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# Numpy Polyfit Plot

xlim(0, 5) plt. poly1d(z1) #多项式系数 print(p1) # 在屏幕上打印拟合多项式 yvals=p1(xxx) plt. _homework2: ========================================== Homework 2 ==========================================. The ability to obtain predictive variance for Effective Quadratures’ polynomial approximations would be useful for a variety of applications. { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Physics Lab 3 ", " ", "## Part 3: Free-fall Analysis ", " ", "names (all lab partners. A simple script will be introduced that opens a spreadsheet of data, computes some simple statistics, creates some simple plots. org or mail your article to [email protected] As can be seen for instance in Fig. C:\Users\My Name>python demo_ml_polynomial_badfit. import numpy. polynomial import Polynomial p = Polynomial. import numpy as np x = np. 15406152e+00 6. If order is greater than 1, use numpy. これは、 numpy. 8775に近いほど、よい補間といえる。 N次曲線でスプライン補間をする. When we try to model the relationship between a single feature variable and a single target variable, it is called simple linear regression. Curve Fitting and Plotting in Python: Two Simple Examples Following are two examples of using Python for curve fitting and plotting. arange(10) y = 5 * x + 10 # Fit with polyfit b, m = polyfit(x, y, 1) plt. Nothing is truly static, especially in data science. Temperature 250 300 350 400 450 500 550 600 650 700 750 800 900 1000 Heat Capacity 0. py # ----- # # PYTHON for DUMMIES 19-20 # Problème 2 # # Script de test # Vincent Legat # # ----- # from matplotlib import pyplot as plt from numpy. Least squares fit to data. Official website for Costsco Wholesale. We gloss over their pros and cons, and show their relative computational complexity measure. linregress # Sample data creation # number of points. 7, há também uma palavra-chave cov que retornará a matriz de covariância para seus coeficientes, que você pode usar para calcular a incerteza dos próprios coeficientes de ajuste. use ( 'ggplot' ) np. Fit the frequencies and returns to a line. choice() to choose an index of a pair of data points. polyfit(xData_A, yData_A, polynomialOrder) fittedParameters_B = numpy. arange(0, 1000) yyy = np. scatter(x_observed,y_observed) coeffs = numpy. 2 for a quadratic, 3 for a cubic, etc. Buongiorno a tutti sono nuova e soprattutto da poco tempo sto usando python per creare grafici. This may require copying data and coercing values, which may be expensive. Pour des raisons pratiques, il est intéressant que les listes U et I soient déclarées comme des tableaux numpy, des “numpy array”. And similarly, the quadratic equation which of degree 2. 06806137, -6. To produce an array for plotting after fitting the original data, the function np. pyplot as plt x = [10,20,30,40,50,60,70,80] x = np. If True, assume that y is a binary variable and use statsmodels to estimate a logistic regression model. Matplotlib and Seaborn provide built in functions to plot scatter plots. seed (12) x = np. polyfit(xVals, yVals, 2) pylab. csv ', delimiter = ', ', usecols = (6,), unpack = True) vale = np. poly1d(kertoimet) # Lasketaan y:n arvot usealle x:n arvolle pol_X = np. Core packages for analysis: NumPy, and SciPy¶ NumPy ¶ NumPy is at the core of nearly every scientific Python application or module since it provides a fast N-d array datatype that can be manipulated in a vectorized form. Solution: Try to install numpy 1. pyplot import plot from matplotlib. # Generate regression polynomial polynomial_coefficients = numpy. polyfit function is the easy thing to use when fitting any polynomial (linear or not). Numpy and Matplotlib. optimize import curve_fit def func (x, a, b, c): return a * x ** 2 + b * x + c x = np. The trick is that we use np. import numpy as np from scipy. fit (x, y, deg, domain=None, rcond=None, full=False, w=None, window=None) [source] ¶. Numeric (typical differences) Python; NumPy, Matplotlib Description; help(); modules [Numeric] List available packages: help(plot) Locate functions. positive; plt. Keyword Research: People who searched polyfit also searched. The fit method can do whatever it wants. arange(10) y = x**2 -3*x + np. The weights apply to (=multiply) the fit residuals, not only to the y-coordinates. polyfit (x, y, deg = 1) line = w * x + b return line line = give_me_a_straight_line (x, y) plt. 72547264e-17, 2. 1926072073491056 Na versão 1. Hi All, I am trying to plot time against mean daily temperature values. Importing the NumPy module There are several ways to import NumPy. And here is the result. poly1d(trend) and then plt. import numpy as np x = np. T,y)[0] # obtaining the parameters # plotting the line line = w[0]*xi+w[1] # regression line plot(xi,line,'r-',xi,y,'o') show(). SciPy and NumPy Travis Oliphant SIAM 2011 Mar 2, 2011 2. 将你凌乱的数据划分成整齐好看的数据. Following section 1. polyfit (X, y, 6) poly1d = np. py is a Python package that interfaces to gnuplot, the popular open-source plotting program. Note: The code below has been amended to do multivariate fitting, but the plot image was part of the earlier, non-multivariate answer. def plot_question7(): """ graph of total resources generated as a function of time, for upgrade_cost_increment == 1 """ data = resources_vs_time(1. Matplotlib is a plotting library for the Python programming language and its numerical mathematics extension NumPy. Fit Curve To Scatter Plot Python. """ xmax = 5. Polyfit erklärt dir damit nicht die Welt, sondern schätzt lediglich diese Parameter p_i. Currently I'm looking through numpy but I don't think the function exists to fit a function like this: y = ax**4 + bx**3 + cx**2 + dx + e (I'm not sure what thats called but one degree up from a cubic curve) Also, I'm sure it'll take alot of time to brute force it like. How will a small start-up brand in the fashion-beauty cosmetics industry grow over the next few years on Instagram?. The ability to obtain predictive variance for Effective Quadratures’ polynomial approximations would be useful for a variety of applications. 42660977e-01 -1. Return a series instance that is the least squares fit to the data y sampled at x. poly1d(z) pylab. 15406152e+00 -2. plot(x_new, ffit) 或者，創建多項式函數：. pyplot as pp import numpy as np xNDArray. by Tirthajyoti Sarkar In this article, we discuss 8 ways to perform simple linear regression using Python code/packages. Time and space complexity are both O(n) where n is the size of your sample. plot(spectrum) Since most of the sum is in the background region there is a lot of noise and cosmic-ray contamination. polyfit (x, log_ISI_values, 1) ISI_semilog_slope = slope LibV5 : ISI_log_slope The slope of a linear fit to a loglog plot of the ISI values. 116], 'bo') plt. Fit Curve To Scatter Plot Python. 2 Release Notes¶ This is a bugfix release in the 1. polyfit(x, y, degree). array([(1, 1), (2, 4), (3, 1), (9, 3)]) # get x and y vectors x = points[:,0] y = points[:,1] # calculate polynomial z = np. Use of polyfit np. これは、 numpy. linspace to generate a number of points for us. In the plot, x-axis is for residue number and y-axis is for mean B-factor in. import numpy import matplotlib. pyplot as plt # Change the line plot below to a scatter plot plt. I convert that image to a scatter plot and then do a fit. plot(x,y, 'yo', x, fit_fn(x), '--k') plt. plot_pos when computing the plotting positions. 54464720615 \times 10^{-6} \\  The plot of the polynomial with the plot of data looks like: Here, red is the polynomial function and the blue is a plot of the data. For example UQ of surrogate models, and guiding selection of suitable polynomial orders. The NumPy polyfit function can fit a set of data points to a polynomial even if the underlying function is not continuous. polynomial import polyfit import matplotlib. array(y, dtype=float) #so the curve_fit can work """ create a function to fit with your data. 8 Manual」の、 「numpy. rand (n) # Plots best-fit line via polyfit plt. Here's a link to NumPy's open source repository on GitHub. polyfit(x, y, 1) f = np. Use the NumPy utilities polyfit and poly1d, as explained in Exercise 18: Fit a polynomial to data points, to fit polynomials of degree 1, 2, and 3 to the $$L$$ and $$T$$ data. 最小二乘多项式拟合。 拟合多项式 p（x） = p [0] * x **度 + + p [deg] > deg到点（x，y）。 返回使平方误差最小的系数p的向量。. table("data. 3D Plot with a colormap - Python I am trying to make a 3D surface plot showing voltage vs. polyfit (x, y, 1) trend = np. Specific Command References. Buongiorno a tutti sono nuova e soprattutto da poco tempo sto usando python per creare grafici. poly1d(fit) # fit_fn is now a function which takes in x and returns an estimate for y plt. polyval helpful for calculating the predicted y values based on the model. Moreover, we discussed example of Histogram in Python and Python bar Plotting example. Parameters : p : [array_like or poly1D] polynomial coefficients are given in decreasing order of powers. 从一阶到九阶拟合多项式拟合正弦函数. polyfit(x,y,1) fit_fn = np. y values we plot a Regression Line and to check that all the point are near line, for linear regression we use polyfit() function as numpy. A table is an array of tuples, each of the same length and type. exp(-b * x) + c x = np. In R this data type is called a data frame. linspacey(0,10,10) 由0,10之間產生十個數值，畫出該曲線，. legend() plt. 本系列文章將透過系統介紹資料科學（Data Science）相關的知識，透過 Python 帶領讀者從零開始進入資料科學的世界。. 67K GitHub forks. Linear regression is defined as a linear approach which is used to model the relationship between dependent variable and one or more independent variable(s). (2) Scatter Plot # Import package import matplotlib. Gossamer Mailing List Archive. import csv, numpy, scipy, scipy. import numpy as np import matplotlib. Felix Ste enhagen (Uni Freiburg) Using Python for Scienti c Computing 2011 8 / 37 Functions on numpy arrays The worst thing you can do is iterating with a for-loop over a numpy array. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. However the numpy. Parameters: x: array_like, shape (M,). MATLAB's built-in polyfit command can determine the coefficients of a polynomial fit. シンプルな多項式フィットで始めることをお勧めしますscipy. Numpy point Numpy point. 93626493e-01 1. Example: populations. rcParams['axes. Fit Curve To Scatter Plot Python. Fit Curve To Scatter Plot Python. Set objects also support mathematical operations like union, intersection, difference, and symmetric difference. This tells numpy that this is an integer. lmplot (x, Plot data and regression model fits across a FacetGrid. The polyfit() function from the NumPy module is another curve fitting tool which is essentially a least squares polynomial fit. figure() plt. Felix Ste enhagen (Uni Freiburg) Using Python for Scienti c Computing 2011 8 / 37 Functions on numpy arrays The worst thing you can do is iterating with a for-loop over a numpy array. normal(size=npoints). lmplot ¶ seaborn. polynomial import polyfit import matplotlib. comment :: Not yet assigned and may change. arange ([start,] stop, [step,], dtype = None)-> numpy. pyplot as plt # Change the line plot below to a scatter plot plt. polyfit(x,y,1) fit_fn = np. polyfit」と線形行列方程式の最小二乗解を得る「numpy. plot(x_pts, y_pts, 'o') # plot known data points Both linear and non-linear polynomial regression can be done with Numpy's polyfitfunction: numpy. pyplot as plt # show plots in notebook % matplotlib inline #parameters a = 0. The scatter plot is to contain a regression line. using matplotlib we can plot dirrerent scatter plots, line graphs. This chapter covers (in-depth) Matplotlib, a very useful Python plotting library. polyfit (x, y, deg, full = True) Quindi, il p sono i tuoi parametri di stima, e la res sarà residui, come descritto sopra. 6, 60) pol_Y = polynomi(pol_X. a, b, c and d are the. Clipping is done about median, but mean is returned. show() Instead of using range, we could also use numpy's np. png" that looks like this: Not bad! Let's add a trend line to the plot based on a simple linear model of the data. random(10) p, res, _, _, _ = numpy. I have used the exact same script on a similar dataset and there it works. I want to be able to ignore this and continue plotting. polyfit¶ numpy. Temperature 250 300 350 400 450 500 550 600 650 700 750 800 900 1000 Heat Capacity 0. py should create a "plots" folder and put a file inside called "day_vs_temp. [Start, Stop) Parameters : start : [optional] start of interval range. interpolationTest. When the polyfit function is called with an additional parameter: polyfit(t,y,2,cov=True) it returns the A,B,C coefficients as before and also a "covariance matrix" which gives the variance in each of the fit coefficients. What polyfit does is, given an independant and dependant variable (x & y) and a degree of polynomial, it applies a least-squares estimation to fit a curve to the data. Let's use numpy to compute the regression line: from numpy import arange,array,ones,linalg from pylab import plot,show xi = arange(0,9) A = array([ xi, ones(9)]) # linearly generated sequence y = [19, 20, 20. numpyでの配列操作(indexing, broadcasting)を理解する。 numpyで使える代表的な関数を理解する。 numpyで線形代数計算、統計処理を行う理解する。 なお慣習にしたがって、numpytは別名npとしてimportします。. Ejecutando Python en segundo plano en OS X Cómo pivotar en Google BigQuery La métrica de distancia por pares más rápida en python Mostrar mensaje cuando se desplaza sobre algo con el cursor del mouse en Python Averigüe si / que biblioteca BLAS es utilizada por Numpy ¿Cómo obtener la expresión de llamada al rastrear una función de Python?. filterwarnings('ignore') import pandas as pd import numpy as np import matplotlib. Fit Curve To Scatter Plot Python. plot(profile) Now plot the spectrum by summing along the spatial direction:: spectrum = img. poly1d(z) pylab. poly1d(self. poly1d(trend) plt. If order is greater than 1, use numpy. png" that looks like this: Not bad! Let's add a trend line to the plot based on a simple linear model of the data. Hint: You might find numpy. 総合演習では Numpy, Scipy そして Matplotlib を主に使います. by Tirthajyoti Sarkar In this article, we discuss 8 ways to perform simple linear regression using Python code/packages. The van der waal equation is a cubic polynomial $$f(V) = V^3 - \frac{p n b + n R T}{p} V^2 + \frac{n^2 a}{p}V - \frac{n^3 a b}{p} = 0$$, where $$a$$ and $$b$$ are constants, $$p$$ is the pressure, $$R$$ is the gas constant, $$T$$ is an absolute temperature and $$n$$ is the number of moles. plot(x,y,'o') # calc the trendline z = numpy. power(x,N-i)*c[i]. Singular values smaller than this relative to the largest singular value will be ignored. Following section 1. plot and pylab. 从一阶到九阶拟合多项式拟合正弦函数. array(y, dtype=float) #so the curve_fit can work """ create a function to fit with your data. py, which is not the most recent version. It will then output a continous value. The trick is that we use np. import numpy as np import matplotlib. pyplot as plt # show plots in notebook % matplotlib inline #parameters a = 0. polyval(coeffs, x_full). 0 , num = 5 ) b = np. plot (X, my) plt. y=ax**2+bx+c. Linear Regression with numpy Compare LSE from numpy. normal(size=len(x)) popt, pcov = curve_fit(func, x, yn) plt. So far, it is. fit(x, y, 4) plt. polyfit 和 np. polyfit (X, y, 6) poly1d = np. NumPy module has a number of functions for searching inside an array. pyplot as plt # Change the line plot below to a scatter plot plt. We use the same dataset as with polyfit: npoints = 20 slope = 2 offset = 3 x = np. Nonlinear solver: failed to converge, residual norm too large. array (x) #this will convert a list in to an array y = np. reshape(4, 5) print(a) plt. Hi All, I am trying to plot time against mean daily temperature values. You may do so in any reasonable manner, but. Computes an iteratively sigma-clipped mean on a data set. poly1d which can do the y = mx + b calculation for us. Ask Question Asked 12 months ago. polyfit return coefficients in different order #7478. Numpy and Matplotlib. pyplot as plt % matplotlib inline matplotlib. 577304577155 \times 10^{-1} \\ 2. arange(10,30). Parameters : -> arr : [array_like] The polynomial coefficients are given in decreasing order of powers. For example if you want to fit an exponential function (from the documentation):. What I am trying to is the following:. I have simple x,y data from a csv file of which I want to plot a linear fit. table("data. 15406152e+00 6. plot(x, yf, linestyle = "-"); La función polyfit La función numpy. Relative condition number of the fit. pyplot as plt x = [1,2,3,4] y = [3,5,7,10] # 10, not 9, so the fit isn't perfect fit = np. polyfit¶ numpy. Matplotlib trendline Drawing a trendline of a scatter plot in matplotlib is very easy thanks to numpy’s polyfit function. Most of the code below is taken from. ones( (2,3,4), dtype=np. You can store data as 8, 16 or 32 bits. figure() plt. polyfit (x, log_ISI_values, 1) ISI_semilog_slope = slope LibV5 : ISI_log_slope The slope of a linear fit to a loglog plot of the ISI values. Numpy polyfit () method is used to fit our data inside a polynomial function. poly1d¶ class numpy. array(y, dtype=float) #so the curve_fit can work """ create a function to fit with your data. Intro to Numpy and Simple Plotting¶. Dec 24, 2018 · Plotting Moving averages in python for trend following strategies: Before we plot the moving averages, we will first define a time period and choose a company stock so that we can analyse it. By default, the dtype of the returned array will be the common NumPy dtype of all types in the DataFrame. polyfit(x,y,1) fit_fn = np. Matplotlib trendline Drawing a trendline of a scatter plot in matplotlib is very easy thanks to numpy's polyfit function. Paige and Z. While Matlab’s syntax for some array manipulations is more compact than NumPy’s, NumPy (by virtue of being an add-on to Python) can do many things that Matlab just cannot, for instance dealing properly with stacks of matrices. Generator, or numpy. 9 numpy; 10 logical masks; 11 printing; 12 plotting; 13 functions; 14 program flow; 15 dictionaries; 16 binary; 17 linear algebra; 18 Not on test; ints and floats. 81349206, 1. Vi skapar ett dataset som vi sedan passar med en rak linje $f (x) = mx + c$. Numpy has a function polyfit(x, y, deg) for finding a “best fit" of a polynomial of degree deg to a set of data points given by the array arguments x and y. multipolyfit as mpf data = [. We create a dataset that we then fit with a straight line $f (x) = m x + c$. pyplot as plt x = [1,2,3,4] y = [3,5,7,10] # 10, not 9, so the fit isn't perfect fit = np. linspace to generate a number of points for us. linspace(1, 22, 100). loadtxt(' VALE. Die Ausgabe von Polyfit sind die Parameter des (durch n) vorgegebenen Polynoms (Gleichung in doc polyfit). Linear Regression with numpy Compare LSE from numpy. csv ', delimiter = ', ', usecols = (6,), unpack = True) # polyfit 用于多项式拟合 # 参数为训练. Plotting model residuals¶. Official website for Costsco Wholesale. array(y, dtype=float) #so the curve_fit can work """ create a function to fit with your data. py, which is not the most recent version. plot(profile) Now plot the spectrum by summing along the spatial direction:: spectrum = img. In this article we will show you some examples of legends using matplotlib. Software Development Tidbits Notes, commentary, and other drivel on all aspects of software development, with the occasional off topic bits thrown in. Do you know about Python Matplotlib 3. In this lesson you will be introduced to Numpy, and some simple plotting using pylab. { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Physics Lab 3 ", " ", "## Part 3: Free-fall Analysis ", " ", "names (all lab partners. normal (size=npoints) p = np. GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. This returns the coefficients which you can then use for plotting using numpy's polyval. I have been learning python for a few months, albeit slowly, because I can only do it in my free time and profession is something else. random(10) p, res, _, _, _ = numpy. However the numpy. 15 manual at NumPy v1. The ability to obtain predictive variance for Effective Quadratures' polynomial approximations would be useful for a variety of applications. When you have a huge number of points and you want just a polynomial fit, I found that it is (numerically) better to use the polyfit function from numpy: sage: import numpy as np sage: a,b=np. polynomial as poly coefs = poly. polyfit and poly1d, the first performs a least squares polynomial fit and the second calculates the new points:. linspace()) In the above code, p is used to scaled and shifted x values for numerical stability. 2] # Read the bandstructures objects calculated at different tb09 parameters and calculate the # indirect band gap. R/S-Plus Python Description; f <- read. The simplest polynomial is a line which is a polynomial degree of 1. b) We shall assume that $$L$$ as a function of $$T$$ is a polynomial. 116], 'bo') plt. - 2D surface plot, and 3D height field and scatter plot (under developing) - Can use numpy and scipy special functions to generate and plot 1d and 2d data - Column by column plotting. import numpy as np import numpy. array(x, dtype=float) #transform your data in a numpy array. array (y) m, b = polyfit (x, y, 1) plot (x, y, 'yo', x, m * x + b, '--k') show (). Polynomial fitting using numpy. import numpy as np x = np. Plotly Express is the easy-to-use, high-level interface to Plotly, which operates on a variety of types of data and produces easy-to-style figures. normal (size = len (x)) return x, y #main program n = 10 x, y = create_data (n) #use finer and regular mesh for plot xfine = np. import numpy as np import matplotlib. exp(x) """ Plot your data """ plt. linspace(0, 3, 50) y = np. NumPy has a good and systematic basic tutorial available. plot(i, f(i), 'go')plt. python code examples for numpy. ylim(0, 12). How will a small start-up brand in the fashion-beauty cosmetics industry grow over the next few years on Instagram?. scatter(x, y, c = colors, alpha = 0. 0] should be considered as valid. predicted values (two series, scatterplot). ylabel("y axis caption") plt. # many functions are avaible in modules or libraries # in this example we will load the numpy module of functions import numpy as np # this command loads all of the functions in numpy and labels them np import pandas as pd # data organization module import matplotlib. See related question on stackoverflow. random (10) p, res, _, _, _ = numpy. We're living in the era of large amounts of data, powerful computers, and artificial intelligence. Follow 292 views (last 30 days) Johan Lilliestråle on 19 Feb 2015. plot(spectrum) Since most of the sum is in the background region there is a lot of noise and cosmic-ray contamination. splrep(x,y) scipy. plot(profile) Now plot the spectrum by summing along the spatial direction:: spectrum = img. GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. 5, 22, 23, 23, 25. Hint: You might find numpy. pyplot as plt import numpy as np from random import random X = np. Returns: _来自Numpy 1. (4 replies) Hi, I have 2 points in 3D space and a bunch of points in-between them. 返回系数向量 p 这样可以最大限度地减少顺序中的平方误差。 deg ， deg-1 ，… 0. Use of polyfit np. linspace (0, 30, 100) y = np. For example UQ of surrogate models, and guiding selection of suitable polynomial orders. # many functions are avaible in modules or libraries # in this example we will load the numpy module of functions import numpy as np # this command loads all of the functions in numpy and labels them np import pandas as pd # data organization module import matplotlib. This gets rid of a few lines of code. 4 in [1], for 1D training data with N inputs (x_1,\\dots,x_N) and outputs (y_1,\\dots,y_N), a polynomial approximation can be obtained via. normal (size=npoints) p = np. 00000000e+00]) The first term is x**2, second term x in the coefficient is 2, and the constant term is 5. import numpy as np x = np. polyfit to estimate a polynomial regression. By the time you finish this book, you'll be able to write clean and fast code with NumPy. While Matlab’s syntax for some array manipulations is more compact than NumPy’s, NumPy (by virtue of being an add-on to Python) can do many things that Matlab just cannot, for instance dealing properly with stacks of matrices. MATLAB's built-in polyfit command can determine the coefficients of a polynomial fit. Python - fitting data with numpy - Stack Overflow. Temperature 250 300 350 400 450 500 550 600 650 700 750 800 900 1000 Heat Capacity 0. Code faster with the Kite plugin for your code editor, featuring Line-of-Code Completions and cloudless processing. I followed the example in the first answer to this question: Linear regression with matplotlib / numpy My code looks. popt and pcov are the out puts of the polynomials we define in order to fit the curve. Tukey, 1965, “An algorithm for the machine calculation of complex Fourier series,” Math. polyfit (x, y, deg, rcond=None, full=False, w=None, cov=False) [source] ¶. poly1d(z) pylab. Examples: A very simple example of using the numpy zeros function; Create a numpy zeros array with a specific data type. Sewing provides a way to join panels that can be structurally superior to glues and welding processes for woven materials. plot (x, y) plt. But I found no such functions for expo. plot(x, func(x, *popt), 'r-', label="Fitted Curve") plt. linregress 7 8 #Sample data creation 9 #number of points 10 n = 50 11 t = linspace (-5, 5, n) 12 #. ˇ model2 = pylab. The DFT is defined, with the conventions used in this implementation, in the documentation for the numpy. Plotting a scatter plot; Step #1: Import pandas, numpy and matplotlib! Just as we have done in the histogram article, as a first step, you'll have to import the libraries you'll use. polyfit(x, y, 3)) t = np. This file is licensed under the Creative Commons Attribution-Share Alike 4. plot (x, line, 'r--') There: if all you wanted was the code for a straight line through. log2(x)*p[0] + p[1]) return y_fit, p[0], p[1]. NumPy; csv file; polyfit() 2 Abstract Python libraries to be used in this tutorial. unique(x))) Using np. 00000000e+02 1. normal(size=npoints). polyfit (x, price. 4: 3526: 84. pyplot as plt # Sample data x = np. polyfitとpoly1dを使った簡単な3次の多項式近似poly1d 。 最初は最小二乗多項式近似を実行し、2番目は新しい点を計算します。. scatter绘制的点图？ 如 here所述 在numpy的帮助下,可以计算出一个线性拟合. According to the users manua SciPy minimize example - Fitting IDF Curves SciPy (pronounced “Sigh Pie”) is an open source Python library used by scientists, analysts, and engineers doing scientific computing and t. Numpy Tutorial - Features of Numpy. pyplot as plt The modelling class is derived from ModellingBase, a constructor is defined and the response function is defined. 085-350-7540 , 084-88-00-255 , [email protected] polyfit 不確実性を明示的に指定することはできません。 fmt="ro") # fit a polynomial of degree 1, no explicit uncertainty a1, b1 = np. poly1d(kertoimet) # Lasketaan y:n arvot usealle x:n arvolle pol_X = np. polyfit (x, y, 1))(np. # -*- coding: utf-8 -*- #from numpy import* from matplotlib. numpy documentation: Using np. import numpy as np. filterwarnings('ignore') import pandas as pd import numpy as np import matplotlib. csv ', delimiter = ', ', usecols = (6,), unpack = True) vale = np. Buongiorno a tutti sono nuova e soprattutto da poco tempo sto usando python per creare grafici. One of which is extremely useful for the topic at hand: the polyfit function. Recall that pylab. Do you know about Python Matplotlib 3. Report the final value of each state as t \to \infty. The vector output of polyfit() is used as input to poly1d(), which calculates the actual y-axis data points. Polynomial curve fitting now we will see how to find a fitting polynomial for the data using the function polyfit provided by numpy: filtering fitting forecast histogram image linear algebra machine learning math matplotlib natural language NLP numpy pandas plotly plotting probability random regression scikit-learn sorting statistics. pyplot as plt from matplotlib import font_manager, rc plt. 당신이 polyfit로 호출 full=True을 지정하면. pyplot as plt from scipy import stats import numpy as np x = np. Just like Numpy, you most probably won’t use Scipy itself, but the above-mentioned Scikit-Learn library highly relies on it. 5,rep) # cos(0. plot(x,y,'o') import matplotlib. Numpy polyfit() method is used to fit our data inside a polynomial function. ylabel("y axis caption") plt. But we don’t just want to do this for a single point, instead we want to compute the trend at every single pixel inside our analysis area. 08703704, -0. plot(x,y,'o') # calc the trendline z = numpy. plot(x_new, ffit) 或者，創建多項式函數：. To measure if the model is good enough, we can use a method called Train/Test. pyplot as plt import multipolyfit. polyval helpful for calculating the predicted y values based on the model. by Tirthajyoti Sarkar In this article, we discuss 8 ways to perform simple linear regression using Python code/packages. Note: The code below has been amended to do multivariate fitting, but the plot image was part of the earlier, non-multivariate answer. Pour déterminer le polynôme de degré n le plus proche d'un jeu de données (x, y), nous disposons de la fonction numpy. eg, ra ndom walk import numpy as np z10 = np. polyfit (x, y, deg, rcond=None, full=False, w=None, cov=False) [source] ¶. Vi skapar ett dataset som vi sedan passar med en rak linje $f (x) = mx + c$. import platform import matplotlib. Kemian tekniikassa haluamme usein sovittaa polynomeja mittausdataan. 0 , num = 20 ) print ( a ) print ( b ). arange(10) y = 5 * x + 10 # Fit with polyfit b, m = polyfit(x, y, 1) plt. Linear regression is defined as a linear approach which is used to model the relationship between dependent variable and one or more independent variable(s). We return the result from polyfit - Glue doesn't care what fit returns, it just passes that to other methods (as we will now see) Line 13 overrides the predict() method. Numpy permet la manipulations des vecteurs, matrices et polynômes. polyfit(x[j:j+window_length], y[j:j+window_length], 1)[0] for j in range(n - window_length)] x_mids = [x[j+window_length/2] for j in range(n - window_length)] plt. Examples: A very simple example of using the numpy zeros function; Create a numpy zeros array with a specific data type. txt: # year hare lynx carrot 1900 30e3 4e3 48300 1901 47. 最小二乘多项式拟合。 拟合多项式 p（x） = p [0] * x **度 + + p [deg] > deg到点（x，y）。 返回使平方误差最小的系数p的向量。. polyfit(x,y,1) fit_fn = np. We gloss over their pros and cons, and show their relative computational complexity measure. Navigation. Pythonの行列演算ライブラリnumpyを用いて,単一変数xによって説明されるyについて 最小二乗フィッティングを行いたいと思います。 まず、numpyを用いて確率的に3次のグラフを描画します。 グラフの描画 #モジュールのイ. import numpy as np import matplotlib. coeffs = mpf( 、 coeffs = numpy. numpy documentation: np. I want to be able to ignore this and continue plotting. And you'll also have to make a small tweak in your Jupyter environment. random (100) # Note: polynomialOrder too large will yeild warning: RankWarning: Polyfit may be poorly conditioned # Note: depends on the signal you're fitting. curve_fitうとします。. rand(6) y = random. { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Physics Lab 3 ", " ", "## Part 3: Free-fall Analysis ", " ", "names (all lab partners. com/technologycult/PythonForMachineLearning/tree/master/Part52 ''' Topics to be covered - Polynomial Regression without skle. polyfit과 같은 데이터 세트를 사용합니다. are used to get mode and sigma of the sky on the. Nonlinear solver: failed to converge, residual norm too large. 但是，的文檔狀態顯然是為了避免 np. Pour un polynôme de degré 1, la fonction polyfit renvoie un tableau numpy contenant deux valeurs : [1. In this notebook, we will explore the basic plot interface using pylab. Defining functions 0. Keyword Research: People who searched polyfit also searched. Cooley, James W. by Tirthajyoti Sarkar In this article, we discuss 8 ways to perform simple linear regression using Python code/packages. arange(10) y = 5 * x + 10 # Fit with polyfit b, m = polyfit(x, y, 1) plt. array (y) m, b = polyfit (x, y, 1) plot (x, y, 'yo', x, m * x + b, '--k') show (). pyplot as plt # Sample data x = np. Polynomial fits are those where the dependent data is related to some set of integer powers of the independent variable. polyfit(x,y,3)使用する代わりに、 非多変量データセットの場合、これを行う最も簡単な方法はおそらくnumpyのpolyfitです： numpy. unique(x))) Using np. import numpy as np x = np. # coding: utf-8 ''' Authors: Tyler Reddy and Anna Duncan The purpose of this Python module is to provide utility functions for analyzing the diffusion of particles in molecular dynamics simulation trajectories using either linear or anomalous diffusion models. polyfit(x, y, n). plot(x,y) pylab. Least squares fit to data. Gallery generated by Sphinx-Gallery. array` The linear fit a : float64 Slope of the fit b : float64 Intercept of the fit """ # fig log vs log p = np. NumPy has a good and systematic basic tutorial available. Instead, it is common to import under the briefer name np:. polynomial import polyfit import matplotlib. In this Python NumPy Tutorial, we are going to study the feature of NumPy: NumPy stands on CPython, a non-optimizing bytecode interpreter. coefficients = numpy. Generator, or numpy. 主要用的numpy里面的函数是polyfit，这个函数有三个参量（x,y,n),x和y是要输入的数据，n是要进行要拟合的多项式的最高次数，比如此次用的就是线性拟合，n=1，其返回值是多项式拟合的系数，对于线性拟合就是斜率和截距，另外要调用的函数就是poly1d,拟合出这个. It is highly recommended that you read this tutorial to fill in the gaps left by this workshop, but on its own it's a. 15 manual at NumPy v1. Keyword CPC PCC Volume Score; polyfit: 1. plot(x,y, 'yo', x, fit_fn(x), '--k') plt. dtype, optional. print(c) --#plot the matrix import numpy as np import matplotlib. plot(xxx, yvals, 'r',label. NumPy N-dimensional Array. For example UQ of surrogate models, and guiding selection of suitable polynomial orders. About : arange([start,] stop[, step,][, dtype]) : Returns an array with evenly spaced elements as per the interval. If you have a nice notebook you’d like to add here, or you’d like to make some other edits, please see the SciPy-CookBook repository. polyfit (x, price. SoX stands for Sound eXchange. Plotting Parabola (y = x 2) using Python and Matplotlib. pyplot as plt import pygimli as pg The modelling class is derived from ModellingBase, a constructor is defined and the response function is defined. Pour un polynôme de degré 1, la fonction polyfit renvoie un tableau numpy contenant deux valeurs : [1. polyfit(x, y, degree). I followed the example in the first answer to this question: Linear regression with matplotlib / numpy My code looks. Keyword Research: People who searched polyfit also searched. Chapter 9, Plotting with Matplotlib, discusses how NumPy on its own cannot be used to create graphs and plots. 我建议你从简单的多项式拟合开始，scipy. Background There are several good tutorials on linear regression and curve fitting using python already available. linspace (0, 1, 200) y1 = np. 035 seconds) Download Python source code: plot_polyfit. linspacey(0,10,10) 由0,10之間產生十個數值，畫出該曲線，. To produce an array for plotting after fitting the original data, the function np. In order to use the polynomial like a more standard f(x) function, you can pass your coefficients to poly1d () and use its return value as a function directly!. This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm [CT]. polyfit(x,y,5) ypredLearn more about plot, polyfit. 805] # the polyfit functions does the nth degree polynomial best fit on the data, # returning the polynomial coefficients n = 4 # 4th degree polynomial, you can change for whatever. call of duty black ops code redemption. plot([4,8,12,16,20,24], [0. Numpy and Matplotlib. polyfit(x, y, 3)) t = np. OK, I Understand. pyplot as plt points = np. In the below code, we move the left and bottom spines to the center of the graph applying set_position('center') , while the right and top spines are hidden by setting their colours to none with set_color('none'). 70710678 -0. show() Total running time of the script: (0 minutes 0. array([(1, 1), (2, 4), (3. 4 in [1], for 1D training data with N inputs (x_1,\\dots,x_N) and outputs (y_1,\\dots,y_N), a polynomial approximation can be obtained via. The model has a value of 𝑅² that is satisfactory in many cases and shows trends nicely. 2 Car Nd Masking and Colouring a Region of the Image May 9, 2017 • Python Machine-Learning Computer-Vision • 4 minutes to read In this part of Car-ND, we will look at how to mask and colour the region. poly1d(z) pylab. polyfit issues a RankWarning when the least-squares fit is badly conditioned. We gloss over their pros and cons, and show their relative computational complexity measure. n is the number of samples of the curve used to fit the parabola. import numpy as np. Generator, or numpy. picktest() print "You finally picked a location, at: ",coords """ import matplotlib. from QuantumATK import * import pylab import numpy as np from pylab import * # List of c(tb09) parameters used in the MGGA bandstructure calculations tb09=[0. We now have two sets of data: Tx and Ty, the time series, and tX and tY, sinusoidal data with noise. pyplot import (clf, plot, show, xlim, ylim, get_current_fig_manager, gca, draw, connect) Run this cell to play with the node placement toy:. ''' import numpy import scipy import scipy. We can fit a simple linear regression model using libraries such as Numpy or Scikit-learn. 👍 55 🎉 10 😕 11  13. Creating trend maps from spatio-temporal datasets A common task in the analysis of remotely-sensed datasets is to calculate rates of change over time in, for example, ice motion or melt rates. More Python libraries and packages for data science…. View license def draw_linear_regression(x, y, x_label, y_label, title, body): "Plot a linear regression chart" # x and y are matplotlib pylab arrays, body is a StringIO import pylab import matplotlib # clear graph matplotlib. 00472693, -0. py, which is not the most recent version. normal(size=npoints). poly1d (c_or_r, r=False, variable=None) [source] ¶. Polyfit erklärt dir damit nicht die Welt, sondern schätzt lediglich diese Parameter p_i. curve_fitは、あなたが知る必要がある関数fを点集合にscipy. normal (size = len (x)) popt, pcov = curve_fit (func, x, yn) La suite de la documentation, de la pcov donne: L'estimation de la covariance de popt. NumPy มีวิธีการที่ให้เราสร้างแบบจำลองพหุนาม mymodel = numpy. BayesianCNNBase method) (astroNN. array (matchdata) # Create Trendline x, y = data [:, 0], data [:, 1] polyfit = np. There: if all you wanted was the code for a straight line through your data, you should be all set! …. Chapter 9, Plotting with Matplotlib, discusses how NumPy on its own cannot be used to create graphs and plots. Simple and multiple linear regression with Python. 512622 Iteration 5, residual norm 0. 15 manual at NumPy v1. C / C++ Forums on Bytes. Same is the case with a cubic polynomial of the form y=ax**3+bx**2+cx+d; we need to have four constant-coefficient values for a, b, c, and d, which is calculated using the numpy. Numpy and Matplotlib¶These are two of the most fundamental parts of the scientific python "ecosystem". Software Development Tidbits Notes, commentary, and other drivel on all aspects of software development, with the occasional off topic bits thrown in. y-coordinates of the sample points. The legend() method adds the legend to the plot. title("Circular Motion") plt. import numpy as np: import matplotlib. optimize import curve_fit def func (x, a, b, c): return a * x ** 2 + b * x + c x = np. Die Ausgabe von Polyfit sind die Parameter des (durch n) vorgegebenen Polynoms (Gleichung in doc polyfit). print (data) plt. polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False) x:要拟合点的横坐标 y:要拟合点的纵坐标 deg:自由度. exp(x) """ Plot your data """ plt. Even though this model is quite rigid and often does not reflect the true relationship, this still remains a popular approach for several reasons. But when there is more than one independent […]. plot(xs,ys,'o') trendpoly = np. We ignore dy and constraints. 2016/02/05 - Matplotlib trendline Drawing a trendline of a scatter plot in matplotlib is very easy thanks to numpy’s polyfit function. poly1d(z)fori in range(min(x), max(x)): plt. Fit the frequencies and returns to a line. Project 1: Business Instagram Account Growth Model. Ho la necessità di creare due plot sovrapposti di dati (dati che leggo da due differenti files di testo) e fare un fit lineare su entrambi, con una legenda che riporti i valori del fit medesimo. polyfit(x, y, d) Where x is the x-axis data, y is the y-axis data, and d is the degree of the polynomial. lstsq」 への3件のフィードバック ピンバック: 線形回帰で切片を気にする意味は無い | 粉末@それは風のように (日記). 예제1 ¶ import numpy as np a = np. legend(['data to fit', '4th degree poly', '5th degree poly']) pylab. A one-dimensional polynomial class. Welcome to pure python polyfit, the polynomial fitting without any third party module like numpy, scipy, etc. 26633786, 0. Keyword Research: People who searched polyfit also searched. Creating trend maps from spatio-temporal datasets A common task in the analysis of remotely-sensed datasets is to calculate rates of change over time in, for example, ice motion or melt rates. choice() to choose an index of a pair of data points. plot(rets, p[0] * rets + p[1]) matplotlib. 4 in [1], for 1D training data with N inputs (x_1,\\dots,x_N) and outputs (y_1,\\dots,y_N), a polynomial approximation can be obtained via. And similarly, the quadratic equation which of degree 2. pyplot as plt import time. The example below plots a polynomial line on top of the collected data. polyfit 输出的时候没有R或者R-Square，是我哪里输错了么？. import numpy as np import matplotlib. arange (10) y = 5 * x + 10 # Fit with polyfit b, m = polyfit (x, y, 1) plt. plot_surface(x, y, z) 三次元プロット; fig. polyval(x_new, coefs) plt.
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# Three-Slit Interference: A Duality Relation

Authors
Type
Published Article
Publication Date
Aug 27, 2015
Submission Date
May 22, 2014
Identifiers
DOI: 10.1093/ptep/ptv112
Source
arXiv
Yellow

## Abstract

The issue of interference and which-way information is addressed in the context of 3-slit interference experiments. A new path distinguishability ${\mathcal D_Q}$ is introduced, based on Unambiguous Quantum State Discrimination (UQSD). An inequality connecting the interference visibility and path distinguishability, ${\mathcal V} + {2{\mathcal D_Q}\over 3- {\mathcal D_Q}} \le 1$, is derived which puts a bound on how much fringe visibility and which-way information can be simultaneously obtained. It is argued that this bound is tight. For 2-slit interference, we derive a new duality relation which reduces to Englert's duality relation and Greenberger-Yasin's duality relation, in different limits.

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# Is this a typo in my math book?

I am doing homework which is submitted online. I came across a question asking if two functions are equal. $f(x)=3x+4$ and $g(x)=14+(8/x)+b(x-4)$. I set the two equations together and got 7 for an answer but the book says the answer is $\frac {7}{3}$.

Here is an image of the solution in the book:

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The homework tag should never be used alone. Please add at least another one explaining the topic(s) covered by this homework –  M Turgeon Sep 5 '12 at 2:46
Perhaps you should edit your query here to state the question that was asked clearly. The two functions ae not equal as functions. The solution seems to suggest that what was asked is something like "Find all values of $b$ suc that $f(-3) = g(-3)$ and $f(4) = g(4)$", that is the functions have equal value for two specific values of $x$. This is quite different from saying that the functions are equal. –  Dilip Sarwate Sep 5 '12 at 2:47

If you set $f(-3) = g(-3)$, you end up with:

$$-5 = 14 -(8/3) -7b$$

If you multiply by 3 to remove fractions, you are left with

$$-15 = 42 - 8 -21b$$

Collecting like terms, we are left with

$$21b = 49$$

reducing yields: $$b = 7/3$$

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The book is correct. Setting $f(-3)=g(-3)$ gives us

$-5=14-\frac{8}{3}-7b \Rightarrow -19=-\frac{8}{3}-7b \Rightarrow 49=21b \Rightarrow b=\frac{7}{3}$.

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Well, at least to your specific question "Is this a typo?", the answer is "No." You are making an arithmetic error somewhere, because: $$\frac{1}{7}\bigg(\frac{34}{3} + \frac{15}{3}\bigg) = \frac{1}{7}\frac{49}{3} = \frac{7}{3}.$$

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$$-5 = \left(\frac{34}{3}\right) - 7B$$

Multiply both sides by 3:

$$-15= 3\cdot\left(\frac{34}{3}\right) - 3\cdot(7B)$$

that is:

$$-15 = 34 - 21B$$

so

$$-15-34 = -21B$$

$$B=\frac{49}{21}=\frac{7}{3}$$

The book is correct on this one.

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@Joe, appreciate the Tex work. –  Emmad Kareem Sep 5 '12 at 4:09
No problem - was just minor while I was reading this problem. :) –  Joe Sep 5 '12 at 4:13


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# Cost-effectiveness of malaria intermittent preventive treatment in infants (IPTi) in Mozambique and the United Republic of Tanzania

Hutton, G. and Schellenberg, D. and Tediosi, F. and Macete, E. and Kahigwa, E. and Sigauque, B. and Mas, X. and Trapero, M. and Tanner, M. and Trilla, A. and Alonso, P. and Menendez, C.. (2009) Cost-effectiveness of malaria intermittent preventive treatment in infants (IPTi) in Mozambique and the United Republic of Tanzania. Bulletin of the World Health Organization = Bulletin de l'Organisation mondiale de la santé, Vol. 87, H. 2. pp. 123-129.

Full text not available from this repository.

Official URL: http://edoc.unibas.ch/dok/A5843178

OBJECTIVE: To estimate the cost-effectiveness of malaria intermittent preventive treatment in infants (IPTi) using sulfadoxine-pyrimethamine (SP). METHODS: In two previous IPTi trials in Ifakara (United Republic of Tanzania) and Manhica (Mozambique), SP was administered three times to infants before 9 months of age through the Expanded Programme on Immunization. Based on the efficacy results of the intervention and on malaria incidence in the target population, an estimate was made of the number of clinical malaria episodes prevented. This number and an assumed case-fatality rate of 1.57% were used, in turn, to estimate the number of disability-adjusted life years (DALY) averted and the number of deaths averted. The cost of the intervention, including start-up and recurrent costs, was then assessed on the basis of these figures. FINDINGS: The cost per clinical episode of malaria averted was US$1.57 (range: US$ 0.8-4.0) in Ifakara and US$4.73 (range: US$ 1.7-30.3) in Manhica; the cost per DALY averted was US$3.7 (range: US$ 1.6-12.2) in Ifakara and US$11.2 (range: US$ 3.6-92.0) in Manhica; and the cost per death averted was US$100.2 (range: US$ 43.0-330.9) in Ifakara and US$301.1 (range: US$ 95.6-2498.4) in Manhica. CONCLUSION: From the health system and societal perspectives, IPTi with SP is expected to produce health improvements in a cost-effective way. From an economic perspective, it offers good value for money for public health programmes


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4.12.2. Legendre¶

beamIntegration('Legendre', tag, secTag, N)

Create a Gauss-Legendre beamIntegration object. Gauss-Legendre integration is more accurate than Gauss-Lobatto; however, it is not common in force-based elements because there are no integration points at the element ends.

Places N Gauss-Legendre integration points along the element. The location and weight of each integration point are tabulated in references on numerical analysis. The force deformation response at each integration point is defined by the section. The order of accuracy for Gauss-Legendre integration is 2N-1.

Arguments and examples see Lobatto.


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Question

# Prove the rule of exponents (ab)n=anbn by using principle of mathematical induction for every natural number.

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Solution

## Step (1): Assume given statement Let the given statement be P(n), i. e., P(n):(ab)n=anbn Step (2): Checking statement P(n) for n=1 Put n=1 in P(n), we get P(1):(ab)1=a1b1 ⇒ab=ab Thus P(n) is true for n=1. Step (3): P(n) for n=1 Put n=K in P(n) and assume this is true for some natural number K i.e., P(K):(ab)K=aKbK ⋯(1) Step (4): Checking statement P(n) for n=K+1 Now we shall prove that P(K+1) is true whenever P(K) is true. Now, we have (ab)(K+1) =(ab)K(ab) =(aKbK)(ab) (Using (1)) =(aK⋅a1)(bK⋅b1) =aK+1⋅bK+1 Therefore, P(K+1) is true whenever P(K) is true. Final Answer: Hence, by principle of mathematical induction, P(n) is true for all n∈N.

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# Dielectric

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in a conductor, but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced toward the field and negative charges shift in the opposite direction. This creates an internal electric field which reduces the overall field within the dielectric itself.[1] If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axis aligns to the field.[1]

While the term "insulator" implies low electrical conduction, "dielectric" is typically used to describe materials with a high polarizability. The latter is expressed by a number called the dielectric constant. The term insulator is generally used to indicate electrical obstruction while the term dielectric is used to indicate the energy storing capacity of the material (by means of polarization). A common example of a dielectric is the electrically insulating material between the metallic plates of a capacitor. The polarization of the dielectric by the applied electric field increases the capacitor's surface charge.[1]

The study of dielectric properties is concerned with the storage and dissipation of electric and magnetic energy in materials.[2] It is important to explain various phenomena in electronics, optics, and solid-state physics.

The term "dielectric" was coined by William Whewell (from "dia-electric") in response to a request from Michael Faraday.[3][4]

## Electric susceptibility

The electric susceptibility χe of a dielectric material is a measure of how easily it polarizes in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.

It is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that

${\mathbf P}=\varepsilon_0\chi_e{\mathbf E},$

where $\, \varepsilon_0$ is the electric permittivity of free space.

The susceptibility of a medium is related to its relative permittivity $\, \varepsilon_r$ by

$\chi_e\ = \varepsilon_r - 1.$

So in the case of a vacuum,

$\chi_e\ = 0.$

The electric displacement D is related to the polarization density P by

$\mathbf{D} \ = \ \varepsilon_0\mathbf{E} + \mathbf{P} \ = \ \varepsilon_0 (1+\chi_e) \mathbf{E} \ = \ \varepsilon_r \varepsilon_0 \mathbf{E}.$

### Dispersion and causality

In general, a material cannot polarize instantaneously in response to an applied field. The more general formulation as a function of time is

$\mathbf{P}(t)=\varepsilon_0 \int_{-\infty}^t \chi_e(t-t') \mathbf{E}(t')\, dt'.$

That is, the polarization is a convolution of the electric field at previous times with time-dependent susceptibility given by $\chi_e(\Delta t)$. The upper limit of this integral can be extended to infinity as well if one defines $\chi_e(\Delta t) = 0$ for $\Delta t < 0$. An instantaneous response corresponds to Dirac delta function susceptibility $\chi_e(\Delta t) = \chi_e \delta(\Delta t)$.

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product,

$\mathbf{P}(\omega)=\varepsilon_0 \chi_e(\omega) \mathbf{E}(\omega).$

Note the simple frequency dependence of the susceptibility, or equivalently the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. $\chi_e(\Delta t) = 0$ for $\Delta t < 0$), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility $\chi_e(0)$.

## Dielectric polarization

### Basic atomic model

Electric field interaction with an atom under the classical dielectric model.

In the classical approach to the dielectric model, a material is made up of atoms. Each atom consists of a cloud of negative charge (Electrons) bound to and surrounding a positive point charge at its center. In the presence of an electric field the charge cloud is distorted, as shown in the top right of the figure.

This can be reduced to a simple dipole using the superposition principle. A dipole is characterized by its dipole moment, a vector quantity shown in the figure as the blue arrow labeled M. It is the relationship between the electric field and the dipole moment that gives rise to the behavior of the dielectric. (Note that the dipole moment is shown to be pointing in the same direction as the electric field. This isn't always correct, and it is a major simplification, but it is suitable for many materials.)

When the electric field is removed the atom returns to its original state. The time required to do so is the so-called relaxation time; an exponential decay.

This is the essence of the model in physics. The behavior of the dielectric now depends on the situation. The more complicated the situation the richer the model has to be in order to accurately describe the behavior. Important questions are:

• Is the electric field constant or does it vary with time?
• If the electric field does vary, at what rate?
• What are the characteristics of the material?
• Is the direction of the field important (isotropy)?
• Is the material the same all the way through (homogeneous)?
• Are there any boundaries/interfaces that have to be taken into account?
• Is the system linear or do nonlinearities have to be taken into account?

The relationship between the electric field E and the dipole moment M gives rise to the behavior of the dielectric, which, for a given material, can be characterized by the function F defined by the equation:

$\mathbf{M} = \mathbf{F}(\mathbf{E})$.

When both the type of electric field and the type of material have been defined, one then chooses the simplest function F that correctly predicts the phenomena of interest. Examples of phenomena that can be so modeled include:

### Dipolar polarization

Dipolar polarization is a polarization that is either inherent to polar molecules (orientation polarization), or can be induced in any molecule in which the asymmetric distortion of the nuclei is possible (distortion polarization). Orientation polarization results from a permanent dipole, e.g. that arising from the 104.45° angle between the asymmetric bonds between oxygen and hydrogen atoms in the water molecule, which retains polarization in the absence of an external electric field. The assembly of these dipoles forms a macroscopic polarization.

When an external electric field is applied, the distance between charges, which is related to chemical bonding, remains constant in orientation polarization; however, the polarization itself rotates. This rotation occurs on a timescale which depends on the torque and the surrounding local viscosity of the molecules. Because the rotation is not instantaneous, dipolar polarizations lose the response to electric fields at the lowest frequency in polarizations. A molecule rotates about 1ps per radian in a fluid, thus this loss occurs at about 1011 Hz (in the microwave region). The delay of the response to the change of the electric field causes friction and heat.

When an external electric field is applied in the infrared, a molecule is bent and stretched by the field and the molecular moment changes in response. The molecular vibration frequency is approximately the inverse of the time taken for the molecule to bend, and the distortion polarization disappears above the infrared.

### Ionic polarization

Ionic polarization is polarization which is caused by relative displacements between positive and negative ions in ionic crystals (for example, NaCl).

If crystals or molecules do not consist of only atoms of the same kind, the distribution of charges around an atom in the crystals or molecules leans to positive or negative. As a result, when lattice vibrations or molecular vibrations induce relative displacements of the atoms, the centers of positive and negative charges might be in different locations. These center positions are affected by the symmetry of the displacements. When the centers don't correspond, polarizations arise in molecules or crystals. This polarization is called ionic polarization.

Ionic polarization causes ferroelectric transition as well as dipolar polarization. The transition, which is caused by the order of the directional orientations of permanent dipoles along a particular direction, is called order-disorder phase transition. The transition which is caused by ionic polarizations in crystals is called displacive phase transition.

## Dielectric dispersion

In physics, dielectric dispersion is the dependence of the permittivity of a dielectric material on the frequency of an applied electric field. Because there is always a lag between changes in polarization and changes in an electric field, the permittivity of the dielectric is a complicated, complex-valued function of frequency of the electric field. It is very important for the application of dielectric materials and the analysis of polarization systems.

This is one instance of a general phenomenon known as material dispersion: a frequency-dependent response of a medium for wave propagation.

When the frequency becomes higher:

1. it becomes impossible for dipolar polarization to follow the electric field in the microwave region around 1010 Hz;
2. in the infrared or far-infrared region around 1013 Hz, ionic polarization and molecular distortion polarization lose the response to the electric field;
3. electronic polarization loses its response in the ultraviolet region around 1015 Hz.

In the frequency region above ultraviolet, permittivity approaches the constant ε0 in every substance, where ε0 is the permittivity of the free space. Because permittivity indicates the strength of the relation between an electric field and polarization, if a polarization process loses its response, permittivity decreases.

## Dielectric relaxation

Dielectric relaxation is the momentary delay (or lag) in the dielectric constant of a material. This is usually caused by the delay in molecular polarization[disambiguation needed] with respect to a changing electric field in a dielectric medium (e.g. inside capacitors or between two large conducting surfaces). Dielectric relaxation in changing electric fields could be considered analogous to hysteresis in changing magnetic fields (for inductors or transformers). Relaxation in general is a delay or lag in the response of a linear system, and therefore dielectric relaxation is measured relative to the expected linear steady state (equilibrium) dielectric values. The time lag between electrical field and polarization implies an irreversible degradation of free energy(G).

In physics, dielectric relaxation refers to the relaxation response of a dielectric medium to an external electric field of microwave frequencies. This relaxation is often described in terms of permittivity as a function of frequency, which can, for ideal systems, be described by the Debye equation. On the other hand, the distortion related to ionic and electronic polarization shows behavior of the resonance or oscillator type. The character of the distortion process depends on the structure, composition, and surroundings of the sample.

The number of possible wavelengths of emitted radiation due to dielectric relaxation can be equated using Hemmings' first law (named after Mark Hemmings)

$n = \frac{l^2-l}{2}$

where

n is the number of different possible wavelengths of emitted radiation
$l$ is the number of energy levels (including ground level).

### Debye relaxation

Debye relaxation is the dielectric relaxation response of an ideal, noninteracting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity $\varepsilon\,\!$ of a medium as a function of the field's frequency $\omega$:

$\hat{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{\Delta\varepsilon}{1+i\omega\tau},$

where $\varepsilon_{\infty}$ is the permittivity at the high frequency limit, $\Delta\varepsilon = \varepsilon_{s}-\varepsilon_{\infty}$ where $\varepsilon_{s}$ is the static, low frequency permittivity, and $\tau$ is the characteristic relaxation time of the medium.

This relaxation model was introduced by and named after the physicist Peter Debye (1913).[5]

## Paraelectricity

Paraelectricity is the ability of many materials (specifically ceramic crystals) to become polarized under an applied electric field. Unlike ferroelectricity, this can happen even if there is no permanent electric dipole that exists in the material, and removal of the fields results in the polarization in the material returning to zero.[6] The mechanisms which give rise to paraelectric behaviour are the distortion of individual ions (displacement of the electron cloud from the nucleus) and the polarization of molecules or combinations of ions or defects.

Paraelectricity occurs in crystal phases in which electric dipoles are unaligned (i.e., unordered domains that are electrically charged) and thus have the potential to align in an external electric field and strengthen it. In comparison to the ferroelectric phase, the domains are unordered and the internal field is weak.

The LiNbO3 crystal is ferroelectric below 1430 K, and above this temperature it transforms into a disordered paraelectric phase. Similarly, other perovskites also exhibit paraelectricity at high temperatures.

Paraelectricity has been explored as a possible refrigeration mechanism; polarizing a paraelectric by applying an electric field under adiabatic process conditions raises the temperature, while removing the field lowers the temperature.[7] A heat pump that polarizes the paraelectric, allows it to return to ambient temperature, then brings it into contact with the object to be cooled, and depolarizes it, would result in refrigeration.

## Applications

### Capacitors

Charge separation in a parallel-plate capacitor causes an internal electric field. A dielectric (orange) reduces the field and increases the capacitance.

Commercially manufactured capacitors typically use a solid dielectric material with high permittivity as the intervening medium between the stored positive and negative charges. This material is often referred to in technical contexts as the "capacitor dielectric".[8]

The most obvious advantage to using such a dielectric material is that it prevents the conducting plates on which the charges are stored from coming into direct electrical contact. More significant, however, a high permittivity allows a greater charge to be stored at a given voltage. This can be seen by treating the case of a linear dielectric with permittivity ε and thickness d between two conducting plates with uniform charge density σε. In this case the charge density is given by

$\sigma_{\varepsilon}=\varepsilon\frac{V}{d}$

and the capacitance per unit area by

$c=\frac{\sigma_{\varepsilon}}{V}=\frac{\varepsilon}{d}$

From this, it can easily be seen that a larger ε leads to greater charge stored and thus greater capacitance.

Dielectric materials used for capacitors are also chosen such that they are resistant to ionization. This allows the capacitor to operate at higher voltages before the insulating dielectric ionizes and begins to allow undesirable current.

### Dielectric resonator

A dielectric resonator oscillator (DRO) is an electronic component that exhibits resonance for a narrow range of frequencies, generally in the microwave band. It consists of a "puck" of ceramic that has a large dielectric constant and a low dissipation factor. Such resonators are often used to provide a frequency reference in an oscillator circuit. An unshielded dielectric resonator can be used as a Dielectric Resonator Antenna (DRA).

## Some practical dielectrics

Dielectric materials can be solids, liquids, or gases. In addition, a high vacuum can also be a useful, lossless dielectric even though its relative dielectric constant is only unity.

Solid dielectrics are perhaps the most commonly used dielectrics in electrical engineering, and many solids are very good insulators. Some examples include porcelain, glass, and most plastics. Air, nitrogen and sulfur hexafluoride are the three most commonly used gaseous dielectrics.

• Industrial coatings such as parylene provide a dielectric barrier between the substrate and its environment.
• Mineral oil is used extensively inside electrical transformers as a fluid dielectric and to assist in cooling. Dielectric fluids with higher dielectric constants, such as electrical grade castor oil, are often used in high voltage capacitors to help prevent corona discharge and increase capacitance.
• Because dielectrics resist the flow of electricity, the surface of a dielectric may retain stranded excess electrical charges. This may occur accidentally when the dielectric is rubbed (the triboelectric effect). This can be useful, as in a Van de Graaff generator or electrophorus, or it can be potentially destructive as in the case of electrostatic discharge.
• Specially processed dielectrics, called electrets (which should not be confused with ferroelectrics), may retain excess internal charge or "frozen in" polarization. Electrets have a semipermanent external electric field, and are the electrostatic equivalent to magnets. Electrets have numerous practical applications in the home and industry.
• Some dielectrics can generate a potential difference when subjected to mechanical stress, or change physical shape if an external voltage is applied across the material. This property is called piezoelectricity. Piezoelectric materials are another class of very useful dielectrics.
• Some ionic crystals and polymer dielectrics exhibit a spontaneous dipole moment which can be reversed by an externally applied electric field. This behavior is called the ferroelectric effect. These materials are analogous to the way ferromagnetic materials behave within an externally applied magnetic field. Ferroelectric materials often have very high dielectric constants, making them quite useful for capacitors.


text:
# Creating an empty iterator of a certain type in Rust

2018-12-15

I am working these days on the development of offst's Index server. I needed to implement a basic directed graph structure, allowing to run the BFS algorithm to find routes with a certain amount of capacity.

The Index server has the job of collecting capacity reports from nodes in the network. Nodes can then query the Index server for routes of certain capacities.

Whenever a node A wants to send k credits to node B, the node A should first send a request to the Index server, requesting a route from A to B of capacity at least k credits. When A obtains a response from the Index server containing a route, A can push credits along the provided route all the way to B.

Example:

A requests a route:

RouteRequest  (A -> IndexServer):  (A -> B, capacity >= 20)
RouteResponse (IndexServer -> A):  (A -- M1 -- M2 -- M3 -- B, capacity = 35)

Note that in the example A requested a route of send capacity at least 20 credits, and he obtained as response a route with capacity 35, which is more than what A asked for.

Now A can use the obtained route A -- M1 -- M2 -- M3 -- B to push credits all the way to B.

## Returning an empty iterator

During the work on the Index server I had the problem of wanting to return an empty iterator in an early flow of a function.

Let's begin with some context to the problem. The graph we am working with has the following structure:

pub struct SimpleCapacityGraph<N> {
nodes: HashMap<N, HashMap<N,(u128, u128)>>,
}

This is a map, mapping every node in the graph to a map of his neighbors. The map of his neighbors maps each neighbor to a pair of capacities, send capacity and receive capacity.

Example:

The graph:

 0 -- 1
|
|
2

Could be described with the following map:

0 -> {1 -> (10, 20), 2 -> (30, 15)}
1 -> {0 -> (20, 10)}
2 -> {0 -> (15, 35)}

In the example above, note that the node 0 reported a send capacity of 10 credits to the node 1 and a receive capacity of 20 credits to the node 1. The report of node 1 about capacities to node 0 matches the report of node 0.

However, note that this will not always be the case. For example, node 2 reported receive capacity of 35 credits from node 0, but node 0 reported send capacity of 30 credits to node 2.

The details about the send and receive capacities are not very relevant to the discussion here, but I felt obligated to explain the non matching numbers in the example above.

Going back to the code, somewhere inside the implementation of SimpleCapacityGraph there is the neighbors_with_send_capacity(&self, a: N, capacity: u128). This function returns all the neighbors of the node a with capacity which is at least the given capacity argument:

impl<N> SimpleCapacityGraph<N>
where
N: cmp::Eq + hash::Hash + Clone + std::fmt::Debug,
{
/* ... */

fn neighbors_with_send_capacity(&self, a: N, capacity: u128) -> ??? {
// ???
}

/* ... */
}

## Returning a Vec

The steps we should follow are roughly as follows:

1. Try to obtain the node a's map, using self.nodes.get(&a). If this node is not present, a has no matching neighbors.

2. Iterate over a_map, the map of a's neighbors. For every neighbor make sure that it is of a large enough capacity. Return all the neighbors that have large enough capacity.

What should be the return value for neighbors_with_send_capacity? We could choose something like Vec, obtaining this implementation:

fn neighbors_with_send_capacity_vec(&self, a: N, capacity: u128) -> Vec<&N> {
let a_map = match self.nodes.get(&a) {
Some(a_map) => a_map,
None => return vec![],
};

a_map
.keys()
.filter(move |b| self.get_send_capacity(&a,b) >= capacity)
.collect::<Vec<&N>>()
}

This is a working solution, however, note that we allocate a vector every time we call this function, and where is the fun in using Rust if we don't have zero cost abstractions?

Another important detail is that every usage of neighbors_with_send_capacity_vec() in our code uses it to iterate over the resulting neighbors. It would have been nice if we could somehow return an Iterator from the function.

## Returning Option of impl Iterator

Using the impl trait feature we can do the following:

fn neighbors_with_send_capacity_option(&self, a: N, capacity: u128) -> Option<impl Iterator<Item=&N>> {
let a_map = match self.nodes.get(&a) {
Some(a_map) => a_map,
None => return None,
};
let iter = a_map.keys().filter(move |b| self.get_send_capacity(&a,b) >= capacity);
Some(iter)
}

However, this solution is not very ergonomic, as it requires the user of the function to first check if the returned value is Some(iterator) and only then iterate over the iterator. The user code will look like this:

if let Some(neighbors) = self.neighbors_with_send_capacity_option(a, capacity) {
for neighbor in neighbors {
// Do something here
}
}

## std::iter::empty() attempt

After some searching in Rust's std documentation we can find std::iter::empty(). This is a function that creates an empty iterator. A naive attempt to use it will look like this:

fn neighbors_with_send_capacity_iter_empty(&self, a: N, capacity: u128) -> impl Iterator<Item=&N> {
let a_map = match self.nodes.get(&a) {
Some(a_map) => a_map,
None => return std::iter::empty(),
};
let iter = a_map.keys().filter(move |b| self.get_send_capacity(&a,b) >= capacity);
iter
}

This function will not compile. The compiler throws the following error:

error[E0308]: mismatched types
--> components/index_server/src/graph/simple_capacity_graph.rs:87:9
|
87 |         iter
|         ^^^^ expected struct std::iter::Empty, found struct std::iter::Filter
|
= note: expected type std::iter::Empty<&N>
found type std::iter::Filter<std::collections::hash_map::Keys<'_, N, (u128, u128)>, [closure@components/index_server/src/graph/simple_capacity_graph.rs:86:40: 86:89 self:_, a:_, capacity:_]>

The compiler points out that std::iter::Empty<&N> and our other returned filter are not the same. The compiler has to choose a single return value for the function, and therefore the function could not be compiled.

## Boxing

To make this function compile we can try to box the the return values:

fn neighbors_with_send_capacity_iter_empty_box(&self, a: N, capacity: u128) -> Box<dyn Iterator<Item=&N>> {
let a_map = match self.nodes.get(&a) {
Some(a_map) => a_map,
None => return Box::new(std::iter::empty()),
};
let iter = a_map.keys().filter(move |b| self.get_send_capacity(&a,b) >= capacity);
Box::new(iter)
}

This function will not compile. The compilation error:

error: unsatisfied lifetime constraints
--> components/index_server/src/graph/simple_capacity_graph.rs:98:9
|
92 |     fn neighbors_with_send_capacity_iter_empty_box(&self, a: N, capacity: u128) -> Box<dyn Iterator<Item=&N>> {
|                                                    - let's call the lifetime of this reference '1
...
98 |         Box::new(iter)
|         ^^^^^^^^^^^^^^ returning this value requires that '1 must outlive 'static

But with some anonymous lifetime annotations magic this function will compile:

fn neighbors_with_send_capacity_iter_empty_box(&self, a: N, capacity: u128) -> Box<dyn Iterator<Item=&N> + '_> {
let a_map = match self.nodes.get(&a) {
Some(a_map) => a_map,
None => return Box::new(std::iter::empty()),
};
let iter = a_map.keys().filter(move |b| self.get_send_capacity(&a,b) >= capacity);
Box::new(iter)
}

## OptionIterator

The Boxing solution still feels like some kind of a compromise.

To solve this problem, let's introduce a helper type called OptionIterator. OptionIterator will be able to wrap an Option<I>, where I is some Iterator. If the Option is None, the wrapper iterator will finish immediately. Otherwise, if the Option is Some(iterator), the wrapping iterator will do exactly what the internal iterator is doing.

pub struct OptionIterator<I> {
opt_iterator: Option<I>,
}

impl<I,T> Iterator for OptionIterator<I>
where
I: Iterator<Item=T>,
{
type Item = T;
fn next(&mut self) -> Option<T> {
match &mut self.opt_iterator {
Some(iterator) => iterator.next(),
None => None,
}
}
}

impl<I> OptionIterator<I> {
pub fn new(opt_iterator: Option<I>) -> OptionIterator<I> {
OptionIterator {
opt_iterator,
}
}
}

As can be seen, the only field of OptionIterator is opt_iterator, which either contains an iterator or nothing. Next, let's look at the next() method of the Iterator implementation for OptionIterator. If there exists an internal iterator, the internal iterator's next() method is called. Otherwise, None is returned, which means that the iterator has finished.

In other words, for some iterator, OptionIterator::new(Some(iterator)) will behave exactly like iterator. However, OptionIterator::new(None) will behave like an empty iterator.

The advantage of using the OptionIterator wrapper is that both the existing internal iterator and the empty internal iterator cases produce the same type: OptionIterator. Recall our non compiling attempt of neighbors_with_send_capacity_iter_empty() above. We can now rewrite this function as follows:

fn neighbors_with_send_capacity(&self, a: N, capacity: u128) -> OptionIterator<impl Iterator<Item=&N>> {
let a_map = match self.nodes.get(&a) {
Some(a_map) => a_map,
None => return OptionIterator::new(None),
};
let iter = a_map.keys().filter(move |b| self.get_send_capacity(&a,b) >= capacity);
OptionIterator::new(Some(iter))
}

Zero cost abstractions, as promised (:

This is the actual implementation currently chosen for the code in offst. If you are interested, you can view the full code of SimpleCapacityGraph implementation on github.

• Discussion on reddit. Contains some very interesting advice for other possible solutions. For example, using either::Either or res.into_iter().flat_map(|iter| iter).


text:
# Standard production of alignment objects

While the use of the constructors of the AliAlignObj derived classes has been explained in the pages describing the functionality of the alignment framework, we give here a brief description of the macros producing alignment objects for each detector. These macros include

• the detector macros $ALICE_ROOT/DET/MakeDETMisAlignment and • the steering macros$ALICE_ROOT/macros/MakeAllDETsMisAlignment

where type is Zero, Res or Full
The detector macros define the expected deviation of the position of alignable volumes w.r.t. their ideal position (full misalignment) and the expected resolution in the knowledge of these position after survey and alignment procedures will be run (residual misalignment).

The user can run either one of the steering macros or directly the macro for his/her detector only.
The steering macros call the macros of all detectors for the corresponding type; they take care of setting the needed environment variables and running the needed additional macros, if needed. They are launched in the following way:

.x $ALICE_ROOT/macros/MakeAllDETsMisAlignment("cdburi"); or: .x$ALICE_ROOT/macros/MakeAllDETsMisAlignment("cdburi",kTRUE);

the first argument specifying the uri for the CDB storage where one wants to create the alignment objects (as the STORAGE variable explained below), the second argument, which defaults to kFALSE, corresponding to the STORAGE environment variable for partial geometry (also explained in more detail below). In case the user is running his/her detector macro directly, it will needed to consider those preliminary settings by hand, as explained below.

The environment variables used by the macros are three; if not explicitly set to kTRUE they are considered false.

1. TOCDB is to be set to kTRUE when the user wants to produce the alignment objects in a CDB storage; in this case the user will have to set also the next environment variable STORAGE variable to the uri of the storage. For example, supposing you are in a bash environment:
export TOCDB=kTRUE
2. STORAGE corresponds to the uri of the storage
export STORAGE="local://$HOME/MyCDB" 3. PARTGEOM is to be set to kTRUE when the user wants to produce objects for partial geometry (some TOF TRD sectors are switched off): export PARTGEOM=kTRUE Note that in this case the config-file passed to the macros updating the geometry in the given CDB storage (UpdateCDBIdealGeom.C) must be consistent. As an example the Config.C produces a full geometry, while the Config_PDC06.C produces a partial geometry. If the objects are produced on file, the ideal geometry is loaded from$ALICE_ROOT/GRP/Geometry/Data; when saving to CDB instead the macro requires to load the ideal geometry from the same CDB storage. This implies that if you are running the detector macro directly and you are saving the objects to a CDB storage, you have to first crate the ideal geometry file there:

aliroot ALICE_ROOT/GRP/UpdateCDBIdealGeom("cdburi","configfile"); An additional care is to be taken when running directly the macros for TOF or TRD, since they are made dependent from the alignment objects for FRAME (the space-frame holding their super-modules). Thus in this case the user needs to run before: alirootALICE_ROOT/GRP/MakeSTRUCTMisAlignment.C


text:
# Difference between revisions of "human-level control through deep reinforcement learning"

## Introduction

Reinforcement learning is "the study of how animals and artificial systems can learn to optimize their behaviour in the face of rewards and punishments" <ref>Dayan, Peter and Watkins, Christopher. "Reinforcement Learning." Encyclopedia of Cognitive Science.</ref>. Attempting to teach a dog a new trick is an example of this type of learning in animals, as the dog will (hopefully) learn that when it does the desired trick, it obtains a reward. The process is similar in artificial systems. For a given task, the algorithm for reinforcement learning selects an action that maximizes the expected cumulative future reward at every step in time, given the current state of the process. Then, it will carry out this action, observe the reward, and update the model to incorporate information about the reward generated by the action. The natural connection to neuroscience and animal behaviour makes reinforcement learning an attractive machine learning paradigm.

When creating artificial systems based on machine learning, however, we run into the problem of finding efficient representations from high-dimensional inputs. Furthermore, these representations need to use high-dimensional information from past experiences and generalize to new situations. The human brain is adept at this type of learning, using systems involving dopamine in the neurons with a similar structure to reinforcement learning algorithms <ref> Schultz, W., Dayan, P. & Montague, P. R. A neural substrate of prediction and reward. Science 275, 1593–1599 (1997) </ref>. Unfortunately, previous models have only been able replicate the success of humans, as they have only performed well on fully-observable, low-dimensional problems, such as backgammon <ref> Tesauro, Gerald. TD-Gammon, a self-teaching backgammon program, achieves master's play. AAAI Techinical Report (1993)</ref>.

In the paper that is summarized below <ref name = "main">Mnih, Volodymyr et. al. "Human-level control through deep reinforcement learning." Nature 518, 529-533 (2015) </ref>, a new structure called a deep Q-network (DQN) is proposed to handle high-dimensional inputs directly. The task that the DQN model is tested on is playing Atari 2600 games, where the only input data are the pixels displayed on the screen and the score of the game. It turns out that the DQN produces state-of-the-art results in this particular task, as we will see.

## Problem Description

The goal of this research is to create a framework that would be able to excel at a variety of challenging learning tasks - a common goal in artificial intelligence. To that end, the DQN network is proposed, which combines reinforcement learning with deep neural networks <ref name = "main"></ref>. In particular, a deep convolutional network will be used to approximate the so-called optimal action-value function

$Q^*\left(s,a\right) = \max_\pi \mathop{\mathbb{E}}\left[r_t + \gamma r_{t+1} + \gamma^2 r_{t+2} + \ldots | s_t=s, a_t=a, \pi\right]$

which is the maximum cumulative sum of rewards $r_t\,$ discounted by $\,\gamma$ at each timestep $t\,$. This sum can be achieved from a policy $\pi = P\left(a|s\right)$ after making an observation $\,s$ and taking an action $\,a$ <ref name = "main"></ref>.

### Instability of Neural Networks as Function Estimate

Unfortunately, current methods which use deep networks to estimate $Q\,$suffer from instability or divergence for the following reasons:

1. Correlation within sequence of observations
2. Small updates to $Q\,$can significantly change the policy, and thus the data distribution
3. The action values $Q\,$are correlated with the target values $\,y = r_t + \gamma \max_{a'}Q(s', a')$

### Overcoming Instability

One method of overcoming this instability is through the use of a biologically-inspired mechanism called experience replay. This involves storing $e_t = \left(s_t, a_t, r_t, s_{t+1}\right)$ - known as the "experiences" - at each time step in a dataset $D_t = \left(e_1, e_2, \ldots, e_t\right)$. Then, during each learning update, minibatch updates are performed, where points are sampled uniformly from $\,D_t$. In practice, only $N$ experiences are stored, where $N$ is some large, finite number (e.g. $N=10^6$). Incorporating experience replay into the algorithm removes the correlation between observations and smooths out the data distribution, since it takes an average over the past $N$ states. This makes it much more unlikely that instability or divergence will occur.

Another method used to combat instability is to use a separate network for generating the targets $y_i$ as opposed to the same network. This is implemented by cloning the network $\,Q$ every $\,C$ iterations, and using this static, cloned network to generate the next $\,C$ target values. This additional modification reduces correlation between the target values and the action values, providing more stability to the network. In this paper, $\,C = 10^4$.

### Data & Preprocessing

The data used for this experiment is initially frame-by-frame pixel data from the Atari 2600 emulator, along with the game score (and the number of lives in applicable games). The frames are 210 x 160 images with colour, so some preprocessing is performed to simplify the data. The first step to encode a single frame is to take the maximum value for each pixel colour value over the frame being encoded and the previous frame <ref name = "main"></ref>. This removes flickering between frames, as sometimes images are only shown on every even or odd frame. Then, the image is converted to greyscale and downsampled to 84 x 84. This process is applied to the $m$ most recent frames (here $m=4$), and these 84x84x4 images are the inputs to the network.

## Model Architecture

The framework for building an agent that can learn from environmental inputs is an iterative reward-based system. The agent will perform actions and constantly re-assess how its actions have affected its environment. To model this type of behaviour, the agent attempts to build up a model that relates actions to rewards over time. The underlying model relating actions to rewards, $Q$, is estimated by a deep convolutional network, and is updated at every step in time.

The structure of the network itself is as follows. There are separate output units for each possible action, and the only input to the network is the state representation. The outputs are the predicted Q-values for each action performed on the input state. The first hidden layer convolves 32 8x8 filters with stride 4, then applies a rectified nonlinear function <ref>Jarrett K. et. al. What is the best multi-stage architecture for object recognition? Proc. IEEE Int. Conf. Comput. Vis. 2146-2153 (2009)</ref>. In the next hidden layer, 64 4x4 filters with stride 2 are convolved and again followed by rectified non-linearity. The next layer is the final convolutional layer, with 64 3x3 filters of stride 1, followed by the rectifier. The final hidden layer in the network is fully-connected, with 512 rectifying units. The output layer is a fully-connected linear layer with a single output for each valid action, of which there ranged from 4 to 18 in any particular game<ref name = "main"></ref>.

## Training

### Framework and Additional Setup Details

Forty-nine Atari games were considered as experiments. A unique DQN is trained for each game, but the same structure, algorithm, and global parameters (e.g. $C$ or $m$ as above, among others) were used throughout. The value of the global parameters was selected by performing an informal search on a small subset of the 49 games. The goal is to use minimal prior knowledge and perform end-to-end training of these models based on game experience.

The reward structure for games was slightly changed, clipping negative rewards at -1 and positive rewards at 1, since score varies from game to game. This could be problematic since the agent may not properly prioritize higher-scoring actions, but it also helps stabilize the network and allows it to generalize to more games. However, the game itself is not otherwise changed.

There is also a frame-skipping technique employed, in which the agent only performs an action every $k^{th}$ frame to allow the agent to play $k$ times more games, as the network does not have to be trained on the skipped frames ($k=4$ here). Furthermore, I believe this creates a more realistic experience for the agent, as human players would not be able to change their own actions every single frame.

The agents are trained on 50 million frames of game play, which is about 38 days. The RMSProp <ref>Hinton, Geoffrey et al.Overview of Minibatch Gradient Descent. University of Toronto.</ref> algorithm, which performs stochastic gradient descent in small batches, is used to train the network.

### Algorithm Background

At each step in time, the agent selects an action $a_t\,$ from the set of legal game actions $\mathbb{A}$. The agent observes an image $x_t \in \mathbb{R}^d$ from the emulator, along with a reward $r_t\,$. It is impossible to fully understand the current game situation from a single screen, so a sequence of actions and observations $s_t = x_1, a_1, x_2, \ldots, a_{t-1}, x_t$ is the input state.

Recall that if we define $R_t = \sum_{t'=t}^T \gamma^{t'-t}r_t$, where $\gamma\,$ is the discount factor and $\,T$ is the step in which the game terminates, then

$Q^*\left(s,a\right) = \max_\pi \mathop{\mathbb{E}}\left[R_t| s_t=s, a_t=a, \pi\right]$

is the optimal action-value function.

#### The Bellman Equation in the Loss Framework

The optimal action-value function obeys the Bellman Equation:

$Q^*\left(s,a\right) = \mathop{\mathbb{E}_{s'}}\left[r + \gamma \max_{a'}Q^*\left(s',a'\right) | s, a\right]$

The intuition behind this identity is as follows: if the optimal value $Q^*(s',a')\,$ at the next time step was known for all possible actions $a'\,$, then the optimal strategy is to select the action $a'$ maximizing the expected value above <ref name = "main"> </ref>. Using the Bellman Equation as an iterative update formula is impractical, however, since the action-value function is estimated separately for each sequence and cannot generalize.

It is necessary, in practice, to operate with an approximation of the action-value function. When a neural network with weights $\,\theta$ is used, it is referred to as a Q-Network. A Q-Network is trained by adjusting $\,\theta_t$ to reduce the mean-squared error in the Bellman Equation. The new target values for training are given by $y = r + \gamma\max_{a'} Q\left(s', a'; \theta_t^-\right)$, where $\theta_t^-\,$ are the parameters from some previous iteration.

### The Full Algorithm

Now, with all the background behind us, the full Deep Q-Learning with Experience Replay algorithm is presented below:

• Replay memory is used to implement the experience replay technique described above
• An episode is one game
• Correlations between target values and the action function $Q\,$are mitigated by using $\hat{Q}$ for the target values
• Only updated every $\,C$ steps

## Results

### Evaluation Procedure

The trained networks played each game 30 times, up to 5 minutes at a time. The random agent, which is the baseline comparison, chooses a random action every 6 frames (10 Hz). The human player uses the same emulator as the agents, and played under controlled conditions (most notably without sound). The human performance is the average reward from around 20 episodes of the game lasting up to 5 minutes, after 2 hours of practice playing each game. The human performance is set to be 100%, and the random agent has performance set to 0%.

### Raw Score Results

The DQN agent outperforms the best existing reinforcement learning methods on 43 of the games without incorporating prior knowledge about Atari games. Furthermore, the agent scores at least 75% of the human score on more than half of the games. Also, DQN performs well across a number of types of games. However, games which involve extended planning strategies still pose a major problem to DQN (e.g. Montezuma's Revenge). These results are visualized in the figure below:

### Results with model components removed

Two important advances in this paper were presented: experience replay and creating a separate network to evaluate the targets. To visualize the impact of these advances, the network was trained with and without both of these concepts, and evaluated on its performance in each case. The results are shown in the table below, in percentage form as above:

Game With Replay and Target Q With Replay, Without Target Q Without Replay, With Target Q Without Replay, Without Target Q
Breakout 316.8 240.7 10.2 3.2
Enduro 1006.3 831.4 141.9 29.1
River Raid 7446.6 4102.8 2867.7 1453.0
Seaquest 2894.4 822.6 1003.0 275.8
Space Invaders 1088.9 826.3 373.2 302.0

Clearly, experience replay and maintaining a secondary network for computing target values are important. From these results, it seems that experience replay is more important on its own, except in Seaquest.

## Conclusion

The framework presented has demonstrated the ability to learn how to play Atari games, given minimal prior knowledge of the game and very basic inputs. Using reinforcement learning with the Q-network architecture was more effective than previous similar attempts, since experience replay and a separate target network were utilized in training. These two modifications removed correlations between sequential inputs, which improved stability in the network. Future work should be undertaken to improve the experience replay algorithm: instead of sampling uniformly from the replay memory, the sampling should be biased towards high-reward events. However, this may add a layer of instability to the network, but it is certainly worth investigating.

<references />


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# Boxes and multiples

For an integer $$n \geq 3$$, John has $$n-1$$ cards numbered from 2 to $$n$$ and he wants to put them in two boxes according to the following rule:
If $$a$$ and $$b$$ are the numbers of two cards and $$b \mid 2^{a}-1$$, then the cards with these numbers are in different boxes.

What's the biggest number $$n$$ for which it is possible for John to keep following this rule?

Note by John Smith
1 year, 3 months ago

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*italics* or _italics_ italics
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Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

## Comments

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@Jon Haussmann You're right. I made a mistake at n=7. Do you think there's any better way to prove this? I can't come up with anything other than this brute force method.

- 1 year, 2 months ago

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I don't see any other way. And it's pretty easy to go up to $$n = 7$$, so I wouldn't call it brute force.

- 1 year, 2 months ago

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I can only get up to $$n = 6$$.

Call the boxes box A and box B. The number 2 has to go somewhere, so let's just put it in box A.

Since 3 divides $$2^2 - 1$$, the number 3 has to go in box B. Since 3 divides $$2^4 - 1$$ and $$2^6 - 1$$, the numbers 4 and 6 have to go in box A. Since 5 divides $$2^4 - 1$$, the number 5 has to go in box B. Thus, we can place the numbers 2, 3, 4, 5, and 6. Furthermore, the placement of these numbers is forced, i.e. this is the only possible way. (We can swap the numbers in box A and box B, but it's basically the same.)

Now, 7 divides $$2^3 - 1$$ and $$2^6 - 1$$, so we cannot place the number 7 in either box. This means $$n = 6$$ is as high as we can go.

- 1 year, 2 months ago

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I tried this until n=15 and it failed there so I think the answer is 14 but it isn't a very elegant proof (brute force never is ahah)

- 1 year, 2 months ago

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# How to Find the value of Z, Using the X and Y, between a three point plane

2018-02-17 06:47:17

I would like to know how to work out the value $z$, using $x$ and $y$ values, between 3 points.

For example

if point A is (2,4,0), B is (10,3,3) and C is (2,10,1)

D is (6,5,?) the ? being the unknown $z$ value i want to find

Thank you

The equation of a plane is $ax+by+cz=1$.

We have $2a+4b=1$, $10a+3b+3c=1$, and $2a+10b+c=1$.

Solving gives us $a=\frac{19}{70}, b=\frac{8}{70}, c=-\frac{48}{70}$.

Plugging this into $ax+by+cz=1$ and (x,y)=(6,5) yields z=$\boxed{\frac{7}{4}}$.

• The equation of a plane is $ax+by+cz=1$.

We have $2a+4b=1$, $10a+3b+3c=1$, and $2a+10b+c=1$.

Solving gives us $a=\frac{19}{70}, b=\frac{8}{70}, c=-\frac{48}{70}$.

Plugging this into $ax+by+cz=1$ and (x,y)=(6,5) yields z=$\boxed{\frac{7}{4}}$.

2018-02-17 08:27:01


text:
# Caching Ruby Gems

Ruby gems are not cached by default on the bitrise.io virtual machines. Ruby gems are installed into a location depending on the current rbenv version. The version can be checked by running rbenv version.

You can get the location of this directory with gem environment gemdir. However, it is not enough to cache this directory as rbenv sets up link to ruby version specific gems: the whole ruby version specific directory - for example, /Users/vagrant/.rbenv/versions/2.5.3 - has to be cached.

It is not recommended to set the value of the $GEM_HOME Enviroment Variable, as this can result in installed gems not being found. 1. Open your app on Bitrise. 2. Go to the Workflow tab. 3. Select a Workflow from the WORKFLOW dropdown menu. 4. Add a Script Step to the Workflow. 5. Set the $GEM_CACHE_PATH Environment Variable in the Script Step.

 - script:
title: Set GEM_CACHE_PATH env var
inputs:
- content: |-
#!/bin/bash
set -ex
RBENV_DIR="cd $(rbenv which ruby)/../..;pwd" echo "Gem cache directory:$RBENV_DIR"
envman add --key GEM_CACHE_PATH --value $RBENV_DIR 6. Insert the Cache:Pull Step after the Git Clone but before the Android Build Steps. 7. Insert the Cache:Push Step to the very end of your Workflow. 8. Open the input Cache paths of the Step Cache:Push and add $GEM_CACHE_PATH in a new line as an additional cache dir.

And you’re done!