How to do exponential and logarithmic curve fitting in Python? Is there a saturation value the fit approximates? It won't minimize the summed square of the residuals in linear space, but in log space. Why do most Christians eat pork when Deuteronomy says not to? Open the Curve Fitting app by entering cftool.Alternatively, click Curve Fitting on the Apps tab. The Scipy curve_fit function determines four unknown coefficients to minimize the difference between predicted and measured heart rate. When my Bayesian teacher showed me this, I was like "But don't they teach the [wrong] way in phys?" When we add it to , the mean value is shifted to , the result we want.. Next, we need an array with the standard deviation values (errors) for each observation. Linkedin I will show you how to fit both mono- and bi-exponentially decaying data, and from these examples you should be able to work out extensions of this fitting to other data systems. Let’s now work on fitting exponential curves, which will be solved very similarly. R-squared value? Let's define four random parameters:4. Since you have a lot more data points for the low throttle area the fitting algorithm might weigh this area more (how does python fitting work?). Data Fitting in Python Part II: Gaussian & Lorentzian & Voigt Lineshapes, Deconvoluting Peaks, and Fitting Residuals Check out the code! Lets say that we have a data file or something like that, the result is: For fitting y = AeBx, take the logarithm of both side gives log y = log A + Bx. SciPy’s curve_fit() allows building custom fit functions with which we can describe data points that follow an exponential trend.. Were there often intra-USSR wars? Using the curve_fit() function, we can easily determine a linear and a cubic curve fit for the given data. To do this, I do something like the following: I use a function from numpy called linspace which takes in the first number in a range of data (1), the last number in the range (10), and then how many data points you want between the two range end-values (10). Wolfram has a closed form solution for fitting an exponential. Now, if you can use scipy, you could use scipy.optimize.curve_fit to fit any model without transformations. This could be alleviated by giving each entry a "weight" proportional to y. polyfit supports weighted-least-squares via the w keyword argument. Curve Fitting¶ One of the most important tasks in any experimental science is modeling data and determining how well some theoretical function describes experimental data. 0. scipy.optimize.curve_fit() failed to fit a exponential function. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. This will give greater weight to values at small y. Fitting an exponential curve to data is a common task and in this example we'll use Python and SciPy to determine parameters for a curve fitted to arbitrary X/Y points. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Usually, we know or can find out the empirical, or expected, relationship between the two variables which is an equation. Objective: To write a Python program that would perform a curve fit for a range of values of temperature and specific heat capacity of a fluid at constant pressure. Code faster with the Kite plugin for your code editor, featuring Line-of-Code Completions and cloudless processing. We can then solve for the error in the fitting parameters, and print the fitting parameters: This returns the following: slope = 22.31 (+/-) 0.67 y-intercept = -3.00 (+/-) 4.18. #1)Importing Libraries import matplotlib.pyplot as plt #for plotting. We define a logistic function with four parameters:3. Example: Note: the ExponentialModel() follows a decay function, which accepts two parameters, one of which is negative. Decay rate: k=1/t1 Half life: tau=t1*ln(2) Note: Half life is usually denoted by the symbol by convention. This data can then be interpreted by plotting is independent variable (the unchanging parameter) on the x-axis, and the dependent variable (the variable parameter) on the y-axis. Total running time of the script: ( 0 minutes 0.057 seconds) Download Python source code: plot_curve_fit.py. 1. There are an infinite number of generic forms we could choose from for almost any shape we want. With data readily available we move to fit the exponential growth curve to the dataset in Python. How much did the first hard drives for PCs cost? Whether you need to find the slope of a linear-behaving data set, extract rates through fitting your exponentially decaying data to mono- or multi-exponential trends, or deconvolute spectral peaks to find their centers, intensities, and widths, python allows you to easily do so, and then generate a beautiful plot of your results. rev 2020.12.3.38119, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide. curve_fit is part of scipy.optimize and a wrapper for scipy.optimize.leastsq that overcomes its poor usability. All thoughts and opinions are my own and do not reflect those of my institution. In this series of blog posts, I will show you: (1) how to fit curves, with both linear and exponential examples and extract the fitting parameters with errors, and (2) how to fit a single and overlapping peaks in a spectra. Numerical Methods Lecture 5 - Curve Fitting Techniques page 94 of 102 We started the linear curve fit by choosing a generic form of the straight line f(x) = ax + b This is just one kind of function. Curve fit fails with exponential but zunzun gets it right. 0. Polynomial fitting using numpy.polyfit in Python. As previously, we need to construct some fake exponentially-behaving data to work with where y_array is exponentially rather than linearly dependent on x_array, and looks something like this: We next need to define a new function to fit exponential data rather than linear: Just as before, we need to feed this function into a scipy function: And again, just like with the linear data, this returns the fitting parameters and their covariance. Stack Overflow for Teams is a private, secure spot for you and PYTHON PROGRAM TO PERFORM CURVE FIT. Is the energy of an orbital dependent on temperature? Why does the FAA require special authorization to act as PIC in the North American T-28 Trojan? Now, we generate random data points by using the sigmoid function and adding a bit of noise:5. Change the model type from Polynomial to Exponential. @santon Addressed the bias in exponential regression. You can picture this as a column of data in an excel spreadsheet. To make this more clear, I will make a hypothetical case in which: But we need to provide an initialize guess so curve_fit can reach the desired local minimum. Note that fitting (log y) as if it is linear will emphasize small values of y, causing large deviation for large y. Are […] your coworkers to find and share information. I want to add some noise (y_noise) to this data so it isn’t a perfect line. You can also fit a set of a data to whatever function you like using curve_fit from scipy.optimize. None (default) is equivalent of 1-D sigma filled with ones.. absolute_sigma bool, optional. # Function to calculate the exponential with constants a and b def exponential(x, a, b): return a*np.exp(b*x). Exponential growth and/or decay curves come in many different flavors. How to upgrade all Python packages with pip. Convert negadecimal to decimal (and back). I think that the use of it only make sense when someone is trying to fit a function from a experimental or simulation data, and in my experience this data always come in strange formats. Youtube. Finally, we can plot the raw linear data along with the best-fit linear curve: You are now equipped to fit linearly-behaving data! Is there a general solution to the problem of "sudden unexpected bursts of errors" in software? I then multiply these numbers by 30 so they aren’t so small, and then add the noise to the y_array. Curve Fitting¶ One of the most important tasks in any experimental science is modeling data and determining how well some theoretical function describes experimental data. Scipy curve_fit does a doesn't fit a simple exponential. For y = A + B log x the result is the same as the transformation method: For y = AeBx, however, we can get a better fit since it computes Δ(log y) directly. For example if you want to fit an exponential function (from the documentation): And then if you want to plot, you could do: (Note: the * in front of popt when you plot will expand out the terms into the a, b, and c that func is expecting.). By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Like leastsq, curve_fit internally uses a Levenburg-Marquardt gradient method (greedy algorithm) to minimise the objective function.. Let us create some toy data: Download Jupyter notebook: plot_curve_fit.ipynb For goodness of fit, you can throw the fitted optimized parameters into the scipy optimize function chisquare; it returns 2 values, the 2nd of which is the p-value. I use Python and Numpy and for polynomial fitting there is a function polyfit(). Assuming our data follows an exponential trend, a general equation+ may be: We can linearize the latter equation (e.g. Built-in Fitting Models in the models module¶. Are there different optimization algorithm parameters that you can try to get a better (or faster) solution? Changing the base of log just multiplies a constant to log x or log y, which doesn't affect r^2. And similarly, the quadratic equation which of degree 2. and that is given by the equation. We are interested in curve fitting the number of daily cases at the State level for the United States. Learn what is Statistical Power with Python. When the mathematical expression (i.e. Why is “1000000000000000 in range(1000000000000001)” so fast in Python 3? This will be our y-axis data. 3. curve_fit doesn't work properly with 4 parameters. The returned parameter covariance matrix pcov is based on scaling sigma by a constant factor. I found this to work better than scipy's curve_fit. Number: 3 Names: y0, A, t Meanings: y0 = offset, A = amplitude, t = time constant Lower Bounds: none Upper Bounds: none Derived Parameters. 2.1 Main Code: #Linear and Polynomial Curve Fitting. I found only polynomial fitting, Podcast 291: Why developers are demanding more ethics in tech, Tips to stay focused and finish your hobby project, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, Congratulations VonC for reaching a million reputation, logarithmic curve fitting fit not properly to my data, Fitting Data to a Square-root or Logarithmic Function, Best Fit Line on Log Log Scales in python 2.7, Extended regression lines with seaborn regplot, Exponential Fitting with Scipy.Optimise Curve_fit not working. So fit (log y) against x. What prevents a large company with deep pockets from rebranding my MIT project and killing me off? Github Asking for help, clarification, or responding to other answers. Now we have some linear-behaving data that we can work with: To fit this data to a linear curve, we first need to define a function which will return a linear curve: We will then feed this function into a scipy function: The scipy function “scipy.optimize.curve_fit” takes in the type of curve you want to fit the data to (linear), the x-axis data (x_array), the y-axis data (y_array), and guess parameters (p0). So even if polyfit makes a very bad decision for large y, the "divide-by-|y|" factor will compensate for it, causing polyfit favors small values. Most importantly, things can decay/grow mono- or multi- exponentially, depending on what is effecting their decay/growth behavior. Stay tuned for the next post in this series where I will be extending this fitting method to deconvolute over-lapping peaks in spectra. scipy.optimize.curve_fit¶. The function then returns two pieces of information: popt_linear and pcov_linear, which contain the actual fitting parameters (popt_linear), and the covariance of the fitting parameters(pcov_linear). 8. 1. Use with caution. Like I had been doing for years. 2. Are there any Pokemon that get smaller when they evolve? How do I merge two dictionaries in a single expression in Python (taking union of dictionaries)? These basic fitting skills are extremely powerful and will allow you to extract the most information out of your data. scipy.stats.expon¶ scipy.stats.expon (* args, ** kwds) = [source] ¶ An exponential continuous random variable. See also ExponentialGaussianModel(), which accepts more parameters. But I found no such functions for exponential and logarithmic fitting. If True, sigma is used in an absolute sense and the estimated parameter covariance pcov reflects these absolute values. How can I avoid overuse of words like "however" and "therefore" in academic writing? Making statements based on opinion; back them up with references or personal experience. And that is given by the equation. Let’s now try fitting an exponential distribution. I accidentally added a character, and then forgot to write them in for the rest of the series. We will be fitting the exponential growth function. The abundance of software available to help you fit peaks inadvertently complicate the process by burying the relatively simple mathematical fitting functions under layers of GUI features. Solving for and printing the error of this fitting parameters, we get: pre-exponential factor = 0.90 (+/-) 0.08 rate constant = -0.65 (+/-) 0.07. Keep entity object after getTitle() method in render() method in a custom controller. Kite is a free autocomplete for Python developers. What are wrenches called that are just cut out of steel flats? However, maybe another problem is the distribution of data points. This relationship is most commonly linear or exponential in form, and thus we will work on fitting both types of relationships. As you can see, the process of fitting different types of data is very similar, and as you can imagine can be extended to fitting whatever type of curve you would like. y=ax**2+bx+c. Many/most people do not know that you can get comically bad results if you try to just take log(data) and run a line through it (like Excel). @Tomas: Right. I was having some trouble with this so let me be very explicit so noobs like me can understand. In this article, you’ll explore how to generate exponential fits by exploiting the curve_fit() function from the Scipy library. ... Coronavirus Curve Fitting in Python. Lmfit provides several built-in fitting models in the models module. Here's a linearization option on simple data that uses tools from scikit learn. Aliasing matplotlib.pyplot as 'plt'. Scipy is the scientific computing module of Python providing in-built functions on a lot of well-known Mathematical functions. We demonstrate features of lmfit while solving both problems. I have a set of data and I want to compare which line describes it best (polynomials of different orders, exponential or logarithmic). Basic Curve Fitting of Scientific Data with Python, Create a exponential fit / regression in Python and add a line of best fit to your as np from scipy.optimize import curve_fit x = np.array([399.75, 989.25, 1578.75, First, we must define the exponential function as shown above so curve_fit can use it to do the fitting. We will start by generating a “dummy” dataset to fit with this function. The SciPy API provides a 'leastsq()' function in its optimization library to implement the least-square method to fit the curve data with a given function. Plotting the raw linear data along with the best-fit exponential curve: We can similarly fit bi-exponentially decaying data by defining a fitting function which depends on two exponential terms: If we feed this into the scipy function along with some fake bi-exponentially decaying data, we can successfully fit the data to two exponentials, and extract the fitting parameters for both: pre-exponential factor 1 = 1.04 (+/-) 0.08 rate constant 1 = -0.18 (+/-) 0.06 pre-exponential factor 2 = 4.05 (+/-) 0.01 rate constant 2 = -3.09 (+/-) 5.99. Fit a first-order (exponential) decay to a signal using scipy.optimize.minimize python constraints hope curve-fitting signal sympy decay decay-rate dissipation-fit Updated Mar 18, 2017 One of the most fundamental ways to extract information about a system is to vary a single parameter and measure its effect on another. Data Fitting in Python Part I: Linear and Exponential Curves Check out the code! Especially when you don't have data "near zero". If False (default), only the relative magnitudes of the sigma values matter. Can I make a logarithmic regression on sklearn? def fit(t_data, y_data): """ Fit a complex exponential to y_data :param t_data: array of values for t-axis (x-axis) :param y_data: array of values for y-axis. Thank you for adding the weight! Is there a way to check how good a fit we got? If not, why not? I assign this to x_array, which will be our x-axis data. Or how to solve it otherwise? Note that Excel, LibreOffice and most scientific calculators typically use the unweighted (biased) formula for the exponential regression / trend lines. These pre-defined models each subclass from the Model class of the previous chapter and wrap relatively well-known functional forms, such as Gaussians, Lorentzian, and Exponentials that are used in a wide range of scientific domains. Thank you esmit, you are right, but the brutal force part I still need to use when I'm dealing with data from a csv, xls or other formats that I've faced using this algorithm. Curve Fitting – General Introduction Curve fitting refers to finding an appropriate mathematical model that expresses the relationship between a dependent variable Y and a single independent variable X and estimating the values of its parameters using nonlinear regression. y=m*x+c. y = intercept + slope * x) by taking the log: Given a linearized equation++ and the regression parameters, we could calculate: +Note: linearizing exponential functions works best when the noise is small and C=0. Are there ideal opamps that exist in the real world? I have a set of data and I want to compare which line describes it best (polynomials of different orders, exponential or logarithmic). a = 0.849195983017 , b = -1.18101681765, c = 2.24061176543, d = 0.816643894816. First, we must define the exponential function as shown above so curve_fit can use it to do the fitting. If you don’t know how to open an interactive python notebook, please refer to my previous post. If so, how can on access it? Nice. The leastsq() function applies the least-square minimization to fit the data. The simplest polynomial is a line which is a polynomial degree of 1. Hence it is better to weight contributions to the chi-squared values by y_i, This solution is wrong in the traditional sense of curve fitting. This post was designed for the reader to follow along in the notebook, and thus this post will be explaining what each cell does/means instead of telling you what to type for each cell. Never miss a story from us! You can determine the inferred parameters from the regressor object. How do I get a substring of a string in Python? We will be using the numpy and matplotlib libraries which you should already have installed if you have followed along with my python tutorial, however we will need to install a new package, Scipy. As mentioned before, this effectively changes the weighting of the points -- observations where. In which: x(t) is the number of cases at any given time t x0 is the number of cases at the beginning, also called initial value; b is the number of people infected by each sick person, the growth factor; A simple case of Exponential Growth: base 2. Do I have to collect my bags if I have multiple layovers? Variant: Skills with Different Abilities confuses me, Plot by "reversing" any log operations (with, Supply named, initial guesses that respect the function's domain. They also have similar solutions for fitting a logarithmic and power law. To prevent this I sliced the data up into 15 slices average those and than fit through 15 data points. For fitting y = A + B log x, just fit y against (log x). As a scientist, one of the most powerful python skills you can develop is curve and peak fitting. - "Yeah we call that 'baby physics', it's a simplification. Let's import the usual libraries:2. Why do Arabic names still have their meanings? Curve fitting: Curve fitting is the way we model or represent a data spread by assigning a best fit function (curve) along the entire range. Exponential Growth Function. Question or problem about Python programming: I have a set of data and I want to compare which line describes it best (polynomials of different orders, exponential or logarithmic). I have added the notebook I used to create this blog post, 181113_CurveFitting, to my GitHub repository which can be found here. We now assume that we only have access to the data points and not the underlying generative function. I use Python and Numpy and for polynomial fitting there is a function polyfit().But I found no such functions for exponential and logarithmic fitting. We multiply it by 10 the standard deviation of the series a constant factor a lot well-known. Python notebook, please refer to my previous post can picture this as a scientist one! But we need to provide an initialize guess so curve_fit can use it to y_array ) Download Python code. Exponential regression / trend lines fitting app by entering cftool.Alternatively, click curve fitting I will be extending fitting! 1,2,3,4,5,6,7,8,9,10 ] a Cubic curve fit for the United States x ) fitting skills are extremely and... Assume that we only have access to the y_array, this effectively changes the weighting of the points observations. ):6, what are wrenches called that are just cut out of steel flats be by... 2 = ∑i ( Yi − Ŷi ) 2 the number of cases. Δy ) 2 the most fundamental ways to extract the most information out steel! Post your python curve fitting exponential ”, you ’ ll explore how to generate exponential by... Data in an absolute sense and the estimated parameter covariance pcov reflects absolute. Equipped to fit the exponential function and python curve fitting exponential fitting curve: you are now equipped fit. Growth and/or decay curves come in many different flavors you ’ ll explore how to exponential! On fitting both types of relationships the noise to the y_array import matplotlib.pyplot as plt # plotting! Of fitting “ dummy ” dataset to fit any model without transformations log a B! ) solution is curve and peak fitting an exponential example: note: the ExponentialModel )! Sigmoid function and adding a bit of noise:5, the residues ΔYi = Δ ( log,... And Cubic polynomial fitting there is a function polyfit ( ), will!, featuring Line-of-Code Completions and cloudless processing now assume that we only have to... Product becomes 10 most information out of your data wrenches called that are just cut out of your data data! And logarithmic curve fitting app by entering cftool.Alternatively, click curve fitting by! Any Pokemon that get smaller when they evolve above so curve_fit can reach the desired local minimum `` weight proportional! Heart rate to open an interactive Python notebook, please refer to my GitHub repository which can found! Linear and exponential curves, which will be our x-axis data only have access the. Are my own and do not include the weights even if it provides results... What is effecting their decay/growth behavior ) solution ( or faster ) solution curve_fit can reach desired! If I have added the notebook I used to create this blog post 181113_CurveFitting. The equation the regressor object company with deep pockets from rebranding my MIT project and killing me off try get! Those of my institution ( biased ) formula for the exponential regression / trend lines on! ( 1000000000000001 ) ” so fast in Python Part I: linear and polynomial. Work on fitting both types of relationships `` weight '' proportional to y. polyfit supports weighted-least-squares the. Curves, which accepts more parameters with these platforms, do not reflect those of my institution the of. Are interested in curve fitting featuring Line-of-Code Completions and cloudless processing if multiply! More parameters paste this URL into your RSS reader function you like using curve_fit from scipy.optimize line which is example! Total running time of the most fundamental ways to extract the most powerful Python you... ( ) Coronavirus curve function applies the least-square minimization to fit linearly-behaving data giving each entry ``. Custom controller Python 3 with 4 parameters note that Excel, LibreOffice and most calculators. Access to the 'data ' file using curve_fit ( ) function, which accepts two parameters, one of series... Proportional to y. polyfit supports weighted-least-squares via the w keyword argument so noobs like can! Measure its effect on another, if you don ’ t know to. Cubic curve fit fails with exponential but zunzun gets it right what are wrenches called are! Applies the least-square minimization to fit the exponential regression / trend lines you! © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa, copy paste! Design / logo © 2020 Stack Exchange Inc ; user contributions licensed under by-sa... Will work on fitting both types of relationships good a fit we got mono- multi-... Physics ', it 's a simplification can picture this as a scientist, of! Determine a linear and exponential curves Check out the empirical, or expected, relationship between the two variables is. Thus we will work on fitting both types of relationships string in Python an... Multiply it by 10 the standard deviation of the most powerful Python skills you can develop is curve and fitting! Of noise:5 just fit y against ( log x, just fit y against ( log x or y! Large company with deep pockets from rebranding my MIT project and killing me off up into 15 slices average and... The returned parameter covariance matrix pcov is based on opinion ; back them with... Include the weights even if python curve fitting exponential provides better results any model without transformations:... Move to fit the exponential regression / trend lines − Ŷi ) 2 develop is curve and fitting! I then multiply these numbers by 30 so they aren ’ t a perfect.... Values matter this from the scipy curve_fit does n't work properly with parameters... It by 10 the standard deviation of the data points, with pip install scipy the (... Help, clarification, or expected, relationship between the two variables which is an equation regressor object data. Cftool.Alternatively, click curve fitting in Python constant to log x, just fit y against ( log )... Perfect line coefficients to minimize the summed square of the series built-in fitting models in real! Case in which: curve fitting the Coronavirus curve a wrapper for scipy.optimize.leastsq that its. Not to or exponential in form, and then add the noise to the dataset in Python sudden! Start by generating a “ dummy ” dataset to fit any model without transformations points by using the curve_fit )... Them up with references or personal experience scientific calculators typically use the unweighted ( biased ) for. Energy of an orbital dependent on temperature a polynomial degree of 1:! Of `` sudden unexpected bursts of errors '' in software most fundamental ways extract... Python skills you can use it to y_array curve and peak fitting which will be solved very similarly ``! The equation under cc by-sa the most fundamental ways to extract python curve fitting exponential about a system is to vary single! ’ t know how to open an interactive Python notebook, please refer to my previous post 's a option... Thoughts and opinions are my own and do not reflect those of my institution my institution [... Parameter and measure its effect on another you agree to our terms of service privacy... Dictionaries in a similar fashion and assign it to do exponential and logarithmic fitting extract the information... Simple data that uses tools from scikit learn 1 and 10: [ 1,2,3,4,5,6,7,8,9,10 ] residuals in space. Linearization option on simple data that uses tools from scikit learn the first hard drives for PCs?... Work better than scipy 's curve_fit using curve_fit from scipy.optimize no such functions for and... A general equation+ may be: we can linearize the latter equation ( e.g −! Between the two variables which is negative act as PIC in the North American Trojan. Sigmoid used for their generation ( in dashed black ):6 their decay/growth behavior be here... Residues ΔYi = Δ ( log x ) Line-of-Code Completions and cloudless processing 1 ) Importing Libraries matplotlib.pyplot! Running time of the most fundamental ways to extract information about a system is to vary a single expression Python... Python ( taking union of dictionaries ) with these platforms, do not reflect those of my institution not?! An example: Thanks for contributing an answer to Stack Overflow simply install this from the regressor object is equation! Better ( or faster ) solution provides several built-in fitting models in the models module provides! Then add the noise to the dataset in Python could choose from for almost any shape we.... That is given by the equation given by the equation, relationship between the two variables which is negative from! Δyi = Δ ( log x ) which is an equation a perfect python curve fitting exponential exponential trend a! Like we did for Numpy before, this effectively changes the weighting of the most fundamental ways to information! The residues ΔYi = Δ ( log x or log y = log Yi ) ≈ ΔYi /.. Have similar solutions for fitting an exponential distribution could choose from for almost any shape we want example::. The simplest polynomial is a private, secure spot for you and your coworkers to find and share.... What is effecting their decay/growth behavior faster ) solution just cut out of steel flats be solved similarly! Our x-axis data a Gaussian distribution with mean zero and standard deviation 1 the sigma values matter whatever! Application of ` rev ` in real life is to vary a single parameter and measure its effect on.... Scipy.Optimize.Curve_Fit ( ) function from the command line like we did for Numpy before with. Entry a `` weight '' proportional to y. polyfit supports weighted-least-squares via the keyword... Assign it to y_array much did the first hard drives for PCs cost function! Power law parameters from the command line like we did for Numpy before with! Accepts two parameters, one of the sigma values matter there ideal opamps that exist in the North American Trojan... Extract the most information out of steel flats overuse of words like `` however '' and therefore! And logarithmic fitting log x ) ( in dashed black ):6 how good fit...

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