(alpha = 0.5, bw="sj") + facet_wrap(~ Feature, scales = "free") + labs(title = "Distribution of Sepal and Petal Lengths by Species", x = "Length (cm)", y = "Density") + theme_minimal() Density distributions for Petal and Sepal length. The univariate densities appear to be unimodal and symmetric across all species for both features. Wrap-up Thanks for reading! We dove into the theory behind the histogram and kernel density estimators and how to apply them in context.. Let’s briefly summarize what we discussed: Density estimation is a fundamental tool in Statistical Analysis for analyzing the distribution of a variable or as an intermediate tool for deeper statistical analysis. Density estimation approaches may be broadly categorized as parametric or non-parametric. Histograms and KDEs are two popular approaches for non-parametric density estimation. Histograms produce density estimates by computing the normalized frequency of points within each distinct bin of the data. KDEs are “smoothed” histograms that estimate the density at a given point by computing a weighted sum of its surrounding points, where neighbors are weighted in proportion to their distance. Non-parametric density estimation can be applied to classification algorithms that require modeling the feature densities for each target class (Bayesian classification). Classifiers built using non-parametric density estimates may be able to define more flexible decision boundaries at the cost of higher variance. Check out the sources below if you’re interested in learning more! The author has created all images in this article. Sources Learning Resources: García-Portugués, E. (2025). Notes for Nonparametric Statistics. Version 6.12.0. ISBN 978-84-09-29537-1. Available at https://bookdown.org/egarpor/NP-UC3M/. García-Portugués, E. (2022). A Short Course on Nonparametric Curve Estimation. Version 2.1.1. Available at https://bookdown.org/egarpor/NP-EAFIT/. UW Stat 425: Introduction to Nonparametric Statistics (Winter 2018) James et al., An Introduction to Statistical Learning Hastie et al., The Elements of Statistical Learning Andrey Akinshin, The importance of kernel density estimation bandwidth Datasets: Fisher, R. (1936). Iris [Dataset]. UCI Machine Learning Repository. https://doi.org/10.24432/C56C76. (CC BY 4.0) The post Non-Parametric Density Estimation: Theory and Applications appeared first on Towards Data Science.

We believe that the world is a better place with more creators in it. We make tools and services that help creators succeed, from individuals building their first games to professional studios working on the next great franchise.That’s why we continue to be excited by the promise of AI- and ML-driven techniques to reduce complexity, speed up creation, and, most importantly, unlock new ideas. Simply put, we think that this technology’s accessibility will help more people to become creators.We’ve worked for years, both internally and with partners, to explore how AI can be used in simulation, content creation, and game optimization. We see the present moment’s Cambrian explosion of generative AI as an opportunity to go even further.Unity is uniquely positioned to help you succeed while adopting generative AI because of the Unity Editor, runtime, data, and the Unity Network.More people use the Unity Editor to create games and other real-time 3D (RT3D) experiences than any other workflow in the world. Over the last 18 years, the Unity Editor has helped to democratize game development while contributing to a massive proliferation of new games across countless devices.Today, we strongly believe that the power of generative AI will enable Unity creators to be much more productive while ushering in scores of new creators who will face lower barriers to building RT3D games and experiences. We think that these AI tools will complement rather than replace existing tools and workflows. They offer the promise to help creators do more for and by themselves by filling the gaps in skill sets and resources so they can achieve what scarcely seems possible today.Just as a student might use a generative pre-trained transformer (GPT) tool to jumpstart research or even create a first draft before refining and finalizing a paper in Microsoft Word or Google Docs, Unity creators will be able to use natural-language generative tools together with deterministic, non-AI tools to create code, animations, physical effects, or other real-time content. Creators will move back and forth from rough approximations and text to fine-grained controls and code to iterate and refine the experience they envision.What’s better, we’re building the technology in the Unity Editor to better define what AI draws from. This not only means using appropriate and licensable datasets for generating content but also integrating AI techniques that are customized to their specific content (for example, by using Low-Rank Adaptation, or LoRA, language models during asset builds to deliver new content that’s trained on their existing work).The Unity runtime powers the most real-time applications in the world, with billions of downloads on billions of devices every month, in well over 100 countries. This means that Unity is the predominant way that content created with AI tools will come alive for consumers and users, since the output of any generative AI creation tools made available in the Unity Editor get delivered via the Unity runtime. The Unity runtime makes 3D content interactive and available on almost any device, ensuring that it responds to user input, as well as simulating effects like lighting or physics.But we see an even bigger opportunity. We believe that AI is not just the domain of creation tools, but that it offers the opportunity for new forms of interaction by moving inference – the process of feeding data through a machine learning model – to runtime.We’ve been working on this technology – code-named “Barracuda” – for more than five years. What will it mean when designers can build game loops that rely on inference on devices from mobile to console to web and PC? What happens when that AI capability is fast, efficient, scalable, and does not require expensive cloud compute?We have some ideas – NPCs that come to life, diffusion content as a gameplay mechanism, boundaryless user-generated content – but we know that our creators will do far more with this technology than we could ever even dream.Most of the digital content in the world today is 2D and linear – think sprites, photos, a set of film frames, a rendering of a building floor plan, or source code. AI data models train on this information to learn and, in the case of generative AI, to create content.Unity enables the real-time training of models based on unique datasets produced in the creation and operation of RT3D experiences. Through this training, we can build ever-richer services on top of Unity and provide extraordinary capabilities for our partners to leverage Unity as a data creation, simulation, and training engine for their own needs. Natural-language AI models incorporated into the Unity Editor and runtime train on real code and images. That real-usage training data is abstracted from its initial use (it’s not captured or recorded as-is), however this learning enables Unity’s customers to substantially increase their productivity.The Unity Network, whichconsists of our analytics tools, ad networks, publishing systems, and cloud services, reach a combined total of more than 4B users each month. Each of these service fields yields data that we can use to help our customers massively improve how they attract new users, increase engagement, or drive greater revenue from that base. Unity has been using the power of neural networks to help continuously optimize systems to support user acquisition, engagement and monetization for over three years.Generative AI has been used in some form or another for much of the history of video games, and it has tremendous potential as a tool to help developers achieve more with fewer resources. We’ll be sharing more over the coming months about our vision for AI at Unity, what we’re working on, and how this technology can help you achieve your vision.Stay tuned to the blog for more about Unity and AI, and, if you haven’t already, sign up for the AI Beta Program to be the first to hear about new tools and services.

A federal court held the very first hearing on President Donald Trump’s wide-ranging, so-called Liberation Day tariffs on Tuesday, offering the earliest window into whether those tariffs — and potentially all of the shifting tariffs Trump has imposed since he retook office — will be struck down. The case is V.O.S. Selections v. Trump.It is unclear how the three-judge panel that heard the case will rule, but it appears somewhat more likely than not that they will rule that the tariffs are unlawful. All three of the judges, who sit on the US Court of International Trade, appeared troubled by the Trump administration’s claim that the judiciary may not review the legality of the tariffs at all. But Jeffrey Schwab, the lawyer representing several small businesses challenging the tariffs, also faced an array of skeptical questions.Many of the judges’ questions focused on United States v. Yoshida International (1975), a federal appeals court decision which upheld a 10 percent tariff President Richard Nixon briefly imposed on nearly all foreign goods. That is understandable: Yoshida remains binding on the trade court, and the three judges must take it into account when they make their decision. It is not, however, binding upon the Supreme Court, whose justices will be free to ignore Yoshida if they want. Ultimately, that means it is unclear how much influence the trade court’s eventual decision will have over the Supreme Court, which is likely to have the final word on the tariffs. At the heart of V.O.S. Selections are four key words in the International Emergency Economic Powers Act of 1977 (IEEPA), the statute Trump relied on when he imposed these tariffs. That statute permits the president to “regulate” transactions involving foreign goods — a verb which Yoshida held is expansive enough to permit tariffs — but only “to deal with an unusual and extraordinary threat with respect to which a national emergency has been declared.” It is likely that the trade court’s decision will turn on what the words “unusual and extraordinary threat” means. While Yoshida offered guidance on “regulate,” there appears to be few, if any, precedents interpreting what those four words mean. In his executive order laying out the rationale for these tariffs, Trump claimed they are needed to combat “large and persistent annual US goods trade deficits” — meaning that the United States buys more goods from many countries than it sells to them. But it’s far from clear how this trade deficit, which has existed for decades, qualifies as either “unusual” or “extraordinary.”Schwab seemed to flub several direct questions from the judges asking him to come up with a universal rule they could apply to determine which “threats” are “unusual” or “extraordinary.” When Judge Gary Katzmann, an Obama appointee, asked Schwab to name the best case supporting his argument that a trade deficit is neither unusual nor extraordinary, for example, Schwab was unable to do so.That said, some of the judges sounded outright offended when Eric Hamilton, the lawyer for the Trump administration, claimed that the question of what constitutes an unusual or extraordinary threat is a “political question” — a legal term meaning that the courts aren’t allowed to decide that matter. As Judge Jane Restani, a Reagan appointee, told Hamilton, his argument suggests that there is “no limit” to the president’s power to impose tariffs, even if the president claims that a shortage of peanut butter is a national emergency.The overall picture presented by the argument is that all three judges (the third is Judge Timothy Reif, a Trump appointee) are troubled by the broad power Trump claims in this case. But they were also frustrated by a lack of guidance — both from existing case law and from Schwab and Hamilton’s arguments — on whether Trump can legally claim the power to issue such sweeping tariffs.Early in the argument, Schwab appeared to be in trouble, as he faced a barrage of questions about how the Yoshida decision cuts against some of his arguments. As Restani told him at one point, the argument that a statute permitting the president to “regulate” does not include the power to impose tariffs is a nonstarter, because Yoshida held the opposite.That said, all three judges proposed ways to distinguish the Nixon tariffs upheld by Yoshida from the Trump tariffs now before the trade court.Restani, for her part, argued that the Nixon tariffs involved a “very different situation” that was both “new” and “extraordinary.” For several decades, US dollars could be readily converted into gold at a set exchange rate. Nixon ended this practice in 1971, in an event many still refer to as the “Nixon shock.” When he did so, he briefly imposed tariffs to protect US goods from fluctuating exchange rates.Yoshida, in other words, upheld temporary tariffs that were enacted in order to mitigate the impact of a sudden and very significant shift in US monetary policy, albeit a shift that Nixon caused himself. That’s a very different situation than the one surrounding Trump’s tariffs, which were enacted in response to ongoing trade deficits that have existed for many years.Restani and Katzmann also pointed to a footnote in Yoshida that said Congress enacted a new law, the Trade Act of 1974, after the Nixon shock. This footnote states a future attempt to impose similar tariffs “must, of course, comply with the statute now governing such action.” Whatever power Nixon might have had in 1971, in other words, may now be limited by newer laws.Reif also made a similar argument, pointing out that there is a separate federal statute dealing with trade practices such as “dumping,” when an exporter sells goods below their normal value. He questioned whether the president could bypass the procedures laid out in that anti-dumping statute by simply declaring an emergency, and then imposing whatever trade barriers the president wanted to impose under IEEPA.That said, none of the judges — and neither of the lawyers — were able to articulate a rule that would allow future courts to determine which presidential actions are “unusual” or “extraordinary.” Hamilton’s suggestion that courts can’t decide this question at all sunk like a pair of concrete shoes, with Katzmann arguing that the IEEPA’s “unusual and extraordinary” provision would be entirely “superfluous” if Congress hadn’t intended courts to enforce it.Schwab, meanwhile, earned a scolding from Restani when he kept trying to argue that Trump’s tariffs are such an obvious violation of the statute that there’s no need to come up with a broader legal rule. “You know it when you see it doesn’t work,” she told him — a reference to Justice Potter Stewart’s infamously vague standard for determining what constitutes pornography.The three judges, in other words, expressed serious concerns about the Trump administration’s argument for the tariffs. But it’s not clear that they have figured out how to navigate the uncertain legal landscape looming over this case.Though the bulk of the argument focused on the four key words in the IEEPA, it’s not clear that a narrow decision holding that this law does not permit these tariffs will have much staying power. Trump could potentially try to impose the tariffs again, using the somewhat more drawn out process laid out in the 1974 Trade Act, which permits the government to “impose duties or other import restrictions” after the US Trade Representative makes certain findings. So if the courts issue a narrow ruling against these tariffs, they may have to go through a very similar dog and pony show in a few months.There are, however, two controversial legal doctrines popular with conservatives — known as “major questions” and “nondelegation” — which could lead to a more permanent reduction of Trump’s authority. Broadly speaking, both of these doctrines empower the courts to strike down a presidential administration’s actions even if those actions appear to be authorized by statute.Late in the argument, Restani seemed to latch onto the nondelegation theory. Under current law, Congress may delegate power to the president or a federal agency so long as it “shall lay down by legislative act an intelligible principle to which the person or body authorized to [exercise the delegated authority] is directed to conform.” This “intelligible principle” test is famously very deferential to Congress.Nevertheless, Restani asked some questions indicating that she may think that the IEEPA is the rare law which provides so little guidance to the president that it must be struck down. She noted that the law does permit Congress to pass a resolution canceling tariffs after the fact, but argued that this kind of after-the-fact review is not a substitute for an intelligible principle letting the president know how to act before he takes action.The major questions doctrine, meanwhile, establishes that Congress must “speak clearly” if it wants to give the executive branch authority over matters of “vast ‘economic and political significance.’” By some estimates, Trump’s tariffs are expected to reduce real family income by $2,800, so that’s certainly a matter of vast economic importance. Thus, to the extent that the IEEPA’s language is unclear, the major questions doctrine suggests that the law should be construed to not permit these tariffs.Hamilton’s primary argument against this line of reasoning is that the major questions doctrine does not apply to the president at all, only to actions by federal agencies that are subordinate to the president. But none of the three judges appeared sympathetic to this argument. Restani, in particular, seemed incredulous at the suggestion.Overall, the judges seemed interested in exploring the nondelegation and major questions factors, and repeatedly rebutted suggestions that ruling on the tariffs was beyond their power. And that suggests the trade court will likely rule against the tariffs. That outcome is far from certain, however, and the trade court is highly unlikely to have the final word on this question. But the legal case for the tariffs appeared weak before Tuesday’s hearing, and nothing that happened on Tuesday changes that.See More:

Choose wisely! The correct answer, the explanation, and an intriguing story await. Correct Answer: $20,000 A little background Although it wasn't initially intended for the general public, the first commercial audio CD recorder made its debut in 1991 at the National Association of Broadcasters (NAB) Convention in Las Vegas (later simply known as the NAB Show). Japanese electronics maker Denon introduced the DN-770R, a pro-grade Compact Disc recorder designed to appeal to radio stations and audio production facilities. These institutions could use the device to master demo discs, archive broadcasts, or distribute pre-recorded content to affiliate stations more efficiently than analog tape allowed. According to an article written by The Syracuse Newspapers at the time, the Denon DN-770R was the first of its kind and was being tendered by the electronics maker for $20,000: "Denon DN-770R audio CD recorder debuts as the first of its kind...The CDs it produces can be played on any CD player." However, the DN-770R was far from a plug-and-play solution. Contemporary reports noted that to make the system fully functional, users also needed around $100,000 worth of ancillary equipment, such as digital audio processors, mastering consoles, and synchronization gear. The cost of consumables added further barriers: blank CDs sold for approximately $35 to $40 each and could only be recorded once. Unlike cassette tapes, which were cheap, reusable, and ubiquitous in both professional and consumer audio environments, CD-Rs were a costly and rigid medium at the time. Despite the early limitations, the introduction of CD recording technology marked a turning point in digital audio. Until then, the compact disc – first launched commercially in 1982 – had been a strictly read-only format used for music distribution. The arrival of CD recording devices opened new possibilities for content creation and data storage. Denon's innovation was clearly aimed at a niche audience and focused solely on audio playback. The first modern CD burners were introduced between 1991 and 1992, when Sony, Philips, (the two creators of de CD format) and Yamaha each released CD recorders around the same time. The Philips CDD521, Sony CDW-900E, and Yamaha YPR-201 were some of the entry-level professional models still sold for over $10,000. Throughout the early 1990s, CD recorders remained out of reach for most consumers, however the landscape began to change rapidly in the years that followed. In 1995, Hewlett-Packard introduced the HP SureStore 4020i, one of the first CD-R drives marketed to consumers. Priced at $995, it was a significant milestone in making CD recording accessible to small businesses, hobbyists, and eventually the mass market. The 4020i could write both data and audio CDs. This democratization of CD recording paved the way for a surge in home media production during the late 1990s and early 2000s. It enabled users to burn custom music CDs, back up data, and share multimedia content in a way that had previously required professional studio equipment. As the price of CD-R drives and blank media continued to fall, the format became an essential part of personal computing and digital media until it was gradually overtaken by DVD, flash storage, and eventually cloud-based solutions.

True emotional attunement involves about noticing, responding and showing up when it counts. Here ... More are three things emotionally responsive partners do.getty Some individuals are naturally more attuned to their partner’s emotional cues. They notice changes in tone, body language or energy prior to being explicitly told that something’s wrong. Sometimes it’s a pause, a sigh or a shorter reply than usual. Others rely more on direct communication, expecting their partner to verbalize what they need. They may think, “If it’s important, they’ll tell me.” While it’s impossible and unrealistic to expect a partner to read your mind, emotional attunement supplements healthy, overt communication. Without it, there may be disconnects in otherwise stable relationships. One partner may interpret the other’s lack of responsiveness as indifference, while the other may simply not realize anything is amiss. And that’s where most people get it wrong — not because they don’t care, but because they’re looking for something more direct. Being emotionally receptive isn’t about fixing things for the other person or reading minds. It’s about tuning in. Here are three signs you’re emotionally in tune with your partner, making them feel truly seen, heard and cared for. 1. You Notice Subtle Changes In Their Nonverbal Cues Most people think emotional support means always saying the right thing. But in relationships, it often starts with recognition. Noticing a shift in your partner’s energy, or picking up on a silence that wasn’t there yesterday. How you respond in that moment — whether you lean in, pull back or overlook it — shapes how emotionally safe your relationship feels. A 2023 review published in Current Opinion in Psychology highlights how perceived partner responsiveness is a core component of emotional intimacy. When individuals feel that their partner understands, validates and cares for their internal experience, especially during everyday interactions, they’re more likely to feel secure and satisfied in the relationship. That responsiveness often begins with something as simple as noticing when the mood shifts. At the same time, nonverbal communication isn’t always clear. Another 2023 study published in Perspectives on Psychological Science cautions against the assumption that facial expressions or body language reliably convey emotional truth. According to the researchers, nonverbal cues are highly context-dependent and shaped by individual, cultural and relational factors. In other words, tuning in emotionally is less about decoding signals and more about staying attentive to changes within the patterns you’ve come to understand over time. Noticing a shift is only the first step. What you do next — how you respond to that shift — matters just as much. 2. You Validate Their Emotions Instead Of Minimizing Or Fixing Emotional receptivity isn’t limited to noticing when something is wrong. It also involves creating space for it without trying to control or dismiss it. When your partner expresses frustration, sadness or discomfort, the instinct might be to offer a solution or downplay the issue. But often, what they need most is to feel seen and heard. Offering validation means acknowledging what your partner is feeling without judgment or immediate redirection. It’s the difference between saying, “You shouldn’t feel that way,” and saying, “That sounds really difficult.” The first creates defensiveness. The second builds trust. This emotional availability also hinges on how a partner manages their own reactions during stressful moments. A 2020 study published in Emotion followed over 100 couples across 12 months and found that individuals with either highly inert or highly erratic emotional patterns were consistently perceived as less responsive by their partners. These perceptions, in turn, predicted steeper declines in relationship satisfaction. In other words, when someone struggles to respond to emotional cues in a balanced, context-sensitive way, either by shutting down or overreacting, it becomes harder for their partner to feel supported. Validation, then, is not just about what you say. It’s about how well your emotional presence matches the moment. Imagine you come home after a brutal day. You’re drained and frustrated. You sit down, and your partner, still in their own rhythm, says, “Can you help me with something real quick?” You say, “I’m really tired.” They don’t slow down. They stay upbeat, maybe a little impatient. “You’ll feel better if you get up. It’ll only take a second.” They mean well. But they haven’t shifted. They’re still in task mode while you’re barely holding it together. That kind of mismatch — where one person’s emotional state doesn’t adjust to the other’s — doesn’t just lead to hurt feelings. It leads to a partner feeling unseen, unsupported and eventually, disconnected. The fix isn’t dramatic. It starts with emotional awareness. It helps to pause long enough to register where your partner is emotionally and adjust your energy to meet that moment. If they seem overwhelmed, it might mean softening your tone, holding off on a request or simply saying, “You seem worn out. Want to sit for a bit?” This doesn’t require suppressing your own needs. It means making space for theirs too, especially in moments when they can’t carry both. Over time, these small adjustments signal something big: I care about you. I see where you are and I’m adjusting because I want you to feel seen. This willingness to shift and respond instead of react is at the heart of emotional responsiveness. 3. You Adjust To Meet Their Emotional Needs In The Moment Emotional support is often visible in our actions, through intentional shifts in energy, tone and timing. Receptive partners make small adjustments in real time based on what the other person seems to need. It could be pausing a story because your partner looks overwhelmed, skipping the joke when they’re quiet or holding off on a conversation when they’re not in a space to engage. None of this is about walking on eggshells. It’s about reading the room and respecting it. This kind of moment-to-moment flexibility, grounded in what the other person appears to need, is strongly linked to healthier relationships. A 2020 meta-analysis published in the Journal of Contextual Behavioral Science reviewed data from over 43,000 individuals across 174 studies and found that psychological flexibility — the ability to stay present and adjust behavior in emotionally challenging moments — is associated with greater relationship satisfaction, lower conflict and stronger emotional support. In contrast, inflexibility, or sticking to your own emotional rhythm regardless of your partner’s state, is linked to disconnection and higher levels of negative interaction. This kind of flexibility isn’t a path to losing yourself. It simply means recognizing that emotional needs aren’t always aligned in a given moment and choosing to respond accordingly. Partners who consistently do this are often described as easy to be around. Not because they avoid conflict, but because they create a sense of emotional safety. Their presence feels responsive, rather than rigid. The most responsive partners aren’t the ones who always get it right. They’re the ones who choose to be in tune with their partner, consistently and intentionally. Do you find your partner emotionally responsive? Take the science-backed Perceived Responsiveness Scale to find out.

Since the DMA took effect in Europe, Apple has been forced to loosen its grip by allowing third-party access to its platforms, devices, and operating systems. One key example is the iPhone’s NFC chip, which was once locked to Apple Pay but is now required to support third-party payment providers. It took longer than one would have guessed, but now PayPal is taking advantage of this wider access. Enn Eff Tseh, bitte Earlier this month, PayPal announced that German iPhone users would soon start earning cashback by making contactless payments using the PayPal app in select stores. As the company explained: In the coming weeks, PayPal’s first-ever contactless mobile wallet will launch nationwide, available to PayPal consumers in Germany before anywhere else in the world. The new contactless feature will be accessed through the latest version of the popular PayPal App (both iOS and Android). Consumers will be able to choose PayPal to pay safely and easily with a simple tap of their phone at any location that accepts Mastercard contactless payments. And, for the first time, the PayPal App will bring together a consolidated view of both online and offline purchases for the consumer.  Now, the system appears to be going live. As German site iPhone Ticker reported (via The Verge), PayPal’s direct contactless system has been intermittently appearing for users, in what looks like a soft rollout or limited trial ahead of an official launch. The system works on Mastercard-enabled payment terminals, allowing users to simply tap their iPhone and pay using the PayPal app. No Apple Pay required. Back in 2022, PayPal was one of the companies that backed the EU’s antitrust complaints against Apple’s control over the iPhone’s NFC chip, and now it appears to be one of the first to benefit from the outcome. At the time, as reported by Bloomberg, the company “was one of multiple companies making informal complaints about the situation to the commission,” and “helped spur a formal antitrust complaint,” that arguably (at least partially) led to the outcome of the DMA. Add 9to5Mac to your Google News feed.  FTC: We use income earning auto affiliate links. More.You’re reading 9to5Mac — experts who break news about Apple and its surrounding ecosystem, day after day. Be sure to check out our homepage for all the latest news, and follow 9to5Mac on Twitter, Facebook, and LinkedIn to stay in the loop. Don’t know where to start? Check out our exclusive stories, reviews, how-tos, and subscribe to our YouTube channel

There’s a routine, but profoundly telling moment when many visitors complete their first walkthrough at Future Pastime, the exhibition of paintings by the visionary artist and visual futurist Syd Mead currently on display in Manhattan’s Chelsea neighborhood. “His outlook on the future is so positive. I thought Syd Mead was dystopian,” many attendees exclaim, as if on cue.  Space Wheel Interior (1979) This perception is heavily guided by the work that—-for many—is the single most widely known reference point for Mead’s impact as a visual futurist: Director Ridley Scott’s 1982 rain soaked, techno-dystopian masterpiece, Blade Runner, whose visual landscape was largely Mead’s creation. In Blade Runner, Mead’s dystopian Los Angeles is so sleek, so complex, so rich with detail, one cannot blame audience members for confusing him with being a master of the morose. However, to quote Blade Runner 2049 director Denis Villeneuve (and another Mead collaborator): “[Syd] traveled in dystopia only once, and it was because of Ridley Scott. Syd’s first drawings of Los Angeles for Blade Runner were pure, bright and peaceful, but Ridley wanted his new world to be more claustrophobic and oppressive. And Syd dived into the darkness.”  One would reasonably expect Mead, with his fluent visual language of darkness so ably on display in Blade Runner, to be a committed future-cynic. However, it was quite the opposite. Mead was, in Vileneuve’s words, “one of the last great utopians,” a tidbit often lost on audiences raised on a steady diet of apocalyptic scenarios, dismal futures, and collapsed civilizations—the same narrative requests made often by Mead’s commissioning directors. In fact, Mead’s output outside the world of cinema—which in fact represents the lionshare of his more than 65-year professional career—is a replete and uniform world of robust optimism and hopeful aspiration. The future of Syd Mead is a bright and gilded one; the fulfillment of our greatest hopes and aspirations. It exists on the distant end of realism, yet is still somehow within reach.  The son of a Baptist minister (and part time art teacher), Mead gathered the strands of a childhood crowded by poverty and increasing world strife and instead formulated a unique worldview and artistic direction that was irrepressibly optimistic, often in spite of (and in stark contrast to) the current affairs and fortunes of the time.  To Mead, the prospect of an optimistic future was not a question of chance but of preparation. “Why wouldn’t you rehearse for a good future?” he often said. “I think that we should celebrate and rehearse for a bright future, and maybe it will come true. I don’t have time to illustrate misery or dystopian scenarios because they’ll happen. If you let everything go, they’ll happen anyway.” This outlook was successfully channeled into a singular career as an industrial futurist, becoming one of a rare group of individuals kept on speed dial by titans of industry to predict and illuminate likely future outcomes across myriad disciplines: architecture, urban planning, engineering, automotive, aeronautics, mass transit, spaceflight, technology, innovation, consumer goods, media, and more. Mead took it a step further by assuming the mantle of “visual futurist,” delivering his results not in research papers or dissertations, but via vivid, dynamic artworks, most principally paintings in his medium of choice: gouache.  Running of the 200th Kentucky Derby (1975) What he never anticipated was that the language of his industrial forecasting work would escape its enclosure and go on to define the visual identity of modern science fiction storytelling and cinematic futurism.   Mead entered professional life with nary a thought nor desire to work in the movies. In fact, movies were banned in the Mead household until he was 13 years old, with his earlier years filled with a potent mixture of his father’s chief obsessions: end-times religion, study and practice of painting and fine arts, and the pulp science fiction adventures of Buck Rogers and Flash Gordon (which Kenneth, his father, would buy family friendly editions so they could read together). Syd’s personal contributions to this percolating artistic concoction—an obsession with automobiles, the vibrant creations of contemporary illustrators as diverse as Maxfield Parrish and Chesley Bonestell, and the constant desire to innovate and improve one’s artistic craft—provided the ingredients to a simmering brew that one day would synthesize into the poetic future seer we celebrate today. By his mid-teens, Mead distanced himself from the old time religion of his parents as a new devotion emerged:bright future. The works of his adolescence and early adulthood burst with an almost unbridled gusto of color and life as potential futures were considered. Cowboys draped in pastel tour rural ranch grounds on horseback and bi-copter. Flying transport, early prototypes of Blade Runner-esque vehicles to come, twenty five years before the fact, are wrapped in chrome and almost seem to lift off the page. Join our mailing list Get the best of Den of Geek delivered right to your inbox! After his 1959 graduation from the Art Center in Los Angeles where he majored in industrial design, word quickly spread of this young man with a “future touch:” the uncanny ability to conceptualize and render achievable futures with the dexterity of an industrial engineer, the functionality of a city planner, and the poetry of a master painter. His first post-college berth at Ford’s Advanced Car Styling studio—where he worked on futuristic concept cars like the Gyron—lasted all of 26 months. The “thinness of purpose” got to him, he said. “If the company decided to stop making cars and start making washing machines next week, the process would not change at all—you just start doing sketches of washing machines.”  Mead departed Ford in 1961, and it was a providential overture made by industrial giant U.S. Steel shortly prior that which truly set the stage for the budding visual futurist to prove his worth. Aluminum, both lighter and cheaper, had been steadily eating into U.S. Steel’s profits, as well as the public’s romance with the once iconic alloy. The task for Mead was clear: to portray steel as a material of the future. Fifty years. One hundred years. Beyond. To make it relevant again today by showcasing its dreamy successes of tomorrow.  It was the assignment he had been practicing for his whole life, and Mead did not disappoint. Essentially given carte blanche, he remarkably completed the entire book—text and images—in just 30 days. While first intended as a product marketing catalogue aimed squarely at U.S. Steel customers, clients, and manufacturing partners, his work was such a sensation that the company commissioned four more books through 1969, and expanded the distribution of the volumes, running adverts for free copies in young people’s magazines, and seeding editions to all of the major art and design schools across the country. The result was the 1960s equivalent of a viral sensation; the books “went horizontal through the design community,” as Syd put it himself. Blossoming directors, designers, technologists, and innovators sought out the books and cherished them as cult objects. Sacred windows into the future of both technology and design. Acclaimed genre director Joe Johnston, who cut his teeth as art director on the original Star Wars Wheeless Truck (1969) As a result, the “Syd Mead look” is all over Star Wars, very specifically in the case of the AT-ATs, the iconic tall legged imperial transports featured in The Empire Strikes Back, which were derived by Johnston and the Lucasfilm team directly from Mead’s “Wheelless Truck” Despite his overwhelming influence on the series, Mead was not involved directly in Star Wars. He wasn’t even asked. It’s an understandable oversight. The concept of calling up Syd Mead in 1975 (by then an acclaimed industrial designer and futurist) to work on a motion picture is equivalent to asking Zaha Hadid in her prime to design buildings for an episode of Dynasty. Requests like these were simply not made, and as the 20th anniversary of Syd Mead’s professional career approached, he had yet to work on a single motion picture, let alone even be asked. This all changed when a fortuitous thought entered the mind of Star Wars VFX director John Dykstra in early 1979. Dykstra, who had attended Long Beach State alongside Johnston, and maintained a personal collection of Mead’s U.S. Steel books, now found himself as the visual effects lead on Star Trek: The Motion Picture. With the design of the film’s central antagonist—a leviathan entity of “unimaginable scale” named V’ger—suddenly in jeopardy, Dykstra took a leap of faith and reached out to Mead, uttering 12 words that would change both film and futurism forever:  “Syd, would you like to work on a science fiction motion picture?” Syd replied, “Sure.” In this one moment, the circuit was completed, and the board lit up. Suddenly alerted to the fact that their favorite futurist was down to do film, the entire town pivoted to Mead in unison. Ridley Scott, Steven Lisberger, Peter Hyams, James Cameron, and John Badham all reached out in quick succession. By 1986, via Blade Runner, Tron, 2010, Aliens, and Short Circuit, the very aesthetic of science fiction was altered forever… not by the hand of an imaginative production designer, but an actual futurist trained on developing real world futures to be built.     Elon Solo is the co-curator of Syd Mead: Future Pastime, an exhibition exploring the original artwork of Syd Mead, currently on display in NYC through May 21.  City on Wheels (1969)

Maybe your gaming laptop doesn’t have enough storage. Or you simply want an easy way to make your game library portable. An external SSD can fix both of these issues (and more) by providing an easy way to expand storage that you can take on the go. But choosing an external SSD means sorting through a dizzying array of options, and making a poor choice can leave you feeling hard done by. Lucky for you, we’ve done the testing and can offer some sure-fire recommendations that are guaranteed to help, and not hinder, your gaming setup. Why you should trust us: We are PCWorld. Our reviewers have been putting computer hardware through its paces for decades. Our external drive evaluations are thorough and rigorous, testing the limits of every product — from performance benchmarks to the practicalities of regular use. As consumers ourselves, we know what makes a product exceptional. For more about our testing process, scroll to the bottom of this article. Scroll below our recommendations to learn about other external SSDs for gaming that did not make our list. Lexar SL600 Blaze – Best 20Gbps external SSD for gaming Pros Good 20Gbps performer Top bang for the buck Five-year warranty Cons 4TB model not yet available The competition is very close in the top tier of 20Gbps external drives, with the big-name contenders trading wins up and down the benchmark charts. But a winner is a winner, and in the end, the Lexar overtook our previous champ, Crucial’s X10 Pro, even if only by a hair. The upshot is that you can expect excellent performance from the Lexar SL600. It also comes in a uniquely shaped form factor, complete with an opening to accommodate a lanyard, for easy toting. Gamers might appreciate that you can even add some bling by opting for the SL660 variant, which features RGB lighting within its miniature handle. The drive comes with the standard five year warranty. When performance is this closely matched among products, the determining factor should be price. And here, too, the SL600 is neck-and-neck with the Crucial X10 Pro, and priced slightly to significantly cheaper than some of its competitors, at least as of this writing — particularly at the 2TB level. Read our full Lexar SL600 20Gbps USB SSD review Teamgroup M200 – Best budget 20Gbps external SSD for gaming Pros Fast everyday performance Available in up to 8TB (eventually) capacity Attractively styled Cons No TBW rating Company will change components if shortages demand Writes slow to 200MBps off cache The Teamgroup M200 provides excellent bang for your buck with 20Gbps transfer rates and up to 4TB of storage for a very reasonable price. It has great everyday performance, too. Its slick military-style design is based on the CheyTac M200 sniper rifle—a perfect fit for those late night frag sessions. Unfortunately, Teamgroup doesn’t provide a TBW rating or official IP rating for the M200 so it’s more difficult to compare it as a whole to its competitors. However the M200 is a fast, extremely well priced external SSD with a gamer-friendly design that will look good and perform well in almost any setup. Read our full Teamgroup T-Force M200 20Gbps USB SSD review PNY RP60 20Gbps USB SSD – Best rugged 20Gbps external SSD for gaming Pros Handsome, IP65-rated design (dust-proof, water-resistant) Good 20Gbps performance Nice flat Type-C USB ribbon cable Cons USB port plug tether is difficult to reinsert. Best Prices Today: Retailer Price PNY $99.99 View Deal Price comparison from over 24,000 stores worldwide Product Price Price comparison from Backmarket If you tend to take your gaming drive everywhere, and/or you’re not the most careful person with hardware, a ruggedized external SSD is a practical answer. The PNY RP60 offers more than just a rugged IP65-rated exterior that protects against dust and water droplets; and while very handsome and lightweight to boot; the drive is also an admirable performer, even besting one of the fastest 20Gbps drives we’ve tested — the Crucial X10 Pro — in a couple of our tests. The RP60 is also competitively priced at $100 and $180 for 1TB and 2TB, respectively. Read our full PNY RP60 20Gbps USB SSD review Sabrent Rocket Nano V2 – Most portable 20Gbps external SSD for gaming Pros Extremely small profile Shock-absorbing silicone jacket Top-flight packaging Good overall performance Cons A tad behind the 20Gbps curve performance-wise Best Prices Today: Retailer Price Sabrent $119.99 View Deal Price comparison from over 24,000 stores worldwide Product Price Price comparison from Backmarket If you’re after a very small SSD that you can easily fit into a pocket, the Sabrent Rocket Nano V2 is that. This USB 3.2×2 20Gbps drive measures a petite 2.73 inches long, 1.16-inches wide, and 0.44-inches thick. It weighs a dainty 1.7 ounces. Of course, you’ll probably want to slide on its included shock-absorbing silicone jacket (shown in picture), which will add .06 inches to all its dimensions, while giving it a badass look. But looks aside, the Nano V2 is a solid performer. It wasn’t quite at the same level as our top picks in everything, but it traded wins and losses within the pack. For instance it was second only to the Crucial X10 Pro in our 450GB write test. And it took the top spot in CrystalDiskMark 8’s random writes, and was very competitive in random reads. This wee drive also comes in up to 4TB capacities, making it an all-around good choice if you’re looking to get a lot of storage and performance in a tiny package. We’re also fond of its five-year warranty and the nifty metal box it comes in, which can be repurposed for other uses. Read our full Sabrent Rocket Nano V2 review Adata SE920 EX – Best USB 4 external SSD for gaming Pros Fastest external storage we’ve tested (at 40Gbps) Affordable for the ilk Stylish enclosure Available up to 4TB Also fast on the Mac Cons Pricier than USB 3.2×2 (20Gbps) A USB 4 external SSD isn’t for everyone. Not only does your PC need to support the spec in order to take advantage of the 40Gbps transfers, but you’ll also need to be willing to pay a premium for the privilege. If you’re ready to enter the club, however, the Adata SE920 EX will reward you with the fastest USB 4 performance that we’ve experienced, and at a much more affordable price than our previous pick for USB 4, the OWC Express 1M2 — we’re talking $180 for a 1TB Adata SE920 versus $250 for the OWC drive. And if you want a lot of capacity, the Adata SE920 EX comes with up to 4TB, for a reasonable $500 (compared to OWC’s $598.99). In almost every benchmark, the SE920 EX beat the OWC Express 1M2 at 40Gbps performance, albeit by small margins. It also comes with a nifty built-in fan, which is activated by sliding open the enclosure. This kept our drive noticeably free of heat during our benchmark tests. The SE920 EX is also quite portable at 4.13 inches long by 2.52-inches wide by 0.62-inches thick, and weighing 7 ounces — another advantage it has over the bulkier OWC Express 1M2. In the end, the choice for a USB 4 external SSD is clear. Alternative option: You can get record-breaking performance by rolling your own USB 4 external SSD, using Ugreen’s CM642 enclosure. In our tests, the $110 enclosure combined with an NVMe SSD bested the performance of the two external USB 4 SSDs above. Read our full Adata SE920 USB4 SSD review Other external drive reviews: Sandisk Extreme Pro SSD with USB4: There’s a lot to like about this rugged, handsome design and relative affordability. But connection issues and comparably lackluster 40Gbps performance left me unamazed. PNY Pro Elite V3: This 10Gbps, Type-C USB stick is fast, but what really won me over is its clever, retracting physical design and svelte form-factor. Addlink P21: Good 20Gbps performance and a striking design are a winning formula for this external SSD. Seagate Ultra Compact SSD: This 10Gbps USB thumb drive delivers far better performance than generic 5Gbps or 400Mbps types, as well as free data recovery and other software perks. Corsair EX400U: While Corsair’s EX400U is on the slow side for a USB4 external SSD, it’s also less expensive than the competition. Ugreen CM642 USB4: Fast, handsome, rugged, and easy to use, Ugreen’s CM642 USB4 SSD enclosure is a great way to roll your own high-performance external storage. Corsair Flash Survivor: If you’re after an eminently portable USB stick with 10Gbps/NVMe speed that’s also weatherproofed, ruggedized, and looks burly as hell, this is the drive for you. Seagate Game Drive SSD: Targeted at PS4/PS5 owners, the 10Gbps SSD is very fast for its class, and attractively styled, complete with a Playstation logo, but it’s relatively pricey. Lexar Armor 700: Like the PNY RP60 above, the Lexar Armor 700 is a 20Gbps drive that can withstand some abuse — perfect for the gamer on the go, or the accident-prone. Its IP66 weatherized body is attractive, its performance is competitive. It’s also a bit pricey. Teamgroup PD20M: This little 20Gbps drive comes with a handy travel case and is one of the lightest portable drives we’ve tested. The only problem is, its performance slows considerably once its 20GB of cache has been tapped. Best for light-duty chores. Adata SD810: This is a solid 20Gbps drive, as long as you aren’t in the habit of writing very large amounts of data to it on a regular basis, because in our tests, the drive slowed down considerably in that scenario. That being said, the 4TB capacity is a particularly good value at just $300. Lexar SL500: A stablemate of the Lexar SL600 — our pick for best 20Gbps external drive — the SL500 stands out for its very slim and attractive form factor. It has almost identical performance to the SL600, with the exception of performing slower than its sibling in our 48GB file writes. How we test Drive tests currently utilize Windows 11, 64-bit running on an X790 (PCIe 4.0/5.0) motherboard/i5-12400 CPU combo with two Kingston Fury 32GB DDR5 4800MHz modules (64GB of memory total). Both 20Gbps USB and Thunderbolt 4 are integrated to the back panel and Intel CPU/GPU graphics are used. The 48GB transfer tests utilize an ImDisk RAM disk taking up 58GB of the 64GB of total memory. The 450GB file is transferred from a 2TB Samsung 990 Pro which also runs the OS. Each test is performed on a newly formatted and TRIM’d drive so the results are optimal. Note that in normal use, as a drive fills up, performance may decrease due to less NAND for secondary caching, as well as other factors. This can be less of a factor with the current crop of SSDs with far faster late-generation NAND. Caveat: The performance numbers shown apply only to the drive we were shipped and to the capacity tested. SSD performance can and will vary by capacity due to more or fewer chips to shotgun reads/writes across and the amount of NAND available for secondary caching. Vendors also occasionally swap components. If you ever notice a large discrepancy between the performance you experience and that which we report, by all means, let us know. To learn more about our testing methodology see PCWorld’s article on how we test external SSDs. Further reading: How to make your own external SSD and save some cash

Rust has become a popular programming language in recent years as it combines security and high performance and can be used in many applications. It combines the positive characteristics of C and C++ with the modern syntax and simplicity of other programming languages such as Python. In this article, we will take a step-by-step look at the installation of Rust on various operating systems and set up a simple command line interface to understand Rust’s structure and functionality. Installing Rust — step by step Regardless of the operating system, it is quite easy to install Rust thanks to the official installer rustup, which is available for free on the Rust website. This means that the installation only takes a few steps and only differs slightly for the various operating systems. Installing Rust under Windows In Windows, the installer completely controls the installation, and you can follow the steps below: Go to the “Install” subsection on the official Rust website (https://www.rust-lang.org/tools/install) and download the rustup-init.exe file there. The website recognizes the underlying operating system so that the appropriate suggestions for the system used are made directly. As soon as the download is complete, the rustup-init.exe file can be executed. A command line with various installation instructions then opens. Press the Enter key to run the standard installation to install Rust. This also includes the following tools: rustc is the compiler, which compiles the code and checks for errors before execution. cargo is Rust’s build and package management tool. rustup is the version manager. After successful installation, Rust should automatically be available in your PATH. This can be easily checked in PowerShell or CMD using the following commands: rustc --version cargo --version If “rustc” and “cargo” are then displayed in the output with the respective version numbers, then the installation was successful. However, if the command is not found, it may be due to the environment variables. To check these, you can follow the path “This PC –> Properties –> Advanced system settings –> Environment variables”. Once there, you should make sure that the path to Rust, for example “C:\Users\UserName\.cargo\bin”, is present in the PATH variable. Installing Rust under Ubuntu/Linux In Linux, Rust can be installed completely via the terminal without having to download anything from the Rust website. To install Rust, the following steps must be carried out: Open the terminal, for example, with the key combination Ctrl + Alt + T. To install Rust, the following command is executed: curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh 3. You will then be asked whether the installation should be started. This can be confirmed by entering “1” (default), for example. All required packages are then downloaded, and the environment is set up.4. You may have to set the path manually. In this case, you can use this command, for example: source $HOME/.cargo/env After the installation has been completed, you can check whether everything has worked properly. To do this, you can explicitly display the versions of rustc and cargo: rustc --version cargo --version Installing Rust under macOS There are several ways to install Rust on macOS. If you have installed Homebrew, you can simply use this to install Rust by executing the following command: brew install rustup rustup-init Alternatively, you can also install Rust directly using this script: curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh During the installation, you will be asked whether you want to run the standard installation. You can simply confirm this by pressing the Enter key. Regardless of the variant selected, you can then check the installation by displaying the version of Rust to ensure that everything has worked: rustc --version cargo --version Creating a Rust project with cargo During the installation of Rust, you have probably already come across the cargo program. This is the official package manager and build system of Rust and is comparable to pip in Python. cargo performs the following tasks, among others: Initialization of a project Management of dependencies Compiling the code Execution of tests Optimization of builds This allows you to manage complete projects in Rust without having to deal with complicated build scripts. It also helps you to set up new projects quickly and easily, which can then be filled with life. For our example, we will create a new project. To do this, we go to the terminal and navigate to a folder in which we want to save it. We then execute the following command to create a new Rust project: cargo new json_viewer --bin We call this project json_viewer because we are building a CLI tool that can be used to open and process JSON files. The --bin option indicates that we want to create an executable program and not a library. You should now be able to see the following folder structure in your directory after executing the command: json_viewer/ ├── Cargo.toml # Project configuration └── src └── main.rs # File for Rust Code Every new project has this structure. Cargo.toml contains all the dependencies and metadata of your project, such as the name, the libraries used, or the version. The src/main.rs, on the other hand, later contains the actual Rust code, which then defines the steps that are executed when the program is started. First, we can define a simple function here that generates an output in the terminal: fn main() { println!("Hello, Rust CLI-Tool!"); } The program can be easily called up from the terminal using cargo: cargo run For this call to work, it must be ensured that you are in the main directory of the project, i.e. where the Cargo.toml file is stored. If everything has been set up and executed correctly, you will receive this output: Hello, Rust CLI-Tool! With these few steps, you have just created your first successful Rust project, which we can build on in the next section. Building a CLI tool: Simple JSON parser Now we start to fill the project with life and create a program that can read JSON files and output their content in the terminal in a structured way. The first step is to define the dependencies, i.e., the libraries that we will use in the course of the project. These are stored in the Cargo.toml file. In Rust, the so-called crates are comparable to libraries or modules that offer certain predefined functionalities. For example, they can consist of reusable code written by other developers. We need the following crates for our project: serde enables the serialization and deserialization of data formats such as JSON or YAML. serde_json, on the other hand, is an extension that was developed specifically for working with JSON files. For your project to access these crates, they must be stored in the Cargo.toml file. This looks like this immediately after creating the project: [package] name = "json_viewer" version = "0.1.0" edition = "2021" [dependencies] We can now add the required crates in the [dependencies] section. Here we also define the version to be used: [dependencies] serde = "1.0" serde_json = "1.0" To ensure that the added dependencies are available in the project, they must first be downloaded and built. To do this, the following terminal command can be executed in the main directory: cargo build During execution, cargo searches the central Rust repository crates.io for the dependencies and the specified versions to download them. These crates are then compiled together with the code and cached so that they do not have to be downloaded again for the next build. If these steps have worked, we are now ready to write the actual Rust code that opens and processes the JSON file. To do this, you can open the src/main.rs file and replace the existing content with this code: use std::fs; use serde_json::Value; use std::env; fn main() { // Check arguments let args: Vec<String> = env::args().collect(); if args.len() < 2 { println!(“Please specify the path to your file.”); return; } // Read in File let file_path = &args[1]; let file_content = fs::read_to_string(file_path) .expect(“File could not be read.”); // Parse JSON let json_data: Value = serde_json::from_str(&file_content) .expect(“Invalid JSON format.”); // Print JSON println!(" JSON-content:\n{}”, json_data); } The code follows these steps: Check arguments: We read the arguments from the command line via env::args(). The user must specify the path to the JSON file at startup. Read the file: With the help of fs::read_to_string(), the content of the file is read into a string. Parse JSON: The crate serde_json converts the string into a Rust object with the type Value. Format output: The content is output legibly in the console. To test the tool, you can, for example, create a test file in the project directory under the name examples.json: { "name": "Alice", "age": 30, "skills": ["Rust", "Python", "Machine Learning"] } The program is then executed using cargo run and the path to the JSON file is also defined: cargo run ./example.json This brings us to the end of our first project in Rust and we have successfully built a simple CLI tool that can read JSON files and output them to the command line. This is what you should take with you Installing Rust is quick and easy in many operating systems. The other components that are required are already installed. With the help of cargo, an empty project can be created directly, which contains the necessary files and in which you can start writing Rust code Before you start Programming, you should enter the dependencies and download them using the build command. Now that everything is set up, you can start with the actual programming. The post Get Started with Rust: Installation and Your First CLI Tool – A Beginner’s Guide appeared first on Towards Data Science.