2d curve fitting

2d curve fitting. The output is a single value of size 1. Nearest neighbor. S. A smaller residual means a better fit. Built into the Wolfram Language are state-of-the-art constrained nonlinear fitting capabilities, conveniently accessed with models given directly in symbolic form. pyplot as plt data = np. The actual one-line code to carry out the fit of the data in myExpData to the function myExpDecay is the following. This method fits a different linear polynomial between each pair of data points for curves, or between sets of three points for surfaces. For instance, a linear fit would use a function like. $ 449. May 21, 2024 · The purpose of curve fitting is to find a function f ( x) in a function class Φ for the data ( xi, yi) where i =0, 1, 2,…, n –1. In the Select Fitting Data dialog box, select temp as the X data value and thermex as Mar 31, 2015 · How to fit a 2d curve to a matrix. 7 Function File. 87992. x_dummy = np. A reduced size data set with min, max, and (hopefully) evenly spaced additional data points in between are used. Use interpolation to smooth observed data, fill in missing data, and make predictions. # variables = 5. By Jason Brownlee on November 14, 2021 in Optimization 76. The algorithms are translated from MINPACK-1, which is a rugged minimization routine found on Netlib, and distributed with permission. #. Nadim El Hayek, O. Smoothing is a method of reducing the noise within a data set. 2 Fitting 3D Surfaces (Pro Only) Origin supports 3D Surface Fitting, which is an extension of ordinary nonlinear fitting, for both XYZ and matrix data. To use this you have to flatten the array as scipy's curve_fit only takes a 1d array. Learn more about statistics, 3d, 2d, surface, data analysis, fitting, curve fitting, lsqcurvefit, nlinfit, fit regression surface to 3d data MATLAB I have some numbers as a function of 2 variables: _( x, y ) ↦ z_. You can use the curve fitting toolbox (cftool) or the statistics toolbox (regress, LinearModel. The idea is to make this extensible and allow users to easily add other fitters. Case 2: 2-dimensional stres Nov 14, 2021 · Curve Fitting With Python. Nov 1, 2004 · Abstract. up to and including x^kx*y^ky, are considered. The method of least squares can be generalized to allow fitting more complex functions to data. I'd like to fit z with a function, like in scipy. Oct 1, 2017 · The experiments evaluate the performance of the adaptive fitting method in real 2D and 3D scenarios. for functions with k predictors. Or there is skimage's blob detection. In this Part, we review the method while adapting it to the problem of Explore how to fit curves to data points using different methods and tools. 5 Derived Parameters. e. Overview of Curve Fitting In curve fitting we have raw data and a function with unknown coefficients. 3 Sample Curve. To generate a set of points for our x values that are evenly distributed over a specified interval, we can use the np. 7. Add to cart. So if the curve you want to fit is "a log (x^2) + b y", where a and b are free parameters, then you Apr 22, 2020 · 1. The one easy solution is to use the for loop. Jan 3, 2017 · IDL Curve Fitting and Function Optimization. Clear. Nonlinear fit and SciPy curve_fit. The program generates a 2D Gaussian. All Fitters can be called as functions. This flexibility in managing multiple input variables makes it a unique multivariable curve fitting tool on the market. To fit a 2 dimensional curve your network should be fed with vectors of size 2, that is a vector of x and y coordinates. In the module Least Squares, we learned how to find the best fit of a straight line to a set of data points. # Generate dummy dataset. On the Curve Fitter tab, in the Data section, click Select Data. Population Data. The method is effective in removing far-from-the-shape noise and in handling a shape of changing density. A minimization process in which location parameters and control points are calculated simultaneously and topological distances that can be assimilated to the Hausdorff distances are proposed Because lifetime data often follows a Weibull distribution, one approach might be to use the Weibull curve from the previous curve fitting example to fit the histogram. then ranked by a fit statistic such as AIC or SSQ errors. ie, if we are fitting a 2D polynomial, xs will be of length numpts * 2. providing initial parameters with suitable estimates of the true parameters. The following is what I tried and failed. Example of 2D Feb 6, 2016 · func(X, p1, p2) is now of the form return p1*X[0] + p2*X[1] and I am calling the fitting function as curve_fit(func, xdata, ydata) where xdata is a 2x111 array and ydata is a length 111 array. Note that we must supply starting guesses. The result is shown below. ndarray, 2d Surface to fit. the point to which the closest midpoint of the elements will be computed. 0. The goal is to find a curve (or function) that models the relationship in the data as accurately as possible. 1. Datasets. scipy. Curve Fitting & Approximate Functions. You can also add up to 100 external C or FORTRAN language functions to the TableCurve 2D equation set. Gibaru, +3 authors. Lee discussed the deficiencies of direct application of MLS in curve fitting and proposed some improvements by introducing the Minimal Spanning Tree (MST) in a pre-processing phase [14]. Let’s generate 200 datapoints from 0 to 20 with numpy linspace function. 1. along each rows, which gives as fit parameters for each T as given below (as an example): T a b c. Vote. Nearest neighbor interpolation. The algorithm should generate curves and surfaces that are interpretable and similar to the original 2D and 3D objects. Smooth data interactively using the Curve Fitter app or at the command line using the smooth function. The algorithm consists of the following steps: Initialization of the B-spline surface by using the Principal Component Analysis (PCA). I'd like to say divide the curve into 10 and fit the best polynomial to Sep 5, 2016 · c2y = 283. Curve fitting is the process of finding a mathematical function that closely matches a set of data points. Unfortunately, this results in an incorrect peak position measurement, due to the overlapping signal of the neighbouring peaks. We assume that you have theoretical reasons for picking a function of a certain form. m” with not input parameters. The whole T dependence should be taken care of in the fit, but I don't need it displayed with splot. A fit function with already three Gaussians in it is used. We cater Jan 1, 2016 · A parametric curve is often used for shape reconstruction of a point cloud. 1774. TableCurve 2D's built-in library includes a wide array of linear and nonlinear models for any application including equations that the function x(t) is determined solely by the x coordinates of the control points. fit, NonLinearModel. Definition at line 710 of file fitting_curve_2d. To try this approach, convert the histogram to a set of points (x,y), where x is a bin center and y is a bin height, and then fit a curve to those points. 00. c - the point of inflection (i. The zero-set of the polynomial is determined by the IP coefficients and . Open the Curve Fitter app. # function evals = 87. curve_fit also requires a function that provides the type of fit you would like. The The fit constructor takes a set of points and fits them with a polynomial of the requested order. (1) cartesian distance between the endpoints of each curve. By default, the app fits a linear polynomial to the data. cpp. Some common use cases for curve fitting include: Fit Surface. The Basics of Curve Fitting. Mar 31, 2014 · You'll get the end result as a "spline," a smooth sequence of piecewise cubic polynomials. Notice that we are weighting by positional uncertainties during the fit. Arguments x and y correspond to the values of the data points that we want to fit, on the x and y axes, respectively. Load some example data at the MATLAB command line. And, unlike most curve fitting programs, TableCurve 2D’s user-defined functions are compiled so custom curve fitting can be performed quickly, at nearly the speed as with the built-in equations. If I feed 3000 points into LoessInterpolator it generates 2999 polynomial functions. Curve Fitting Functions. Fitting a two-dimensional polynomial to a surface is, in principle, a linear least-squares problem, since the fitting function is linear in the fit coefficients, ci,j c i, j : zfit(x, y) = c0,0 +c1 A 1-D sigma should contain values of standard deviations of errors in ydata. The calculator below uses the linear least squares method for curve fitting, in other words, to approximate A common use of least-squares minimization is curve fitting, where one has a parametrized model function meant to explain some phenomena and wants to adjust the numerical values for the model so that it most closely matches some data. For the problem-based approach, create problem variables, and then represent the objective function and constraints in terms of these symbolic variables. I tested kPCA, LLE and isomap. Further more, the numerical problem associated with constrained fitting is tackled in our paper. The app plots the data points as you select variables. h. optimize. The operation to perform a surface fit is similar to that of nonlinear fitting. TableCurve 2D’s built-in library includes a wide array of linear and nonlinear models for any application including equations that may never have been considered, from simple linear equations to high order Chebyshev polynomials. Polynomial::Polynomial(const vector<double>& xs, const vector<double>& ys, int in_dims, int in_order). setting the fit tolerance and maximum number of iterations. 4 Parameters. We use the term “parameters” to talk about the values that you pass to operations and functions. E. # fitting method = leastsq. Both classes have the draw method making use the parameter-based representation: # t being the coordinate parameter xp, yp = model. load hahn1. The third parameter specifies the degree of our polynomial function. Sometimes we are interested in relationships which are not linear, in such case we wonder how can we approximate our data. Contents. Interpolation is a method of estimating values between known data points. Feb 6, 2013 · f(h,T) = a * h * (1 + alpha * T) + f0. curve_fit. Adjust parameters, compare models, and test your predictions. Scipy is the scientific computing module of Python providing in-built functions on a lot of well-known Mathematical functions. This assumes that the point-cloud has two main orientations, i. More accurate result would require more refined method. Often we end up “fitting” a dataset to some sort of mathematical function like a line, a sinusoid wave, or an exponentially decaying function. The algorithm takes O(nlog n) time and O(n) space for a set of n points. where: a - the minimum value that can be obtained ( y at x = 0) b - Hill’s slope of the curve. TLDR. curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, check_finite=True, bounds=(- inf, inf), method=None, jac=None, full_output=False, **kwargs constraints. Use non-linear least squares to fit a function, f, to data. The residual is the distance between the data samples and f ( x ). exp(b*x) We will start by generating a “dummy” dataset to fit with this function. With scipy, such problems are typically solved with scipy. If additional constraints on the approximating function are entered, the calculator uses Lagrange multipliers to find the solutions. The method is an extension to the published methods for 2D constrained fitting. # data points = 10000. In the Select Fitting Data dialog box, select x as the X data value, y as the Y data value, and z as the Z data value. To interactively fit a curve, follow the steps in this simple example: Load some data at the MATLAB ® command line. curve_fit(func, x, y) will return a numpy array containing two arrays: the first will contain values for a and b that best fit your data, and the second will be Apr 12, 2020 · def exponential(x, a, b): return a*np. fit) to perform multiple linear regression. optimize import curve_fit import matplotlib. Fit in python with curve_fit. The coefficients c1,c2,c3,c4 c 1, c 2, c 3, c 4 were adjusted by linear regression (least squares fitting) The coefficients in exponent were roughly adjusted in a first provisional approach. On fitting a 2d Gaussian, read here. Jan 30, 2022 · This earlier blog post presented a way of performing a non-linear least squares fit on two-dimensional data using a sum of (2D) Gaussian functions. Overview of Curve Fitting. Levin proposed the moving least squares method (LMS) for curve fitting [13]. Method. curve_fit, which is a wrapper around scipy Jul 18, 2019 · Link. 00 2. Boundary fitting for 2D curve reconstruction. this means 1D, 2D, 3D, curves are all really the same. Ive tried looking through a few examples provided by altenbach To plot this, we can interpolate the data onto a grid. 08-16-2019 10:59 AM. optimize and have created a parabolic function, which I then try to fit with using: coeff, cov = curve_fit(parabolic, data[:,0], data[:,1], p0) #Return co-effs for fit and covariance, p0 = user given seed fitting parameters y_values = np. Definition at line 55 of file fitting_curve_2d_apdm. 3. Also, the best-fit parameters uncertainties are estimated from the variance-covariance matrix. return closest midpoint in parametric domain. kx, ky: int, default is 3 Polynomial order in x and y, respectively. From these ten arrays for every element (36000000), I need to do the least If your data is well-behaved, you can fit a power-law function by first converting to a linear equation by using the logarithm. order: int or None, default is None If None, all coefficients up to maxiumum kx, ky, ie. 10 10. optimize that apply non-linear least squares to fit the data to a function. Then you can use a function such as lsqnonlin (nonlinear least squares) to fit a 2d curve to your matrix z. Assuming you want to process the plot as an image, this is a possible workflow: Binarize the grayscale image ("rgb2gray" and "imbinarize") Find the connected components in the binarized image ("bwconncomp") Use the 'PixelIdxList' that you get for these components to fit a curve (Interactively using "cftool" or programmatically using R S S = ∑ ( o b s − p r e d) 2. In the end I need a plot with f on the y-axis and h on the x-axis. So I have a 2d array that z = f (x,y) that I simply want to fit it is completely random in shape but I want to fit it as accurately as possible using. LabVIEW. Options. My dream would be something like : curve_fit2D(custom_2D_function, (x_values,y_values), z) I couldn't find anything in scipy documentation This online calculator builds a regression model to fit a curve using the linear least squares method. processing the output data. normal(loc=5. stats import norm import matplotlib. M_new[i,] = apply(M[which(distance_M[i,] < eps),], 2, mean) 2/ Manifold identification to transform points to a time-series. However, because squares of the Fast B-Spline 2D Curve Fitting for unorganized Noisy Datasets. This works a little better, although it is very sensitive to the chosen parameter. Add a comment. Some 8) Curve Fitting (nonlinear regression - least squares method, Levenberg-Marquardt algorithm -, almost 500 functions at the library with one and two independent variables, functions finder, option that let you write your own fitting function with up to 150 characters, 6 independent variables and 10 parameters). Nov 28, 2015 · z: np. z (x,y) = a + bx + cy + dxy + ex^2 + fy^2. A neural network can approximate an arbitrary function of any number of parameters to a space of any dimension. ) 1. Link. Curve fitting is a type of optimization that finds an optimal set of parameters for a defined function that best fits a given set of observations. 000000; p2y = 200. MatDeck has a variety of unique curve fitting features, methods and functions. Curve Fitting Toolbox™ functions allow you to perform interpolation by fitting a curve Sep 25, 2021 · The way that I determined collision was through extrapolation of each curve against each other curve. import numpy as np from scipy. 099363; p2x = 40. Note that for an initial guesstimate of parameter values, not all data need be used. Jun 23, 2010 · For example: [x,y] = meshgrid(1:size(z,2),1:size(z,1)); Trace x and y in your command window to see their structure, it'll make sense. The formula used for fitting is following: y=d+1+(cx)ba−d. Since curve fitting is done in a canvas, it allows the user to utilize the nearly all MatDeck Features. But it works fine. 6 Script Access. that it is roughly planar. In this case, we can use a built-in model to fit. In the Select Fitting Data dialog box, select x as the X data value and p as the Y data value. norm as follows: import numpy as np from scipy. The idea of curve fitting is to find a mathematical model that fits your data. This work deals with fitting 2D and 3D implicit polynomials (IPs) to 2D curves and 3D surfaces, respectively. Oct 22, 2009 · In this paper we present a 3-step algorithm for reconstructing curves from unorganized points: data clustering to filter out the noise, data confining to get the boundary, and region thinning to find the skeleton curve. Linear interpolation. A minimization process in which location parameters and control points are calculated simultaneously and topological distances that can be assimilated to the Hausdorff distances are proposed Nov 28, 2015 · z: np. linspace(start=5, stop=15, num=50) start Auto-fit thousands of curve equations to the selected data, discover the best equation to model the data and graphically review the fitting results; Include over 5300 2D built-in curve equations that are divided into 19 equation families with support for combined equations (i. Assumes ydata = f (xdata, *params) + eps. 1 Function. May 15, 2013 · Accepted Answer. Expand. Parameters: [in] nurbs. Curve Fitting Software- TableCurve 2D® curve fitting software gives engineers and researchers the power to find the ideal model for even the most complex data, by putting thousands of equations at their fingertips. I have ten arrays of the size 6000 x 6000. Graphics Gems has a simple C code example of Bezier curve fitting with no other library dependencies: (The code is public domain; see the readme in the repo. the point on the curve halfway between a and d) d - the maximum value that can be obtained ( y at x tending to infinity) Fast B-Spline 2D Curve Fitting for unorganized Noisy Datasets. The Wolfram Language also supports unique symbolic interpolating functions that can immediately be used throughout the system to efficiently Apr 1, 2017 · 1/ Doing a simple non-parametric smoothing do not give desired outputs. 2000+ functions, GUIs, 2D Graphs, various toolkits, formula templates, 3D Graphs, databases and other specialized features such as ArrayFire acceleration functions which allows the users Feb 1, 2021 · 3. That's almost it but I would like to specify how many sections the time sequence is divided into. Origin provides 22 built-in surface fitting functions, such as Gauss2D, DoseReponse2D, Exponential2D, and We use the term “coefficients” for the numbers that the curve fit is to find. x(t), for fixed t, is a linear combination (weighted Aug 23, 2022 · What is Curve Fit in Scipy. The first trial is made with two exponentials. Order Now! License Type. xs is a multidimensional array of the axes of the polynomial points. algebraic as opposed to parametric curves (in the 2D case), and on one-step application of matrix-inversion-style least-squares methods, which we refer to as direct methods in distinction to slower iterative methods. Akaike info crit = 7245. In this case, the optimized function is chisq = sum((r / sigma) ** 2). Use polyfit with three outputs to fit a 5th-degree polynomial using centering and scaling, which improves the numerical properties of the problem. stats. Circular initialization of the B-spline curve. Curve Fitting. ¶. Published 1 June 2014. Feb 1, 2006 · This paper describes a 2D constrained fitting method for profile curve reconstruction in reverse engineering. reduced chi-square = 2. Execute “mainD2GaussFitRot. The toolbox lets you perform exploratory data analysis, preprocess and post-process data, compare candidate models, and remove outliers. 02. load franke. linspace(y_min,y_max,n) And a measured 2D array z (n x n Matrix). Nyiri. Aug 10, 2018 · Then just remove the unwanted distribution from the image and fit to it. chi-square = 20618. Then use the optimize function to fit a straight line. A 2-D sigma should contain the covariance matrix of errors in ydata. fit tries to fit the parameters of a normal distribution based on Apr 27, 2013 · I’m trying to use curve_fit from scipy. And, unlike most curve fitting programs, TableCurve 2D's user-defined functions are compiled so custom curve fitting can be performed quickly — at nearly the speed as with the built-in equations. Fitting a 2D B-Spline curve to 2D point-clouds using asymmetric point-distance-minimization Based on paper: TODO. polyfit centers the data in year at 0 and scales it to have a standard deviation of 1, which avoids an ill-conditioned Vandermonde matrix in the fit calculation. Smoothing. Interpolation. In the Curve Fitter app, on the Curve Fitter tab, in the Data section, click Select Data. As expected, this software exhibits Drawing the fitted shapes. The problem can have bounds, linear constraints, or nonlinear constraints. xdata_ = xdata[i] ydata_ = ydata[i] coeff, var = curve_fit(func,xdata_,ydata_) But to use a for loop is a very slow process because in the actual case the array size is 6000 x 6000. Jul 10, 2014 · I use the following code to attempt a curve_fit of data drawn from a bivariate gaussian that is converted to a 2D histogram. draw ( t) The nature of the parameter t above depends on the fitted shape. def func(x, a, b): return a*x + b. This module provides wrappers, called Fitters, around some Numpy and Scipy fitting functions. We want to find values for the Jun 11, 2017 · You can use fit from scipy. In particular, the parabola is some of the most employed shape Curve Fitting Toolbox™ supports the interpolation methods described in the following table. Member. using a data variable containing the data values to be fit. The scipy. If you don't have any of those toolboxes then you can set up your own optimization problem to fit your function as below: Oct 16, 2013 · I have a 2D cloud of points, i imported them into matlab and i want to have a smoothing spline function that passes through them. Fit function to experimental data using curve_fit with 15. May 24, 2020 · I use three examples to show the procedures for MATLAB curve fitting. Fitting Models to Data. polyfit() function, accepts three different input values: x, y and the polynomial degree. 06284916. 0, scale=2. May 24, 2024 · A mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets ("the residuals") of the points from the curve. The function f ( x) minimizes the residual under the weight W. In the following sections, examples are provided in which Gpufit is used to fit simulated datasets. [in] pt. Linear. The syntax is given below. This technique is captured in the pyeq3 open source fitting code. But if I'm correct, this is only working for 1D functions. MPFIT - Robust non-linear least squares curve fitting. Jewsus. For example, to obtain a linear fit, use degree 1. fit(data) norm. linspace function. optimize package equips us with multiple optimization procedures. Nov 17, 2020 · The np. At first it is curve fit using a quadratic expression Z = a * x ^ 2 + b * x + c along a constant value of T i. Our main goal is to evaluate the quality of the solutions. The problem is that putting a print function inside func still shows X being equal to xdata (a 2x111 array), rather than a particular column of xdata(a Jun 10, 2015 · I want to fit an 2D sum of gaussians to this data: After failing at fitting a sum to this initially I instead sampled each peak separately and returned a fit by find it's moments (essentially using this code). After each step of extrapolation, 3 estimates of fitness were assessed to determine a local minima (and subsequently best fitness for curve combination). Curve Fitting Toolbox provides an app and functions for fitting curves and surfaces to data. The curve fit finds the specific coefficients (parameters) which make that function match your data as closely as possible. Engineering. Shashank Prasanna on 15 May 2013. An online curve-fitting solution making it easy to quickly perform a curve fit using various fit methods, make predictions, export results to Excel, PDF, Word and PowerPoint, perform a custom fit through a user defined equation and share results online. Oct 21, 2013 · scipy. Specifying a fit. Then we create a sinusoidal signal with an offset, a multiplication factor and some noise. From our visual inspection above, we'll use ampl=10 and tau=5. Refinement and fitting of the B-spline surface. Description. We wish to ﬁt an algebraic curve or surface of a given shape to m points in d-dimensional space Rd. Jul 23, 2023 · 2D data fitting - Surface. random. ndCurveMaster supports curve fitting across multiple dimensions (or input variables), from 2D up to an unlimited number of dimensions (nD), and is capable of handling data of any complexity. moving a control point two inches to the right moves x(t) twice as far as moving it by one inch. A 3-step algorithm for reconstructing curves from unorganized points: data clustering to filter out the noise, data confining to get the boundary, and region thinning to find the skeleton curve. To fit a sigmoidal model, click Sigmoidal in the Fit Type gallery of the Curve Fitter tab. 100. For training you must generate ground truth data, that is a mapping Jun 12, 2012 · Fits a 2D Gaussian function to simulated data. Unlike supervised learning, curve fitting requires that you define the function that maps examples of inputs to Aug 16, 2019 · 2D Non-Linear Curve Fitting. 0, size=1000) mean,std=norm. where a and f0 are to be determined by the fit, alpha is known. They take an instance of FittableModel as input and modify its parameters attribute. 2 Brief Description. Author: Thomas Mörwald . I tried with the "curve fitting" app and the "smoothing spline" option but in the extremes the function seems to not follow much the cloud and anyway in the middle of the cloud does not pass trough the points but Aug 6, 2022 · We can get a single line using curve-fit () function. Jan 1, 2014 · Fitting discrete data obtained by image acquisition devices to a curve is a common task in many fields of science and engineering. I am receiving broadcast errors and I am not sure as to why this is happening. Some tips are in the docstring for the draw method. SKU: N/A Categories: Engineering, New Tag: TableCurve2D. 2 degree multi variable curve fit. 00 1. Fit interpolating curves or surfaces, estimate values between known data points. pyplot as plt import gauss import plotutils # Produce a number of Given a point pt, the function finds the closest midpoint of the elements of the curve. Another approach is described here. Case 1: 1-dimensional distribution probability curve fitting. Mar 31, 2015 · The ultimate aim is to develop a correlation of the form Z = f (X, T). The model function, f (x, ). The 2D and 3D datasets are selected as follows. These IDL routines provide a robust and relatively fast way to perform least-squares curve and surface fitting. Basic Curve Fitting# Lesson overview# Analyzing measured data with a theoretical model is a common task for a scientist and engineer. Problem Statement. Spline curves are linear functions of their controls. , poly-functions); Nov 16, 2017 · Gpufit was developed to meet the need for a high performance, general-purpose nonlinear curve fitting library which is publicly available and open source. It must take the independent variable as the first argument and the parameters to fit as separate remaining arguments. the B-Spline curve. The program then attempts to fit the data using the MatLab function “lsqcurvefit “ to find the position, orientation and width of the two-dimensional Gaussian. curveFitter. The curve_fit() method of module scipy. Curve Fitting Toolbox™ allows you to smooth data using methods such as moving average, Savitzky-Golay filter and Lowess models or by fitting a smoothing spline. The sum of the squares of the offsets is used instead of the offset absolute values because this allows the residuals to be treated as a continuous differentiable quantity. Part 1: Example: Quadratic Fit to U. 000000; And here's a visualisation showing my points and the attempt to fit them: Add a comment. 5. The described method has been implemented in RE-SOFT, which Nonlinear least-squares solves min (∑|| F ( xi ) - yi || 2 ), where F ( xi ) is a nonlinear function and yi is data. mt oh zn wy xh lb ma vp yf pg