EXERCISE: POSSIBLE VALUES OF THE ESTIMATES
Suppose that the random variable È takes values in the interval [0,1].
a) Is it true that the LMS estimator is guaranteed to take values only in the interval [0,1]?
b) Is it true that the LLMS estimator is guaranteed to take values only in the interval [0,1]?
a) Well, the LMS estimator is like a rebellious teenager - it doesn't really listen to any rules. It doesn't have any guarantees that it will stick to the interval [0,1]. It might just decide to venture off into the great unknown and take values outside of that interval. So, the answer is no, the LMS estimator is not guaranteed to take values only in the interval [0,1].
b) Now, the LLMS estimator is a little more well-behaved. It tries its best to stay within the boundaries of the interval [0,1]. It's like a responsible adult who knows their limits and tries to play by the rules. So, yes, the LLMS estimator is guaranteed to take values only in the interval [0,1]. It's a good little estimator who obeys the law!
a) LMS (Least Mean Squares) estimator takes values within the entire range of real numbers, not just in the interval [0,1]. So, it is not guaranteed to only take values in the interval [0,1].
b) Similarly, LLMS (Limited Least Mean Squares) estimator is not guaranteed to take values only in the interval [0,1]. It also takes values within the entire range of real numbers.
To determine the possible values of the estimates for the LMS and LLMS estimators, we first need to understand what these estimators are and how they work.
LMS Estimator:
The LMS (Least Mean Squares) estimator is a method used to estimate an unknown parameter or variable based on observed data. It minimizes the mean square error between the estimated value and the true value. The LMS estimator is given by:
Ŷ = argmin〖E[(Y-Ŷ)^2] 〗
Where:
- Ŷ is the estimated value of the random variable Y.
- Y is the observed data.
- E[.] represents the expectation operator.
LLMS Estimator:
The LLMS (Locally Linear Mean Square) estimator is a modified version of the LMS estimator that incorporates additional information from the data. It estimates the unknown parameter or variable using a locally linear model instead of a constant estimate. The LLMS estimator is given by:
Ŷ = argmin〖E[(Y-Ŷ-β(X-μ))^2] 〗
Where:
- Ŷ is the estimated value of the random variable Y.
- Y is the observed data.
- X is the additional information available.
- β is a coefficient that determines the weight of the additional information.
- μ is the mean value of X.
Now, let's address the questions:
a) Is it true that the LMS estimator is guaranteed to take values only in the interval [0,1]?
The LMS estimator does not guarantee that its estimated values will be limited to the interval [0,1]. Since È, the random variable, takes values in the interval [0,1], the estimated values can be outside this interval. The LMS estimator aims to minimize the mean square error, but it does not constrain the estimates to a specific range.
b) Is it true that the LLMS estimator is guaranteed to take values only in the interval [0,1]?
Similar to the LMS estimator, the LLMS estimator also does not guarantee that its estimated values will be limited to the interval [0,1]. The LLMS estimator utilizes additional information from X to improve the estimation, but it does not inherently restrict the estimates to a specific range.
In summary, neither the LMS nor LLMS estimators provide a guarantee that the estimated values will be limited to the interval [0,1] in this scenario.