In this article, you’ll explore how to generate exponential fits by exploiting the
curve_fit() function from the Scipy library. SciPy’s
curve_fit() allows building custom fit functions with which we can describe data points that follow an exponential trend.
- In the first part of the article, the
curve_fit()function is used to fit the exponential trend of the number of COVID-19 cases registered in California (CA).
- The second part of the article deals with fitting histograms, characterized, also in this case, by an exponential trend.
Disclaimer: I’m not a virologist, I suppose that the fitting of a viral infection is defined by more complicated and accurate models; however, the only aim of this article is to show how to apply an exponential fit to model (to a certain degree of approximation) the increase in the total infection cases from the COVID-19.
Exponential fit of COVID-19 total cases in California
Data related to the COVID-19 pandemic have been obtained from the official website of the “Centers for Disease Control and Prevention” (https://data.cdc.gov/Case-Surveillance/United-States-COVID-19-Cases-and-Deaths-by-State-o/9mfq-cb36) and downloaded as a .csv file. The first thing to do is to import the data into a Pandas dataframe. To do this, the Pandas functions pandas.read_csv() and pandas.Dataframe() were employed. The created dataframe is made up of 15 columns, among which we can find the submission_date, the state, the total cases, the confirmed cases and other related observables. To gain an insight into the order in which these categories are displayed, we print the header of the dataframe; as can be noticed, the total cases are listed under the voice “tot_cases”.
Since in this article we are only interested in the data related to the California, we create a sub-dataframe that contains only the information related to the California state. To do that, we exploit the potential of Pandas in indexing subsections of a dataframe. This dataframe will be called df_CA (from California) and contains all the elements of the main dataframe for which the column “state” is equal to “CA”. After this step, we can build two arrays, one (called tot_cases) that contains the total cases (the name of the respective header column is “tot_cases”) and one that contains the number of days passed by the first recording (called days). Since the data were recorded daily, in order to build the “days” array, we simply build an array of equally spaced integer number from 0 to the length of the “tot_cases” array, in this way, each number refers to the n° of days passed from the first recording (day 0).
At this point, we can define the function that will be used by
curve_fit() to fit the created dataset. An exponential function is defined by the equation:
y = a*exp(b*x) +c
where a, b and c are the fitting parameters. We will hence define the function
exp_fit() which return the exponential function, y, previously defined. The
curve_fit() function takes as necessary input the fitting function that we want to fit the data with, the x and y arrays in which are stored the values of the datapoints. It is also possible to provide initial guesses for each of the fitting parameters by inserting them in a list called
p0 = […] and upper and lower boundaries for these parameters (for a comprehensive description of the
curve_fit() function, please refer to https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html ). In this example, we will only provide initial guesses for our fitting parameters. Moreover, we will only fit the total cases of the first 200 days; this is because for the successive days, the number of cases didn’t follow an exponential trend anymore (possibly due to a decrease in the number of new cases). To refer only to the first 200 values of the arrays “days” and “tot_cases”, we exploit array slicing (e.g. days[:200]).
The output of
curve_fit() are the fitting parameters, presented in the same order that was used during their definition, within the fitting function. Keeping this in mind, we can build the array that contains the fitted results, calling it
Now that we built the fitting array, we can plot both the original data points and their exponential fit.
The final result will be a plot like the one in Figure 1:
Application of an exponential fit to histograms
Now that we know how to define and use an exponential fit, we will see how to apply it to the data displayed on a histogram. Histograms are frequently used to display the distributions of specific quantities like prices, heights etc…The most common type of distribution is the Gaussian distribution; however, some types of observables can be defined by a decaying exponential distribution. In a decaying exponential distribution, the frequency of the observables decreases following an exponential[A1] trend; a possible example is the amount of time that the battery of your car will last (i.e. the probability of having a battery lasting for long periods decreases exponentially). The exponentially decaying array will be defined by exploiting the Numpy function random.exponential(). According to the Numpy documentation, the random.exponential() function draws samples from an exponential distribution; it takes two inputs, the “scale” which is a parameter defining the exponential decay and the “size” which is the length of the array that will be generated. Once obtained random values from an exponential distribution, we have to generate the histogram; to do this, we employ another Numpy function, called histogram(), which generates an histogram taking as input the distribution of the data (we set the binning to “auto”, in this way the width of the bins is automatically computed). The output of histogram() is a 2D array; the first array contains the frequencies of the distribution while the second one contains the edges of the bins. Since we are only interested in the frequencies, we assign the first output to the variable “hist”. For this example, we will generate the array containing the bin position by using the Numpy arange() function; the bins will have a width of 1 and their number will be equal to the number of elements contained in the “hist” array.
At this point, we have to define the fitting function and to call curve_fit() for the values of the just created histogram. The equation describing an exponential decay is similar to the one defined in the first part; the only difference is that the exponent has a negative sign, this allows the values to decrease according to an exponential fashion. Since the elements in the “x” array, defined for the bin position, are the coordinates of the left edge of each bin, we define another x array that stores the position of the center of each bin (called “x_fit”); this allows the fitting curve to pass through the center of each bin, leading to a better visual impression. This array will be defined by taking the values of the left side of the bins (“x” array elements) and adding half the bin size; which corresponds to half the value of the second bin position (element of index 1). Similar to the previous part, we now call curve_fit(), generate the fitting array and assign it to the varaible “fit_eq”.
Once the distribution has been fitted, the last thing to do is to check the result by plotting both the histogram and the fitting function. In order to plot the histogram, we will use the matplotlib function bar(), while the fitting function will be plotted using the classical plot() function.
The final result is displayed in Figure 2:
In these two examples, the
curve_fit() function was used to apply to different exponential fits to specific data points. However, the power of the
curve_fit() function, is that it allows you defining your own custom fit functions, being them linear, polynomial or logarithmic functions. The procedure is identical to the one shown in this article, the only difference is in the shape of the function that you have to define before calling
import pandas as pd import numpy as np import matplotlib.pyplot as plt from scipy.optimize import curve_fit url = "United_States_COVID-19_Cases_and_Deaths_by_State_over_Time" #url of the .csv file file = pd.read_csv(url, sep = ';', thousands = ',') # import the .csv file df = pd.DataFrame(file) # build up the pandas dataframe print(df.columns) #visualize the header df_CA = df[df['state'] == 'CA'] #initialize a sub-dataframe for storing only the values for the California tot_cases = np.array((df_CA['tot_cases'])) #create an array with the total n° of cases days = np.linspace(0, len(tot_cases), len(tot_cases)) # array containing the n° of days from the first recording #DEFINITION OF THE FITTING FUNCTION def exp_fit(x, a, b, c): y = a*np.exp(b*x) + c return y #----CALL THE FITTING FUNCTION---- fit = curve_fit(exp_fit,days[:200],tot_cases[:200], p0 = [0.005, 0.03, 5]) fit_eq = fit*np.exp(fit*days[:200])+fit # #----PLOTTING------- fig = plt.figure() ax = fig.subplots() ax.scatter(days[:200], tot_cases[:200], color = 'b', s = 5) ax.plot(days[:200], fit_eq, color = 'r', alpha = 0.7) ax.set_ylabel('Total cases') ax.set_xlabel('N° of days') plt.show() #-----APPLY AN EXPONENTIAL FIT TO A HISTOGRAM-------- data = np.random.exponential(5, size=10000) #generating a random exponential distribution hist = np.histogram(data, bins="auto") #generating a histogram from the exponential distribution x = np.arange(0, len(hist), 1) # generating an array that contains the coordinated of the left edge of each bar #---DECAYING FIT OF THE DISTRIBUTION---- def exp_fit(x,a,b): #defining a decaying exponential function y = a*np.exp(-b*x) return y x_fit = x + x/2 # the point of the fit will be positioned at the center of the bins fit_ = curve_fit(exp_fit,x_fit,hist) # calling the fit function fit_eq = fit_*np.exp(-fit_*x_fit) # building the y-array of the fit #Plotting plt.bar(x,hist, alpha = 0.5, align = 'edge', width = 1) plt.plot(x_fit,fit_eq, color = 'red') plt.show()