I.I NTRODUCCIÓN
5. Patrones alimentarios: Dieta Mediterránea
Although inverse-exponential functions capture a wide variety of context measure- ments that occur in practice, there are other interesting scenarios that cannot be explained with this class of functions. In particular, a major limitation of inverse-
exponential functions is that they are symmetric around their θ parameter; thus,
they would not be well suited for modeling inherently non-symmetric context mea- surements such as a vital sign crossing a predefined threshold (e.g., the blood-oxygen
saturation is less than 90%). Similarly, inverse-exponential functions cannot be used to model a scenario in which a building can only be detected from certain angles (e.g., because of occlusions).
In order to overcome these limitations, in this section we investigate a second broad class of context detection functions, namely sigmoid functions.
Assumption. Suppose the probability of context detection functions are sigmoid functions that are defined as the probit logistic function [153]:
pdk(ykb |xk) = Φ(ybk(b T
kxk+ak)), (3.7)
where Φ is the cumulative distribution function of the standard Normal distribution,
bk ∈ Rd is a vector of known weight parameters, and ak ∈ R is a known parameter
offset. Note that pd
k(ykb = 1|xk) = 1−pkd(ykb =−1|xk) due to the rotational sym-
metry of Φ, i.e., Φ(−x) = 1−Φ(x). We assume there is a finite set of size C of context weights and offsets V ={(b1, a1), . . . ,(bC, aC)}.
Note that the inner function in (3.7) defines a hyperplane, determined by the
values of ak and bk, that can be intuitively considered as the detection threshold,
i.e., the probability of getting a detection is very low when the state xk is below the
“threshold” and increases rapidly asxkcrosses the “threshold”. To explain the name
of this class of context detection functions, note that in the one-dimensional case,
this function greatly resembles the classical sigmoid function: f(x) = 1/(1 +e−x),
which also exhibits this pattern of values close to 0 as x approaches −∞ and close
to 1 for largex, with a very quick transition period in between. Due to this step-like
shape, sigmoid functions are well suited for modeling the scenarios presented above – it is expected that once a signal exceeds a certain threshold, even inaccurate sensors will be able to detect the event and raise an alarm.
presented in this section hold in the presence of continuous (linear) measurements as well.
Developing an exact filter incorporating probit-based measurements is not straight- forward, however, due to the fact that the posterior distribution, once context mea- surements have been received, is not the same as the prior (even if the prior is Gaussian). At the same, as argue below, a Gaussian distribution with the same mean and covariance matrix is a good approximation for the posterior distribution. We now present the phases of the sigmoid-based filter, in a similar fashion to the GM-based one (excluding the continuous update). In this case we assume the prior
pk−1|k−1, at time k ≥ 1, is a single Gaussian distribution with mean µk−1|k−1
and covariance matrix Σk−1|k−1.
3.4.1
Predict
The predict phase is the classical Kalman filter prediction:
pk|k−1(x) =
Z
φ(x;Ak−1z, Q)φ(z;µk−1|k−1,Σk−1|k−1)dz
=φ(x;Ak−1µk−1|k−1, Ak−1Σk−1|k−1ATk−1+Q)
=φ(x;µk|k−1,Σk|k−1),
whereφ(x;µ,Σ) denotes the pdf of a Gaussian distribution with meanµand covari-
ance matrix Σ.
3.4.2
Update
The posterior distribution after the receipt of a binary measurement yb
k is shown in
Proposition 2 below (all proofs are given in the Appendix).
Proposition 2. Upon receipt of a discrete measurement yb
update is as follows: pk|k(x) = Φ(yb k(bTkx+ak))φ(x;µk|k−1,Σk|k−1) Zk , (3.8) where Zk = Φ yb k(bTkµk|k−1+ak) q bT kΣk|k−1bk+ 1 .
Approximation. We approximate the posterior distribution in (3.8) with a Gaus- sian distribution with the same mean and covariance matrix.
Note that the posterior distribution after incorporating context measurements is no longer Gaussian. However, a Gaussian still seems to be a good approximation for (3.8). In particular, as shown in Proposition 3 below, the distribution in (3.8) is log-concave; log-concavity, in turn, implies unimodality, as discussed in Corollar- ies 1 and 2. Thus, we approximate the posterior in (3.8) with a Gaussian with the same mean and covariance matrix as the distribution in (3.8) – these quantities are computed in Proposition 4 below.
Proposition 3. The distribution in (3.8) is log-concave, i.e., the function
g(x) = ln(pk|k(x)) (3.9)
is concave.
Corollary 1 ([64]). In one dimension, the distribution in (3.8) is unimodal, i.e., there exists a point x∗ such that pk|k(x) is increasing for x ≤ x∗ and pk|k(x) is
decreasing for x≥x∗.
Corollary 2([64]). In many dimensions, the distribution in (3.8)isstar-unimodal
ery bounded non-negative Borel measurable function f on Rn, tn
E[f(tX)] is non-
decreasing for t∈[0,∞)).3
Proposition 4. The mean of the distribution in (3.8) is:
µk|k =µk|k−1 + Σk|k−1bk(bTkΣk|k−1bk+χk)−1ykb, (3.10) where χk = q bT kΣk|k−1bk+ 1−bkTΣk|k−1bkα(Mk) α(Mk) (3.11) α(x) = φ(x; 0,1)/Φ(x) (3.12) Mk = ykb(bTkµk|k−1+ak) q bT kΣk|k−1bk+ 1 . (3.13)
The covariance matrix of the distribution in (3.8) is:
Σk|k = Σk|k−1−Σk|k−1bk(bTkΣk|k−1bk+γk)−1bTkΣk|k−1 (3.14) where γk = (1−h(Mk))bkTΣk|k−1bk+ 1 h(Mk) (3.15) h(x) = α(x)(x+α(x)). (3.16)
Remark. Note that the context-aware filter is similar to Kalman filtering with in- termittent observations [190] in that measurements arrive in a stochastic manner.
Thus (3.14) resembles a standard Riccati equation (update), where the non-linear
term γk could be considered as the equivalent of measurement noise.
Note also that the functions α and h defined in (3.12) and (3.16), respectively,
3Note that while there is a standard definition of unimodality in one dimension, many definitions
have been studied extensively in the statistics community. The ratio α is known as the inverse Mills ratio; some properties of the inverse Mills ratio that are used throughout this dissertation are summarized below.
Definition. The inverse Mills ratio is defined as the ratio of the pdf and cdf of a standard Normal distribution, respectively, i.e.,
α(x) =φ(x; 0,1)/Φ(x).
Proposition 5 ([173]). The following facts are true about the inverse Mills ratio: 1. h(x) :=−α0(x) = α(x)(x+α(x))
2. 0< h(x)<1,∀x∈R
3. h0(x)<0,∀x∈R.
Remark. Since 0< h(x)<1, we can conclude that γk >1.