text: (cid:90) (cid:90) − − (cid:90) 1 (cid:90) 1 T T = σq 2 | | − − φ(z)dzdF (b) 1 3 0 I˜c\Icγ ηq(b/σ + z; χ) 2 q + χq(q 1) ≤ − 1 (cid:90) (cid:90) 1 σq 2 | − | − − φ(z)dzdF (b) 0 I˜c\Icγ ηq(bc/σ1 γ; χ) 2 q + χq(q 1) − (≤b) − − − 1 2bc(q 2) σq 2+(1 γ)(2 q) (cid:90) −(cid:90) φ(z)dzdF (b) 0 √I˜c\Icγ q 1 − (cid:124) − − − C˜σ log(1/σ(cid:123))(cid:122) 2bc(q 2) (cid:125) = σq 2+(1 γ)(2 q) − φ(z)dzdF (b) 0(cid:90) (cid:90)I˜c\Icγ q 1 T4 − (cid:124) − − − 1 √ (cid:123)(cid:122) 2bc(q 2) (cid:125) + σq 2+(1 γ)(2 q) − φ(z)dzdF (b), C˜σ log(1/σ) I˜c\Icγ q 1 T5 where (b) is the result of Lemma 14 and C˜ is a positive constant. We ﬁrst bound T in the following: (cid:90) (cid:18) (cid:19) 5 (cid:112) − − − − − − − T ≤ 2σq 2+(−1 γ)(2 q) 1 √ bc(q 1) φ b dF (b) ≤ 2σq 3+(−1 γ)(2 q) φ(C˜ log(1/σ)/2). 5 q 1 C˜σ log(1/σ) σ 2σ q 1 It is then easily seen that T goes to zero by choosing large enough C˜. For the remaining term T , 5 (cid:90) (cid:18) (cid:19) 4 √ − − − ≤ 2σq 3+(1 γ)(2 q) C˜σ log(1/σ) − b T − (cid:112) bc(q 1)φ dF (b) (cid:112) 4 q 1 2σ 0 − − − ≤ 2σq 3+(1 γ)(2 q) − P | | ≤ − (C˜σ log(1/σ))c(q 1)φ(0) ( B C˜σ log(1/σ)) q 1 ≤ − − − − − − → O(1)σc(q 1) γ(2 q)+1 q+(cid:15)(log(1/σ))(c(q 1)+2 q+(cid:15))/2 0. To obtain the last statement, we can choose γ close enough to zero and c close to 1. Hence we can conclude → → T 0 as σ 0. This combined with the results in (37) and (38) gives us 1 − − − E 1 − E| | − σli→m0 σ 2S2 = 2Cq(q 1) |ηq(|B| + σZ; Cσ2)|2−q + Cσ2q(q − 1) = 2Cq(q 1) B q 2. The above result together with (36) ﬁnishes the proof. ∗ As stated in the roadmap of the proof, our ﬁrst goal is to characterize the convergence rate of χ (σ). ∗ q Towards this goal, we ﬁrst show that χ (σ) cannot be either too large or too small. In particular, in Lemmas ∗ q− ∗ 16 and 17, we show that χ (σ) = O(σq 1) and χ (σ) = Ω(σq). We then utilize such result in Lemma 18 to ∗ q q conclude that χ (σ) = Θ(σq). q Lemma 16. Suppose E|B|2 < ∞, if χσ1−q = ∞ and χ = o(1), then R (χ, σ) → ∞, as σ → 0. q 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T

text: T

text: σq

text: φ(z)dzdF

text: (b)

text: I˜c\Icγ

text: ηq(b/σ

text: z;

text: χ)

text: q

text: χq(q

text: σq

text: φ(z)dzdF

text: (b)

text: I˜c\Icγ

text: γ;

text: χ)

text: q

text: χq(q

text: (≤b)

text: σq

text: q)

text: φ(z)dzdF

text: (b)

text: √I˜c\Icγ

text: q

text: C˜σ

text: σq

text: q)

text: φ(z)dzdF

text: (b)

text: q

text: σq

text: q)

text: φ(z)dzdF

text: (b),

text: C˜σ

text: I˜c\Icγ

text: q

text: where

text: (b)

text: is

text: the

text: result

text: of

text: Lemma

text: and

text: C˜

text: is

text: a

text: positive

text: constant.

text: We

text: ﬁrst

text: bound

text: T

text: in

text: the

text: following:

text: T

text: q)

text: bc(q

text: φ

text: b

text: dF

text: (b)

text: q)

text: φ(C˜

text: q

text: C˜σ

text: σ

text: 2σ

text: q

text: It

text: is

text: then

text: easily

text: seen

text: that

text: T

text: goes

text: to

text: zero

text: by

text: choosing

text: large

text: enough

text: C˜.

text: For

text: the

text: remaining

text: term

text: T

text: q)

text: C˜σ

text: b

text: T

text: bc(q

text: dF

text: (b)

text: q

text: 2σ

text: q)

text: P

text: (C˜σ

text: B

text: C˜σ

text: q

text: To

text: obtain

text: the

text: last

text: statement,

text: we

text: can

text: choose

text: γ

text: close

text: enough

text: to

text: zero

text: and

text: c

text: close

text: to

text: Hence

text: we

text: can

text: conclude

text: T

text: as

text: σ

text: This

text: combined

text: with

text: the

text: results

text: in

text: and

text: gives

text: us

text: E

text: E|

text: σ

text: |ηq(|B|

text: σZ;

text: B

text: q

text: The

text: above

text: result

text: together

text: with

text: ﬁnishes

text: the

text: proof.

text: As

text: stated

text: in

text: the

text: roadmap

text: of

text: the

text: proof,

text: our

text: ﬁrst

text: goal

text: is

text: to

text: characterize

text: the

text: convergence

text: rate

text: of

text: χ

text: (σ).

text: q

text: Towards

text: this

text: goal,

text: we

text: ﬁrst

text: show

text: that

text: χ

text: (σ)

text: cannot

text: be

text: either

text: too

text: large

text: or

text: too

text: small.

text: In

text: particular,

text: in

text: Lemmas

text: q−

text: and

text: we

text: show

text: that

text: χ

text: (σ)

text: O(σq

text: and

text: χ

text: (σ)

text: Ω(σq).

text: We

text: then

text: utilize

text: such

text: result

text: in

text: Lemma

text: to

text: q

text: q

text: conclude

text: that

text: χ

text: (σ)

text: Θ(σq).

text: q

text: Lemma

text: Suppose

text: if

text: χσ1−q

text: and

text: χ

text: then

text: R

text: (χ,

text: σ)

text: as

text: σ

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof. Consider the formula of R (χ, σ) in (35). Since χ = o(1), it is straightforward to apply Dominated q Convergence Theorem to obtain − − E| | − lim χ 2σ2q 2S = q2 B 2q 2. → 1 σ 0 − → ∞ → ∞ Because χ2σ2 2q , we know S . Also note 1 | | ≤ − · 1 S 2χq(q 1) − = 2. 2 χq(q 1) → ∞ Hence, R (χ, σ) . q Lemma 17. Suppose that the same conditions for B in Lemma 15 hold, if χ = o(σq), then − R (χ, σ) 1 → → q 0, as σ 0. σ2 − Proof. Consider the expression of R (χ, σ) 1 in (35). First note that q − E| − | − S1 χ2σ2 2qq2 ηq(B + σZ; χσ(cid:104)2 q) 2q 2 (cid:105) lim = lim = 0. → → σ 0 σ2 σ 0 σ2 (cid:2) (cid:3) I − − − − − I Now we study the behavior of S . Recall that we deﬁned − = b σq (cid:15) , b + σq (cid:15) and γ = 2 1 σ log( 1 ) σ log( 1 ) − − − σ σ b α− , b + α− in (26) and (33), respectively. It is straightforward to use the same argument as for σ σ1 γ σ σ1 γ boundin(cid:90)g P−(cid:90)1 in the proof of Lemma 12 (see the derivations in (28(cid:90))) to (cid:90)have ∞ ∞ χ φ(z) ≤ χ φ(z) → | | − − dzdF (b) − dzdF (b) 0. σ2 I η (b/σ + z; χ) 2 q + χq(q 1) σ2 I χq(q 1) 0 −1 q 0 −1 Moreover, since χ < Cσq for small enough σ, Lemma 6 part (v) implies | | ≥ | | η (b/σ + z; χ) η (b/σ + z; Cσq) . q q → Therefore, as σ 0 (cid:90) (cid:90) ∞ χ (cid:90) (cid:90) φ(z) | | − dzdF (b) σ2 I \I η (b/σ + z; χ) 2 q 0 γ −1∞ q ≤ χ · 1 φ(z) → − | | − dzdF (b) 0, σq σ2 q I \I (cid:82) η(cid:82)(b/σ + z; Cσq) 2 q 0 γ −1 q ∞ → where the last statement holds because of 1− I \I | φ(z) | − dzdF (b) 0 that has already σ2 q 0 γ −1 ηq(b/σ+z;Cσq) 2 q been shown in the proof of Lemmas 12 and 13. Above all we have proved (cid:90) (cid:90) ∞ χ φ(z) → | | − − dzdF (b) 0. σ2 I η (b/σ + z; χ) 2 q + χq(q 1) 0 γ q 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

label: paragraph

text: Consider

text: the

text: formula

text: of

text: R

text: (χ,

text: σ)

text: in

text: Since

text: χ

text: it

text: is

text: straightforward

text: to

text: apply

text: Dominated

text: q

text: Convergence

text: Theorem

text: to

text: obtain

text: E|

text: lim

text: χ

text: B

text: σ

text: Because

text: χ2σ2

text: we

text: know

text: S

text: Also

text: note

label: equation

text: S

text: χq(q

text: Hence,

text: R

text: (χ,

text: σ)

text: q

text: Lemma

text: Suppose

text: that

text: the

text: same

text: conditions

text: for

text: B

text: in

text: Lemma

text: hold,

text: if

text: χ

text: o(σq),

text: then

text: R

text: (χ,

text: σ)

text: q

text: as

text: σ

text: σ2

text: Proof.

text: Consider

text: the

text: expression

text: of

text: R

text: (χ,

text: σ)

text: in

text: First

text: note

text: that

text: q

text: E|

text: χ2σ2

text: ηq(B

text: σZ;

text: q)

text: lim

text: lim

text: σ

text: σ2

text: σ

text: σ2

text: I

text: I

text: Now

text: we

text: study

text: the

text: behavior

text: of

text: S

text: Recall

text: that

text: we

text: deﬁned

text: b

text: σq

text: b

text: σq

text: and

text: γ

text: σ

text: log(

text: σ

text: log(

text: σ

text: σ

text: b

text: α−

text: b

text: α−

text: in

text: and

text: respectively.

text: It

text: is

text: straightforward

text: to

text: use

text: the

text: same

text: argument

text: as

text: for

text: σ

text: σ1

text: γ

text: σ

text: σ1

text: γ

text: in

text: the

text: proof

text: of

text: Lemma

text: (see

text: the

text: derivations

text: in

text: to

text: χ

text: φ(z)

text: χ

text: φ(z)

text: dzdF

text: (b)

text: dzdF

text: (b)

text: σ2

text: I

text: η

text: (b/σ

text: z;

text: χ)

text: q

text: χq(q

text: σ2

text: I

text: χq(q

text: −1

text: q

text: −1

text: Moreover,

text: since

text: χ

text: Cσq

text: for

text: small

text: enough

text: σ,

text: Lemma

text: part

text: (v)

text: implies

text: η

text: (b/σ

text: z;

text: χ)

text: η

text: (b/σ

text: z;

text: Cσq)

text: q

text: q

text: Therefore,

text: as

text: σ

text: χ

text: φ(z)

text: dzdF

text: (b)

text: σ2

text: I

text: \I

text: η

text: (b/σ

text: z;

text: χ)

text: q

text: γ

text: −1∞

text: q

text: χ

text: φ(z)

text: dzdF

text: (b)

text: σq

text: σ2

text: q

text: I

text: \I

text: z;

text: Cσq)

text: q

text: γ

text: −1

text: q

text: where

text: the

text: last

text: statement

text: holds

text: because

text: of

text: 1−

text: I

text: \I

text: φ(z)

text: dzdF

text: (b)

text: that

text: has

text: already

text: σ2

text: q

text: γ

text: −1

text: ηq(b/σ+z;Cσq)

text: q

text: been

text: shown

text: in

text: the

text: proof

text: of

text: Lemmas

text: and

text: Above

text: all

text: we

text: have

text: proved

text: χ

text: φ(z)

text: dzdF

text: (b)

text: σ2

text: I

text: η

text: (b/σ

text: z;

text: χ)

text: q

text: χq(q

text: γ

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:90) ∞ | ∗ | − 1 η (b/σ + z; χ ) 4 2q − (cid:90) | (cid:90)q ∗ | − ∗ − φ(z)dzdF (b) σ2 q 0 Im¯ \K (√ηq(b/σ + z; χ ) 2 q + χ q(q 1))3 ≤ 1 (cid:90)C3σ log(1/σ) (cid:90) 1 − ∗ − φ(z)dzdF (b) σ2 q I \K χ η2 q(1; 1) 0 ∞ m¯ q 1 √ (cid:112) 1 (cid:112) + − ∗ − φ(z)dzdF (b) σ2 q I \K χ η2 q(1; 1) C3σ log(1/σ) m¯ q |I |P | | ≤ ≤ φ(0) ( B C σ log(1/σ)) φ(C log(1/σ)/2) m¯ 3 3 − − ∗ + − − ∗ . σ2 qη2 q(1; 1)χ σ2 qη2 q(1; 1)χ q q ∗ Since χ = Ω(σq) from Corollary 7, it is clear that the second term in the above upper bound goes to zero by choosing large enough C . Regarding the ﬁrst term we have 3 (cid:112) |Im¯ |P(|B| ≤ C3σ log(1/σ)) (≤a) (cid:40) σ(cid:15)(log(1/σ))2−2q+(cid:15) σ (cid:15)+q−q−11(2−q)m¯ −q−(cid:15)1 − ∗ O(1) ∗ S − σ2 qχ χ log(1/σ) 0m¯ (2 q) − (≤b) O(1)(log(1/σ)) q2+(cid:15) = o(1) if m¯ > 0, − − O(1)(log(1/σ))(cid:15)+42 3q σq2 (2+(cid:15))q+3(cid:15)+1 = o(1) if m¯ = 0, (cid:18) (cid:19) ∗ − − − ≤ ∗ where (a) holds even when m¯ = m +1 since (cid:15)+−q 1(2 q)m¯ −(cid:15) 0 by the deﬁnition of m ; and (b) is q 1 q 1 due to the fact that (χ∗)1/(2−q) = Ω σ((cid:15)l+oqg−q(−111/(σ2)−)qS)0m¯m¯−−11(−2−q−q(cid:15))1 when m¯ > 0 and (χ∗)1/(2−q) = Ω(loσg(q1−/(cid:15)σ)) when m¯ ∗=|I0,|a−cco→rding to the choice of m¯ . For the second integral in (46), note that (χ∗)2−1q = o(|Im¯ |), hence χ q 2 0. It implies that the arguments in calculating the second integral in Case 1 hold m¯ here as well. − − ∗ So far we have been able to derive the limit of σq 2U . We next analyze the term σq 2χ U and show → 2 3 that it goes to zero as σ 0. We have (cid:90) (cid:90) ∗ ∞ | ∗ | − χ (cid:90) (cid:90) η (b/σ + z; χ ) 2 q (cid:90) (cid:90) − | q ∗ | − ∗ − φ(z)dzdF (b) σ2 q K ( η (b/σ + z; χ ) 2 q + χ q(q 1))3 0 q ∗ ∞ ∗ − ∞ − ≤ χ χ η2 q(1; 1) 1 η2 q(1; 1) q q − ∗ − φ(z)dzdF (b) = − ∗ − φ(z)dzdF (b) σ2 q K (χ q(q 1))3 σ2 q K χ (q(q 1))3 0 0 The upper bound above has been shown to be zero in the preceding calculations regarding the ﬁrst term ∈ K in (45). Furthermore, note that when z / , | ∗ | − ≥ ∗ − η (b/σ + z; χ ) 2 q χ η2 q(1; 1). q q We can then obtain 36 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: η

text: (b/σ

text: z;

text: χ

text: φ(z)dzdF

text: (b)

text: σ2

text: q

text: Im¯

text: \K

text: (√ηq(b/σ

text: z;

text: χ

text: q

text: χ

text: q(q

text: φ(z)dzdF

text: (b)

text: σ2

text: q

text: I

text: \K

text: χ

text: η2

text: m¯

text: q

text: φ(z)dzdF

text: (b)

text: σ2

text: q

text: I

text: \K

text: χ

text: η2

text: m¯

text: q

text: |I

text: |P

text: B

text: C

text: σ

text: φ(C

text: m¯

text: σ2

text: σ2

text: q

text: q

text: Since

text: χ

text: Ω(σq)

text: from

text: Corollary

text: it

text: is

text: clear

text: that

text: the

text: second

text: term

text: in

text: the

text: above

text: upper

text: bound

text: goes

text: to

text: zero

text: by

text: choosing

text: large

text: enough

text: C

text: Regarding

text: the

text: ﬁrst

text: term

text: we

text: have

text: |Im¯

text: |P(|B|

text: (≤a)

text: σ

text: S

text: σ2

text: qχ

text: χ

text: q)

text: (≤b)

text: if

text: m¯

text: if

text: m¯

text: where

text: (a)

text: holds

text: even

text: when

text: m¯

text: m

text: since

text: q)m¯

text: by

text: the

text: deﬁnition

text: of

text: m

text: and

text: (b)

text: is

text: q

text: q

text: due

text: to

text: the

text: fact

text: that

text: Ω

text: when

text: m¯

text: and

text: when

text: m¯

text: to

text: the

text: choice

text: of

text: m¯

text: For

text: the

text: second

text: integral

text: in

text: note

text: that

text: o(|Im¯

text: hence

text: χ

text: q

text: It

text: implies

text: that

text: the

text: arguments

text: in

text: calculating

text: the

text: second

text: integral

text: in

text: Case

text: hold

text: m¯

text: here

text: as

text: well.

text: So

text: far

text: we

text: have

text: been

text: able

text: to

text: derive

text: the

text: limit

text: of

text: σq

text: We

text: next

text: analyze

text: the

text: term

text: σq

text: 2χ

text: U

text: and

text: show

text: that

text: it

text: goes

text: to

text: zero

text: as

text: σ

text: We

text: have

text: χ

text: η

text: (b/σ

text: z;

text: χ

text: q

text: q

text: φ(z)dzdF

text: (b)

text: σ2

text: q

text: K

text: η

text: (b/σ

text: z;

text: χ

text: q

text: χ

text: q(q

text: q

text: χ

text: χ

text: η2

text: η2

text: q

text: q

text: φ(z)dzdF

text: (b)

text: φ(z)dzdF

text: (b)

text: σ2

text: q

text: K

text: (χ

text: q(q

text: σ2

text: q

text: K

text: χ

text: (q(q

text: The

text: upper

text: bound

text: above

text: has

text: been

text: shown

text: to

text: be

text: zero

text: in

text: the

text: preceding

text: calculations

text: regarding

text: the

text: ﬁrst

text: term

text: K

text: in

text: Furthermore,

text: note

text: that

text: when

text: z

text: η

text: (b/σ

text: z;

text: χ

text: q

text: χ

text: η2

text: q

text: q

text: We

text: can

text: then

text: obtain

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: optimal choice of λ deﬁned in (3). It turns out that the distribution p impacts AMSE(λ∗ , q, σ ) in a subtle β ,q w way. For analysis purposes we ﬁrst study the signals whose elements are bounded away from zero in Theorem 3 and then study other distributions in Theorem 4. Theorem 3. Consider the region δ > 1. Suppose that P(|B| > µ) = 1 with µ being a positive constant and E| | ∞ ∈ → B 2 < . Then, for q (1, 2], as σ 0 w − E| | − σ2 − δ3(q 1)2( B q 2)2 AMSE(λ∗,q, q, σw) = 1 −w1/δ (δ − 1)3E|B|2q−2 σw4 + o(σw4 ), → and for q = 1, as σ 0 w σ2 − | −µ˜2(δ−1) | AMSE(λ∗,q, q, σw) = −w o(e δσw2 ) , 1 1/δ where µ˜ is any positive number smaller than µ. The proof can be found in Section . We observe that the ﬁrst dominant term in the expansion of AMSE(λ∗ , q, σ ) − ,q w is exactly the same for all values of q, including q = 1 and is equal to σ2 /(1 1/δ). This is also the same as w the AMSE of the OLS. We may consider this term as the ‘phase transition’ term, since it will go to zero only when δ > 1. In a nutshell, the ﬁrst term in the expansion provides the phase transition information. However, we are able to derive the second order term for AMSE(λ∗ , q, σ ). This term gives us what is beyond phase ,q w transition analysis. The impact of the signal distribution p and the regularizer (cid:96) , that is omitted in PT diagram, β q is revealed in the second order term. As a result, to compare the performance of LQLS with different values of q in the low noise regime, we can compare their second order terms. First note that all the regularizers that are studied in Theorem 3 improve the performance of OLS. When the distribution of the coefﬁcients is bounded away from 0, no signiﬁcant gain is obtained from LASSO since the second dominant term in the expansion of AMSE is exponentially small. However, the rate of the second order term exhibits an interesting transition from exponential to a polynomial decay when q increases from 1. In fact, it seems that bridge regularizers with q > 1 offer more substantial improvements over OLS. Even though LASSO is suboptimal, it is not clear which value of q provides the best performance here. Among other LQLS ∈ with q (1, 2], the optimality is determined by the constant involved in the second order term (they all have the same orders). To simplify our discussions, deﬁne − E| | − (q 1)2( B q 2)2 ∗ Cq = E|B|2q−2 , q = arg m≤ax Cq. 1<q 2 ∗ ∗ Then LQLS with q = q will perform the best. To provide some insights on q , we focus on a special family of distributions. | | ∼ − Lemma 2. Consider the two-point mixture B α∆ + (1 α)∆ , where ∆ denotes the probability ≤ ∈ µ1 ∗ µ2 µ ∗ → measure putting mass 1 at µ, 0 < µ µ , α (0, 1). Then q = 2 when µ = µ , and q 1 as → ∞ 1 2 1 2 µ /µ . 2 1 − − ∗ Proof. When µ = µ , it is clear that C = (q 1)2µ 2 and thus q = 2. We now consider 0 < µ < µ . 1 2 q 1 1 2 Denote κ = µ /µ . We can then write C as: 2 1 q 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: optimal

text: choice

text: of

text: λ

text: deﬁned

text: in

text: It

text: turns

text: out

text: that

text: the

text: distribution

text: p

text: impacts

text: AMSE(λ∗

text: q,

text: σ

text: in

text: a

text: subtle

text: β

text: ,q

text: w

text: way.

text: For

text: analysis

text: purposes

text: we

text: ﬁrst

text: study

text: the

text: signals

text: whose

text: elements

text: are

text: bounded

text: away

text: from

text: zero

text: in

text: Theorem

text: and

text: then

text: study

text: other

text: distributions

text: in

text: Theorem

text: Theorem

text: Consider

text: the

text: region

text: δ

text: Suppose

text: that

text: P(|B|

text: µ)

text: with

text: µ

text: being

text: a

text: positive

text: constant

text: and

text: E|

text: B

text: Then,

text: for

text: q

text: as

text: σ

text: w

text: E|

text: σ2

text: B

text: q

text: AMSE(λ∗,q,

text: q,

text: σw)

text: (δ

text: and

text: for

text: q

text: as

text: σ

text: w

text: σ2

text: −µ˜2(δ−1)

text: AMSE(λ∗,q,

text: q,

text: σw)

text: −w

text: o(e

text: δσw2

text: where

text: µ˜

text: is

text: any

text: positive

text: number

text: smaller

text: than

text: µ.

text: The

text: proof

text: can

text: be

text: found

text: in

text: Section

text: We

text: observe

text: that

text: the

text: ﬁrst

text: dominant

text: term

text: in

text: the

text: expansion

text: of

text: AMSE(λ∗

text: q,

text: σ

text: ,q

text: w

text: is

text: exactly

text: the

text: same

text: for

text: all

text: values

text: of

text: q,

text: including

text: q

text: and

text: is

text: equal

text: to

text: σ2

text: This

text: is

text: also

text: the

text: same

text: as

text: w

text: the

text: AMSE

text: of

text: the

text: OLS.

text: We

text: may

text: consider

text: this

text: term

text: as

text: the

text: ‘phase

text: transition’

text: term,

text: since

text: it

text: will

text: go

text: to

text: zero

text: only

text: when

text: δ

text: In

text: a

text: nutshell,

text: the

text: ﬁrst

text: term

text: in

text: the

text: expansion

text: provides

text: the

text: phase

text: transition

text: information.

text: However,

text: we

text: are

text: able

text: to

text: derive

text: the

text: second

text: order

text: term

text: for

text: AMSE(λ∗

text: q,

text: σ

text: This

text: term

text: gives

text: us

text: what

text: is

text: beyond

text: phase

text: ,q

text: w

text: transition

text: analysis.

text: The

text: impact

text: of

text: the

text: signal

text: distribution

text: p

text: and

text: the

text: regularizer

text: that

text: is

text: omitted

text: in

text: PT

text: diagram,

text: β

text: q

text: is

text: revealed

text: in

text: the

text: second

text: order

text: term.

text: As

text: a

text: result,

text: to

text: compare

text: the

text: performance

text: of

text: LQLS

text: with

text: different

text: values

text: of

text: q

text: in

text: the

text: low

text: noise

text: regime,

text: we

text: can

text: compare

text: their

text: second

text: order

text: terms.

text: First

text: note

text: that

text: all

text: the

text: regularizers

text: that

text: are

text: studied

text: in

text: Theorem

text: improve

text: the

text: performance

text: of

text: OLS.

text: When

text: the

text: distribution

text: of

text: the

text: coefﬁcients

text: is

text: bounded

text: away

text: from

text: no

text: signiﬁcant

text: gain

text: is

text: obtained

text: from

text: LASSO

text: since

text: the

text: second

text: dominant

text: term

text: in

text: the

text: expansion

text: of

text: AMSE

text: is

text: exponentially

text: small.

text: However,

text: the

text: rate

text: of

text: the

text: second

text: order

text: term

text: exhibits

text: an

text: interesting

text: transition

text: from

text: exponential

text: to

text: a

text: polynomial

text: decay

text: when

text: q

text: increases

text: from

text: In

text: fact,

text: it

text: seems

text: that

text: bridge

text: regularizers

text: with

text: q

text: offer

text: more

text: substantial

text: improvements

text: over

text: OLS.

text: Even

text: though

text: LASSO

text: is

text: suboptimal,

text: it

text: is

text: not

text: clear

text: which

text: value

text: of

text: q

text: provides

text: the

text: best

text: performance

text: here.

text: Among

text: other

text: LQLS

text: with

text: q

text: the

text: optimality

text: is

text: determined

text: by

text: the

text: constant

text: involved

text: in

text: the

text: second

text: order

text: term

text: (they

text: all

text: have

text: the

text: same

text: orders).

text: To

text: simplify

text: our

text: discussions,

text: deﬁne

text: E|

text: (q

text: B

text: q

text: Cq

text: q

text: arg

text: m≤ax

text: Cq.

text: Then

text: LQLS

text: with

text: q

text: q

text: will

text: perform

text: the

text: best.

text: To

text: provide

text: some

text: insights

text: on

text: q

text: we

text: focus

text: on

text: a

text: special

text: family

text: of

text: distributions.

text: Lemma

text: Consider

text: the

text: two-point

text: mixture

text: B

text: α∆

text: α)∆

text: where

text: denotes

text: the

text: probability

text: µ1

text: µ2

text: µ

text: measure

text: putting

text: mass

text: at

text: µ,

text: µ

text: µ

text: α

text: Then

text: q

text: when

text: µ

text: µ

text: and

text: q

text: as

text: µ

text: /µ

text: Proof.

text: When

text: µ

text: µ

text: it

text: is

text: clear

text: that

text: C

text: (q

text: and

text: thus

text: q

text: We

text: now

text: consider

text: µ

text: µ

text: q

text: Denote

text: κ

text: µ

text: /µ

text: We

text: can

text: then

text: write

text: C

text: as:

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: As a result, we can derive E| | − ≤ ( B q+q0 2)2 ≤ E | | − Cq τ2q02 (E|B|q−1)2 τ2q02 ( B q0 1)2 = C1. ∗ We can then conclude q = 1 = max(1, q ). 0 Note that Lemma 3 studies a family of distributions whose probability density function exists and is non- zero at zero, but exhbit very different tail behaviors. As conﬁrmed by this lemma, in this case the tail behavior ∈ has an inﬂuence on the performance of LQLS. In particular, LQLS with q = q (1, 2] is optimal for distri- 0 − butions with the exponential decay tail e τbq0. Since βˆ(λ, q ) can be considered as the maximum a posterior 0 estimate (MAP), our result suggests that MAP offers the best performance in the low noise regime (among the bridge estimators). This is in general not true. See Zheng et al. (2017) for a counterexample in large noise cases. It is also interesting to observe that as the tail becomes heavier than that of Laplacian distribution, the ∗ optimal q approaches 1. Again, this observation is consistent with the fact that ridge often penalizes large signal values more aggressively than the other estimators. Hence, if the tail of the distribution is light (like Gaussian distributions), then ridge offers the best performance, otherwise, other values of q offer better results. In the next lemma, we further support this claim by considering a special family of distributions with light tails. | | ≤ ≤ Lemma 4. Consider B follows a uniform distribution with density function f(b) = 11(0 b θ), where ∗ θ θ > 0 is the location parameter. Then q = 2. Proof. It is clear that (cid:90) (cid:90) − − E| | − 1 θ − θq 2 E| | − 1 θ − θ2q 2 B q 2 = bq 2db = − , B 2q 2 = b2q 2db = − . θ q 1 θ 2q 1 0 0 ∗ − E| | − − Hence, q = arg max1<q≤2 (q 1E)2|B( |2Bq−q2 2)2 = arg max1<q≤2 2qθ2 1 = 2. 3.3 Implications for classical asymptotics In this section, we would like to show that the results we have derived in the previous sections, has, perhaps → ∞ surprisingly, some interesting connections to the classical asymptotic setting in which n , while p is ﬁxed. → ∈ ∞ Our analysis so far has been focused on the high-dimensional setting in which n/p δ (0, ). Furthermore, we assumed that the noise variance is small. In the classical asymptotics, it is assumed that the signal-to-noise → ∞ ratio of each observation is ﬁxed and n/p . Note that having more measurements is at the intuitive level equivalent to less noise. Hence, we expect our low-noise sensitivity to have some implications for the classical asymptotics too. Our goal below is to formalize this connection and explain the implications of our low-noise analysis framework for the classical asymptotics. Towards that goal, we will consider the scenarios where the sample size n is much larger than the dimension p. Analytically, we let δ go to inﬁnity and calculate the expansions for AMSE in terms of large δ (similar to what we did in Section for low noise). In this section, we write AMSE(λ∗ , q, δ) for AMSE(λ∗ , q, σ ) to ,q ,q w make it clear that the expansion is derived in terms of δ. Before getting to the results, we should clarify an important issue. Recall the deﬁnition of a converging sequence in Deﬁnition 1. It is straightforward to conﬁrm 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: As

text: a

text: result,

text: we

text: can

text: derive

text: E|

text: B

text: E

text: Cq

text: B

text: We

text: can

text: then

text: conclude

text: q

text: q

text: Note

text: that

text: Lemma

text: studies

text: a

text: family

text: of

text: distributions

text: whose

text: probability

text: density

text: function

text: exists

text: and

text: is

text: non-

text: zero

text: at

text: zero,

text: but

text: exhbit

text: very

text: different

text: tail

text: behaviors.

text: As

text: conﬁrmed

text: by

text: this

text: lemma,

text: in

text: this

text: case

text: the

text: tail

text: behavior

text: has

text: an

text: inﬂuence

text: on

text: the

text: performance

text: of

text: LQLS.

text: In

text: particular,

text: LQLS

text: with

text: q

text: q

text: is

text: optimal

text: for

text: distri-

text: butions

text: with

text: the

text: exponential

text: decay

text: tail

text: e

text: Since

text: βˆ(λ,

text: q

text: can

text: be

text: considered

text: as

text: the

text: maximum

text: a

text: posterior

text: estimate

text: (MAP),

text: our

text: result

text: suggests

text: that

text: MAP

text: offers

text: the

text: best

text: performance

text: in

text: the

text: low

text: noise

text: regime

text: (among

text: the

text: bridge

text: estimators).

text: This

text: is

text: in

text: general

text: not

text: true.

text: See

text: Zheng

text: et

text: al.

text: for

text: a

text: counterexample

text: in

text: large

text: noise

text: cases.

text: It

text: is

text: also

text: interesting

text: to

text: observe

text: that

text: as

text: the

text: tail

text: becomes

text: heavier

text: than

text: that

text: of

text: Laplacian

text: distribution,

text: the

text: optimal

text: q

text: approaches

text: Again,

text: this

text: observation

text: is

text: consistent

text: with

text: the

text: fact

text: that

text: ridge

text: often

text: penalizes

text: large

text: signal

text: values

text: more

text: aggressively

text: than

text: the

text: other

text: estimators.

text: Hence,

text: if

text: the

text: tail

text: of

text: the

text: distribution

text: is

text: light

text: (like

text: Gaussian

text: distributions),

text: then

text: ridge

text: offers

text: the

text: best

text: performance,

text: otherwise,

text: other

text: values

text: of

text: q

text: offer

text: better

text: results.

text: In

text: the

text: next

text: lemma,

text: we

text: further

text: support

text: this

text: claim

text: by

text: considering

text: a

text: special

text: family

text: of

text: distributions

text: with

text: light

text: tails.

text: Lemma

text: Consider

text: B

text: follows

text: a

text: uniform

text: distribution

text: with

text: density

text: function

text: f(b)

text: b

text: θ),

text: where

text: θ

text: θ

text: is

text: the

text: location

text: parameter.

text: Then

text: q

text: Proof.

text: It

text: is

text: clear

text: that

text: E|

text: θ

text: θq

text: E|

text: θ

text: B

text: q

text: bq

text: B

text: θ

text: q

text: θ

text: E|

text: Hence,

text: q

text: arg

text: (q

text: arg

text: Implications

text: for

text: classical

text: asymptotics

text: In

text: this

text: section,

text: we

text: would

text: like

text: to

text: show

text: that

text: the

text: results

text: we

text: have

text: derived

text: in

text: the

text: previous

text: sections,

text: has,

text: perhaps

text: surprisingly,

text: some

text: interesting

text: connections

text: to

text: the

text: classical

text: asymptotic

text: setting

text: in

text: which

text: n

text: while

text: p

text: is

text: ﬁxed.

text: Our

text: analysis

text: so

text: far

text: has

text: been

text: focused

text: on

text: the

text: high-dimensional

text: setting

text: in

text: which

text: n/p

text: δ

text: Furthermore,

text: we

text: assumed

text: that

text: the

text: noise

text: variance

text: is

text: small.

text: In

text: the

text: classical

text: asymptotics,

text: it

text: is

text: assumed

text: that

text: the

text: signal-to-noise

text: ratio

text: of

text: each

text: observation

text: is

text: ﬁxed

text: and

text: n/p

text: Note

text: that

text: having

text: more

text: measurements

text: is

text: at

text: the

text: intuitive

text: level

text: equivalent

text: to

text: less

text: noise.

text: Hence,

text: we

text: expect

text: our

text: low-noise

text: sensitivity

text: to

text: have

text: some

text: implications

text: for

text: the

text: classical

text: asymptotics

text: too.

text: Our

text: goal

text: below

text: is

text: to

text: formalize

text: this

text: connection

text: and

text: explain

text: the

text: implications

text: of

text: our

text: low-noise

text: analysis

text: framework

text: for

text: the

text: classical

text: asymptotics.

text: Towards

text: that

text: goal,

text: we

text: will

text: consider

text: the

text: scenarios

text: where

text: the

text: sample

text: size

text: n

text: is

text: much

text: larger

text: than

text: the

text: dimension

text: p.

text: Analytically,

text: we

text: let

text: δ

text: go

text: to

text: inﬁnity

text: and

text: calculate

text: the

text: expansions

text: for

text: AMSE

text: in

text: terms

text: of

text: large

text: δ

text: (similar

text: to

text: what

text: we

text: did

text: in

text: Section

text: for

text: low

text: noise).

text: In

text: this

text: section,

text: we

text: write

text: AMSE(λ∗

text: q,

text: δ)

text: for

text: AMSE(λ∗

text: q,

text: σ

text: to

text: ,q

text: ,q

text: w

text: make

text: it

text: clear

text: that

text: the

text: expansion

text: is

text: derived

text: in

text: terms

text: of

text: δ.

text: Before

text: getting

text: to

text: the

text: results,

text: we

text: should

text: clarify

text: an

text: important

text: issue.

text: Recall

text: the

text: deﬁnition

text: of

text: a

text: converging

text: sequence

text: in

text: Deﬁnition

text: It

text: is

text: straightforward

text: to

text: conﬁrm

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3.5.2. Tree Cover The Tree Cover model is y(s) = β + β x(s) + (cid:15) (s), (2) 0y 1y y where y(s) and (cid:15) (s) are deﬁned as in model (1). The intercept, β , and regression slope parameter, y 0y β , together describe the linear relationship between the Landsat-based tree cover product, x(s), and y(s). 1y 3.5.3. Spatial Incorporating a spatial random eﬀect into model (1) establishes the Spatial model and is y(s) = β + w (s) + (cid:15) (s), (3) 0y y y where y(s), β and (cid:15) (s) are deﬁned as in model (1). Here, w (s) is modeled as a Gaussian process (GP) 0y y y with a zero mean and a spatial covariance function that captures the covariance between any pair of locations ∗ ∼ ∗ ∗ ∗ ∗ s and s , i.e., w (s) GP(0, C(s, s ; θ )). We specify C(s, s ; θ ) = σ2ρ(s, s ; φ ) where ρ(s, s ; φ ) is y y { } y y y y a valid spatial correlation function and θ = σ2, φ , where φ is a correlation decay parameter and y y y y || − ∗|| σ2 = V ar(w ). For this analysis, we assume an exponential correlation function, i.e., ρ( s s ; φ ) = y − || −y ∗|| || − ∗|| ∗ y exp( φ s s ), where s s is the Euclidean distance between locations s and s in kilometers (km). y This speciﬁcation for the spatial correlation function requires the modeler to assume the spatially structured error variability to be stationary and isotropic, meaning that spatial variability left after accounting for the trend does not depend on location or direction. This is true for the spatial random eﬀects deﬁned in subsequent sections as well. To ease interpretation of the φ estimates, corresponding eﬀective range y estimates, labeled er , are presented in the results tables. We deﬁne er as the distance (km) where the y y spatial correlation between locations drops to 0.05. See Gelfand et al. (2004) for speciﬁcs on how to calculate eﬀective spatial ranges using coregionalization models. 3.5.4. Spatial+Tree Cover Introducing a spatial random eﬀect in model (2) deﬁnes the Spatial+Tree Cover candidate written as y(s) = β + β x(s) + w (s) + (cid:15) (s), (4) 0y 1y y y with all terms deﬁned previously. 3.5.5. Coregionalization A bivariate coregionalization m(cid:20)odel(cid:21)for (cid:20)the j(cid:21)oint pr(cid:20)edictio(cid:21)n of(cid:20)squar(cid:21)e-root transformed AGB and lidar- derived 90th percentile height is y(s) β u (s) (cid:15) (s) = 0y + A y + y , (5) z(s) β u (s) (cid:15) (s) 0z z z where y(s), β and (cid:15) (s) are deﬁned previously. z(s) is 90th percentile height recorded at location 0y y s and, because no other covariates appear in the sub-model for z(s), β approximates the overall mean 0z of the sampled 90th percentile heights. Here, u (s) and u (s) are modeled as GPs with zero means and y z spatial correlation functions (rather than covariance functions as with w (s) in Spatial and Spatial+Tree ∼ ∗ ∼ y ∗ Cover models), i.e., u (s) GP(0, ρ(s, s ; φ )) and u (s) GP(0, ρ(s, s ; φ )), where φ and φ are y y z z y z spatial decay parameters. We again assume an exponential correlation function for both random eﬀects, i.e., || − ∗|| − || − ∗|| || − ∗|| − || − ∗|| ρ( s s ; φ ) = exp( φ s s ) and ρ( s s ; φ ) = exp( φ s s ). Eﬀective range estimates (er y y z z (cid:48) y and er ) are again presented in the results tables to facilitate interpretation. The matrix K = AA models z the cross-covariance between the spatial random eﬀects u (s) and u (s). The 2 x 2 parameter matrix A y z is the lower triangular square-root of K. The diagonal and oﬀ-diagonal elements of K capture the spatial processes variances and covariances, respectively. We assume the errors associated with the lidar sub-model 12

label: section

text: Tree

text: Cover

text: The

text: Tree

text: Cover

text: model

text: is

text: y(s)

text: β

text: β

text: x(s)

text: (s),

text: y

text: where

text: y(s)

text: and

text: (s)

text: are

text: deﬁned

text: as

text: in

text: model

text: The

text: intercept,

text: β

text: and

text: regression

text: slope

text: parameter,

text: y

text: β

text: together

text: describe

text: the

text: linear

text: relationship

text: between

text: the

text: Landsat-based

text: tree

text: cover

text: product,

text: x(s),

text: and

text: y(s).

text: Spatial

text: Incorporating

text: a

text: spatial

text: random

text: eﬀect

text: into

text: model

text: establishes

text: the

text: Spatial

text: model

text: and

text: is

text: y(s)

text: β

text: w

text: (s)

text: (s),

text: y

text: y

text: where

text: y(s),

text: β

text: and

text: (s)

text: are

text: deﬁned

text: as

text: in

text: model

text: Here,

text: w

text: (s)

text: is

text: modeled

text: as

text: a

text: Gaussian

text: process

text: (GP)

text: y

text: y

text: with

text: a

text: zero

text: mean

text: and

text: a

text: spatial

text: covariance

text: function

text: that

text: captures

text: the

text: covariance

text: between

text: any

text: pair

text: of

text: locations

text: s

text: and

text: s

text: i.e.,

text: w

text: (s)

text: C(s,

text: s

text: θ

text: We

text: specify

text: C(s,

text: s

text: θ

text: s

text: φ

text: where

text: ρ(s,

text: s

text: φ

text: is

text: y

text: y

text: y

text: y

text: y

text: y

text: a

text: valid

text: spatial

text: correlation

text: function

text: and

text: θ

text: φ

text: where

text: φ

text: is

text: a

text: correlation

text: decay

text: parameter

text: and

text: y

text: y

text: y

text: y

text: σ2

text: V

text: ar(w

text: For

text: this

text: analysis,

text: we

text: assume

text: an

text: exponential

text: correlation

text: function,

text: i.e.,

text: ρ(

text: s

text: s

text: φ

text: y

text: −y

text: y

text: exp(

text: φ

text: s

text: s

text: where

text: s

text: s

text: is

text: the

text: Euclidean

text: distance

text: between

text: locations

text: s

text: and

text: s

text: in

text: kilometers

text: (km).

text: y

text: This

text: speciﬁcation

text: for

text: the

text: spatial

text: correlation

text: function

text: requires

text: the

text: modeler

text: to

text: assume

text: the

text: spatially

text: structured

text: error

text: variability

text: to

text: be

text: stationary

text: and

text: isotropic,

text: meaning

text: that

text: spatial

text: variability

text: left

text: after

text: accounting

text: for

text: the

text: trend

text: does

text: not

text: depend

text: on

text: location

text: or

text: direction.

text: This

text: is

text: true

text: for

text: the

text: spatial

text: random

text: eﬀects

text: deﬁned

text: in

text: subsequent

text: sections

text: as

text: well.

text: To

text: ease

text: interpretation

text: of

text: the

text: φ

text: estimates,

text: corresponding

text: eﬀective

text: range

text: y

text: estimates,

text: labeled

text: er

text: are

text: presented

text: in

text: the

text: results

text: tables.

text: We

text: deﬁne

text: er

text: as

text: the

text: distance

text: (km)

text: where

text: the

text: y

text: y

text: spatial

text: correlation

text: between

text: locations

text: drops

text: to

text: See

text: Gelfand

text: et

text: al.

text: for

text: speciﬁcs

text: on

text: how

text: to

text: calculate

text: eﬀective

text: spatial

text: ranges

text: using

text: coregionalization

text: models.

text: Spatial+Tree

text: Cover

text: Introducing

text: a

text: spatial

text: random

text: eﬀect

text: in

text: model

text: deﬁnes

text: the

text: Spatial+Tree

text: Cover

text: candidate

text: written

text: as

text: y(s)

text: β

text: β

text: x(s)

text: w

text: (s)

text: (s),

text: y

text: y

text: with

text: all

text: terms

text: deﬁned

text: previously.

text: Coregionalization

text: A

text: bivariate

text: coregionalization

text: transformed

text: AGB

text: and

text: lidar-

text: derived

text: percentile

text: height

text: is

text: y(s)

text: β

text: u

text: (s)

text: (s)

text: A

text: y

text: y

text: z(s)

text: β

text: u

text: (s)

text: (s)

text: z

text: z

text: where

text: y(s),

text: β

text: and

text: (s)

text: are

text: deﬁned

text: previously.

text: z(s)

text: is

text: percentile

text: height

text: recorded

text: at

text: location

text: y

text: s

text: and,

text: because

text: no

text: other

text: covariates

text: appear

text: in

text: the

text: sub-model

text: for

text: z(s),

text: β

text: approximates

text: the

text: overall

text: mean

text: of

text: the

text: sampled

text: percentile

text: heights.

text: Here,

text: u

text: (s)

text: and

text: u

text: (s)

text: are

text: modeled

text: as

text: GPs

text: with

text: zero

text: means

text: and

text: y

text: z

text: spatial

text: correlation

text: functions

text: (rather

text: than

text: covariance

text: functions

text: as

text: with

text: w

text: (s)

text: in

text: Spatial

text: and

text: Spatial+Tree

text: y

text: Cover

text: models),

text: i.e.,

text: u

text: (s)

text: ρ(s,

text: s

text: φ

text: and

text: u

text: (s)

text: ρ(s,

text: s

text: φ

text: where

text: φ

text: and

text: φ

text: are

text: y

text: y

text: z

text: z

text: y

text: z

text: spatial

text: decay

text: parameters.

text: We

text: again

text: assume

text: an

text: exponential

text: correlation

text: function

text: for

text: both

text: random

text: eﬀects,

text: i.e.,

text: ρ(

text: s

text: s

text: φ

text: exp(

text: φ

text: s

text: s

text: and

text: ρ(

text: s

text: s

text: φ

text: exp(

text: φ

text: s

text: s

text: Eﬀective

text: range

text: estimates

text: (er

text: y

text: y

text: z

text: z

text: y

text: and

text: er

text: are

text: again

text: presented

text: in

text: the

text: results

text: tables

text: to

text: facilitate

text: interpretation.

text: The

text: matrix

text: K

text: AA

text: models

text: z

text: the

text: cross-covariance

text: between

text: the

text: spatial

text: random

text: eﬀects

text: u

text: (s)

text: and

text: u

text: (s).

text: The

text: x

text: parameter

text: matrix

text: A

text: y

text: z

text: is

text: the

text: lower

text: triangular

text: square-root

text: of

text: K.

text: The

text: diagonal

text: and

text: oﬀ-diagonal

text: elements

text: of

text: K

text: capture

text: the

text: spatial

text: processes

text: variances

text: and

text: covariances,

text: respectively.

text: We

text: assume

text: the

text: errors

text: associated

text: with

text: the

text: lidar

text: sub-model


text: corresponding models exceed the average ﬂight-line separation distances of < 1.5 km at BCEF and CPCRW (2.15 km < er < 2.41 km). On average, grid cell-level predictions at BCEF and CPCRW are borrowing z more proximate lidar data to inform prediction because there is more available within the eﬀective range of spatial dependence at these sites. Since ﬁeld plots are also closer together at CPCRW and BCEF compared to TVSF and TNWR, grid cell predictions are potentially tapping into more information provided by nearby AGB observations as well. Any AGB explanatory power oﬀered by the Landsat tree cover product is already explained by simply borrowing strength from nearby ﬁeld and lidar data to inform prediction at BCEF and CPCRW. However, since ﬁeld and lidar data are much farther apart at TVSF and TNWR, we do see beneﬁt from the wall-to-wall information provided by the Landsat-based tree cover product. 4.3. Comparing Coregionalization and Coregionalization+Tree Cover mapped predictions Figs. 4, 5, 6 and 7 show maps of PPD median point estimates and standard deviations for AGB density using the Coregionalization and Coregionalization+Tree Cover models. The ﬂight-lines where lidar was collected are easily discernible in the Coregionalization model prediction maps (Figs. 4 and 5). We see diﬀerentiation of AGB within lidar strips and near ﬁeld plot locations. As one moves away from regions where information was collected, predicted AGB densities retreat to the global mean. Additionally, grid cell-level uncertainty maps show a trend of higher predictive precision in close proximity to ﬁeld and lidar observations compared to areas farther away. Even though the Coregionalization models provide reliable estimates of total (mean) AGB and show favorable CV-RMSE accuracy assessments, the resulting prediction maps leave something to be desired. Having no information outside the ﬂight-lines, the coregionalization models only rely on borrowing strength from nearby observations to inform prediction between strips. The addition of the wall-to-wall Landsat tree cover product allows for better diﬀerentiation of median point predictions outside lidar strips, leading to maps that are potentially more useful for assessing the spatial distribution of AGB across the landscape. At TVSF and TNWR, CV-RMSE estimates suggest that the addition of the Landsat tree cover product leads to better map accuracy. However, at BCEF and CPCRW, the inclusion of Landsat tree cover appears to only oﬀer cosmetic adjustments to resulting maps, not actual increases in prediction accuracy. It seems that the intensity of the ﬁeld and lidar sample at BCEF and CPCRW mixed with constructing models to borrow information from neighboring locations oﬀered more predictive advantage than simply including Landsat tree cover as a covariate. 4.4. Watershed unit AGB estimates at Caribou-Poker Creeks Research Watershed Fig. 8 shows estimated mean AGB densities for the 11 watershed units at CPCRW predicted using the Coregionalization+Tree Cover model with a permafrost polygon layer overlaid. Average AGB density was negatively correlated with permafrost presence at the watershed unit level. By appropriately summarizing the model’s PPD surface, it is possible to obtain an estimate of the correlation between AGB density and permafrost proportion at the watershed unit scale while accounting for uncertainty in the AGB density predictions. By calculating the correlation between the proportion of the watershed unit covered in per- mafrost with each MCMC sample of average AGB density, we can obtain a PPD summary characterizing their degree of correlation. The median correlation point estimate representing the relationship between − permafrost and watershed-unit AGB density is 0.685. The 95 percent upper and lower credible bounds for − − this estimate are 0.587 and 0.764, respectively. Since this interval does not include zero, we can conclude that permafrost presence is negatively related to AGB density with 95 percent certainty. We can also explore potential causes for departure from the overall trend between watershed unit- level AGB density and permafrost. We see that the watershed units furthest below the trend-line (points highlighted in red in Fig. 8) correspond to watersheds with the highest proportion of area burned during the 2004 Boundary Fire, suggesting that these areas are still in a state of recovery from ﬁre after a decade. We may expect these two watershed units to continue to accrue AGB until they begin to align with the overall trend between AGB density and permafrost. Fig. 8 captures the eﬀect of ﬁre history on watershed-unit AGB variability beyond what would be expected due to variation in permafrost coverage at CPCRW. 5. Comparing Coregionalization model and cokriging predictions Table 5 presents holdout prediction accuracies and 95% coverage probabilities for the cokriging inter- polations and Coregionalization models at BCEF, CPCRW, TNWR and TVSF. The results in Table 5 indicate that cokriging and Coregionalization holdout prediction accuracies were similar at all four study 19

text: corresponding

text: models

text: exceed

text: the

text: average

text: ﬂight-line

text: separation

text: distances

text: of

text: km

text: at

text: BCEF

text: and

text: CPCRW

text: km

text: er

text: km).

text: On

text: average,

text: grid

text: cell-level

text: predictions

text: at

text: BCEF

text: and

text: CPCRW

text: are

text: borrowing

text: z

text: more

text: proximate

text: lidar

text: data

text: to

text: inform

text: prediction

text: because

text: there

text: is

text: more

text: available

text: within

text: the

text: eﬀective

text: range

text: of

text: spatial

text: dependence

text: at

text: these

text: sites.

text: Since

text: ﬁeld

text: plots

text: are

text: also

text: closer

text: together

text: at

text: CPCRW

text: and

text: BCEF

text: compared

text: to

text: TVSF

text: and

text: TNWR,

text: grid

text: cell

text: predictions

text: are

text: potentially

text: tapping

text: into

text: more

text: information

text: provided

text: by

text: nearby

text: AGB

text: observations

text: as

text: well.

text: Any

text: AGB

text: explanatory

text: power

text: oﬀered

text: by

text: the

text: Landsat

text: tree

text: cover

text: product

text: is

text: already

text: explained

text: by

text: simply

text: borrowing

text: strength

text: from

text: nearby

text: ﬁeld

text: and

text: lidar

text: data

text: to

text: inform

text: prediction

text: at

text: BCEF

text: and

text: CPCRW.

text: However,

text: since

text: ﬁeld

text: and

text: lidar

text: data

text: are

text: much

text: farther

text: apart

text: at

text: TVSF

text: and

text: TNWR,

text: we

text: do

text: see

text: beneﬁt

text: from

text: the

text: wall-to-wall

text: information

text: provided

text: by

text: the

text: Landsat-based

text: tree

text: cover

text: product.

text: Comparing

text: Coregionalization

text: and

text: Coregionalization+Tree

text: Cover

text: mapped

text: predictions

text: Figs.

text: and

text: show

text: maps

text: of

text: PPD

text: median

text: point

text: estimates

text: and

text: standard

text: deviations

text: for

text: AGB

text: density

text: using

text: the

text: Coregionalization

text: and

text: Coregionalization+Tree

text: Cover

text: models.

text: The

text: ﬂight-lines

text: where

text: lidar

text: was

text: collected

text: are

text: easily

text: discernible

text: in

text: the

text: Coregionalization

text: model

text: prediction

text: maps

text: (Figs.

text: and

text: We

text: see

text: diﬀerentiation

text: of

text: AGB

text: within

text: lidar

text: strips

text: and

text: near

text: ﬁeld

text: plot

text: locations.

text: As

text: one

text: moves

text: away

text: from

text: regions

text: where

text: information

text: was

text: collected,

text: predicted

text: AGB

text: densities

text: retreat

text: to

text: the

text: global

text: mean.

text: Additionally,

text: grid

text: cell-level

text: uncertainty

text: maps

text: show

text: a

text: trend

text: of

text: higher

text: predictive

text: precision

text: in

text: close

text: proximity

text: to

text: ﬁeld

text: and

text: lidar

text: observations

text: compared

text: to

text: areas

text: farther

text: away.

text: Even

text: though

text: the

text: Coregionalization

text: models

text: provide

text: reliable

text: estimates

text: of

text: total

text: (mean)

text: AGB

text: and

text: show

text: favorable

text: CV-RMSE

text: accuracy

text: assessments,

text: the

text: resulting

text: prediction

text: maps

text: leave

text: something

text: to

text: be

text: desired.

text: Having

text: no

text: information

text: outside

text: the

text: ﬂight-lines,

text: the

text: coregionalization

text: models

text: only

text: rely

text: on

text: borrowing

text: strength

text: from

text: nearby

text: observations

text: to

text: inform

text: prediction

text: between

text: strips.

text: The

text: addition

text: of

text: the

text: wall-to-wall

text: Landsat

text: tree

text: cover

text: product

text: allows

text: for

text: better

text: diﬀerentiation

text: of

text: median

text: point

text: predictions

text: outside

text: lidar

text: strips,

text: leading

text: to

text: maps

text: that

text: are

text: potentially

text: more

text: useful

text: for

text: assessing

text: the

text: spatial

text: distribution

text: of

text: AGB

text: across

text: the

text: landscape.

text: At

text: TVSF

text: and

text: TNWR,

text: CV-RMSE

text: estimates

text: suggest

text: that

text: the

text: addition

text: of

text: the

text: Landsat

text: tree

text: cover

text: product

text: leads

text: to

text: better

text: map

text: accuracy.

text: However,

text: at

text: BCEF

text: and

text: CPCRW,

text: the

text: inclusion

text: of

text: Landsat

text: tree

text: cover

text: appears

text: to

text: only

text: oﬀer

text: cosmetic

text: adjustments

text: to

text: resulting

text: maps,

text: not

text: actual

text: increases

text: in

text: prediction

text: accuracy.

text: It

text: seems

text: that

text: the

text: intensity

text: of

text: the

text: ﬁeld

text: and

text: lidar

text: sample

text: at

text: BCEF

text: and

text: CPCRW

text: mixed

text: with

text: constructing

text: models

text: to

text: borrow

text: information

text: from

text: neighboring

text: locations

text: oﬀered

text: more

text: predictive

text: advantage

text: than

text: simply

text: including

text: Landsat

text: tree

text: cover

text: as

text: a

text: covariate.

text: Watershed

text: unit

text: AGB

text: estimates

text: at

text: Caribou-Poker

text: Creeks

text: Research

text: Watershed

text: Fig.

text: shows

text: estimated

text: mean

text: AGB

text: densities

text: for

text: the

text: watershed

text: units

text: at

text: CPCRW

text: predicted

text: using

text: the

text: Coregionalization+Tree

text: Cover

text: model

text: with

text: a

text: permafrost

text: polygon

text: layer

text: overlaid.

text: Average

text: AGB

text: density

text: was

text: negatively

text: correlated

text: with

text: permafrost

text: presence

text: at

text: the

text: watershed

text: unit

text: level.

text: By

text: appropriately

text: summarizing

text: the

text: model’s

text: PPD

text: surface,

text: it

text: is

text: possible

text: to

text: obtain

text: an

text: estimate

text: of

text: the

text: correlation

text: between

text: AGB

text: density

text: and

text: permafrost

text: proportion

text: at

text: the

text: watershed

text: unit

text: scale

text: while

text: accounting

text: for

text: uncertainty

text: in

text: the

text: AGB

text: density

text: predictions.

text: By

text: calculating

text: the

text: correlation

text: between

text: the

text: proportion

text: of

text: the

text: watershed

text: unit

text: covered

text: in

text: per-

text: mafrost

text: with

text: each

text: MCMC

text: sample

text: of

text: average

text: AGB

text: density,

text: we

text: can

text: obtain

text: a

text: PPD

text: summary

text: characterizing

text: their

text: degree

text: of

text: correlation.

text: The

text: median

text: correlation

text: point

text: estimate

text: representing

text: the

text: relationship

text: between

text: permafrost

text: and

text: watershed-unit

text: AGB

text: density

text: is

text: The

text: percent

text: upper

text: and

text: lower

text: credible

text: bounds

text: for

text: this

text: estimate

text: are

text: and

text: respectively.

text: Since

text: this

text: interval

text: does

text: not

text: include

text: zero,

text: we

text: can

text: conclude

text: that

text: permafrost

text: presence

text: is

text: negatively

text: related

text: to

text: AGB

text: density

text: with

text: percent

text: certainty.

text: We

text: can

text: also

text: explore

text: potential

text: causes

text: for

text: departure

text: from

text: the

text: overall

text: trend

text: between

text: watershed

text: unit-

text: level

text: AGB

text: density

text: and

text: permafrost.

text: We

text: see

text: that

text: the

text: watershed

text: units

text: furthest

text: below

text: the

text: trend-line

text: (points

text: highlighted

text: in

text: red

text: in

text: Fig.

text: correspond

text: to

text: watersheds

text: with

text: the

text: highest

text: proportion

text: of

text: area

text: burned

text: during

text: the

text: Boundary

text: Fire,

text: suggesting

text: that

text: these

text: areas

text: are

text: still

text: in

text: a

text: state

text: of

text: recovery

text: from

text: ﬁre

text: after

text: a

text: decade.

text: We

text: may

text: expect

text: these

text: two

text: watershed

text: units

text: to

text: continue

text: to

text: accrue

text: AGB

text: until

text: they

text: begin

text: to

text: align

text: with

text: the

text: overall

text: trend

text: between

text: AGB

text: density

text: and

text: permafrost.

text: Fig.

text: captures

text: the

text: eﬀect

text: of

text: ﬁre

text: history

text: on

text: watershed-unit

text: AGB

text: variability

text: beyond

text: what

text: would

text: be

text: expected

text: due

text: to

text: variation

text: in

text: permafrost

text: coverage

text: at

text: CPCRW.

text: Comparing

text: Coregionalization

text: model

text: and

text: cokriging

text: predictions

text: Table

text: presents

text: holdout

text: prediction

text: accuracies

text: and

text: coverage

text: probabilities

text: for

text: the

text: cokriging

text: inter-

text: polations

text: and

text: Coregionalization

text: models

text: at

text: BCEF,

text: CPCRW,

text: TNWR

text: and

text: TVSF.

text: The

text: results

text: in

text: Table

text: indicate

text: that

text: cokriging

text: and

text: Coregionalization

text: holdout

text: prediction

text: accuracies

text: were

text: similar

text: at

text: all

text: four

text: study


text: Meng, Q., Cieszewski, C., & Madden, M. (2009). Large area forest inventory using landsat etm+: A geostatistical approach. ISPRS Journal of Photogrammetry and Remote Sensing, 64, 27 – 36. Mutanga, O., & Rugege, D. (2006). Integrating remote sensing and spatial statistics to model herbaceous biomass distribution in a tropical savanna. International Journal of Remote Sensing, 27, 3499–3514. Næsset, E. (2004). Practical large-scale forest stand inventory using a small-footprint airborne scanning laser. Scandinavian Journal of Forest Research, 19, 164–179. Næsset, E. (2011). Estimating above-ground biomass in young forests with airborne laser scanning. International Journal of Remote Sensing, 32, 473–501. Neigh, C., Nelson, R., Ranson, K., Margolis, H., Montesano, P., Sun, G., Kharuk, V., Næsset, E., Wulder, M., & Andersen, H. (2013). Taking stock of circumboreal forest carbon with ground measurements, airborne and spaceborne lidar. Remote Sensing of Environment, 137, 274–287. Nelson, R. (2010). Model eﬀects on GLAS-based regional estimates of forest biomass and carbon. International Journal of Remote Sensing, 31, 1359–1372. Nelson, R., Gobakken, T., Næsset, E., Gregoire, T., St˚ahl, G., Holm, S., & Flewelling, J. (2012). Lidar sampling – using an airborne proﬁler to estimate forest biomass in Hedmark County, Norway. Remote Sensing of Environment, 123, 563–578. Nelson, R., Short, A., & Valenti, M. (2004). Measuring biomass and carbon in Delaware using an airborne proﬁling lidar. Scandinavian Journal of Forest Research, 19, 500–511. Pan, Y., Birdsey, R.A., Fang, J., Houghton, R., Kauppi, P.E., Kurz, W.A., Phillips, O.L., Shvidenko, A., Lewis, S.L., Canadell, J.G., Ciais, P., Jackson, R.B., Pacala, S.W., McGuire, A.D., Piao, S., Rautiainen, A., Sitch, S., & Hayes, D. (2011). A large and persistent carbon sink in the world’s forests. Science, 333, 988–993. Pattison, R., Andersen, H.E., Gray, A.N., Schulz, B.K., Smith, K., Jovan, S., & Manies, K.L. (In prep). Forests of the Tanana Valley State Forest and Tetlin National Wildlife Refuge Alaska - results of the 2014 pilot inventory. US Department of Agriculture Forest Service, Paciﬁc Northwest Research Station. Pfeﬀermann, D. (2013). New important developments in small area estimation. Statist. Sci., 28, 40–68. Pﬂugmacher, D., Cohen, W.B., Kennedy, R.E., & Yang, Z. (2014). Using Landsat-derived disturbance and recovery history and lidar to map forest biomass dynamics. Remote Sensing of Environment, 151, 124–137. Powell, S.L., Cohen, W.B., Healey, S.P., Kennedy, R.E., Moisen, G.G., Pierce, K.B., & Ohmann, J.L. (2010). Quantiﬁcation of live aboveground forest biomass dynamics with Landsat time-series and ﬁeld inventory data: A comparison of empirical modeling approaches. Remote Sensing of Environment, 114, 1053–1068. Rieger, S., Furbush, C.E., Schoephorster, D.B., Summerﬁeld Jr, H., & Geiger, L.C. (1972). Soils of the Caribou-Poker Creeks Research Watershed, interior Alaska. Technical Report. DTIC Document. Rinehart, A.J., Jones Jr, J.B., & Harms, T.K. (2015). Hydrologic and biogeochemical inﬂuences on carbon processing in the riparian zone of a subarctic stream. Freshwater Science, 34, 222–232. Saarela, S., Grafstr¨om, A., St˚ahl, G., Kangas, A., Holopainen, M., Tuominen, S., Nordkvist, K., & Hyypp¨a, J. (2015). Model- assisted estimation of growing stock volume using diﬀerent combinations of lidar and Landsat data as auxiliary information. Remote Sensing of Environment, 158, 431–440. S¨arndal, C., Swensson, B., & Wretman, J. (1992). Model Assisted Survey Sampling. Springer series in statistics, Springer-Verlag. Schelin, L., & Sjo¨stedt-De Luna, S. (2010). Kriging prediction intervals based on semiparametric bootstrap. Mathematical Geosciences, 42, 985–1000. Sj¨ostedt-De Luna, S., & Young, A. (2003). The bootstrap and kriging prediction intervals. Scandinavian Journal of Statistics, 30, 175–192. Stow, C.A., Reckhow, K.H., & Qian, S.S. (2006). A bayesian approach to retransformation bias in transformed regression. Ecology, 87, 1472–1477. Tanaka-Oda, A., Kenzo, T., Toriyama, J., & Matsuura, Y. (2016). Variability in the growth rates and foliage δ 15 N values of black spruce trees across a slope gradient in the Alaskan interior. Canadian Journal of Forest Research, . Thompson, S. (2002). Sampling. Wiley Series in Probability and Statistics, Wiley. Tsui, O.W., Coops, N.C., Wulder, M.A., & Marshall, P.L. (2013). Integrating airborne lidar and space-borne radar via multivariate kriging to estimate above-ground biomass. Remote Sensing of Environment, 139, 340–352. UN-REDD (2009). The UN-REDD programme. http://www.un-redd.org/. Accessed: 8-11-2016. U.S. Fish and Wildlife Service (2016). Tetlin National Wildlife Refuge, Alaska. https://www.fws.gov/refuge/tetlin/ wildlife_and_habitat/habitat.html. Accessed: 10-16-2016. Ver Hoef, J. (2002). Sampling and geostatistics for spatial data. Ecoscience, 9, 152–161. Ver Hoef, J., Cressie, N., & Barry, R.P. (2004). Flexible spatial models for kriging and cokriging using moving averages and the fast fourier transform (FFT). Journal of Computational and Graphical Statistics, 13, 265–282. Verbyla, D. (2011). Statewide 2-km raster of year since last widlﬁre, Bonanza Creek LTER - University of Alaska Fairbanks. http://www.lter.uaf.edu/data/data-detail/id/464. Wang, G., Oyana, T., Zhang, M., Adu-Prah, S., Zeng, S., Lin, H., & Se, J. (2009). Mapping and spatial uncertainty analysis of forest vegetation carbon by combining national forest inventory data and satellite images. Forest Ecology and Management, 258, 1275 – 1283. White, J.C., Wulder, M.A., Varhola, A., Vastaranta, M., Coops Nicholas, C., Cook, B.D., Pitt, D., & Woods, M. (2013). A best practices guide for generating forest inventory attributes from airborne laser scanning data using an area-based approach. The Forestry Chronicle, 89, 722–723. Woodall, C.W., Coulston, J.W., Domke, G.M., Walters, B.F., Wear, D.N., Smith, J.E., Andersen, H.E., Clough, B.J., Cohen, W.B., Griﬃth, D.M., et al. (2015). The US forest carbon accounting framework: Stocks and stock change, 1990-2016, . Wulder, M.A., White, J.C., Nelson, R.F., Næsset, E., Ørka, H.O., Coops, N.C., Hilker, T., Bater, C.W., & Gobakken, T. 28

text: Meng,

label: reference

text: Q.,

text: Cieszewski,

text: C.,

text: Madden,

text: M.

text: Large

text: area

text: forest

text: inventory

text: using

text: landsat

text: etm+:

text: A

text: geostatistical

text: approach.

text: ISPRS

text: Journal

text: of

text: Photogrammetry

text: and

text: Remote

text: Sensing,

text: Mutanga,

text: O.,

text: Rugege,

text: D.

text: Integrating

text: remote

text: sensing

text: and

text: spatial

text: statistics

text: to

text: model

text: herbaceous

text: biomass

text: distribution

text: in

text: a

text: tropical

text: savanna.

text: International

text: Journal

text: of

text: Remote

text: Sensing,

text: Næsset,

text: E.

text: Practical

text: large-scale

text: forest

text: stand

text: inventory

text: using

text: a

text: small-footprint

text: airborne

text: scanning

text: laser.

text: Scandinavian

text: Journal

text: of

text: Forest

text: Research,

text: Næsset,

text: E.

text: Estimating

text: above-ground

text: biomass

text: in

text: young

text: forests

text: with

text: airborne

text: laser

text: scanning.

text: International

text: Journal

text: of

text: Remote

text: Sensing,

text: Neigh,

text: C.,

text: Nelson,

text: R.,

text: Ranson,

text: K.,

text: Margolis,

text: H.,

text: Montesano,

text: P.,

text: Sun,

text: G.,

text: Kharuk,

text: V.,

text: Næsset,

text: E.,

text: Wulder,

text: M.,

text: Andersen,

text: H.

text: Taking

text: stock

text: of

text: circumboreal

text: forest

text: carbon

text: with

text: ground

text: measurements,

text: airborne

text: and

text: spaceborne

text: lidar.

text: Remote

text: Sensing

text: of

text: Environment,

text: Nelson,

text: R.

text: Model

text: eﬀects

text: on

text: GLAS-based

text: regional

text: estimates

text: of

text: forest

text: biomass

text: and

text: carbon.

text: International

text: Journal

text: of

text: Remote

text: Sensing,

text: Nelson,

text: R.,

text: Gobakken,

text: T.,

text: Næsset,

text: E.,

text: Gregoire,

text: T.,

text: St˚ahl,

text: G.,

text: Holm,

text: S.,

text: Flewelling,

text: J.

text: Lidar

text: sampling

text: using

text: an

text: airborne

text: proﬁler

text: to

text: estimate

text: forest

text: biomass

text: in

text: Hedmark

text: County,

text: Norway.

text: Remote

text: Sensing

text: of

text: Environment,

text: Nelson,

text: R.,

text: Short,

text: A.,

text: Valenti,

text: M.

text: Measuring

text: biomass

text: and

text: carbon

text: in

text: Delaware

text: using

text: an

text: airborne

text: proﬁling

text: lidar.

text: Scandinavian

text: Journal

text: of

text: Forest

text: Research,

text: Pan,

text: Y.,

text: Birdsey,

text: R.A.,

text: Fang,

text: J.,

text: Houghton,

text: R.,

text: Kauppi,

text: P.E.,

text: Kurz,

text: W.A.,

text: Phillips,

text: O.L.,

text: Shvidenko,

text: A.,

text: Lewis,

text: S.L.,

text: Canadell,

text: J.G.,

text: Ciais,

text: P.,

text: Jackson,

text: R.B.,

text: Pacala,

text: S.W.,

text: McGuire,

text: A.D.,

text: Piao,

text: S.,

text: Rautiainen,

text: A.,

text: Sitch,

text: S.,

text: Hayes,

text: D.

text: A

text: large

text: and

text: persistent

text: carbon

text: sink

text: in

text: the

text: world’s

text: forests.

text: Science,

text: Pattison,

text: R.,

text: Andersen,

text: H.E.,

text: Gray,

text: A.N.,

text: Schulz,

text: B.K.,

text: Smith,

text: K.,

text: Jovan,

text: S.,

text: Manies,

text: K.L.

text: (In

text: prep).

text: Forests

text: of

text: the

text: Tanana

text: Valley

text: State

text: Forest

text: and

text: Tetlin

text: National

text: Wildlife

text: Refuge

text: Alaska

text: results

text: of

text: the

text: pilot

text: inventory.

text: US

text: Department

text: of

text: Agriculture

text: Forest

text: Service,

text: Paciﬁc

text: Northwest

text: Research

text: Station.

text: Pfeﬀermann,

text: D.

text: New

text: important

text: developments

text: in

text: small

text: area

text: estimation.

text: Statist.

text: Sci.,

text: Pﬂugmacher,

text: D.,

text: Cohen,

text: W.B.,

text: Kennedy,

text: R.E.,

text: Yang,

text: Z.

text: Using

text: Landsat-derived

text: disturbance

text: and

text: recovery

text: history

text: and

text: lidar

text: to

text: map

text: forest

text: biomass

text: dynamics.

text: Remote

text: Sensing

text: of

text: Environment,

text: Powell,

text: S.L.,

text: Cohen,

text: W.B.,

text: Healey,

text: S.P.,

text: Kennedy,

text: R.E.,

text: Moisen,

text: G.G.,

text: Pierce,

text: K.B.,

text: Ohmann,

text: J.L.

text: Quantiﬁcation

text: of

text: live

text: aboveground

text: forest

text: biomass

text: dynamics

text: with

text: Landsat

text: time-series

text: and

text: ﬁeld

text: inventory

text: data:

text: A

text: comparison

text: of

text: empirical

text: modeling

text: approaches.

text: Remote

text: Sensing

text: of

text: Environment,

text: Rieger,

text: S.,

text: Furbush,

text: C.E.,

text: Schoephorster,

text: D.B.,

text: Summerﬁeld

text: Jr,

text: H.,

text: Geiger,

text: L.C.

text: Soils

text: of

text: the

text: Caribou-Poker

text: Creeks

text: Research

text: Watershed,

text: interior

text: Alaska.

text: Technical

text: Report.

text: DTIC

text: Document.

text: Rinehart,

text: A.J.,

text: Jones

text: Jr,

text: J.B.,

text: Harms,

text: T.K.

text: Hydrologic

text: and

text: biogeochemical

text: inﬂuences

text: on

text: carbon

text: processing

text: in

text: the

text: riparian

text: zone

text: of

text: a

text: subarctic

text: stream.

text: Freshwater

text: Science,

text: Saarela,

text: S.,

text: Grafstr¨om,

text: A.,

text: St˚ahl,

text: G.,

text: Kangas,

text: A.,

text: Holopainen,

text: M.,

text: Tuominen,

text: S.,

text: Nordkvist,

text: K.,

text: Hyypp¨a,

text: J.

text: Model-

text: assisted

text: estimation

text: of

text: growing

text: stock

text: volume

text: using

text: diﬀerent

text: combinations

text: of

text: lidar

text: and

text: Landsat

text: data

text: as

text: auxiliary

text: information.

text: Remote

text: Sensing

text: of

text: Environment,

text: S¨arndal,

text: C.,

text: Swensson,

text: B.,

text: Wretman,

text: J.

text: Model

text: Assisted

text: Survey

text: Sampling.

text: Springer

text: series

text: in

text: statistics,

text: Springer-Verlag.

text: Schelin,

text: L.,

text: Sjo¨stedt-De

text: Luna,

text: S.

text: Kriging

text: prediction

text: intervals

text: based

text: on

text: semiparametric

text: bootstrap.

text: Mathematical

text: Geosciences,

text: Sj¨ostedt-De

text: Luna,

text: S.,

text: Young,

text: A.

text: The

text: bootstrap

text: and

text: kriging

text: prediction

text: intervals.

text: Scandinavian

text: Journal

text: of

text: Statistics,

text: Stow,

text: C.A.,

text: Reckhow,

text: K.H.,

text: Qian,

text: S.S.

text: A

text: bayesian

text: approach

text: to

text: retransformation

text: bias

text: in

text: transformed

text: regression.

text: Ecology,

text: Tanaka-Oda,

text: A.,

text: Kenzo,

text: T.,

text: Toriyama,

text: J.,

text: Matsuura,

text: Y.

text: Variability

text: in

text: the

text: growth

text: rates

text: and

text: foliage

text: δ

text: N

text: values

text: of

text: black

text: spruce

text: trees

text: across

text: a

text: slope

text: gradient

text: in

text: the

text: Alaskan

text: interior.

text: Canadian

text: Journal

text: of

text: Forest

text: Research,

text: Thompson,

text: S.

text: Sampling.

text: Wiley

text: Series

text: in

text: Probability

text: and

text: Statistics,

text: Wiley.

text: Tsui,

text: O.W.,

text: Coops,

text: N.C.,

text: Wulder,

text: M.A.,

text: Marshall,

text: P.L.

text: Integrating

text: airborne

text: lidar

text: and

text: space-borne

text: radar

text: via

text: multivariate

text: kriging

text: to

text: estimate

text: above-ground

text: biomass.

text: Remote

text: Sensing

text: of

text: Environment,

text: UN-REDD

text: The

text: UN-REDD

text: programme.

text: Accessed:

text: U.S.

text: Fish

text: and

text: Wildlife

text: Service

text: Tetlin

text: National

text: Wildlife

text: Refuge,

text: Alaska.

text: Accessed:

text: Ver

text: Hoef,

text: J.

text: Sampling

text: and

text: geostatistics

text: for

text: spatial

text: data.

text: Ecoscience,

text: Ver

text: Hoef,

text: J.,

text: Cressie,

text: N.,

text: Barry,

text: R.P.

text: Flexible

text: spatial

text: models

text: for

text: kriging

text: and

text: cokriging

text: using

text: moving

text: averages

text: and

text: the

text: fast

text: fourier

text: transform

text: (FFT).

text: Journal

text: of

text: Computational

text: and

text: Graphical

text: Statistics,

text: Verbyla,

text: D.

text: Statewide

text: raster

text: of

text: year

text: since

text: last

text: widlﬁre,

text: Bonanza

text: Creek

text: LTER

text: University

text: of

text: Alaska

text: Fairbanks.

text: Wang,

text: G.,

text: Oyana,

text: T.,

text: Zhang,

text: M.,

text: Adu-Prah,

text: S.,

text: Zeng,

text: S.,

text: Lin,

text: H.,

text: Se,

text: J.

text: Mapping

text: and

text: spatial

text: uncertainty

text: analysis

text: of

text: forest

text: vegetation

text: carbon

text: by

text: combining

text: national

text: forest

text: inventory

text: data

text: and

text: satellite

text: images.

text: Forest

text: Ecology

text: and

text: Management,

text: White,

text: J.C.,

text: Wulder,

text: M.A.,

text: Varhola,

text: A.,

text: Vastaranta,

text: M.,

text: Coops

text: Nicholas,

text: C.,

text: Cook,

text: B.D.,

text: Pitt,

text: D.,

text: Woods,

text: M.

text: A

text: best

text: practices

text: guide

text: for

text: generating

text: forest

text: inventory

text: attributes

text: from

text: airborne

text: laser

text: scanning

text: data

text: using

text: an

text: area-based

text: approach.

text: The

text: Forestry

text: Chronicle,

text: Woodall,

text: C.W.,

text: Coulston,

text: J.W.,

text: Domke,

text: G.M.,

text: Walters,

text: B.F.,

text: Wear,

text: D.N.,

text: Smith,

text: J.E.,

text: Andersen,

text: H.E.,

text: Clough,

text: B.J.,

text: Cohen,

text: W.B.,

text: Griﬃth,

text: D.M.,

text: et

text: al.

text: The

text: US

text: forest

text: carbon

text: accounting

text: framework:

text: Stocks

text: and

text: stock

text: change,

text: Wulder,

text: M.A.,

text: White,

text: J.C.,

text: Nelson,

text: R.F.,

text: Næsset,

text: E.,

text: Ørka,

text: H.O.,

text: Coops,

text: N.C.,

text: Hilker,

text: T.,

text: Bater,

text: C.W.,

text: Gobakken,

text: T.


text: IEEE TRANSACTIONS ON SIGNAL PROCESSING 13 (cid:32) (cid:114) (cid:33) (cid:62) B. Theorem Plugging this into the equation deﬁned by ζ Qζ = 0 yields Now we restate the theorem again for convenience. (cid:48) (cid:107) (cid:107)∗ 3λ τ ± 2 Φ = 2 . Theorem 3. Under Assum(cid:32)ptions(cid:114)1-3, th(cid:33)e solution to the F α 2 + τ2 optimi(cid:98)zatio(cid:101)n problem (21) satisﬁes (cid:107) (cid:107) √ √ Since we are seeking an upper bound for Φ , it can be seen F (cid:107) − (cid:107) ≤ 3λγ s(cid:32)A (cid:114) 2 (cid:33) ≤ 9λγ sA that we take the positive (cid:32)root, (cid:114) (cid:33) A A 2+ (cid:98) (cid:101) F α 2+τ2 α √ √ (cid:48) (cid:48) (cid:107) (cid:107) ≤ 3λ τ 2 ≤ 9λ τ (cid:107)L − L(cid:107)F ≤ 3λαrL 2+ 1+2τ22τ2 ≤ 9λαrL . Φ F α 2 + 2 + τ2 α (cid:107) (cid:107) Similarly, for Ψ , Proof. Starting with Proposition 4.1, since our solution is in (cid:18) (cid:19) F (cid:18) (cid:19) tihmisp(cid:98)lrieesdt(cid:98)rbicyteAdssseu(cid:101)tm, wpte(cid:101)iocnan1,u(cid:32)se(cid:20)the(cid:98)(cid:101)stron(cid:21)ge(cid:16)r c(cid:98)onv(cid:98)exity condit(cid:17)io(cid:33)n ∂(cid:107)Ψ(cid:107)F ζ(cid:62)Qζ = 0 ⇒ 2 + τ2 (cid:107) (cid:107)∗ (cid:48) τ (cid:107) (cid:107)∗ (cid:62) α Ψ = 3λ τ + α Φ . ≥ 1 (cid:62) − 2(1 + τ2) F 2(1 + τ2) F m(A+L) m(A+L)+tr ΓX A+L (A+L) K (cid:62) Finally, plugging this (cid:32)into ζ(cid:114)Qζ = 0 (cid:33)and solving for the (cid:107) (cid:107) + α Φ + Ψ 2 upper bound yields F Revisiting the objective functional at optimality and skip- (cid:48) (cid:48) (cid:107) (cid:107) ≤ 3λ 2τ2 ≤ 9λ ping repetitive algebra (see proof for Proposition 4.1), Ψ 2 + F α 1 + 2τ2 α (cid:107) (cid:107) ≤ λ (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) α Φ+Ψ 2 (3(γ Φ + Ψ ∗) (γ Φ + Ψ ∗)) F 2 S 1 R Sc 1 Rc ≤ 3λ (cid:107) (cid:107) (cid:107) (cid:107) (γ Φ + Ψ ∗) 2 √ 1 ≤ (cid:107) (cid:107) (cid:107) (cid:107) 6λ rL(τ Φ F + Ψ F ) REFERENCES Now from Proposition 4.2, we have (cid:107) (cid:107) ≥(cid:107) (cid:107) (cid:107) (cid:107) − | (cid:62) | [1] R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” Φ+Ψ 2F ≥ (cid:107)Φ (cid:107)F + (cid:107)Ψ (cid:107)2F −2 tr((cid:107)Φ (cid:107)Ψ) (cid:107) (cid:107) Journal of the Royal Statistical Society. Series B (Methodological), Φ + Ψ 2 µ(γ Φ + Ψ ∗)2 vol. 58, no. 1, pp. 267–288, 1996. [Online]. Available: http: F F 1 //www.jstor.org/stable/2346178 Combining the previous two inequalities, we have [2] J. Friedman, T. Hastie, and R. Tibshirani, “Sparse inverse covariance √ (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) estimation with the graphical Lasso,” Biostatistics, vol. 9, no. 3, pp. 432– α( Φ 2F + Ψ 2F) αµ(γ Φ 1+ Ψ ∗)2 6λ rL(τ Φ F + Ψ F) 41, Jul. 2008. [Online]. Available: http://www.pubmedcentral.nih.gov/ ⇒α((cid:107)Φ(cid:107)2 +√(cid:107)Ψ(cid:107)2 )−16αµr (τ(cid:107)Φ(cid:107) +(cid:107)Ψ(cid:107) )2 articlerender.fcgi?artid=3019769&tool=pmcentrez&rendertype=abstract ≤F F (cid:107) (cid:107) (cid:107)L (cid:107) F F [3] A. Bolstad, B. Van Veen, and R. Nowak, “Causal network inference via 6λ r (τ Φ + Ψ ) group sparse regularization,” IEEE Transactions on Signal Processing, L F F ⇒ (cid:62) ≤ vol. 59, no. 6, pp. 2628–2641, Jun. 2011. ζ Qζ 0 [4] S. Basu and G. Michailidis, “Regularized estimation in sparse where high-dimensional time series models,” The Annals of Statistics,  (cid:16) (cid:17) (cid:16) (cid:17)  vol. 43, no. 4, pp. 1535–1567, Aug. 2015. [Online]. Available: (cid:107) (cid:107) (cid:107) (cid:107) (cid:62) http://projecteuclid.org/euclid.aos/1434546214 ζ = (Φ F (cid:16) Ψ F (cid:17)1) (cid:16) (cid:17)  [5] C. W. J. Granger, “Investigating causal relations by econometric models  α 2+τ2 −α τ −3λ(cid:48)τ  and cross-spectral methods,” Econometrica, vol. 37, no. 3, pp. 424–438, 2(1+τ2) 2(1+τ2) Aug. 1969. [Online]. Available: http://www.jstor.org/stable/1912791 − − (cid:48) Q = α τ α 1+2τ2 3λ , [6] A. Sandryhaila and J. M. F. Moura, “Discrete signal processing on −2(1+(cid:48) τ2) −2(1+τ(cid:48)2) graphs,” IEEE Transactions on Signal Processing, vol. 61, no. 7, pp. 3λ τ 3λ √ 0 1644–1656, Apr. 2013. (cid:48) [7] ——, “Discrete signal processing on graphs: Frequency analysis,” IEEE which is a conic in standard form, λ = λ rL, and the entries Transactions on Signal Processing, vol. 62, no. 12, pp. 3042–3054, Jun. of Q follow from(cid:0) taking(cid:1) µ given in Assumption 3. 2014. Checking its discriminant, [8] J. Mei and J. M. F. Moura, “Signal processing on graphs: causal mod- eling of unstructured data,” IEEE Transactions on Signal Processing, − 2 + τ2 (1 + 2τ2) τ2 > 0 vol. 65, no. 8, pp. 2077–2092, Apr. 2017. [9] V. Chandrasekaran, P. A. Parrilo, and A. S. Willsky, “Latent variable so we have that the conic equation describes an ellipse, and graphical model selection via convex optimization,” The Annals of (cid:107) (cid:107) (cid:107) (cid:107) Statistics, vol. 40, no. 4, pp. 1935–1967, Aug. 2012. [Online]. there are bounds on the values of Φ and Ψ . F F Available: http://projecteuclid.org/euclid.aos/1351602527 For these individual bounds, we consider the points at which [10] A. Jalali and S. Sanghavi, “Learning the dependence graph of time (cid:62) (cid:107) (cid:107) (cid:107) (cid:107) the gradients of ζ Qζ vanish w.r.t. each of Φ and Ψ . series with latent factors,” arXiv:1106.1887 [cs], Jun. 2011, arXiv: (cid:107) (cid:107) F F 1106.1887. [Online]. Available: http://arxiv.org/abs/1106.1887 For Φ(cid:18) F , (cid:19) (cid:18) (cid:19) [11] M. T. Bahadori, Y. Liu, and E. P. Xing, “Fast structure learning in (cid:62) generalized stochastic processes with latent factors,” in Proceedings ∂(cid:107) (cid:107) ζ Qζ = 0 Φ F of the 19th ACM SIGKDD International Conference on Knowledge ⇒ 2 + τ2 (cid:107) (cid:107)∗ (cid:48) τ (cid:107) (cid:107)∗ Discovery and Data Mining, ser. KDD ’13. New York, NY, USA: α Φ = 3λ τ + α Ψ . ACM, 2013, pp. 284–292. [Online]. Available: http://doi.acm.org/10. 2(1 + τ2) F 2(1 + τ2) F 1145/2487575.2487578 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: IEEE

text: TRANSACTIONS

text: ON

text: SIGNAL

text: PROCESSING

text: B.

text: Theorem

text: Plugging

text: this

text: into

text: the

text: equation

text: deﬁned

text: by

text: ζ

text: Qζ

text: yields

text: Now

text: we

text: restate

text: the

text: theorem

text: again

text: for

text: convenience.

text: 3λ

text: τ

text: Φ

text: Theorem

text: Under

text: solution

text: to

text: the

text: F

text: α

text: τ2

text: problem

text: satisﬁes

text: Since

text: we

text: are

text: seeking

text: an

text: upper

text: bound

text: for

text: Φ

text: it

text: can

text: be

text: seen

text: F

text: 3λγ

text: 9λγ

text: sA

text: that

text: we

text: take

text: the

text: positive

text: A

text: A

text: F

text: α

text: α

text: 3λ

text: τ

text: 9λ

text: τ

text: Φ

text: F

text: α

text: τ2

text: α

text: Similarly,

text: for

text: Ψ

text: Proof.

text: Starting

text: with

text: Proposition

text: since

text: our

text: solution

text: is

text: in

text: F

text: τ2

text: τ

text: α

text: Ψ

text: 3λ

text: τ

text: α

text: Φ

text: F

text: F

text: m(A+L)

text: m(A+L)+tr

text: ΓX

text: A+L

text: (A+L)

text: K

text: Finally,

text: plugging

text: this

text: solving

text: for

text: the

text: α

text: Φ

text: Ψ

text: upper

text: bound

text: yields

text: F

text: Revisiting

text: the

text: objective

text: functional

text: at

text: optimality

text: and

text: skip-

text: 3λ

text: 9λ

text: ping

text: repetitive

text: algebra

text: (see

text: proof

text: for

text: Proposition

text: Ψ

text: F

text: α

text: α

text: λ

text: α

text: Φ+Ψ

text: Φ

text: Ψ

text: (γ

text: Φ

text: Ψ

text: F

text: S

text: R

text: Sc

text: Rc

text: 3λ

text: (γ

text: Φ

text: Ψ

text: 6λ

text: rL(τ

text: Φ

text: F

text: Ψ

text: F

text: REFERENCES

text: Now

text: from

text: Proposition

text: we

text: have

text: R.

text: Tibshirani,

text: “Regression

text: Shrinkage

text: and

text: Selection

text: via

text: the

text: Lasso,”

text: Φ+Ψ

text: −2

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society.

text: Series

text: B

text: (Methodological),

text: Φ

text: Ψ

text: µ(γ

text: Φ

text: Ψ

text: vol.

text: no.

text: pp.

text: [Online].

text: Available:

text: http:

text: F

text: F

text: Combining

text: the

text: previous

text: two

text: inequalities,

text: we

text: have

text: J.

text: Friedman,

text: T.

text: Hastie,

text: and

text: R.

text: Tibshirani,

text: “Sparse

text: inverse

text: covariance

text: estimation

text: with

text: the

text: graphical

text: Lasso,”

text: Biostatistics,

text: vol.

text: no.

text: pp.

text: α(

text: Φ

text: Ψ

text: αµ(γ

text: Φ

text: Ψ

text: 6λ

text: rL(τ

text: Φ

text: F

text: Ψ

text: F)

text: Jul.

text: [Online].

text: Available:

text: ≤F

text: F

text: F

text: F

text: A.

text: Bolstad,

text: B.

text: Van

text: Veen,

text: and

text: R.

text: Nowak,

text: “Causal

text: network

text: inference

text: via

text: 6λ

text: r

text: (τ

text: Φ

text: Ψ

text: group

text: sparse

text: regularization,”

text: IEEE

text: Transactions

text: on

text: Signal

text: Processing,

text: L

text: F

text: F

text: vol.

text: no.

text: pp.

text: Jun.

text: ζ

text: Qζ

text: S.

text: Basu

text: and

text: G.

text: Michailidis,

text: “Regularized

text: estimation

text: in

text: sparse

text: where

text: high-dimensional

text: time

text: series

text: models,”

text: The

text: Annals

text: of

text: Statistics,

text: vol.

text: no.

text: pp.

text: Aug.

text: [Online].

text: Available:

text: ζ

text: (Φ

text: F

text: Ψ

text: F

text: C.

text: W.

text: J.

text: Granger,

text: “Investigating

text: causal

text: relations

text: by

text: econometric

text: models

text: α

text: −α

text: τ

text: and

text: cross-spectral

text: methods,”

text: Econometrica,

text: vol.

text: no.

text: pp.

text: Aug.

text: [Online].

text: Available:

text: Q

text: α

text: τ

text: α

text: 3λ

text: A.

text: Sandryhaila

text: and

text: J.

text: M.

text: F.

text: Moura,

text: “Discrete

text: signal

text: processing

text: on

text: graphs,”

text: IEEE

text: Transactions

text: on

text: Signal

text: Processing,

text: vol.

text: no.

text: pp.

text: 3λ

text: τ

text: 3λ

text: Apr.

text: “Discrete

text: signal

text: processing

text: on

text: graphs:

text: Frequency

text: analysis,”

text: IEEE

text: which

text: is

text: a

text: conic

text: in

text: standard

text: form,

text: λ

text: λ

text: rL,

text: and

text: the

text: entries

text: Transactions

text: on

text: Signal

text: Processing,

text: vol.

text: no.

text: pp.

text: Jun.

text: of

text: Q

text: follow

text: µ

text: given

text: in

text: Assumption

text: Checking

text: its

text: discriminant,

text: J.

text: Mei

text: and

text: J.

text: M.

text: F.

text: Moura,

text: “Signal

text: processing

text: on

text: graphs:

text: causal

text: mod-

text: eling

text: of

text: unstructured

text: data,”

text: IEEE

text: Transactions

text: on

text: Signal

text: Processing,

text: τ2

text: τ2

text: vol.

text: no.

text: pp.

text: Apr.

text: V.

text: Chandrasekaran,

text: P.

text: A.

text: Parrilo,

text: and

text: A.

text: S.

text: Willsky,

text: “Latent

text: variable

text: so

text: we

text: have

text: that

text: the

text: conic

text: equation

text: describes

text: an

text: ellipse,

text: and

text: graphical

text: model

text: selection

text: via

text: convex

text: optimization,”

text: The

text: Annals

text: of

text: Statistics,

text: vol.

text: no.

text: pp.

text: Aug.

text: [Online].

text: there

text: are

text: bounds

text: on

text: the

text: values

text: of

text: Φ

text: and

text: Ψ

text: F

text: F

text: Available:

text: For

text: these

text: individual

text: bounds,

text: we

text: consider

text: the

text: points

text: at

text: which

text: A.

text: Jalali

text: and

text: S.

text: Sanghavi,

text: “Learning

text: the

text: dependence

text: graph

text: of

text: time

text: the

text: gradients

text: of

text: ζ

text: Qζ

text: vanish

text: w.r.t.

text: each

text: of

text: Φ

text: and

text: Ψ

text: series

text: with

text: latent

text: factors,”

text: [cs],

text: Jun.

text: arXiv:

text: F

text: F

text: [Online].

text: Available:

text: For

text: F

text: M.

text: T.

text: Bahadori,

text: Y.

text: Liu,

text: and

text: E.

text: P.

text: Xing,

text: “Fast

text: structure

text: learning

text: in

text: generalized

text: stochastic

text: processes

text: with

text: latent

text: factors,”

text: in

text: Proceedings

text: ζ

text: Qζ

text: Φ

text: F

text: of

text: the

text: ACM

text: SIGKDD

text: International

text: Conference

text: on

text: Knowledge

text: τ2

text: τ

text: Discovery

text: and

text: Data

text: Mining,

text: ser.

text: KDD

text: New

text: York,

text: NY,

text: USA:

text: α

text: Φ

text: 3λ

text: τ

text: α

text: Ψ

text: ACM,

text: pp.

text: [Online].

text: Available:

text: F

text: F

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##