Let’s say you might be in a buyer care middle, and also you wish to know the likelihood distribution of the variety of calls per minute, or in different phrases, you wish to reply the query: what’s the likelihood of receiving zero, one, two, … and many others., calls per minute? You want this distribution so as to predict the likelihood of receiving completely different variety of calls primarily based on which you’ll plan what number of staff are wanted, whether or not or not an enlargement is required, and many others.
With a view to let our resolution ‘knowledge knowledgeable’ we begin by accumulating knowledge from which we attempt to infer this distribution, or in different phrases, we wish to generalize from the pattern knowledge to the unseen knowledge which is also referred to as the inhabitants in statistical phrases. That is the essence of statistical inference.
From the collected knowledge we are able to compute the relative frequency of every worth of calls per minute. For instance, if the collected knowledge over time seems to be one thing like this: 2, 2, 3, 5, 4, 5, 5, 3, 6, 3, 4, … and many others. This knowledge is obtained by counting the variety of calls acquired each minute. With a view to compute the relative frequency of every worth you’ll be able to rely the variety of occurrences of every worth divided by the entire variety of occurrences. This fashion you’ll find yourself with one thing just like the gray curve within the beneath determine, which is equal to the histogram of the info on this instance.
Another choice is to imagine that every knowledge level from our knowledge is a realization of a random variable (X) that follows a sure likelihood distribution. This likelihood distribution represents all of the potential values which can be generated if we have been to gather this knowledge lengthy into the long run, or in different phrases, we are able to say that it represents the inhabitants from which our pattern knowledge was collected. Moreover, we are able to assume that every one the info factors come from the identical likelihood distribution, i.e., the info factors are identically distributed. Furthermore, we assume that the info factors are impartial, i.e., the worth of 1 knowledge level within the pattern shouldn’t be affected by the values of the opposite knowledge factors. The independence and equivalent distribution (iid) assumption of the pattern knowledge factors permits us to proceed mathematically with our statistical inference downside in a scientific and easy approach. In additional formal phrases, we assume {that a} generative probabilistic mannequin is answerable for producing the iid knowledge as proven beneath.
On this specific instance, a Poisson distribution with imply worth λ = 5 is assumed to have generated the info as proven within the blue curve within the beneath determine. In different phrases, we assume right here that we all know the true worth of λ which is mostly not identified and must be estimated from the info.
Versus the earlier technique through which we needed to compute the relative frequency of every worth of calls per minute (e.g., 12 values to be estimated on this instance as proven within the gray determine above), now we solely have one parameter that we intention at discovering which is λ. One other benefit of this generative mannequin strategy is that it’s higher when it comes to generalization from pattern to inhabitants. The assumed likelihood distribution might be mentioned to have summarized the info in a sublime approach that follows the Occam’s razor precept.
Earlier than continuing additional into how we intention at discovering this parameter λ, let’s present some Python code first that was used to generate the above determine.
# Import the Python libraries that we are going to want on this article
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
import math
from scipy import stats
# Poisson distribution instance
lambda_ = 5
sample_size = 1000
data_poisson = stats.poisson.rvs(lambda_,dimension= sample_size) # generate knowledge
# Plot the info histogram vs the PMF
x1 = np.arange(data_poisson.min(), data_poisson.max(), 1)
fig1, ax = plt.subplots()
plt.bar(x1, stats.poisson.pmf(x1,lambda_),
label="Possion distribution (PMF)",colour = BLUE2,linewidth=3.0,width=0.3,zorder=2)
ax.hist(data_poisson, bins=x1.dimension, density=True, label="Knowledge histogram",colour = GRAY9, width=1,zorder=1,align='left')
ax.set_title("Knowledge histogram vs. Poisson true distribution", fontsize=14, loc="left")
ax.set_xlabel('Knowledge worth')
ax.set_ylabel('Chance')
ax.legend()
plt.savefig("Possion_hist_PMF.png", format="png", dpi=800)Our downside now could be about estimating the worth of the unknown parameter λ utilizing the info we collected. That is the place we’ll use the technique of moments (MoM) strategy that seems within the title of this text.
First, we have to outline what is supposed by the second of a random variable. Mathematically, the kth second of a discrete random variable (X) is outlined as follows:
Take the primary second E(X) for instance, which can also be the imply μ of the random variable, and assuming that we accumulate our knowledge which is modeled as N iid realizations of the random variable X. An affordable estimate of μ is the pattern imply which is outlined as follows:
Thus, so as to get hold of a MoM estimate of a mannequin parameter that parametrizes the likelihood distribution of the random variable X, we first write the unknown parameter as a operate of a number of of the kth moments of the random variable, then we substitute the kth second with its pattern estimate. The extra unknown parameters now we have in our fashions, the extra moments we want.
In our Poisson mannequin instance, that is quite simple as proven beneath.
Within the subsequent half, we take a look at our MoM estimator on the simulated knowledge we had earlier. The Python code for acquiring the estimator and plotting the corresponding likelihood distribution utilizing the estimated parameter is proven beneath.
# Technique of moments estimator utilizing the info (Poisson Dist)
lambda_hat = sum(data_poisson) / len(data_poisson)
# Plot the MoM estimated PMF vs the true PMF
x1 = np.arange(data_poisson.min(), data_poisson.max(), 1)
fig2, ax = plt.subplots()
plt.bar(x1, stats.poisson.pmf(x1,lambda_hat),
label="Estimated PMF",colour = ORANGE1,linewidth=3.0,width=0.3)
plt.bar(x1+0.3, stats.poisson.pmf(x1,lambda_),
label="True PMF",colour = BLUE2,linewidth=3.0,width=0.3)
ax.set_title("Estimated Poisson distribution vs. true distribution", fontsize=14, loc="left")
ax.set_xlabel('Knowledge worth')
ax.set_ylabel('Chance')
ax.legend()
#ax.grid()
plt.savefig("Possion_true_vs_est.png", format="png", dpi=800)The beneath determine exhibits the estimated distribution versus the true distribution. The distributions are fairly shut indicating that the MoM estimator is an inexpensive estimator for our downside. In truth, changing expectations with averages within the MoM estimator implies that the estimator is a constant estimator by the regulation of enormous numbers, which is an efficient justification for utilizing such estimator.
One other MoM estimation instance is proven beneath assuming the iid knowledge is generated by a traditional distribution with imply μ and variance σ² as proven beneath.
On this specific instance, a Gaussian (regular) distribution with imply worth μ = 10 and σ = 2 is assumed to have generated the info. The histogram of the generated knowledge pattern (pattern dimension = 1000) is proven in gray within the beneath determine, whereas the true distribution is proven within the blue curve.
The Python code that was used to generate the above determine is proven beneath.
# Regular distribution instance
mu = 10
sigma = 2
sample_size = 1000
data_normal = stats.norm.rvs(loc=mu, scale=sigma ,dimension= sample_size) # generate knowledge
# Plot the info histogram vs the PDF
x2 = np.linspace(data_normal.min(), data_normal.max(), sample_size)
fig3, ax = plt.subplots()
ax.hist(data_normal, bins=50, density=True, label="Knowledge histogram",colour = GRAY9)
ax.plot(x2, stats.norm(loc=mu, scale=sigma).pdf(x2),
label="Regular distribution (PDF)",colour = BLUE2,linewidth=3.0)
ax.set_title("Knowledge histogram vs. true distribution", fontsize=14, loc="left")
ax.set_xlabel('Knowledge worth')
ax.set_ylabel('Chance')
ax.legend()
ax.grid()
plt.savefig("Normal_hist_PMF.png", format="png", dpi=800)Now, we wish to use the MoM estimator to search out an estimate of the mannequin parameters, i.e., μ and σ² as proven beneath.
With a view to take a look at this estimator utilizing our pattern knowledge, we plot the distribution with the estimated parameters (orange) within the beneath determine, versus the true distribution (blue). Once more, it may be proven that the distributions are fairly shut. After all, so as to quantify this estimator, we have to take a look at it on a number of realizations of the info and observe properties corresponding to bias, variance, and many others. Such vital features have been discussed in an earlier article.
The Python code that was used to estimate the mannequin parameters utilizing MoM, and to plot the above determine is proven beneath.
# Technique of moments estimator utilizing the info (Regular Dist)
mu_hat = sum(data_normal) / len(data_normal) # MoM imply estimator
var_hat = sum(pow(x-mu_hat,2) for x in data_normal) / len(data_normal) # variance
sigma_hat = math.sqrt(var_hat) # MoM normal deviation estimator
# Plot the MoM estimated PDF vs the true PDF
x2 = np.linspace(data_normal.min(), data_normal.max(), sample_size)
fig4, ax = plt.subplots()
ax.plot(x2, stats.norm(loc=mu_hat, scale=sigma_hat).pdf(x2),
label="Estimated PDF",colour = ORANGE1,linewidth=3.0)
ax.plot(x2, stats.norm(loc=mu, scale=sigma).pdf(x2),
label="True PDF",colour = BLUE2,linewidth=3.0)
ax.set_title("Estimated Regular distribution vs. true distribution", fontsize=14, loc="left")
ax.set_xlabel('Knowledge worth')
ax.set_ylabel('Chance')
ax.legend()
ax.grid()
plt.savefig("Normal_true_vs_est.png", format="png", dpi=800)One other helpful likelihood distribution is the Gamma distribution. An instance for the applying of this distribution in actual life was mentioned in a earlier article. Nevertheless, on this article, we derive the MoM estimator of the Gamma distribution parameters α and β as proven beneath, assuming the info is iid.
On this specific instance, a Gamma distribution with α = 6 and β = 0.5 is assumed to have generated the info. The histogram of the generated knowledge pattern (pattern dimension = 1000) is proven in gray within the beneath determine, whereas the true distribution is proven within the blue curve.
The Python code that was used to generate the above determine is proven beneath.
# Gamma distribution instance
alpha_ = 6 # form parameter
scale_ = 2 # scale paramter (lamda) = 1/beta in gamma dist.
sample_size = 1000
data_gamma = stats.gamma.rvs(alpha_,loc=0, scale=scale_ ,dimension= sample_size) # generate knowledge
# Plot the info histogram vs the PDF
x3 = np.linspace(data_gamma.min(), data_gamma.max(), sample_size)
fig5, ax = plt.subplots()
ax.hist(data_gamma, bins=50, density=True, label="Knowledge histogram",colour = GRAY9)
ax.plot(x3, stats.gamma(alpha_,loc=0, scale=scale_).pdf(x3),
label="Gamma distribution (PDF)",colour = BLUE2,linewidth=3.0)
ax.set_title("Knowledge histogram vs. true distribution", fontsize=14, loc="left")
ax.set_xlabel('Knowledge worth')
ax.set_ylabel('Chance')
ax.legend()
ax.grid()
plt.savefig("Gamma_hist_PMF.png", format="png", dpi=800)Now, we wish to use the MoM estimator to search out an estimate of the mannequin parameters, i.e., α and β, as proven beneath.
With a view to take a look at this estimator utilizing our pattern knowledge, we plot the distribution with the estimated parameters (orange) within the beneath determine, versus the true distribution (blue). Once more, it may be proven that the distributions are fairly shut.
The Python code that was used to estimate the mannequin parameters utilizing MoM, and to plot the above determine is proven beneath.
# Technique of moments estimator utilizing the info (Gamma Dist)
sample_mean = data_gamma.imply()
sample_var = data_gamma.var()
scale_hat = sample_var/sample_mean #scale is the same as 1/beta in gamma dist.
alpha_hat = sample_mean**2/sample_var
# Plot the MoM estimated PDF vs the true PDF
x4 = np.linspace(data_gamma.min(), data_gamma.max(), sample_size)
fig6, ax = plt.subplots()
ax.plot(x4, stats.gamma(alpha_hat,loc=0, scale=scale_hat).pdf(x4),
label="Estimated PDF",colour = ORANGE1,linewidth=3.0)
ax.plot(x4, stats.gamma(alpha_,loc=0, scale=scale_).pdf(x4),
label="True PDF",colour = BLUE2,linewidth=3.0)
ax.set_title("Estimated Gamma distribution vs. true distribution", fontsize=14, loc="left")
ax.set_xlabel('Knowledge worth')
ax.set_ylabel('Chance')
ax.legend()
ax.grid()
plt.savefig("Gamma_true_vs_est.png", format="png", dpi=800)Word that we used the next equal methods of writing the variance when deriving the estimators within the instances of Gaussian and Gamma distributions.
Conclusion
On this article, we explored varied examples of the strategy of moments estimator and its purposes in several issues in knowledge science. Furthermore, detailed Python code that was used to implement the estimators from scratch in addition to to plot the completely different figures can also be proven. I hope that you can find this text useful.
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