MLE gives a framework that exactly tackles this query. It introduces a probability perform, which is a perform that yields one other perform. This probability perform takes a vector of parameters, typically denoted as theta, and produces a chance density perform (PDF) that will depend on theta.
The chance density perform (PDF) of a distribution is a perform that takes a price, x, and returns its chance inside the distribution. Subsequently, probability features are usually expressed as follows:
The worth of this perform signifies the probability of observing x from the distribution outlined by the PDF with theta as its parameters.
The objective
When establishing a forecast mannequin, now we have knowledge samples and a parameterized mannequin, and our objective is to estimate the mannequin’s parameters. In our examples, resembling Regression and MA fashions, these parameters are the coefficients within the respective mannequin formulation.
The equal in MLE is that now we have observations and a PDF for a distribution outlined over a set of parameters, theta, that are unknown and never instantly observable. Our objective is to estimate theta.
The MLE method includes discovering the set of parameters, theta, that maximizes the probability perform given the observable knowledge, x.
We assume our samples, x, are drawn from a distribution with a recognized PDF that will depend on a set of parameters, theta. This suggests that the probability (chance) of observing x below this PDF is actually 1. Subsequently, figuring out the theta values that make our probability perform worth near 1 on our samples, ought to reveal the true parameter values.
Conditional probability
Discover that we haven’t made any assumptions concerning the distribution (PDF) on which the probability perform relies. Now, let’s assume our statement X is a vector (x_1, x_2, …, x_n). We’ll take into account a chance perform that represents the chance of observing x_n conditional on that now we have already noticed (x_1, x_2, …, x_{n-1}) —
This represents the probability of observing simply x_n given the earlier values (and theta, the set of parameters). Now, we outline the conditional probability perform as follows:
Later, we are going to see why it’s helpful to make use of the conditional probability perform reasonably than the precise probability perform.
Log-Probability
In follow, it’s typically handy to make use of the pure logarithm of the probability perform, known as the log-likelihood perform:
That is extra handy as a result of we frequently work with a probability perform that could be a joint chance perform of impartial variables, which interprets to the product of every variable’s chance. Taking the logarithm converts this product right into a sum.
For simplicity, I’ll show tips on how to estimate essentially the most fundamental shifting common mannequin — MA(1):
Right here, x_t represents the time-series observations, alpha and beta are the mannequin parameters to be estimated, and the epsilons are random noise drawn from a standard distribution with zero imply and a few variance — sigma, which can even be estimated. Subsequently, our “theta” is (alpha, beta, sigma), which we purpose to estimate.
Let’s outline our parameters and generate some artificial knowledge utilizing Python:
import pandas as pd
import numpy as npSTD = 3.3
MEAN = 0
ALPHA = 18
BETA = 0.7
N = 1000
df = pd.DataFrame({"et": np.random.regular(loc=MEAN, scale=STD, measurement=N)})
df["et-1"] = df["et"].shift(1, fill_value=0)
df["xt"] = ALPHA + (BETA*df["et-1"]) + df["et"]
Word that now we have set the usual deviation of the error distribution to three.3, with alpha at 18 and beta at 0.7. The info seems to be like this —
Probability perform for MA(1)
Our goal is to assemble a probability perform that addresses the query: how possible is it to look at our time collection X=(x_1, …, x_n) assuming they’re generated by the MA(1) course of described earlier?
The problem in computing this chance lies within the mutual dependence amongst our samples — as evident from the truth that each x_t and x_{t-1} rely upon e_{t-1) — making it non-trivial to find out the joint chance of observing all samples (known as the precise probability).
Subsequently, as mentioned beforehand, as an alternative of computing the precise probability, we’ll work with a conditional probability. Let’s start with the probability of observing a single pattern given all earlier samples:
That is a lot easier to calculate as a result of —
All that continues to be is to calculate the conditional probability of observing all samples:
making use of a pure logarithm offers:
which is the perform we should always maximize.
Code
We’ll make the most of the GenericLikelihoodModel
class from statsmodels for our MLE estimation implementation. As outlined within the tutorial on statsmodels’ web site, we merely have to subclass this class and embrace our probability perform calculation:
from scipy import stats
from statsmodels.base.mannequin import GenericLikelihoodModel
import statsmodels.api as smclass MovingAverageMLE(GenericLikelihoodModel):
def initialize(self):
tremendous().initialize()
extra_params_names = ['beta', 'std']
self._set_extra_params_names(extra_params_names)
self.start_params = np.array([0.1, 0.1, 0.1])
def calc_conditional_et(self, intercept, beta):
df = pd.DataFrame({"xt": self.endog})
ets = [0.0]
for i in vary(1, len(df)):
ets.append(df.iloc[i]["xt"] - intercept - (beta*ets[i-1]))
return ets
def loglike(self, params):
ets = self.calc_conditional_et(params[0], params[1])
return stats.norm.logpdf(
ets,
scale=params[2],
).sum()
The perform loglike
is important to implement. Given the iterated parameter values params
and the dependent variables (on this case, the time collection samples), that are saved as class members self.endog
, it calculates the conditional log-likelihood worth, as we mentioned earlier.
Now let’s create the mannequin and match on our simulated knowledge:
df = sm.add_constant(df) # add intercept for estimation (alpha)
mannequin = MovingAverageMLE(df["xt"], df["const"])
r = mannequin.match()
r.abstract()
and the output is:
And that’s it! As demonstrated, MLE efficiently estimated the parameters we chosen for simulation.
Estimating even a easy MA(1) mannequin with most probability demonstrates the facility of this technique, which not solely permits us to make environment friendly use of our knowledge but additionally gives a strong statistical basis for understanding and decoding the dynamics of time collection knowledge.
Hope you appreciated it !
[1] Andrew Lesniewski, Time Series Analysis, 2019, Baruch Faculty, New York
[2] Eric Zivot, Estimation of ARMA Models, 2005
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