What does kde do in python?
In the previous section we covered Gaussian mixture models (GMM), which are a kind of hybrid between a clustering estimator and a density estimator. Recall that a density estimator is an algorithm which takes a $D$-dimensional dataset and produces an estimate of the $D$-dimensional probability distribution which that data is drawn from. The GMM algorithm accomplishes this by representing the density as a weighted sum of Gaussian distributions. Kernel density
estimation (KDE) is in some senses an algorithm which takes the mixture-of-Gaussians idea to its logical extreme: it uses a mixture consisting of one Gaussian component per point, resulting in an essentially non-parametric estimator of density. In this section, we will explore the motivation and uses of KDE. We begin with the standard imports: In [1]: Motivating KDE: Histograms¶As already discussed, a density estimator is an algorithm which seeks to model the probability distribution that generated a dataset. For one dimensional data, you are probably already familiar with one simple density estimator: the histogram. A histogram divides the data into discrete bins, counts the number of points that fall in each bin, and then visualizes the results in an intuitive manner. For example, let's create some data that is drawn from two normal distributions: In [2]: def make_data(N, f=0.3, rseed=1): rand = np.random.RandomState(rseed) x = rand.randn(N) x[int(f * N):] += 5 return x x = make_data(1000) We have previously seen that the standard count-based histogram can be created with the In [3]: hist = plt.hist(x, bins=30, normed=True) Notice that for equal binning, this normalization simply changes the scale on the y-axis, leaving the relative heights essentially the same as in a histogram built from counts. This normalization is chosen so that the total area under the histogram is equal to 1, as we can confirm by looking at the output of the histogram function: In [4]: density, bins, patches = hist widths = bins[1:] - bins[:-1] (density * widths).sum() One of the issues with using a histogram as a density estimator is that the choice of bin size and location can lead to representations that have qualitatively different features. For example, if we look at a version of this data with only 20 points, the choice of how to draw the bins can lead to an entirely different interpretation of the data! Consider this example: In [5]: x = make_data(20) bins = np.linspace(-5, 10, 10) In [6]: fig, ax = plt.subplots(1, 2, figsize=(12, 4), sharex=True, sharey=True, subplot_kw={'xlim':(-4, 9), 'ylim':(-0.02, 0.3)}) fig.subplots_adjust(wspace=0.05) for i, offset in enumerate([0.0, 0.6]): ax[i].hist(x, bins=bins + offset, normed=True) ax[i].plot(x, np.full_like(x, -0.01), '|k', markeredgewidth=1) On the left, the histogram makes clear that this is a bimodal distribution. On the right, we see a unimodal distribution with a long tail. Without seeing the preceding code, you would probably not guess that these two histograms were built from the same data: with that in mind, how can you trust the intuition that histograms confer? And how might we improve on this? Stepping back, we can think of a histogram as a stack of blocks, where we stack one block within each bin on top of each point in the dataset. Let's view this directly: In [7]: fig, ax = plt.subplots() bins = np.arange(-3, 8) ax.plot(x, np.full_like(x, -0.1), '|k', markeredgewidth=1) for count, edge in zip(*np.histogram(x, bins)): for i in range(count): ax.add_patch(plt.Rectangle((edge, i), 1, 1, alpha=0.5)) ax.set_xlim(-4, 8) ax.set_ylim(-0.2, 8) The problem with our two binnings stems from the fact that the height of the block stack often reflects not on the actual density of points nearby, but on coincidences of how the bins align with the data points. This mis-alignment between points and their blocks is a potential cause of the poor histogram results seen here. But what if, instead of stacking the blocks aligned with the bins, we were to stack the blocks aligned with the points they represent? If we do this, the blocks won't be aligned, but we can add their contributions at each location along the x-axis to find the result. Let's try this: In [8]: x_d = np.linspace(-4, 8, 2000) density = sum((abs(xi - x_d) < 0.5) for xi in x) plt.fill_between(x_d, density, alpha=0.5) plt.plot(x, np.full_like(x, -0.1), '|k', markeredgewidth=1) plt.axis([-4, 8, -0.2, 8]); The result looks a bit messy, but is a much more robust reflection of the actual data characteristics than is the standard histogram. Still, the rough edges are not aesthetically pleasing, nor are they reflective of any true properties of the data. In order to smooth them out, we might decide to replace the blocks at each location with a smooth function, like a Gaussian. Let's use a standard normal curve at each point instead of a block: In [9]: from scipy.stats import norm x_d = np.linspace(-4, 8, 1000) density = sum(norm(xi).pdf(x_d) for xi in x) plt.fill_between(x_d, density, alpha=0.5) plt.plot(x, np.full_like(x, -0.1), '|k', markeredgewidth=1) plt.axis([-4, 8, -0.2, 5]); This smoothed-out plot, with a Gaussian distribution contributed at the location of each input point, gives a much more accurate idea of the shape of the data distribution, and one which has much less variance (i.e., changes much less in response to differences in sampling). These last two plots are examples of kernel density estimation in one dimension: the first uses a so-called "tophat" kernel and the second uses a Gaussian kernel. We'll now look at kernel density estimation in more detail. Kernel Density Estimation in Practice¶The free parameters of kernel density estimation are the kernel, which specifies the shape of the distribution placed at each point, and the kernel bandwidth, which controls the size of the kernel at each point. In practice, there are many kernels you might use for a kernel density estimation: in particular, the Scikit-Learn KDE implementation supports one of six kernels, which you can read about in Scikit-Learn's Density Estimation documentation. While there are several versions of kernel density estimation implemented in Python (notably in the SciPy and StatsModels packages), I prefer to use Scikit-Learn's version because of its efficiency and flexibility. It is implemented in the Let's first show a simple example of replicating the above plot using the Scikit-Learn In [10]: from sklearn.neighbors import KernelDensity # instantiate and fit the KDE model kde = KernelDensity(bandwidth=1.0, kernel='gaussian') kde.fit(x[:, None]) # score_samples returns the log of the probability density logprob = kde.score_samples(x_d[:, None]) plt.fill_between(x_d, np.exp(logprob), alpha=0.5) plt.plot(x, np.full_like(x, -0.01), '|k', markeredgewidth=1) plt.ylim(-0.02, 0.22) The result here is normalized such that the area under the curve is equal to 1. Selecting the bandwidth via cross-validation¶The choice of bandwidth within KDE is extremely important to finding a suitable density estimate, and is the knob that controls the bias–variance trade-off in the estimate of density: too narrow a bandwidth leads to a high-variance estimate (i.e., over-fitting), where the presence or absence of a single point makes a large difference. Too wide a bandwidth leads to a high-bias estimate (i.e., under-fitting) where the structure in the data is washed out by the wide kernel. There is a long history in statistics of methods to quickly estimate the best bandwidth based on rather stringent assumptions about the data: if you look up the KDE implementations in the SciPy and StatsModels packages, for example, you will see implementations based on some of these rules. In machine learning contexts,
we've seen that such hyperparameter tuning often is done empirically via a cross-validation approach. With this in mind, the In [11]: from sklearn.grid_search import GridSearchCV from sklearn.cross_validation import LeaveOneOut bandwidths = 10 ** np.linspace(-1, 1, 100) grid = GridSearchCV(KernelDensity(kernel='gaussian'), {'bandwidth': bandwidths}, cv=LeaveOneOut(len(x))) grid.fit(x[:, None]); Now we can find the choice of bandwidth which maximizes the score (which in this case defaults to the log-likelihood): Out[12]: {'bandwidth': 1.1233240329780276} The optimal bandwidth happens to be very close to what we used in the example plot earlier, where the bandwidth was 1.0 (i.e., the default width of Example: KDE on a Sphere¶Perhaps the most common use of KDE is in graphically representing distributions of points. For example, in the Seaborn visualization library (see Visualization With Seaborn), KDE is built in and automatically used to help visualize points in one and two dimensions. Here we will look at a slightly more sophisticated use of KDE for visualization of distributions. We will make use of some geographic data that can be loaded with Scikit-Learn: the geographic distributions of recorded observations of two South American mammals, Bradypus variegatus (the Brown-throated Sloth) and Microryzomys minutus (the Forest Small Rice Rat). With Scikit-Learn, we can fetch this data as follows: In [13]: from sklearn.datasets import fetch_species_distributions data = fetch_species_distributions() # Get matrices/arrays of species IDs and locations latlon = np.vstack([data.train['dd lat'], data.train['dd long']]).T species = np.array([d.decode('ascii').startswith('micro') for d in data.train['species']], dtype='int') With this data loaded, we can use the Basemap toolkit (mentioned previously in Geographic Data with Basemap) to plot the observed locations of these two species on the map of South America. In [14]: from mpl_toolkits.basemap import Basemap from sklearn.datasets.species_distributions import construct_grids xgrid, ygrid = construct_grids(data) # plot coastlines with basemap m = Basemap(projection='cyl', resolution='c', llcrnrlat=ygrid.min(), urcrnrlat=ygrid.max(), llcrnrlon=xgrid.min(), urcrnrlon=xgrid.max()) m.drawmapboundary(fill_color='#DDEEFF') m.fillcontinents(color='#FFEEDD') m.drawcoastlines(color='gray', zorder=2) m.drawcountries(color='gray', zorder=2) # plot locations m.scatter(latlon[:, 1], latlon[:, 0], zorder=3, c=species, cmap='rainbow', latlon=True); Unfortunately, this doesn't give a very good idea of the density of the species, because points in the species range may overlap one another. You may not realize it by looking at this plot, but there are over 1,600 points shown here! Let's use kernel density estimation to show this distribution in a more interpretable way: as a smooth indication of density on the map. Because the coordinate system here lies on a spherical surface rather than a flat plane, we will use the There is a bit of boilerplate code here (one of the disadvantages of the Basemap toolkit) but the meaning of each code block should be clear: In [15]: # Set up the data grid for the contour plot X, Y = np.meshgrid(xgrid[::5], ygrid[::5][::-1]) land_reference = data.coverages[6][::5, ::5] land_mask = (land_reference > -9999).ravel() xy = np.vstack([Y.ravel(), X.ravel()]).T xy = np.radians(xy[land_mask]) # Create two side-by-side plots fig, ax = plt.subplots(1, 2) fig.subplots_adjust(left=0.05, right=0.95, wspace=0.05) species_names = ['Bradypus Variegatus', 'Microryzomys Minutus'] cmaps = ['Purples', 'Reds'] for i, axi in enumerate(ax): axi.set_title(species_names[i]) # plot coastlines with basemap m = Basemap(projection='cyl', llcrnrlat=Y.min(), urcrnrlat=Y.max(), llcrnrlon=X.min(), urcrnrlon=X.max(), resolution='c', ax=axi) m.drawmapboundary(fill_color='#DDEEFF') m.drawcoastlines() m.drawcountries() # construct a spherical kernel density estimate of the distribution kde = KernelDensity(bandwidth=0.03, metric='haversine') kde.fit(np.radians(latlon[species == i])) # evaluate only on the land: -9999 indicates ocean Z = np.full(land_mask.shape[0], -9999.0) Z[land_mask] = np.exp(kde.score_samples(xy)) Z = Z.reshape(X.shape) # plot contours of the density levels = np.linspace(0, Z.max(), 25) axi.contourf(X, Y, Z, levels=levels, cmap=cmaps[i]) Compared to the simple scatter plot we initially used, this visualization paints a much clearer picture of the geographical distribution of observations of these two species. Example: Not-So-Naive Bayes¶This example looks at Bayesian generative classification with KDE, and demonstrates how to use the Scikit-Learn architecture to create a custom estimator. In In Depth: Naive Bayes Classification, we took a look at naive Bayesian classification, in which we created a simple generative model for each class, and used these models to build a fast classifier. For Gaussian naive Bayes, the generative model is a simple axis-aligned Gaussian. With a density estimation algorithm like KDE, we can remove the "naive" element and perform the same classification with a more sophisticated generative model for each class. It's still Bayesian classification, but it's no longer naive. The general approach for generative classification is this:
The algorithm is straightforward and intuitive to understand; the more difficult piece is couching it within the Scikit-Learn framework in order to make use of the grid search and cross-validation architecture. This is the code that implements the algorithm within the Scikit-Learn framework; we will step through it following the code block: In [16]: from sklearn.base import BaseEstimator, ClassifierMixin class KDEClassifier(BaseEstimator, ClassifierMixin): """Bayesian generative classification based on KDE Parameters ---------- bandwidth : float the kernel bandwidth within each class kernel : str the kernel name, passed to KernelDensity """ def __init__(self, bandwidth=1.0, kernel='gaussian'): self.bandwidth = bandwidth self.kernel = kernel def fit(self, X, y): self.classes_ = np.sort(np.unique(y)) training_sets = [X[y == yi] for yi in self.classes_] self.models_ = [KernelDensity(bandwidth=self.bandwidth, kernel=self.kernel).fit(Xi) for Xi in training_sets] self.logpriors_ = [np.log(Xi.shape[0] / X.shape[0]) for Xi in training_sets] return self def predict_proba(self, X): logprobs = np.array([model.score_samples(X) for model in self.models_]).T result = np.exp(logprobs + self.logpriors_) return result / result.sum(1, keepdims=True) def predict(self, X): return self.classes_[np.argmax(self.predict_proba(X), 1)] The anatomy of a custom estimator¶Let's step through this code and discuss the essential features: from sklearn.base import BaseEstimator, ClassifierMixin class KDEClassifier(BaseEstimator, ClassifierMixin): """Bayesian generative classification based on KDE Parameters ---------- bandwidth : float the kernel bandwidth within each class kernel : str the kernel name, passed to KernelDensity """ Each estimator in Scikit-Learn is a class, and it is most convenient for this class to inherit from the Next comes the class initialization method: def __init__(self, bandwidth=1.0, kernel='gaussian'): self.bandwidth = bandwidth self.kernel = kernel This is the actual code that is executed when the object is instantiated with Next comes the def fit(self, X, y): self.classes_ = np.sort(np.unique(y)) training_sets = [X[y == yi] for yi in self.classes_] self.models_ = [KernelDensity(bandwidth=self.bandwidth, kernel=self.kernel).fit(Xi) for Xi in training_sets] self.logpriors_ = [np.log(Xi.shape[0] / X.shape[0]) for Xi in training_sets] return self Here we find the unique classes in the training data, train a label = model.fit(X, y).predict(X) Notice that each persistent result of the fit is stored with a trailing underscore (e.g., Finally, we have the logic for predicting labels on new data: def predict_proba(self, X): logprobs = np.vstack([model.score_samples(X) for model in self.models_]).T result = np.exp(logprobs + self.logpriors_) return result / result.sum(1, keepdims=True) def predict(self, X): return self.classes_[np.argmax(self.predict_proba(X), 1)] Because this is a probabilistic classifier, we first implement Finally, the Using our custom estimator¶Let's try this custom estimator on a problem we have seen before: the classification of hand-written digits. Here we will load the digits, and compute the cross-validation score for a range of candidate bandwidths using the In [17]: from sklearn.datasets import load_digits from sklearn.grid_search import GridSearchCV digits = load_digits() bandwidths = 10 ** np.linspace(0, 2, 100) grid = GridSearchCV(KDEClassifier(), {'bandwidth': bandwidths}) grid.fit(digits.data, digits.target) scores = [val.mean_validation_score for val in grid.grid_scores_] Next we can plot the cross-validation score as a function of bandwidth: In [18]: plt.semilogx(bandwidths, scores) plt.xlabel('bandwidth') plt.ylabel('accuracy') plt.title('KDE Model Performance') print(grid.best_params_) print('accuracy =', grid.best_score_) {'bandwidth': 7.0548023107186433} accuracy = 0.966611018364 We see that this not-so-naive Bayesian classifier reaches a cross-validation accuracy of just over 96%; this is compared to around 80% for the naive Bayesian classification: In [19]: from sklearn.naive_bayes import GaussianNB from sklearn.cross_validation import cross_val_score cross_val_score(GaussianNB(), digits.data, digits.target).mean() One benefit of such a generative classifier is interpretability of results: for each unknown sample, we not only get a probabilistic classification, but a full model of the distribution of points we are comparing it to! If desired, this offers an intuitive window into the reasons for a particular classification that algorithms like SVMs and random forests tend to obscure. If you would like to take this further, there are some improvements that could be made to our KDE classifier model:
Finally, if you want some practice building your own estimator, you might tackle building a similar Bayesian classifier using Gaussian Mixture Models instead of KDE. Why is KDE used?In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights.
What is kind KDE in Python?Kernel density estimation (KDE) is a non-parametric method for estimating the probability density function of a given random variable. It is also referred to by its traditional name, the Parzen-Rosenblatt Window method, after its discoverers.
What does a KDE plot tell you?A kernel density estimate (KDE) plot is a method for visualizing the distribution of observations in a dataset, analogous to a histogram. KDE represents the data using a continuous probability density curve in one or more dimensions.
How does a KDE work?The KDE is calculated by weighting the distances of all the data points we've seen for each location on the blue line. If we've seen more points nearby, the estimate is higher, indicating that probability of seeing a point at that location.
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