# Scipy Gaussian

pyplot as plt plt. optimize improvements. Hey guys, I'm trying to implement a 2D parzen window on a cluster of data to estimate the pdf. It's well known that fourier transform of $\exp(-t^2)$ is $\sqrt{\pi}\exp(-\pi^2 k^2)$. Figure 1 1-D Gaussian distribution with mean 0 and =1 In 2-D, an isotropic (i. If this assumption fails, then non-parametric tests are considered for hypothesis testing. (d) A discrete approximation to a Gaussian with σ = 1, for which the mean is computed with n = 273. SciPy is built on the NumPy array framework and takes scientiﬁc programming to a whole new level by supplying advanced mathematical functions like integration, ordinary differential equation solvers, special functions, optimizations, and more. This function is typically several orders of magnitude faster than scipy. The functions scipy. I'm doing this for school and one of the requirements is to use a Gaussian window with covariance σ2=400σ2=400. stats import norm import numpy as np import matplotlib. Calculating the probability under a normal curve is useful for engineers. Mixture models provide a method of describing more complex propability distributions, by combining several probability distributions. gaussian_kde and matplotlib. edu October 30th, 2014. All gists Back to GitHub. For more details its the photopeak of Co60. quantile_gaussianize (x) [source] ¶ Normalize a sequence of values via rank and Normal c. As an example, Gaussian blur is one of the most commonly used filters when dealing with Machine Learning applications. The SciPy (Scientific Python) package extends the functionality of NumPy with a substantial collection of useful algorithms, like minimization, Fourier transformation, regression, and other applied mathematical techniques. Here in this SciPy Tutorial, we will learn the benefits of Linear Algebra, Working of Polynomials, and how to install SciPy. Consider this short program that creates and displays an image with Gaussian noise: # Import the packages you need import numpy as np import matplotlib. A numpy ndarray of shape n,n. By using this site, scipy. Introduction. by Christoph Gohlke, Laboratory for Fluorescence Dynamics, University of California, Irvine. optimize import curve_fit # counts is a numpy array which holds the number of counts for each. Project scipy/scipy pull requests. Here is a list of NumPy / SciPy APIs and its corresponding CuPy implementations. In this post we will see how to fit a distribution using the techniques implemented in the Scipy library. In the SciPy implementation of these tests, you can interpret the p value as follows. 05), that is used to interpret the p-value. The Gaussian distribution is a continuous function which approximates the exact binomial distribution of events. Gaussian approximation to B-spline basis function of order n. As the shape parameters of a beta distribution become large, the probability distribution becomes approximately normal (Gaussian). peaks_x = peakutils. The order of the filter along each axis is given as a sequence of integers, or as a single number. gaussian_kde takes a keyword argument weights. Utilizing SciPy correctly can sometimes be a very tricky proposition. gaussian_kde. Multivariate normal CDF values in Python. An order of 0 would perform convolution with a Gaussian kernel, whereas, an order of 1, 2, or 3 would convolve with first, second, and third derivatives of a Gaussian. This is what I do: import numpy as np from scipy. NumPy, matplotlib and SciPy HPC Python Antonio G omez-Iglesias [email protected] The second row are the values of scipy_data_fitting. The order of the filter along each axis is given as a sequence of integers, or as a single number. pdf ( pos ). Gaussian mixture models¶ sklearn. 683 of being within one standard deviation of the mean. Find file Copy path Ffisegydd Added a curve_fit example to scipy 53dc2cd Mar 27,. stats subpackage which can also be used to obtain the multivariate Gaussian probability distribution function: from scipy. In the example output from your code, $\sigma$ is huge, i. As the shape parameters of a beta distribution become large, the probability distribution becomes approximately normal (Gaussian). gaussian_kde¶ class scipy. This is because the padding is not done correctly, and does not take the kernel size into account (so the convolution "flows out of bounds of the image"). At this point, we have to call one of the Scipy methods. pdf, you will just plot a gaussian bell with. Representation of a kernel-density estimate using Gaussian kernels. Delaunay? python,scipy,delaunay. Figure 2 2-D Gaussian distribution with mean (0,0) and =1 The idea of Gaussian smoothing is to use this 2-D distribution as a point-spread' function, and this is achieved by. Probability Distributions in Python with SciPy and Seaborn March 1, 2018 by cmdline If you are a beginner in learning data science, understanding probability distributions will be extremely useful. distance import pdist, squareform # this is an NxD matrix, where N is number of items and D its dimensionalites X = loaddata() pairwise_dists = squareform. stats import norm import numpy as np import matplotlib. You can vote up the examples you like or vote down the ones you don't like. If not, then. Use the skimage. gaussian taken from open source projects. optimize and a wrapper for scipy. The standard deviations of the Gaussian filter are given for each axis as a sequence, or as a single number, in which case it is equal for all axes. gaussian_kde works for both uni-variate and multi-variate data. A Gaussian KDE can be thought as a non-parametric probability. So if you want the kernel matrix you do from scipy. SCIPY TUTORIAL 1. Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. The order of the filter along each axis is given as a sequence of integers, or as a single number. SciPy - Introduction. Normality Assumption. This can be a string to pass to the nosetests executable with the ‘-A’ option, or one of several special values. The standard-deviation of the Gaussian filter is passed through the parameter sigma. The following are code examples for showing how to use scipy. An order of 0 corresponds to convolution with a Gaussian. You can vote up the examples you like or vote down the ones you don't like. Python image processing libraries performance: OpenCV vs Scipy vs Scikit-Image feb 16, 2015 image-processing python numpy scipy opencv scikit-image. Introduction. I am a newbie in the scientific computing in python. Laplacian filters are derivative filters used to find areas of rapid change (edges) in images. Kite is a free autocomplete for Python developers. Normality Assumption. The gaussian_filter1d function implements a one-dimensional Gaussian filter. sparse improvements ----- - Significant performance improvement in CSR, CSC, and DOK indexing speed. •It uses linear interpolation as the default, but also can use other forms of interpolation. Numerical Routines: SciPy and NumPy¶. Until recently, I didn’t know how this part of scipy works, and the following describes roughly how I figured out what it does. pdf, you will just plot a gaussian bell with. Has anybody here any experience with SciPy? I'm trying to get SciPy to adjust a gaussian function to some data. integrate import quad from scipy. gaussian_kde¶ class scipy. I have this code (pieced together > from a few files) that does a gaussian filter on a single image in both > OpenCV and in SciPy. Another important problem is scattered fitting with smoothing, which differs from interpolation by presence of noise in the data and need for controlled smoothing. Since derivative filters are very sensitive to noise, it is common to smooth the image (e. distance import pdist, squareform # this is an NxD matrix, where N is number of items and D its dimensionalites X = loaddata() pairwise_dists = squareform. Here is a list of NumPy / SciPy APIs and its corresponding CuPy implementations. How to Integrate Gaussian Functions. 参数估计方法简单来讲，即假定样本集符合某一概率分布，然后根据样本集拟合该分布中的参数，例如：似然估计，混合高斯等，由. B: #———————————————– # populate the coefficient arrays #———————————————-from scipy. Matplotlib. This often relies critically on high fidelity numerical simulations, which are prohibitively computationally expensive. show() The above program will generate the following output. As @Jblasco suggested, you can minimize the sum of squares. gaussian_kde and matplotlib. In this Python tutorial, we will use Image Processing with SciPy and NumPy. gaussian_kde¶ class scipy. There are many forms of interpolation (polynomial, spline, kriging, radial basis function, etc. If not, then. After having observed some function values it can be converted into a posterior over functions. If not, then. Python Forums on Bytes. gmm is a package which enables to create Gaussian Mixture Models (diagonal, spherical, tied and full covariance matrices supported), to sample them, and to estimate them from data using Expectation Maximization algorithm. The Getting Started page contains links to several good tutorials dealing with the SciPy stack. 67 in https://. The upshot being that if a numpy array of values are passed in these will be normalized as a side effect. We will # use this for the scipy convolution img_zerod = img. The standard-deviation of the Gaussian filter is passed through the parameter sigma. I have a spectra with multiple gaussian emission lines over a noisy continuum. Fast RBF interpolation/fitting. Much like scikit-learn's gaussian_process module, GPy provides a set of classes for specifying and fitting Gaussian processes, with a large library of kernels that can be combined as needed. This is a key decision point when it comes to choosing statistical methods for your data sample. face() blurred_face = ndimage. order int or sequence of ints, optional. Project scipy/scipy pull requests. Qhull (used to do the Delaunay triangulation) does not center the data set for you under the default options, so it runs to rounding errors far away from origin. pdf (bin_centers) from matplotlib import pyplot as plt. How to Integrate Gaussian Functions. Can process multi-camera videos. mixture is a package which enables one to learn Gaussian Mixture Models (diagonal, spherical, tied and full covariance matrices supported), sample them, and estimate them from data. Kite is a free autocomplete for Python developers. Find file Copy path Ffisegydd Added a curve_fit example to scipy 53dc2cd Mar 27,. Matplotlib. , still is $1$. Since derivative filters are very sensitive to noise, it is common to smooth the image (e. the Gaussian is extremely broad. In statistics, kernel density estimation (KDE) is a non-parametric way to estimate the probability density function of a random variable. NumPy, matplotlib and SciPy HPC Python Antonio G omez-Iglesias [email protected] In the SciPy implementation of these tests, you can interpret the p value as follows. For both fit and data, each row will be scaled by the corresponding inverse prefix if given in scipy_data_fitting. Documentation for core SciPy Stack projects: Numpy. We will try to fit a Gaussian near each previously detected peak. As stated above, we have two arrays: A and B. face() blurred_face = ndimage. fftpack provides fft function to calculate Discrete Fourier Transform on an array. (SCIPY 2012) Fcm - A python library for ﬂow cytometry Jacob Frelinger†, Adam Richards†, Cliburn Chan† F Abstract—Flow cytometry has the ability to measure multiple parameters of a. Sign in Sign up. Consider the following input image: Lets call this image f. Note: Since SciPy 0. For tutorials, reference documentation, the SciPy. pdfx,locparam0,scaleparam1. save_npz and scipy. - It is used in mathematics. Utilizing SciPy correctly can sometimes be a very tricky proposition. As an example, we take a Gaussian pulse and study variation of density with time. They are extracted from open source Python projects. In particular, these are some of the core packages. So it gives the following set of parameters:\n", "\n",. An order of 1, 2, or 3 corresponds to convolution with the first, second or third derivatives of a Gaussian. Multivariate normal CDF values in Python. As the shape parameters of a beta distribution become large, the probability distribution becomes approximately normal (Gaussian). This is because the padding is not done correctly, and does not take the kernel size into account (so the convolution "flows out of bounds of the image"). In the example output from your code, $\sigma$ is huge, i. For image processing with SciPy and NumPy, you will need the libraries for this tutorial. Convolving a noisy image with a gaussian kernel (or any bell-shaped curve) blurs the noise out and leaves the low-frequency details of the image standing out. In this Tutorial about python for data science, You will learn about how to do hierarchical Clustering using scikit-learn in Python, and how to generate dendrograms using scipy in jupyter notebook. The function call scipy. 14, there has been a multivariate_normal function in the scipy. integrate import quad from scipy. The following are code examples for showing how to use scipy. We welcome contributions for these functions. 2 days ago · I still can’t believe this post exists, given its humble beginnings. I want to apply a Gaussian filter of dimension 5x5 pixels on an image of 512x512 pixels. filters module's unsharp_mask() function with different values of the radius and amount parameters to sharpen an image. 2) find the peaks 3) do least square fit of gaussian at the peaks to find the area under each gaussian. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. The function scipy. B: #———————————————– # populate the coefficient arrays #———————————————-from scipy. An order of 0 corresponds to convolution with a Gaussian kernel. 105/impi/2018. So if you want the kernel matrix you do from scipy. Setting order = 0 corresponds to convolution with a Gaussian kernel. Its characteristic bell-shaped graph comes up everywhere from the normal distribution in. Python gaussian noise. Qhull (used to do the Delaunay triangulation) does not center the data set for you under the default options, so it runs to rounding errors far away from origin. gaussian_kde. The prune method of classes bsr_matrix, csc_matrix, and csr_matrix was updated to reallocate backing arrays under certain conditions, reducing memory usage. The packages currently includes functions for linear and non-linear filtering, binary morphology, B-spline interpolation, and object measurements. Technically this is called the null hypothesis, or H0. Got distracted. In statistics, kernel density estimation (KDE) is a non-parametric way to estimate the probability density function of a random variable. kde import gaussian_kde from scipy. The spatial filter employed in this paper is bilateral filter. Distribution fitting with scipy Distribution fitting is the procedure of selecting a statistical distribution that best fits to a dataset generated by some random process. scipy bivariate normal distribution aka kernel density estimator where the Kernel is a normal distribution with stddev. Hierarchical clustering takes the idea of clustering a step further and imposes an ordering on the clusters themselves. misc import imsave. 14, there has been a multivariate_normal function in the scipy. peaks_x = peakutils. Introduction. Statistical functions (scipy. Also: whats a hyperbolic distribution and is it implemented in scipy? Pdffittedexpon. My primary objective is to find areas under all the gaussian peaks. The standard deviations of the Gaussian filter are given for each axis as a sequence, or as a single number, in which case it is equal for all axes. gaussian_filter(). Numerical integration is sometimes called quadrature, hence the name. nanmean, nanmedian and nanstd functions are deprecated in favor of their numpy. from scipy import stats. The Gaussian distribution is a limiting distribution in the sense of the central limit theorem, but also in that many distributions have a Gaussian distribution as a limit. McKinney, Perktold, Seabold (statsmodels) Python Time Series Analysis SciPy Conference 2011 2 / 29. Multivariate normal CDF values in Python. A library for statistical modeling, implementing standard statistical models in Python using NumPy and SciPy Includes: Linear (regression) models of many forms Descriptive statistics Statistical tests Time series analysis and much more. io: Scipy-input output¶ Scipy provides routines to read and write Matlab mat files. Much like scikit-learn's gaussian_process module, GPy provides a set of classes for specifying and fitting Gaussian processes, with a large library of kernels that can be combined as needed. Single Integrals. To aid in the construction of signals with predetermined properties, the scipy. output : array, optional The output parameter passes an array in which to store the filter output. Kernel density estimation using Python, matplotlib. pyplot as plt Beta distribution. We welcome contributions for these functions. from scipy import misc face = misc. stats we can find a class to estimate and use a gaussian kernel density estimator, scipy. In particular, the submodule scipy. 0 is the rotation parameter which is just passed into the gaussian function. interpolate (x, y, ind = indexes) print Related functionality in SciPy. To aid in the construction of signals with predetermined properties, the scipy. info for ppf, that's exactly what it says as well. A numpy ndarray of shape n,n. order int or sequence of ints, optional. In the example output from your code, $\sigma$ is huge, i. Parametric tests are conducted, with an assumption that the data follows a Gaussian distribution. The SciPy library is one of the core packages that make up the SciPy stack. I think the problem is that most of the elements are close to zero, and there not many points to actually be fitted. A histogram is a useful tool for visualization (mainly because everyone understands it), but doesn't use the available data very efficiently. SciPy is a Python library of mathematical routines. The Gaussian distribution is a continuous function which approximates the exact binomial distribution of events. gaussian_filter An order of 1, 2, or 3 corresponds to convolution with the first, second or third derivatives of a Gaussian. Calculating the moments of the distribution ¶. For image processing with SciPy and NumPy, you will need the libraries for this tutorial. This is seen by formally taking limits of, e. SciPy (pronounced "Sigh Pie") is a Python-based ecosystem of open-source software for mathematics, science, and engineering. The nature of the gaussian gives a probability of 0. quantile_gaussianize (x) [source] ¶ Normalize a sequence of values via rank and Normal c. -in CuPy column denotes that CuPy implementation is not provided yet. face() blurred_face = ndimage. So it gives the following set of parameters:\n", "\n",. The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. We then apply Gaussian filtering in-place on this NumPy array using the corresponding method of SciPy. My primary objective is to find areas under all the gaussian peaks. In particular, the submodule scipy. Here in this SciPy Tutorial, we will learn the benefits of Linear Algebra, Working of Polynomials, and how to install SciPy. Note: Since SciPy 0. Search form. We'll leverage the Cholesky decomposition of the covariance matrix to transform standard. gaussian_kde¶ class scipy. We will # use this for the scipy convolution img_zerod = img. second is a width parameter, defining the size of the wavelet (e. > Similar question, but now a bit harder. convolution of the gaussian kernel with a 2D histogram of the data. In this section we will take a look at Gaussian mixture models (GMMs), which can be viewed as an extension of the ideas behind k-means, but can also be a powerful tool for estimation beyond simple clustering. integrate import quad from scipy. Consider the following input image: Lets call this image f. Multiply pdf. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. Application of the convolution theorem. stats we can find a class to estimate and use a gaussian kernel density estimator, scipy. In the original code that you've linked to the _compute_covariance method sets the covariance matrix for the Gaussian kernel as the product of the factor provided by calling either scotts_factor or silverman_factor and the data covariance. pyplot as plt # # Univariate estimation #-----# # We start with a minimal amount of data in order to see how gaussian_kde works, # and what the different options for bandwidth selection do. python-examples / examples / scipy / fitting a gaussian with scipy curve_fit. If this assumption fails, then non-parametric tests are considered for hypothesis testing. What I would like to do is to take two PMFs from discrete gaussian distributions and recover an unknown distribution using deconvolution. That a subset of Alan Genzs multivariate normal CDF functions are available in Scipy. bessel_diff_formula is deprecated. The variable s you define as the pre-factor for the argument of the corresponding exponential is then only $\approx -1\cdot{}10^{-15}$, which is dangerously close to typical double precision limits (adding $10^{-16}$ to $1$ with typical double precision, e. It can also draw confidence ellipsoides for multivariate models, and compute the Bayesian Information. edu October 30th, 2014. linalg import inv. I decided to use the gaussian_kde class provided by scipy. imshow(blurred_face) plt. Interpolation methods in Scipy oct 28, 2015 numerical-analysis interpolation python numpy scipy. fftpack provides fft function to calculate Discrete Fourier Transform on an array. Discrete Filter Design. Example: scipy. pdf = stats. Numerical integration is sometimes called quadrature, hence the name. Higher order. We then fit the data to the same model function. OF THE 11th PYTHON IN SCIENCE CONF. We are going to compare the performance of different methods of image processing using three Python libraries (scipy, opencv and scikit-image). pyplot as plt Beta distribution. Code faster with the Kite plugin for your code editor, featuring Line-of-Code Completions and cloudless processing. pdf ( pos ). We form them once, and then calculate inverse(A). Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. , using a Gaussian filter) before applying the Laplacian. interp1d() •This function takes an array of x values and an array of y values, and then returns a function. Copula, for the definition of the Gaussian or normal copula model. For both fit and data, each row will be scaled by the corresponding inverse prefix if given in scipy_data_fitting. pyplot as plt plt. Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. Matlab curve fitting toolbox - wrong data fit? matlab,correlation,curve-fitting,data-fitting. from scipy import stats. Writing scipy. By voting up you can indicate which examples are most useful and appropriate. This article will explain how to get started with SciPy, survey what the library has to offer, and give some examples of how to use it for common tasks. For that, the following is the algorithm i have in mind. Contour Plots in Pandas How to make a contour plot in pandas. bessel_diff_formula is deprecated. Here are the examples of the python api scipy. Probability distributions in SciPy. SciPy is an enormous Python library for scientific computing. SCIPY TUTORIAL 1. Then, gaussian_filter(g, sigma, order=[0, 1], mode='constant', cval=1) evaluates to This is t. stats; gh-8548: BUG: Fix ellipj when m is near 1. gaussian_kde(). SciPy FFT scipy. Representation of a kernel-density estimate using Gaussian kernels. gaussian_kde(dataset, bw_method=None)¶. Here, the parameter sigma controls the standard-deviation of the Gaussian filter. Scipy has functions that deal with several common probability distributions. We checked in the command prompt whether we already have these: Also, some methods like imsave() did not. Technically this is called the null hypothesis, or H0. pyplot as plt plt. This section addresses basic image manipulation and processing using the core scientific modules NumPy and SciPy. The SciPy (Scientific Python) package extends the functionality of NumPy with a substantial collection of useful algorithms, like minimization, Fourier transformation, regression, and other applied mathematical techniques. Python gaussian noise. As the shape parameters of a beta distribution become large, the probability distribution becomes approximately normal (Gaussian). You can vote up the examples you like or vote down the ones you don't like. They are extracted from open source Python projects. I am having some trouble to fit a gaussian to data. The advantages of Gaussian processes are: The prediction interpolates the observations (at least for regular kernels). Top 20 Pandas, NumPy and SciPy functions on GitHub A few months ago I noticed a blog post listing the most commonly used functions/modules for a few of the most popular python libraries as determined by number of instances on Github. Note: Since SciPy 0. Checking Parseval's Theorem for Gaussian Signal by Using Scipy I'm trying to check Parseval's theorm for Gaussian signal. Project scipy/scipy pull requests. Inference of continuous function values in this context is known as GP regression but GPs can also be used for classification. This cookbook recipe demonstrates the use of scipy. Some of the operations covered by this tutorial may be useful for other kinds of multidimensional array processing than image processing. gaussian_filterメソッドで画像をガウシアンフィルタで平滑化できます。. iirdesign function arguments. Qhull (used to do the Delaunay triangulation) does not center the data set for you under the default options, so it runs to rounding errors far away from origin. This is because the padding is not done correctly, and does not take the kernel size into account (so the convolution "flows out of bounds of the image"). Normality Assumption. gaussian_filter An order of 1, 2, or 3 corresponds to convolution with the first, second or third derivatives of a Gaussian. Fitting gaussian-shaped data¶ Calculating the moments of the distribution¶ Fitting gaussian-shaped data does not require an optimization routine. The fact-checkers, whose work is more and more important for those who prefer facts over lies, police the line between fact and falsehood on a day-to-day basis, and do a great job. Today, my small contribution is to pass along a very good overview that reflects on one of Trump’s favorite overarching falsehoods. Namely: Trump describes an America in which everything was going down the tubes under  Obama, which is why we needed Trump to make America great again. And he claims that this project has come to fruition, with America setting records for prosperity under his leadership and guidance. “Obama bad; Trump good” is pretty much his analysis in all areas and measurement of U.S. activity, especially economically. Even if this were true, it would reflect poorly on Trump’s character, but it has the added problem of being false, a big lie made up of many small ones. Personally, I don’t assume that all economic measurements directly reflect the leadership of whoever occupies the Oval Office, nor am I smart enough to figure out what causes what in the economy. But the idea that presidents get the credit or the blame for the economy during their tenure is a political fact of life. Trump, in his adorable, immodest mendacity, not only claims credit for everything good that happens in the economy, but tells people, literally and specifically, that they have to vote for him even if they hate him, because without his guidance, their 401(k) accounts “will go down the tubes.” That would be offensive even if it were true, but it is utterly false. The stock market has been on a 10-year run of steady gains that began in 2009, the year Barack Obama was inaugurated. But why would anyone care about that? It’s only an unarguable, stubborn fact. Still, speaking of facts, there are so many measurements and indicators of how the economy is doing, that those not committed to an honest investigation can find evidence for whatever they want to believe. Trump and his most committed followers want to believe that everything was terrible under Barack Obama and great under Trump. That’s baloney. Anyone who believes that believes something false. And a series of charts and graphs published Monday in the Washington Post and explained by Economics Correspondent Heather Long provides the data that tells the tale. The details are complicated. Click through to the link above and you’ll learn much. But the overview is pretty simply this: The U.S. economy had a major meltdown in the last year of the George W. Bush presidency. Again, I’m not smart enough to know how much of this was Bush’s “fault.” But he had been in office for six years when the trouble started. So, if it’s ever reasonable to hold a president accountable for the performance of the economy, the timeline is bad for Bush. GDP growth went negative. Job growth fell sharply and then went negative. Median household income shrank. The Dow Jones Industrial Average dropped by more than 5,000 points! U.S. manufacturing output plunged, as did average home values, as did average hourly wages, as did measures of consumer confidence and most other indicators of economic health. (Backup for that is contained in the Post piece I linked to above.) Barack Obama inherited that mess of falling numbers, which continued during his first year in office, 2009, as he put in place policies designed to turn it around. By 2010, Obama’s second year, pretty much all of the negative numbers had turned positive. By the time Obama was up for reelection in 2012, all of them were headed in the right direction, which is certainly among the reasons voters gave him a second term by a solid (not landslide) margin. Basically, all of those good numbers continued throughout the second Obama term. The U.S. GDP, probably the single best measure of how the economy is doing, grew by 2.9 percent in 2015, which was Obama’s seventh year in office and was the best GDP growth number since before the crash of the late Bush years. GDP growth slowed to 1.6 percent in 2016, which may have been among the indicators that supported Trump’s campaign-year argument that everything was going to hell and only he could fix it. During the first year of Trump, GDP growth grew to 2.4 percent, which is decent but not great and anyway, a reasonable person would acknowledge that — to the degree that economic performance is to the credit or blame of the president — the performance in the first year of a new president is a mixture of the old and new policies. In Trump’s second year, 2018, the GDP grew 2.9 percent, equaling Obama’s best year, and so far in 2019, the growth rate has fallen to 2.1 percent, a mediocre number and a decline for which Trump presumably accepts no responsibility and blames either Nancy Pelosi, Ilhan Omar or, if he can swing it, Barack Obama. I suppose it’s natural for a president to want to take credit for everything good that happens on his (or someday her) watch, but not the blame for anything bad. Trump is more blatant about this than most. If we judge by his bad but remarkably steady approval ratings (today, according to the average maintained by 538.com, it’s 41.9 approval/ 53.7 disapproval) the pretty-good economy is not winning him new supporters, nor is his constant exaggeration of his accomplishments costing him many old ones). I already offered it above, but the full Washington Post workup of these numbers, and commentary/explanation by economics correspondent Heather Long, are here. On a related matter, if you care about what used to be called fiscal conservatism, which is the belief that federal debt and deficit matter, here’s a New York Times analysis, based on Congressional Budget Office data, suggesting that the annual budget deficit (that’s the amount the government borrows every year reflecting that amount by which federal spending exceeds revenues) which fell steadily during the Obama years, from a peak of $1.4 trillion at the beginning of the Obama administration, to$585 billion in 2016 (Obama’s last year in office), will be back up to $960 billion this fiscal year, and back over$1 trillion in 2020. (Here’s the New York Times piece detailing those numbers.) Trump is currently floating various tax cuts for the rich and the poor that will presumably worsen those projections, if passed. As the Times piece reported: