The vertical coordinate ("height") of an object in free fall is described by an equation of the form
x(t) = \theta _0 + \theta _1 t + \theta _2 t^2,
where \theta _0,\theta _1, and \theta _2 are some parameters and t stands for time. At certain times t_1,\dots ,t_ n, we make noisy observations Y_1,\dots ,Y_ n, respectively, of the height of the object. Based on these observations, we would like to estimate the object's vertical trajectory.
We consider the special case where there is only one unknown parameter. We assume that \theta _0 (the height of the object at time zero) is a known constant. We also assume that \theta _2 (which is related to the acceleration of the object) is known. We view \theta _1 as the realized value of a continuous random variable \Theta _1. The observed height at time t_ i is Y_ i = \theta _0 + \Theta _1 t_ i +\theta _2 t_ i^2 + W_ i,\, i = 1, \ldots , n, where W_ i models the observation noise. We assume that \Theta _1\sim N(0,1), W_1,\dots ,W_ n \sim N(0,\sigma ^2), and that all these random variables are independent.
In this case, finding the MAP estimate of \Theta _1 involves the minimization of
\theta _1^2+ \frac{1}{\sigma ^2}\sum _{i=1}^ n(y_ i-\theta _0-\theta _1 t_ i -\theta _2 t_ i^2)^2
with respect to \theta _1.
Carry out this minimization and choose the correct formula for the MAP estimate, \hat{\theta }_1, from the options below.
\hat{\theta }_1 = \displaystyle \frac{\sum _{i=1}^ n t_ i(y_ i-\theta _0-\theta _2t_ i^2)}{\sigma ^2}
\hat{\theta }_1 = \displaystyle \frac{\sum _{i=1}^ n t_ i(y_ i-\theta _0-\theta _2 t_ i^2)}{\sigma ^2 + \sum _{i=1}^ n t_ i^2}
\hat{\theta }_1 = \displaystyle \frac{\sum _{i=1}^ n t_ i(y_ i-\theta _0-\theta _2 t_ i^2)}{\sigma ^2 + \sum _{i=1}^ n \theta _2t_ i^2}
none of the above
unanswered
The formula for \hat{\theta }_1 can be used to define the MAP estimator,\hat{\Theta }_1 (a random variable), as a function of t_1,\dots ,t_ n and the random variables Y_1,\dots ,Y_ n. Identify whether the following statement is true:
The MAP estimator \hat{\Theta }_1 has a normal distribution.
Select an option
unanswered
The correct formula for the MAP estimate, θ̂₁, is:
θ̂₁ = ∑ᵢ₌₁-ⁿ tᵢ(yᵢ−θ₀−θ₂tᵢ²) / (σ² + ∑ᵢ₌₁-ⁿ tᵢ²)
The statement that the MAP estimator θ̂₁ has a normal distribution is false.
To find the MAP estimate of Θ1, we need to minimize the following expression:
θ1^2 + (1/σ^2)∑(yi-θ0-θ1ti-θ2ti^2)^2
To find the optimal value of θ1, we take the derivative of the above expression with respect to θ1 and set it equal to zero:
∂/∂θ1 (θ1^2 + (1/σ^2)∑(yi-θ0-θ1ti-θ2ti^2)^2) = 0
Expanding and simplifying this equation, we get:
2θ1 + (2/σ^2)∑(yi-θ0-θ1ti-θ2ti^2)(-ti) = 0
Simplifying further:
2θ1 - (2/σ^2)∑(yi-θ0-θ1ti-θ2ti^2)ti = 0
Multiplying through by σ^2:
2σ^2θ1 - 2∑(yi-θ0-θ1ti-θ2ti^2)ti = 0
Rearranging:
θ1 = (∑ti(yi-θ0-θ2ti^2))/(σ^2)
So, the correct formula for the MAP estimate, θ̂1, is:
θ̂1 = (∑ti(yi-θ0-θ2ti^2))/(σ^2)
Therefore, the correct answer is:
\hat{\theta }_1 = \displaystyle \frac{\sum _{i=1}^ n t_ i(y_ i-\theta _0-\theta _2t_ i^2)}{\sigma ^2}