Measuring predictability with singular vectors

Michael K. Tippett^{1} Timothy DelSole^{2}

1International Research Institute for Climate and Society The Earth Institute, Columbia University

2George Mason University

Center for Ocean-Land-Atmosphere Studies (COLA)

Main ideas

4/17/07 / Applied Mathematics Colloquium Applications of the SVD to: Diagnosing linear dynamics in weather and climate.

SVD used to ▶ find the errors that grow the most (weather) ▶ find the signals that grow the most (ENSO)

How can the singular value decomposition be used to: ▶ Measure predictability ▶ Identify “predictable components”.

Applications of the SVD to: Diagnosing linear dynamics in weather and climate.

SVD used to ▶ find the errors that grow the most (weather) ▶ find the signals that grow the most (ENSO)

How can the singular value decomposition be used to: ▶ Measure predictability ▶ Identify “predictable components”.

Applications of the SVD to: Diagnosing linear dynamics in weather and climate.

SVD used to ▶ find the errors that grow the most (weather) ▶ find the signals that grow the most (ENSO)

How can the singular value decomposition be used to: ▶ Measure predictability ▶ Identify “predictable components”.

Applications of the SVD to: Diagnosing linear dynamics in weather and climate.

SVD used to ▶ find the errors that grow the most (weather) ▶ find the signals that grow the most (ENSO)

How can the singular value decomposition be used to: ▶ Measure predictability ▶ Identify “predictable components”.

Outline

▶ The SVD. ▶ Applications to error and signal growth. ▶ Why are norms important? ▶ How to measure predictability. ▶ How to use the SVD to measure predictability. ▶ Sea surface temperature example.

The singular value decomposition

A= UΣV^{T} m × n

= [u_{1}, …, u_{m}]diag(σ_{1}, σ_{2}, …, σ_{p})[v_{1}, …, v_{n}]^{T}

= ∑_{i=1}^{p}σ_{i}u_{i}v_{i}^{T}

U, V are orthogonal. Σ is diagonal. σ_{1} ≥ σ_{2} ≥ … σ_{p} ≥ 0. Properties

▶ A maps a unit sphere to an ellipsoid. Av_{i} = σ_{i}u_{i}.

▶ ∥A∥ = max_{x} \frac{∥Ax∥}{∥x∥} = σ_{1}

A= UΣV^{T} m × n

= [u_{1}, …, u_{m}]diag(σ_{1}, σ_{2}, …, σ_{p})[v_{1}, …, v_{n}]^{T}

= ∑_{i=1}^{p}σ_{i}u_{i}v_{i}^{T}

U, V are orthogonal. Σ is diagonal. σ_{1} ≥ σ_{2} ≥ … σ_{p} ≥ 0. Properties

▶ A maps a unit sphere to an ellipsoid. Av_{i} = σ_{i}u_{i}.

▶ ∥A∥ = max_{x} \frac{∥Ax∥}{∥x∥} = σ_{1}

A= UΣV^{T} m × n

= [u_{1}, …, u_{m}]diag(σ_{1}, σ_{2}, …, σ_{p})[v_{1}, …, v_{n}]^{T}

= ∑_{i=1}^{p}σ_{i}u_{i}v_{i}^{T}

U, V are orthogonal. Σ is diagonal. σ_{1} ≥ σ_{2} ≥ … σ_{p} ≥ 0. Properties

▶ A maps a unit sphere to an ellipsoid. Av_{i} = σ_{i}u_{i}.

▶ ∥A∥ = max_{x} \frac{∥Ax∥}{∥x∥} = σ_{1}

A= UΣV^{T} m × n

= [u_{1}, …, u_{m}]diag(σ_{1}, σ_{2}, …, σ_{p})[v_{1}, …, v_{n}]^{T}

= ∑_{i=1}^{p}σ_{i}u_{i}v_{i}^{T}

U, V are orthogonal. Σ is diagonal. σ_{1} ≥ σ_{2} ≥ … σ_{p} ≥ 0. Properties

▶ A maps a unit sphere to an ellipsoid. Av_{i} = σ_{i}u_{i}.

▶ ∥A∥ = max_{x} \frac{∥Ax∥}{∥x∥} = σ_{1}

Example

A = UΣV^{T}

A = [\begin{matrix}\frac{3}{4} & 2.3 \\ 0 & \frac{3}{4} \end{matrix}] , Σ = [\begin{matrix}2.52 & 0 \\ 0 & 0.223 \end{matrix} ]

Application: Growth of small errors

(Lorenz, Farrell, Palmer)

σ_{1} ≥ |λ_{1}(A)|. Minimum growth when A is normal Linear dynamics

f = A(a + δ)

f = forecast.

a = initial condition (analysis). δ = perturbation (analysis error).

Which errors are amplified the most?

max_{δ} \frac{∥Aδ∥}{∥δ∥} = σ_{1} , Av_{1} = σ_{1}u_{1}

Linear dynamics

f = A(a + δ)

f = forecast.

a = initial condition (analysis). δ = perturbation (analysis error).

Which errors are amplified the most?

max_{δ} \frac{∥Aδ∥}{∥δ∥} = σ_{1} , Av_{1} = σ_{1}u_{1}

Linear dynamics

f = A(a + δ)

f = forecast.

a = initial condition (analysis). δ = perturbation (analysis error).

Which errors are amplified the most?

max_{δ} \frac{∥Aδ∥}{∥δ∥} = σ_{1} , Av_{1} = σ_{1}u_{1}

Linear dynamics

f = A(a + δ)

f = forecast.

a = initial condition (analysis). δ = perturbation (analysis error).

Which errors are amplified the most?

max_{δ} \frac{∥Aδ∥}{∥δ∥} = σ_{1} , Av_{1} = σ_{1}u_{1}

Examples: Perturbation growth

Weather forecasting model (ψ_{200} 3-day 20x) Buizza and Palmer,

1995 Lanczos method, adjoint ENSO model Xue et al., 1997 finite differencing

Application: Signal growth

(Penland and Sardeshmukh)

Linear dynamics + stochastic forcing.

f = Ai + ϵ

f = forecast i = initial condition (no error) ϵ = stochastic forcing (model error)

Which initial conditions grow “above the noise”?

max_{i} \frac{∥Ai∥}{∥i∥} = σ_{1} , Av_{1} = σ_{1}u_{1} Linear dynamics + stochastic forcing.

f = Ai + ϵ

f = forecast i = initial condition (no error) ϵ = stochastic forcing (model error)

Which initial conditions grow “above the noise”?

max_{i} \frac{∥Ai∥}{∥i∥} = σ_{1} , Av_{1} = σ_{1}u_{1} Linear dynamics + stochastic forcing.

f = Ai + ϵ

f = forecast i = initial condition (no error) ϵ = stochastic forcing (model error)

Which initial conditions grow “above the noise”?

max_{i} \frac{∥Ai∥}{∥i∥} = σ_{1} , Av_{1} = σ_{1}u_{1} Linear dynamics + stochastic forcing.

f = Ai + ϵ

f = forecast i = initial condition (no error) ϵ = stochastic forcing (model error)

Which initial conditions grow “above the noise”?

max_{i} \frac{∥Ai∥}{∥i∥} = σ_{1} , Av_{1} = σ_{1}u_{1}

Signal growth

ENSO model

initial final

Penland and Sardeshmukh, 1995

Observations projected onto the singular vectors

Paradox?

final initial What are the consequences for predictability?

▶ SVD finds maximum error growth.

Large singular value = less predictability. ▶ SVD finds maximums signal growth.

Large singular value = more predictability?

Solution: “different” SVDs for different problems.

▶ Is the “error” SVD consistent with the “signal” SVD?

▶ Same components, opposite ordering.

▶ What is the SVD for measuring predictability?

▶ How is it related to other predictability analysis methods?

What are the consequences for predictability?

▶ SVD finds maximum error growth.

Large singular value = less predictability. ▶ SVD finds maximums signal growth.

Large singular value = more predictability?

Solution: “different” SVDs for different problems.

▶ Is the “error” SVD consistent with the “signal” SVD?

▶ Same components, opposite ordering.

▶ What is the SVD for measuring predictability?

▶ How is it related to other predictability analysis methods?

What are the consequences for predictability?

▶ SVD finds maximum error growth.

Large singular value = less predictability. ▶ SVD finds maximums signal growth.

Large singular value = more predictability?

Solution: “different” SVDs for different problems.

▶ Is the “error” SVD consistent with the “signal” SVD?

▶ Same components, opposite ordering.

▶ What is the SVD for measuring predictability?

▶ How is it related to other predictability analysis methods?

What are the consequences for predictability?

▶ SVD finds maximum error growth.

Large singular value = less predictability. ▶ SVD finds maximums signal growth.

Large singular value = more predictability?

Solution: “different” SVDs for different problems.

▶ Is the “error” SVD consistent with the “signal” SVD?

▶ Same components, opposite ordering.

▶ What is the SVD for measuring predictability?

▶ How is it related to other predictability analysis methods?

What are the consequences for predictability?

▶ SVD finds maximum error growth.

Large singular value = less predictability. ▶ SVD finds maximums signal growth.

Large singular value = more predictability?

Solution: “different” SVDs for different problems.

▶ Is the “error” SVD consistent with the “signal” SVD?

▶ Same components, opposite ordering.

▶ What is the SVD for measuring predictability?

▶ How is it related to other predictability analysis methods?

What are the consequences for predictability?

▶ SVD finds maximum error growth.

Large singular value = less predictability. ▶ SVD finds maximums signal growth.

Large singular value = more predictability?

Solution: “different” SVDs for different problems.

▶ Is the “error” SVD consistent with the “signal” SVD?

▶ Same components, opposite ordering.

▶ What is the SVD for measuring predictability?

▶ How is it related to other predictability analysis methods?

What are the consequences for predictability?

▶ SVD finds maximum error growth.

Large singular value = less predictability. ▶ SVD finds maximums signal growth.

Large singular value = more predictability?

Solution: “different” SVDs for different problems.

▶ Is the “error” SVD consistent with the “signal” SVD?

▶ Same components, opposite ordering.

▶ What is the SVD for measuring predictability?

▶ How is it related to other predictability analysis methods?

SVD and norms

Usual SVD maximizes

\frac{∥Ax∥^{2}}{∥x∥^{2}}

Cannot apply directly to multivariate problems.

Generalized SVD maximizes

\frac{x^{T}M^{T}Mx}{x^{T}L^{T}Lx} ≡ \frac{∥Ax∥_{M}^{2}}{∥x∥_{L}^{2}}

SVD(MAL^{−1})= generalized SVD (A)
max_{x} \frac{∥MAL^{−1}x∥^{2}}{∥x∥^{2}} = max_{x} \frac{x^{T}L^{−T}A^{T}M^{T}MAL^{−1}x}{x^{T}x}

= max_{y} \frac{y^{T}A^{T}M^{T}MAy}{y^{T}L^{T}Ly} (y = L^{−1}x)

SVD and norms

Usual SVD maximizes

\frac{∥Ax∥^{2}}{∥x∥^{2}}

Cannot apply directly to multivariate problems.

Generalized SVD maximizes

\frac{x^{T}M^{T}Mx}{x^{T}L^{T}Lx} ≡ \frac{∥Ax∥_{M}^{2}}{∥x∥_{L}^{2}}

SVD(MAL^{−1})= generalized SVD (A)
max_{x} \frac{∥MAL^{−1}x∥^{2}}{∥x∥^{2}} = max_{x} \frac{x^{T}L^{−T}A^{T}M^{T}MAL^{−1}x}{x^{T}x}

= max_{y} \frac{y^{T}A^{T}M^{T}MAy}{y^{T}L^{T}Ly} (y = L^{−1}x)

SVD and norms

Usual SVD maximizes

\frac{∥Ax∥^{2}}{∥x∥^{2}}

Cannot apply directly to multivariate problems.

Generalized SVD maximizes

\frac{x^{T}M^{T}Mx}{x^{T}L^{T}Lx} ≡ \frac{∥Ax∥_{M}^{2}}{∥x∥_{L}^{2}}

SVD(MAL^{−1})= generalized SVD (A)
max_{x} \frac{∥MAL^{−1}x∥^{2}}{∥x∥^{2}} = max_{x} \frac{x^{T}L^{−T}A^{T}M^{T}MAL^{−1}x}{x^{T}x}

= max_{y} \frac{y^{T}A^{T}M^{T}MAy}{y^{T}L^{T}Ly} (y = L^{−1}x)

SVD and norms

Usual SVD maximizes

\frac{∥Ax∥^{2}}{∥x∥^{2}}

Cannot apply directly to multivariate problems.

Generalized SVD maximizes

\frac{x^{T}M^{T}Mx}{x^{T}L^{T}Lx} ≡ \frac{∥Ax∥_{M}^{2}}{∥x∥_{L}^{2}}

SVD(MAL^{−1})= generalized SVD (A)
max_{x} \frac{∥MAL^{−1}x∥^{2}}{∥x∥^{2}} = max_{x} \frac{x^{T}L^{−T}A^{T}M^{T}MAL^{−1}x}{x^{T}x}

= max_{y} \frac{y^{T}A^{T}M^{T}MAy}{y^{T}L^{T}Ly} (y = L^{−1}x)

Which norms?

Initial norm

f = A(a + δ) , max_{δ} \frac{∥Aδ∥_{M}}{∥δ∥_{L}}

Lorenz (1965) said choose L so that Lδ is “random in that no direction is … preferred over any other direction.” (Sphere) Translation: choose L so that Lδ is “white” in space. Pick L so that

Cov(Lδ) = LCov(δ)L^{T} = LC_{δ}L^{T} = I

L = C_{δ}^{−1/2}. (∼ standardizing)

σ_{1}^{2} = max_{δ} \frac{∥Aδ∥^{2}}{∥δ∥_{L}^{2}} = max_{δ} \frac{∥Aδ∥^{2}}{δ^{T}C_{δ}^{−1}δ} = SVD(AC_{δ}^{1/2})

f = A(a + δ) , max_{δ} \frac{∥Aδ∥_{M}}{∥δ∥_{L}}

Lorenz (1965) said choose L so that Lδ is “random in that no direction is … preferred over any other direction.” (Sphere) Translation: choose L so that Lδ is “white” in space. Pick L so that

Cov(Lδ) = LCov(δ)L^{T} = LC_{δ}L^{T} = I

L = C_{δ}^{−1/2}. (∼ standardizing)

σ_{1}^{2} = max_{δ} \frac{∥Aδ∥^{2}}{∥δ∥_{L}^{2}} = max_{δ} \frac{∥Aδ∥^{2}}{δ^{T}C_{δ}^{−1}δ} = SVD(AC_{δ}^{1/2})

f = A(a + δ) , max_{δ} \frac{∥Aδ∥_{M}}{∥δ∥_{L}}

Lorenz (1965) said choose L so that Lδ is “random in that no direction is … preferred over any other direction.” (Sphere) Translation: choose L so that Lδ is “white” in space. Pick L so that

Cov(Lδ) = LCov(δ)L^{T} = LC_{δ}L^{T} = I

L = C_{δ}^{−1/2}. (∼ standardizing)

σ_{1}^{2} = max_{δ} \frac{∥Aδ∥^{2}}{∥δ∥_{L}^{2}} = max_{δ} \frac{∥Aδ∥^{2}}{δ^{T}C_{δ}^{−1}δ} = SVD(AC_{δ}^{1/2})

f = A(a + δ) , max_{δ} \frac{∥Aδ∥_{M}}{∥δ∥_{L}}

Lorenz (1965) said choose L so that Lδ is “random in that no direction is … preferred over any other direction.” (Sphere) Translation: choose L so that Lδ is “white” in space. Pick L so that

Cov(Lδ) = LCov(δ)L^{T} = LC_{δ}L^{T} = I

L = C_{δ}^{−1/2}. (∼ standardizing)

σ_{1}^{2} = max_{δ} \frac{∥Aδ∥^{2}}{∥δ∥_{L}^{2}} = max_{δ} \frac{∥Aδ∥^{2}}{δ^{T}C_{δ}^{−1}δ} = SVD(AC_{δ}^{1/2})

f = A(a + δ) , max_{δ} \frac{∥Aδ∥_{M}}{∥δ∥_{L}}

Lorenz (1965) said choose L so that Lδ is “random in that no direction is … preferred over any other direction.” (Sphere) Translation: choose L so that Lδ is “white” in space. Pick L so that

Cov(Lδ) = LCov(δ)L^{T} = LC_{δ}L^{T} = I

L = C_{δ}^{−1/2}. (∼ standardizing)

σ_{1}^{2} = max_{δ} \frac{∥Aδ∥^{2}}{∥δ∥_{L}^{2}} = max_{δ} \frac{∥Aδ∥^{2}}{δ^{T}C_{δ}^{−1}δ} = SVD(AC_{δ}^{1/2})

f = A(a + δ) , max_{δ} \frac{∥Aδ∥_{M}}{∥δ∥_{L}}

Lorenz (1965) said choose L so that Lδ is “random in that no direction is … preferred over any other direction.” (Sphere) Translation: choose L so that Lδ is “white” in space. Pick L so that

Cov(Lδ) = LCov(δ)L^{T} = LC_{δ}L^{T} = I

L = C_{δ}^{−1/2}. (∼ standardizing)

σ_{1}^{2} = max_{δ} \frac{∥Aδ∥^{2}}{∥δ∥_{L}^{2}} = max_{δ} \frac{∥Aδ∥^{2}}{δ^{T}C_{δ}^{−1}δ} = SVD(AC_{δ}^{1/2})

Forecast errors

f = A(a + δ)

The forecast covariance is

C_{f} = Cov(f ) = Cov(Aδ) = ACov(δ)A^{T} = AC_{δ}A^{T}
= (AC_{δ}^{1/2})(AC_{δ}^{1/2})^{T} .

Eigenvalue decomposition of C_{f} = SVD(AC_{δ}^{1/2}) Optimal.

(Houtekamer, Palmer, Tribbia, Ehrendorfer)

Eigenvalues of C_{f} = σ_{i}^{2}. Eigenvectors of C_{f} = u_{i} Final norm? Energy? Errors in a particular region? Are there norms so that the SVD measures predictability?

f = A(a + δ)

The forecast covariance is

C_{f} = Cov(f ) = Cov(Aδ) = ACov(δ)A^{T} = AC_{δ}A^{T}
= (AC_{δ}^{1/2})(AC_{δ}^{1/2})^{T} .

Eigenvalue decomposition of C_{f} = SVD(AC_{δ}^{1/2}) Optimal.

(Houtekamer, Palmer, Tribbia, Ehrendorfer)

Eigenvalues of C_{f} = σ_{i}^{2}. Eigenvectors of C_{f} = u_{i} Final norm? Energy? Errors in a particular region? Are there norms so that the SVD measures predictability?

f = A(a + δ)

The forecast covariance is

C_{f} = Cov(f ) = Cov(Aδ) = ACov(δ)A^{T} = AC_{δ}A^{T}
= (AC_{δ}^{1/2})(AC_{δ}^{1/2})^{T} .

Eigenvalue decomposition of C_{f} = SVD(AC_{δ}^{1/2}) Optimal.

(Houtekamer, Palmer, Tribbia, Ehrendorfer)

Eigenvalues of C_{f} = σ_{i}^{2}. Eigenvectors of C_{f} = u_{i} Final norm? Energy? Errors in a particular region? Are there norms so that the SVD measures predictability?

f = A(a + δ)

The forecast covariance is

C_{f} = Cov(f ) = Cov(Aδ) = ACov(δ)A^{T} = AC_{δ}A^{T}
= (AC_{δ}^{1/2})(AC_{δ}^{1/2})^{T} .

Eigenvalue decomposition of C_{f} = SVD(AC_{δ}^{1/2}) Optimal.

(Houtekamer, Palmer, Tribbia, Ehrendorfer)

Eigenvalues of C_{f} = σ_{i}^{2}. Eigenvectors of C_{f} = u_{i} Final norm? Energy? Errors in a particular region? Are there norms so that the SVD measures predictability?

f = A(a + δ)

The forecast covariance is

C_{f} = Cov(f ) = Cov(Aδ) = ACov(δ)A^{T} = AC_{δ}A^{T}
= (AC_{δ}^{1/2})(AC_{δ}^{1/2})^{T} .

Eigenvalue decomposition of C_{f} = SVD(AC_{δ}^{1/2}) Optimal.

(Houtekamer, Palmer, Tribbia, Ehrendorfer)

Eigenvalues of C_{f} = σ_{i}^{2}. Eigenvectors of C_{f} = u_{i} Final norm? Energy? Errors in a particular region? Are there norms so that the SVD measures predictability?

How to measure predictability

Predictability is …

▶ Relative to a baseline

▶ usually climatology (historical frequencies).

▶ Difference between forecast and climatology.

▶ Univariate predictability ∼ \frac{forecast uncertainty}{climatological uncertainty}.

▶ relative error not absolute error

▶ Should be independent of units or variable weighting. ▶ Pred(A,B) = Pred(A)+Pred(B), if A and B are independent. ▶ Pred(A,B) = Pred(B) if A=f(B).

Predictability is …

▶ Relative to a baseline

▶ usually climatology (historical frequencies).

▶ Difference between forecast and climatology.

▶ Univariate predictability ∼ \frac{forecast uncertainty}{climatological uncertainty}.

▶ relative error not absolute error

▶ Should be independent of units or variable weighting. ▶ Pred(A,B) = Pred(A)+Pred(B), if A and B are independent. ▶ Pred(A,B) = Pred(B) if A=f(B).

Predictability is …

▶ Relative to a baseline

▶ usually climatology (historical frequencies).

▶ Difference between forecast and climatology.

▶ Univariate predictability ∼ \frac{forecast uncertainty}{climatological uncertainty}.

▶ relative error not absolute error

▶ Should be independent of units or variable weighting. ▶ Pred(A,B) = Pred(A)+Pred(B), if A and B are independent. ▶ Pred(A,B) = Pred(B) if A=f(B).

Predictability is …

▶ Relative to a baseline

▶ usually climatology (historical frequencies).

▶ Difference between forecast and climatology.

▶ Univariate predictability ∼ \frac{forecast uncertainty}{climatological uncertainty}.

▶ relative error not absolute error

▶ Should be independent of units or variable weighting. ▶ Pred(A,B) = Pred(A)+Pred(B), if A and B are independent. ▶ Pred(A,B) = Pred(B) if A=f(B).

Predictability is …

▶ Relative to a baseline

▶ usually climatology (historical frequencies).

▶ Difference between forecast and climatology.

▶ Univariate predictability ∼ \frac{forecast uncertainty}{climatological uncertainty}.

▶ relative error not absolute error

▶ Should be independent of units or variable weighting. ▶ Pred(A,B) = Pred(A)+Pred(B), if A and B are independent. ▶ Pred(A,B) = Pred(B) if A=f(B).

Predictability is …

▶ Relative to a baseline

▶ usually climatology (historical frequencies).

▶ Difference between forecast and climatology.

▶ Univariate predictability ∼ \frac{forecast uncertainty}{climatological uncertainty}.

▶ relative error not absolute error

▶ Should be independent of units or variable weighting. ▶ Pred(A,B) = Pred(A)+Pred(B), if A and B are independent. ▶ Pred(A,B) = Pred(B) if A=f(B).

Predictability is …

▶ Relative to a baseline

▶ usually climatology (historical frequencies).

▶ Difference between forecast and climatology.

▶ Univariate predictability ∼ \frac{forecast uncertainty}{climatological uncertainty}.

▶ relative error not absolute error

▶ Should be independent of units or variable weighting. ▶ Pred(A,B) = Pred(A)+Pred(B), if A and B are independent. ▶ Pred(A,B) = Pred(B) if A=f(B).

Measures of predictability

Information theory gives some candidates: ▶ Mutual information (Leung and North, 1990) ▶ Predictive information (Schneider and Griffies, 1999) ▶ Relative entropy (Kleeman, 2002)

On average (over forecasts), they are the same.

Normally distributed variables

Mutual information M

M = − log \frac{detC_{f}}{detC_{c}}

C_{f} = Cov(f ) C_{c} = Cov(climatology) Univariate case:

M = log \frac{σ_{f}^{2}}{σ_{c}^{2}} = log \frac{forecast variance}{climatological variance} Mutual information M

M = − log \frac{detC_{f}}{detC_{c}}

C_{f} = Cov(f ) C_{c} = Cov(climatology) Univariate case:

M = log \frac{σ_{f}^{2}}{σ_{c}^{2}} = log \frac{forecast variance}{climatological variance} Mutual information M

M = − log \frac{detC_{f}}{detC_{c}}

C_{f} = Cov(f ) C_{c} = Cov(climatology) Univariate case:

Normalized forecast covariance

Multivariate generalization of \frac{forecast variance}{climatological variance} G = C_{c}^{−1/2}C_{f}C_{c}^{−1/2}

M = log \frac{σ_{f}^{2}}{σ_{c}^{2}} = log \frac{forecast variance}{climatological variance} detG = det(C_{c}^{−1/2}C_{f}C_{c}^{−1/2}) = det(C_{f}C_{c}^{−1}) = \frac{detC_{f}}{detC_{c}}

M = − log \frac{detC_{f}}{detC_{c}} = − log detG

Predictability is measured by the eigenvalues of G. Eigenvectors of G decompose space according to predictability.

maximize  eigenvectors
variance principal components (EOFs) C_{c}
predictability predictable components  G

detG = det(C_{c}^{−1/2}C_{f}C_{c}^{−1/2}) = det(C_{f}C_{c}^{−1}) = \frac{detC_{f}}{detC_{c}}

M = − log \frac{detC_{f}}{detC_{c}} = − log detG

Predictability is measured by the eigenvalues of G. Eigenvectors of G decompose space according to predictability.

maximize  eigenvectors
variance principal components (EOFs) C_{c}
predictability predictable components  G

detG = det(C_{c}^{−1/2}C_{f}C_{c}^{−1/2}) = det(C_{f}C_{c}^{−1}) = \frac{detC_{f}}{detC_{c}}

M = − log \frac{detC_{f}}{detC_{c}} = − log detG

Predictability is measured by the eigenvalues of G. Eigenvectors of G decompose space according to predictability.

maximize  eigenvectors
variance principal components (EOFs) C_{c}
predictability predictable components  G

detG = det(C_{c}^{−1/2}C_{f}C_{c}^{−1/2}) = det(C_{f}C_{c}^{−1}) = \frac{detC_{f}}{detC_{c}}

M = − log \frac{detC_{f}}{detC_{c}} = − log detG

Predictability is measured by the eigenvalues of G. Eigenvectors of G decompose space according to predictability.

maximize  eigenvectors
variance principal components (EOFs) C_{c}
predictability predictable components  G

detG = det(C_{c}^{−1/2}C_{f}C_{c}^{−1/2}) = det(C_{f}C_{c}^{−1}) = \frac{detC_{f}}{detC_{c}}

M = − log \frac{detC_{f}}{detC_{c}} = − log detG

Predictability is measured by the eigenvalues of G. Eigenvectors of G decompose space according to predictability.

maximize  eigenvectors
variance principal components (EOFs) C_{c}
predictability predictable components  G

Linear prediction model

f = Ai + ϵ
i = a + δ

f = forecast. i = initial condition (analysis), ⟨i⟩ = a δ = initial condition error ϵ = model error.

How does predictability (G) depend on ▶ dynamics A? ▶ initial condition errors δ? ▶ model errors ϵ?

f = Ai + ϵ
i = a + δ

f = forecast. i = initial condition (analysis), ⟨i⟩ = a δ = initial condition error ϵ = model error.

How does predictability (G) depend on ▶ dynamics A? ▶ initial condition errors δ? ▶ model errors ϵ?

f = Ai + ϵ
i = a + δ

f = forecast. i = initial condition (analysis), ⟨i⟩ = a δ = initial condition error ϵ = model error.

How does predictability (G) depend on ▶ dynamics A? ▶ initial condition errors δ? ▶ model errors ϵ?

f = Ai + ϵ
i = a + δ

f = forecast. i = initial condition (analysis), ⟨i⟩ = a δ = initial condition error ϵ = model error.

How does predictability (G) depend on ▶ dynamics A? ▶ initial condition errors δ? ▶ model errors ϵ?

f = Ai + ϵ
i = a + δ

f = forecast. i = initial condition (analysis), ⟨i⟩ = a δ = initial condition error ϵ = model error.

How does predictability (G) depend on ▶ dynamics A? ▶ initial condition errors δ? ▶ model errors ϵ?

Linear prediction model

Error dynamics approach

f = A(a + δ) + ϵ
C_{f}  =  A^{T}C_{δ}A  +  C_{ϵ}
forecast uncertainty  uncertainties from IC error  model error

“Whiten”

C_{c}^{−1/2}C_{f}C_{c}^{−1/2} = C_{c}^{−1/2}AC_{δ}A^{T}C_{c}^{−1/2} + C_{c}^{−1/2}C_{ϵ}C_{c}^{−1/2}
G = A_{error}A_{error}^{T} + C_{c}^{−1/2}C_{ϵ}C_{c}^{−1/2}

A_{error} ≡ C_{c}^{−1/2}AC_{δ}^{1/2} ← initial and final norms If there is no stochastic forcing ϵ = 0,

G = A_{error}A_{error}^{T}

Predictability given by SVD of A_{error}.

f = A(a + δ) + ϵ
C_{f}  =  A^{T}C_{δ}A  +  C_{ϵ}
forecast uncertainty  uncertainties from IC error  model error

“Whiten”

C_{c}^{−1/2}C_{f}C_{c}^{−1/2} = C_{c}^{−1/2}AC_{δ}A^{T}C_{c}^{−1/2} + C_{c}^{−1/2}C_{ϵ}C_{c}^{−1/2}
G = A_{error}A_{error}^{T} + C_{c}^{−1/2}C_{ϵ}C_{c}^{−1/2}

A_{error} ≡ C_{c}^{−1/2}AC_{δ}^{1/2} ← initial and final norms If there is no stochastic forcing ϵ = 0,

G = A_{error}A_{error}^{T}

Predictability given by SVD of A_{error}.

f = A(a + δ) + ϵ
C_{f}  =  A^{T}C_{δ}A  +  C_{ϵ}
forecast uncertainty  uncertainties from IC error  model error

“Whiten”

C_{c}^{−1/2}C_{f}C_{c}^{−1/2} = C_{c}^{−1/2}AC_{δ}A^{T}C_{c}^{−1/2} + C_{c}^{−1/2}C_{ϵ}C_{c}^{−1/2}
G = A_{error}A_{error}^{T} + C_{c}^{−1/2}C_{ϵ}C_{c}^{−1/2}

A_{error} ≡ C_{c}^{−1/2}AC_{δ}^{1/2} ← initial and final norms If there is no stochastic forcing ϵ = 0,

G = A_{error}A_{error}^{T}

Predictability given by SVD of A_{error}.

Linear prediction model

Signal approach

Averaging over forecasts

f = A(a + δ) + ϵ
climatology  signal  uncertainties from IC error  model error
C_{c}  = AC_{a}A^{T} +  AC_{δ}A^{T}  +  C_{ϵ}
=  C_{f}

Whiten

C_{c}^{−1/2}C_{c}C_{c}^{−1/2} = C_{c}^{−1/2}AC_{a}A^{T}C_{c}^{−1/2} + C_{c}^{−1/2}C_{f}C_{c}^{−1/2}
I  =  A_{signal}A_{signal}^{T}  +  G

A_{signal} ≡ C_{c}^{−1/2}AC_{a}^{1/2} ← initial and final norms

G = I − A_{signal}A_{signal}^{T}
λ(G) = 1 − σ^{2}(A_{signal})

Averaging over forecasts

f = A(a + δ) + ϵ
climatology  signal  uncertainties from IC error  model error
C_{c}  = AC_{a}A^{T} +  AC_{δ}A^{T}  +  C_{ϵ}
=  C_{f}

Whiten

C_{c}^{−1/2}C_{c}C_{c}^{−1/2} = C_{c}^{−1/2}AC_{a}A^{T}C_{c}^{−1/2} + C_{c}^{−1/2}C_{f}C_{c}^{−1/2}
I  =  A_{signal}A_{signal}^{T}  +  G

A_{signal} ≡ C_{c}^{−1/2}AC_{a}^{1/2} ← initial and final norms

G = I − A_{signal}A_{signal}^{T}
λ(G) = 1 − σ^{2}(A_{signal})

Averaging over forecasts

f = A(a + δ) + ϵ
climatology  signal  uncertainties from IC error  model error
C_{c}  = AC_{a}A^{T} +  AC_{δ}A^{T}  +  C_{ϵ}
=  C_{f}

Whiten

C_{c}^{−1/2}C_{c}C_{c}^{−1/2} = C_{c}^{−1/2}AC_{a}A^{T}C_{c}^{−1/2} + C_{c}^{−1/2}C_{f}C_{c}^{−1/2}
I  =  A_{signal}A_{signal}^{T}  +  G

A_{signal} ≡ C_{c}^{−1/2}AC_{a}^{1/2} ← initial and final norms

G = I − A_{signal}A_{signal}^{T}
λ(G) = 1 − σ^{2}(A_{signal})

Averaging over forecasts

f = A(a + δ) + ϵ
climatology  signal  uncertainties from IC error  model error
C_{c}  = AC_{a}A^{T} +  AC_{δ}A^{T}  +  C_{ϵ}
=  C_{f}

Whiten

C_{c}^{−1/2}C_{c}C_{c}^{−1/2} = C_{c}^{−1/2}AC_{a}A^{T}C_{c}^{−1/2} + C_{c}^{−1/2}C_{f}C_{c}^{−1/2}
I  =  A_{signal}A_{signal}^{T}  +  G

A_{signal} ≡ C_{c}^{−1/2}AC_{a}^{1/2} ← initial and final norms

G = I − A_{signal}A_{signal}^{T}
λ(G) = 1 − σ^{2}(A_{signal})

Averaging over forecasts

f = A(a + δ) + ϵ
climatology  signal  uncertainties from IC error  model error
C_{c}  = AC_{a}A^{T} +  AC_{δ}A^{T}  +  C_{ϵ}
=  C_{f}

Whiten

C_{c}^{−1/2}C_{c}C_{c}^{−1/2} = C_{c}^{−1/2}AC_{a}A^{T}C_{c}^{−1/2} + C_{c}^{−1/2}C_{f}C_{c}^{−1/2}
I  =  A_{signal}A_{signal}^{T}  +  G

A_{signal} ≡ C_{c}^{−1/2}AC_{a}^{1/2} ← initial and final norms

G = I − A_{signal}A_{signal}^{T}
λ(G) = 1 − σ^{2}(A_{signal})

Whitening transformation

2D picture

Paradox?

No model error

G = A_{error}A_{error}^{T} = I − A_{signal}A_{signal}^{T}

σ^{2}(A_{error}) = 1 − σ^{2}(A_{signal})
(rms^{2} = 1 - corr^{2})

Same singular vectors–opposite ordering. Minimizing error = maximizing signal.

SST example

A maps sea surface temperature at time t to SST temperature at time t + 7 months.

A is an empirical model, fit to data. What are the predictable components?

SST example SST example

Predictability of 7-month components at other leads.

Summary

▶ SVD used in weather and climate applications to diagnose

▶ error growth

▶ signal amplification

▶ Seeming paradox. Error growth bad – signal growth good. ▶ Defining predictability , defines SVD initial and final norms ▶ Predictable component analysis

▶ canonical correlation analysis ▶ fingerprint methods ▶ discriminant analysis ▶ S/N EOFs

▶ Future work

▶ Estimating distributions

▶ Short record ▶ Imperfect models

DelSole, T., and M. K. Tippett, 2007: Measuring Predictability with Singular Vectors. J. Atmos. Sci., submitted. DelSole, T., and M. K. Tippett, 2007: Predictability: Recent Insights from Information Theory. Rev. Geophys.,

Summary

▶ SVD used in weather and climate applications to diagnose

▶ error growth

▶ signal amplification

▶ Seeming paradox. Error growth bad – signal growth good. ▶ Defining predictability , defines SVD initial and final norms ▶ Predictable component analysis

▶ canonical correlation analysis ▶ fingerprint methods ▶ discriminant analysis ▶ S/N EOFs

▶ Future work

▶ Estimating distributions

▶ Short record ▶ Imperfect models

DelSole, T., and M. K. Tippett, 2007: Measuring Predictability with Singular Vectors. J. Atmos. Sci., submitted. DelSole, T., and M. K. Tippett, 2007: Predictability: Recent Insights from Information Theory. Rev. Geophys.,