time series analysis via matrix estimationquestions from retail and time series analysis question 1:...

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Time Series Analysis via Matrix Estimation Devavrat Shah Anish Agarwal Muhammad J Amjad Dennis Shen Massachusetts Institute of Technology https://arxiv.org/pdf/1802.09064.pdf https://arxiv.org/pdf/1711.06940.pdf

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Page 1: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Time Series Analysis via Matrix Estimation

Devavrat Shah

Anish Agarwal Muhammad J Amjad Dennis Shen

Massachusetts Institute of Technology

https://arxiv.org/pdf/1802.09064.pdf https://arxiv.org/pdf/1711.06940.pdf

Page 2: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Questions from Retail

Question 1:

Estimate demand (rate) for Umbrellas at Target store in Sunnyvale

Question 2:

Estimate future demand for Umbrellas at Target store in Sunnyvale

Question 3:

What would demand for Umbrellas at Target store in Sunnyvale be if we did (not) introduce the mobile checkout

Page 3: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Questions from Retail and Time Series Analysis

Question 1: demand rate estimation

estimating latent state of a time-series with missing values

Question 2: future demand

forecasting state of time-series using historical (+ other time-series)

Question 3: demand with(out) intervention

comparing with synthetic control for time-series of interest using other time-series

Page 4: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Matrix Estimation (ME)

movie 1 movie j movie m

user 1

user i

user n

3

4

1

1 4

3

2 3

5

1

32

?

Rating Matrix A

Page 5: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Matrix Estimation (ME)

Observation

noisy observation for a subset E of entries:

Goal

so that the prediction error

produce an estimation

Yij , for (i, j) 2 E

subject to some `noise’ model:

for all (i, j) 2 [n]⇥ [m]Aij

MSE(A) =1

nmEhX

i,j

�Aij �Aij

�2i

is small

ground truth:

Yij ⇠ Aij , 8 i, j.

Aij , 8(i, j) 2 [n]⇥ [m]

Page 6: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Matrix Estimation (ME)

Latent Variable Model

entry corresponding to user i and movie j in A

row i has associated latent features x1(i) 2 X1

column j has associated latent features x2(j) 2 X2

Aij = f(x1(i), x2(j)) f : X1 ⇥ X2 ! Rwhere

That is, is such thatYij

E⇥Yij |x1(i), x2(j)

⇤= Aij = f(x1(i), x2(j))

Canonical representation due to Row-Column Exchangeability

[Hoover 79, 82], [Aldous 81, 82, 85], [Lovasz-Szegedy 08], …

Page 7: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Matrix Estimation (ME)

Goal

given (partial) observation of matrix Y = [Yij ]

so that the prediction error is small

produce an estimation A = [Aij ]

MSE(A)

Performance Metric

(random) fraction of matrix that needs to be observed

so that estimator is consistent, i.e.

Y = [Yij ]

limn,m!1

MSE(A) = 0

p

Page 8: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Matrix Estimation (ME)

Feature space:

Model complexity

[0, 1]d

Function space: bilinear, Lipschitz continuous

Noise model

Additive:

Generic:

Sample complexity

Number of samples observed

Yij = f(x1(i), x2(j)) + ⌘ij , E[⌘ij ] = 0

E[Yij ] = f(x1(i), x2(j)) Yij 2 [�B,B]

Page 9: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Matrix Estimation (ME)

Result Sample Complexity Noise Model Function Class Guarantee

KMO10 Additive bilinear(rank d) MSE to 0

C15 Generic Lipschitz MSE to 0

LLSS16 Additive Lipschitz MSE to 0

BCLS17 Generic bilinear(rank d) MSE to 0

⌦(nd log n)

⌦(n3/2)

⌦(n2� 2d+2 )

[very large number of remarkable results are not reported here]

!(nd5)

Page 10: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

This Talk

Answer to all three time-series questions

estimating latent state of a time-series with missing values

forecasting state of time-series using historical (+ other time-series)

comparing with synthetic control for time-series of interest

Via Matrix Estimation (ME)

we’ll assume access to Matrix Estimation (ME) as a black-box (BB-ME)

transform all three questions to Matrix Estimationand some post-processing

Page 11: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Answer 1: Time Series Imputation

“Ground Truth” of interest: , for examplef(t), t 2 R<latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit>

f(t) =KX

k=1

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f(t) =KX

k=1

↵kt�k

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f(t) =KX

k=1

↵kf(t� k)<latexit sha1_base64="EOwrRrym2h8aWJBvFq6Oag9DSYE=">AAACDHicbVDLSsNAFJ3UV62vqEs3g1VoF5ZEBHVRKLoR3FSwttDEMJlO2iEzSZiZCCX0B9z4K25cqLj1A9z5N07bLLT1wIXDOfdy7z1+wqhUlvVtFBYWl5ZXiqultfWNzS1ze+dOxqnApIVjFouOjyRhNCItRRUjnUQQxH1G2n54OfbbD0RIGke3apgQl6N+RAOKkdKSZx4EFVWFdejIlHtZWLdH99fQQSwZIC+E2jwKq9Azy1bNmgDOEzsnZZCj6ZlfTi/GKSeRwgxJ2bWtRLkZEopiRkYlJ5UkQThEfdLVNEKcSDebfDOCh1rpwSAWuiIFJ+rviQxxKYfc150cqYGc9cbif143VcGZm9EoSRWJ8HRRkDKoYjiOBvaoIFixoSYIC6pvhXiABMJKB1jSIdizL8+T1nHtvGbfnJQbF3kaRbAH9kEF2OAUNMAVaIIWwOARPINX8GY8GS/Gu/ExbS0Y+cwu+APj8wdZsplo</latexit><latexit sha1_base64="EOwrRrym2h8aWJBvFq6Oag9DSYE=">AAACDHicbVDLSsNAFJ3UV62vqEs3g1VoF5ZEBHVRKLoR3FSwttDEMJlO2iEzSZiZCCX0B9z4K25cqLj1A9z5N07bLLT1wIXDOfdy7z1+wqhUlvVtFBYWl5ZXiqultfWNzS1ze+dOxqnApIVjFouOjyRhNCItRRUjnUQQxH1G2n54OfbbD0RIGke3apgQl6N+RAOKkdKSZx4EFVWFdejIlHtZWLdH99fQQSwZIC+E2jwKq9Azy1bNmgDOEzsnZZCj6ZlfTi/GKSeRwgxJ2bWtRLkZEopiRkYlJ5UkQThEfdLVNEKcSDebfDOCh1rpwSAWuiIFJ+rviQxxKYfc150cqYGc9cbif143VcGZm9EoSRWJ8HRRkDKoYjiOBvaoIFixoSYIC6pvhXiABMJKB1jSIdizL8+T1nHtvGbfnJQbF3kaRbAH9kEF2OAUNMAVaIIWwOARPINX8GY8GS/Gu/ExbS0Y+cwu+APj8wdZsplo</latexit><latexit sha1_base64="EOwrRrym2h8aWJBvFq6Oag9DSYE=">AAACDHicbVDLSsNAFJ3UV62vqEs3g1VoF5ZEBHVRKLoR3FSwttDEMJlO2iEzSZiZCCX0B9z4K25cqLj1A9z5N07bLLT1wIXDOfdy7z1+wqhUlvVtFBYWl5ZXiqultfWNzS1ze+dOxqnApIVjFouOjyRhNCItRRUjnUQQxH1G2n54OfbbD0RIGke3apgQl6N+RAOKkdKSZx4EFVWFdejIlHtZWLdH99fQQSwZIC+E2jwKq9Azy1bNmgDOEzsnZZCj6ZlfTi/GKSeRwgxJ2bWtRLkZEopiRkYlJ5UkQThEfdLVNEKcSDebfDOCh1rpwSAWuiIFJ+rviQxxKYfc150cqYGc9cbif143VcGZm9EoSRWJ8HRRkDKoYjiOBvaoIFixoSYIC6pvhXiABMJKB1jSIdizL8+T1nHtvGbfnJQbF3kaRbAH9kEF2OAUNMAVaIIWwOARPINX8GY8GS/Gu/ExbS0Y+cwu+APj8wdZsplo</latexit><latexit sha1_base64="EOwrRrym2h8aWJBvFq6Oag9DSYE=">AAACDHicbVDLSsNAFJ3UV62vqEs3g1VoF5ZEBHVRKLoR3FSwttDEMJlO2iEzSZiZCCX0B9z4K25cqLj1A9z5N07bLLT1wIXDOfdy7z1+wqhUlvVtFBYWl5ZXiqultfWNzS1ze+dOxqnApIVjFouOjyRhNCItRRUjnUQQxH1G2n54OfbbD0RIGke3apgQl6N+RAOKkdKSZx4EFVWFdejIlHtZWLdH99fQQSwZIC+E2jwKq9Azy1bNmgDOEzsnZZCj6ZlfTi/GKSeRwgxJ2bWtRLkZEopiRkYlJ5UkQThEfdLVNEKcSDebfDOCh1rpwSAWuiIFJ+rviQxxKYfc150cqYGc9cbif143VcGZm9EoSRWJ8HRRkDKoYjiOBvaoIFixoSYIC6pvhXiABMJKB1jSIdizL8+T1nHtvGbfnJQbF3kaRbAH9kEF2OAUNMAVaIIWwOARPINX8GY8GS/Gu/ExbS0Y+cwu+APj8wdZsplo</latexit>

or or, their combination….

Observation: s. t. if observed (w.p. p)X(t), t 2 {0, 1, . . . , T}<latexit sha1_base64="yohLOkCg8/GZ2+mfaDnBKHv4v4w=">AAACCHicbVBNS8NAFNz4WetX1KOXxSJUKCURQb0VvXis0NhCE8pmu22XbjZh90UooR69+Fe8eFDx6k/w5r9x2+agrQMLw8wb3r4JE8E1OM63tbS8srq2Xtgobm5t7+zae/t3Ok4VZR6NRaxaIdFMcMk84CBYK1GMRKFgzXB4PfGb90xpHssGjBIWRKQveY9TAkbq2LhVhpMKfgDsc4n9zKlgt4L9bgy6ghv+GHfsklN1psCLxM1JCeWod+wvk6ZpxCRQQbRuu04CQUYUcCrYuOinmiWEDkmftQ2VJGI6yKaXjPGxUbq4FyvzJOCp+juRkUjrURSayYjAQM97E/E/r51C7yLIuExSYJLOFvVSgSHGk1pwlytGQYwMIVRx81dMB0QRCqa8oinBnT95kXin1cuqe3tWql3lbRTQITpCZeSic1RDN6iOPETRI3pGr+jNerJerHfrYza6ZOWZA/QH1ucPMsmXDA==</latexit><latexit sha1_base64="yohLOkCg8/GZ2+mfaDnBKHv4v4w=">AAACCHicbVBNS8NAFNz4WetX1KOXxSJUKCURQb0VvXis0NhCE8pmu22XbjZh90UooR69+Fe8eFDx6k/w5r9x2+agrQMLw8wb3r4JE8E1OM63tbS8srq2Xtgobm5t7+zae/t3Ok4VZR6NRaxaIdFMcMk84CBYK1GMRKFgzXB4PfGb90xpHssGjBIWRKQveY9TAkbq2LhVhpMKfgDsc4n9zKlgt4L9bgy6ghv+GHfsklN1psCLxM1JCeWod+wvk6ZpxCRQQbRuu04CQUYUcCrYuOinmiWEDkmftQ2VJGI6yKaXjPGxUbq4FyvzJOCp+juRkUjrURSayYjAQM97E/E/r51C7yLIuExSYJLOFvVSgSHGk1pwlytGQYwMIVRx81dMB0QRCqa8oinBnT95kXin1cuqe3tWql3lbRTQITpCZeSic1RDN6iOPETRI3pGr+jNerJerHfrYza6ZOWZA/QH1ucPMsmXDA==</latexit><latexit sha1_base64="yohLOkCg8/GZ2+mfaDnBKHv4v4w=">AAACCHicbVBNS8NAFNz4WetX1KOXxSJUKCURQb0VvXis0NhCE8pmu22XbjZh90UooR69+Fe8eFDx6k/w5r9x2+agrQMLw8wb3r4JE8E1OM63tbS8srq2Xtgobm5t7+zae/t3Ok4VZR6NRaxaIdFMcMk84CBYK1GMRKFgzXB4PfGb90xpHssGjBIWRKQveY9TAkbq2LhVhpMKfgDsc4n9zKlgt4L9bgy6ghv+GHfsklN1psCLxM1JCeWod+wvk6ZpxCRQQbRuu04CQUYUcCrYuOinmiWEDkmftQ2VJGI6yKaXjPGxUbq4FyvzJOCp+juRkUjrURSayYjAQM97E/E/r51C7yLIuExSYJLOFvVSgSHGk1pwlytGQYwMIVRx81dMB0QRCqa8oinBnT95kXin1cuqe3tWql3lbRTQITpCZeSic1RDN6iOPETRI3pGr+jNerJerHfrYza6ZOWZA/QH1ucPMsmXDA==</latexit><latexit sha1_base64="yohLOkCg8/GZ2+mfaDnBKHv4v4w=">AAACCHicbVBNS8NAFNz4WetX1KOXxSJUKCURQb0VvXis0NhCE8pmu22XbjZh90UooR69+Fe8eFDx6k/w5r9x2+agrQMLw8wb3r4JE8E1OM63tbS8srq2Xtgobm5t7+zae/t3Ok4VZR6NRaxaIdFMcMk84CBYK1GMRKFgzXB4PfGb90xpHssGjBIWRKQveY9TAkbq2LhVhpMKfgDsc4n9zKlgt4L9bgy6ghv+GHfsklN1psCLxM1JCeWod+wvk6ZpxCRQQbRuu04CQUYUcCrYuOinmiWEDkmftQ2VJGI6yKaXjPGxUbq4FyvzJOCp+juRkUjrURSayYjAQM97E/E/r51C7yLIuExSYJLOFvVSgSHGk1pwlytGQYwMIVRx81dMB0QRCqa8oinBnT95kXin1cuqe3tWql3lbRTQITpCZeSic1RDN6iOPETRI3pGr+jNerJerHfrYza6ZOWZA/QH1ucPMsmXDA==</latexit>

E[X(t)] = f(t)<latexit sha1_base64="957RoC9KjYPGJnd/idpIfrEtmAw=">AAACAHicbVDLSsNAFL2pr1pfUTeCm8Ei1E1JRFAXQlEElxWMLaShTKaTdujkwcxEKKFu/BU3LlTc+hnu/BsnbRbaemCYwzn3cu89fsKZVJb1bZQWFpeWV8qrlbX1jc0tc3vnXsapINQhMY9F28eSchZRRzHFaTsRFIc+py1/eJX7rQcqJIujOzVKqBfifsQCRrDSUtfc64RYDXw/ux677Zo68tAFCvRf6ZpVq25NgOaJXZAqFGh2za9OLyZpSCNFOJbSta1EeRkWihFOx5VOKmmCyRD3qatphEMqvWxywRgdaqWHgljoFyk0UX93ZDiUchT6ujLfV856ufif56YqOPMyFiWpohGZDgpSjlSM8jhQjwlKFB9pgolgeldEBlhgonRoeQj27MnzxDmun9ft25Nq47JIowz7cAA1sOEUGnADTXCAwCM8wyu8GU/Gi/FufExLS0bRswt/YHz+ADqLlP0=</latexit><latexit sha1_base64="957RoC9KjYPGJnd/idpIfrEtmAw=">AAACAHicbVDLSsNAFL2pr1pfUTeCm8Ei1E1JRFAXQlEElxWMLaShTKaTdujkwcxEKKFu/BU3LlTc+hnu/BsnbRbaemCYwzn3cu89fsKZVJb1bZQWFpeWV8qrlbX1jc0tc3vnXsapINQhMY9F28eSchZRRzHFaTsRFIc+py1/eJX7rQcqJIujOzVKqBfifsQCRrDSUtfc64RYDXw/ux677Zo68tAFCvRf6ZpVq25NgOaJXZAqFGh2za9OLyZpSCNFOJbSta1EeRkWihFOx5VOKmmCyRD3qatphEMqvWxywRgdaqWHgljoFyk0UX93ZDiUchT6ujLfV856ufif56YqOPMyFiWpohGZDgpSjlSM8jhQjwlKFB9pgolgeldEBlhgonRoeQj27MnzxDmun9ft25Nq47JIowz7cAA1sOEUGnADTXCAwCM8wyu8GU/Gi/FufExLS0bRswt/YHz+ADqLlP0=</latexit><latexit sha1_base64="957RoC9KjYPGJnd/idpIfrEtmAw=">AAACAHicbVDLSsNAFL2pr1pfUTeCm8Ei1E1JRFAXQlEElxWMLaShTKaTdujkwcxEKKFu/BU3LlTc+hnu/BsnbRbaemCYwzn3cu89fsKZVJb1bZQWFpeWV8qrlbX1jc0tc3vnXsapINQhMY9F28eSchZRRzHFaTsRFIc+py1/eJX7rQcqJIujOzVKqBfifsQCRrDSUtfc64RYDXw/ux677Zo68tAFCvRf6ZpVq25NgOaJXZAqFGh2za9OLyZpSCNFOJbSta1EeRkWihFOx5VOKmmCyRD3qatphEMqvWxywRgdaqWHgljoFyk0UX93ZDiUchT6ujLfV856ufif56YqOPMyFiWpohGZDgpSjlSM8jhQjwlKFB9pgolgeldEBlhgonRoeQj27MnzxDmun9ft25Nq47JIowz7cAA1sOEUGnADTXCAwCM8wyu8GU/Gi/FufExLS0bRswt/YHz+ADqLlP0=</latexit><latexit sha1_base64="957RoC9KjYPGJnd/idpIfrEtmAw=">AAACAHicbVDLSsNAFL2pr1pfUTeCm8Ei1E1JRFAXQlEElxWMLaShTKaTdujkwcxEKKFu/BU3LlTc+hnu/BsnbRbaemCYwzn3cu89fsKZVJb1bZQWFpeWV8qrlbX1jc0tc3vnXsapINQhMY9F28eSchZRRzHFaTsRFIc+py1/eJX7rQcqJIujOzVKqBfifsQCRrDSUtfc64RYDXw/ux677Zo68tAFCvRf6ZpVq25NgOaJXZAqFGh2za9OLyZpSCNFOJbSta1EeRkWihFOx5VOKmmCyRD3qatphEMqvWxywRgdaqWHgljoFyk0UX93ZDiUchT6ujLfV856ufif56YqOPMyFiWpohGZDgpSjlSM8jhQjwlKFB9pgolgeldEBlhgonRoeQj27MnzxDmun9ft25Nq47JIowz7cAA1sOEUGnADTXCAwCM8wyu8GU/Gi/FufExLS0bRswt/YHz+ADqLlP0=</latexit>

(+ independence, conditions on “noise”)

Goal: produce estimate so that X(t)<latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit>

MSE(X, f) is small<latexit sha1_base64="NCrZ+qUDUC/CETSpsfdAKxmQwmg=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHaQlPKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSVmNhiKUDZcoJnjAasBBsEYkGfFdwepu/2zo1++YVDwMbmAQsZZPugH3OCWgpba56wC7h+Ty+jwtOT0CSSPdx94efhjrXGHlEyHStlm0ytYIeJrYGSmiDNW2+eV0Qhr7LAAqiFJN24qglRAJnAqWFpxYsYjQPumypqYB8ZlqJaOHUryjlQ72QqkrADxSf08kxFdq4Lu60yfQU5PeUPzPa8bgHbcSHkQxsICOF3mxwBDiYTq4wyWjIAaaECq5vhXTHpGEgs6woEOwJ1+eJrWD8knZvjosVk6zNPJoC22jErLREaqgC1RFNUTRI3pGr+jNeDJejHfjY9yaM7KZTfQHxucP7Aacuw==</latexit><latexit sha1_base64="NCrZ+qUDUC/CETSpsfdAKxmQwmg=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHaQlPKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSVmNhiKUDZcoJnjAasBBsEYkGfFdwepu/2zo1++YVDwMbmAQsZZPugH3OCWgpba56wC7h+Ty+jwtOT0CSSPdx94efhjrXGHlEyHStlm0ytYIeJrYGSmiDNW2+eV0Qhr7LAAqiFJN24qglRAJnAqWFpxYsYjQPumypqYB8ZlqJaOHUryjlQ72QqkrADxSf08kxFdq4Lu60yfQU5PeUPzPa8bgHbcSHkQxsICOF3mxwBDiYTq4wyWjIAaaECq5vhXTHpGEgs6woEOwJ1+eJrWD8knZvjosVk6zNPJoC22jErLREaqgC1RFNUTRI3pGr+jNeDJejHfjY9yaM7KZTfQHxucP7Aacuw==</latexit><latexit sha1_base64="NCrZ+qUDUC/CETSpsfdAKxmQwmg=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHaQlPKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSVmNhiKUDZcoJnjAasBBsEYkGfFdwepu/2zo1++YVDwMbmAQsZZPugH3OCWgpba56wC7h+Ty+jwtOT0CSSPdx94efhjrXGHlEyHStlm0ytYIeJrYGSmiDNW2+eV0Qhr7LAAqiFJN24qglRAJnAqWFpxYsYjQPumypqYB8ZlqJaOHUryjlQ72QqkrADxSf08kxFdq4Lu60yfQU5PeUPzPa8bgHbcSHkQxsICOF3mxwBDiYTq4wyWjIAaaECq5vhXTHpGEgs6woEOwJ1+eJrWD8knZvjosVk6zNPJoC22jErLREaqgC1RFNUTRI3pGr+jNeDJejHfjY9yaM7KZTfQHxucP7Aacuw==</latexit><latexit sha1_base64="NCrZ+qUDUC/CETSpsfdAKxmQwmg=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHaQlPKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSVmNhiKUDZcoJnjAasBBsEYkGfFdwepu/2zo1++YVDwMbmAQsZZPugH3OCWgpba56wC7h+Ty+jwtOT0CSSPdx94efhjrXGHlEyHStlm0ytYIeJrYGSmiDNW2+eV0Qhr7LAAqiFJN24qglRAJnAqWFpxYsYjQPumypqYB8ZlqJaOHUryjlQ72QqkrADxSf08kxFdq4Lu60yfQU5PeUPzPa8bgHbcSHkQxsICOF3mxwBDiYTq4wyWjIAaaECq5vhXTHpGEgs6woEOwJ1+eJrWD8knZvjosVk6zNPJoC22jErLREaqgC1RFNUTRI3pGr+jNeDJejHfjY9yaM7KZTfQHxucP7Aacuw==</latexit>

Page 12: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

An Example:

“Ground Truth” of interest: as described beforef(t), t 2 R<latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit>

Observation: w.p. 0.1, observe X(t) where for some

X(t) ⇠ Poisson(f(t))<latexit sha1_base64="9lqpcfcMh3klbX8Uxme8EdUAW4g=">AAACB3icbVA9SwNBEN2LXzF+nVpauBiEpAl3Iqhd0MYygjGBJIS9zV6yZG/32J0Tw5F0Nv4VGwsVW/+Cnf/GzUehiQ8GHu/NMDMviAU34HnfTmZpeWV1Lbue29jc2t5xd/fujEo0ZVWqhNL1gBgmuGRV4CBYPdaMRIFgtaB/NfZr90wbruQtDGLWikhX8pBTAlZqu4f1AhTxqGl4NGoCe4C0orgxSg4LoXWKbTfvlbwJ8CLxZySPZqi03a9mR9EkYhKoIMY0fC+GVko0cCrYMNdMDIsJ7ZMua1gqScRMK508MsTHVungUGlbEvBE/T2RksiYQRTYzohAz8x7Y/E/r5FAeN5KuYwTYJJOF4WJwKDwOBXc4ZpREANLCNXc3oppj2hCwWaXsyH48y8vkupJ6aLk35zmy5ezNLLoAB2hAvLRGSqja1RBVUTRI3pGr+jNeXJenHfnY9qacWYz++gPnM8fS6aZEw==</latexit><latexit sha1_base64="9lqpcfcMh3klbX8Uxme8EdUAW4g=">AAACB3icbVA9SwNBEN2LXzF+nVpauBiEpAl3Iqhd0MYygjGBJIS9zV6yZG/32J0Tw5F0Nv4VGwsVW/+Cnf/GzUehiQ8GHu/NMDMviAU34HnfTmZpeWV1Lbue29jc2t5xd/fujEo0ZVWqhNL1gBgmuGRV4CBYPdaMRIFgtaB/NfZr90wbruQtDGLWikhX8pBTAlZqu4f1AhTxqGl4NGoCe4C0orgxSg4LoXWKbTfvlbwJ8CLxZySPZqi03a9mR9EkYhKoIMY0fC+GVko0cCrYMNdMDIsJ7ZMua1gqScRMK508MsTHVungUGlbEvBE/T2RksiYQRTYzohAz8x7Y/E/r5FAeN5KuYwTYJJOF4WJwKDwOBXc4ZpREANLCNXc3oppj2hCwWaXsyH48y8vkupJ6aLk35zmy5ezNLLoAB2hAvLRGSqja1RBVUTRI3pGr+jNeXJenHfnY9qacWYz++gPnM8fS6aZEw==</latexit><latexit sha1_base64="9lqpcfcMh3klbX8Uxme8EdUAW4g=">AAACB3icbVA9SwNBEN2LXzF+nVpauBiEpAl3Iqhd0MYygjGBJIS9zV6yZG/32J0Tw5F0Nv4VGwsVW/+Cnf/GzUehiQ8GHu/NMDMviAU34HnfTmZpeWV1Lbue29jc2t5xd/fujEo0ZVWqhNL1gBgmuGRV4CBYPdaMRIFgtaB/NfZr90wbruQtDGLWikhX8pBTAlZqu4f1AhTxqGl4NGoCe4C0orgxSg4LoXWKbTfvlbwJ8CLxZySPZqi03a9mR9EkYhKoIMY0fC+GVko0cCrYMNdMDIsJ7ZMua1gqScRMK508MsTHVungUGlbEvBE/T2RksiYQRTYzohAz8x7Y/E/r5FAeN5KuYwTYJJOF4WJwKDwOBXc4ZpREANLCNXc3oppj2hCwWaXsyH48y8vkupJ6aLk35zmy5ezNLLoAB2hAvLRGSqja1RBVUTRI3pGr+jNeXJenHfnY9qacWYz++gPnM8fS6aZEw==</latexit><latexit sha1_base64="9lqpcfcMh3klbX8Uxme8EdUAW4g=">AAACB3icbVA9SwNBEN2LXzF+nVpauBiEpAl3Iqhd0MYygjGBJIS9zV6yZG/32J0Tw5F0Nv4VGwsVW/+Cnf/GzUehiQ8GHu/NMDMviAU34HnfTmZpeWV1Lbue29jc2t5xd/fujEo0ZVWqhNL1gBgmuGRV4CBYPdaMRIFgtaB/NfZr90wbruQtDGLWikhX8pBTAlZqu4f1AhTxqGl4NGoCe4C0orgxSg4LoXWKbTfvlbwJ8CLxZySPZqi03a9mR9EkYhKoIMY0fC+GVko0cCrYMNdMDIsJ7ZMua1gqScRMK508MsTHVungUGlbEvBE/T2RksiYQRTYzohAz8x7Y/E/r5FAeN5KuYwTYJJOF4WJwKDwOBXc4ZpREANLCNXc3oppj2hCwWaXsyH48y8vkupJ6aLk35zmy5ezNLLoAB2hAvLRGSqja1RBVUTRI3pGr+jNeXJenHfnY9qacWYz++gPnM8fS6aZEw==</latexit>

Goal: produce estimate so that X(t)<latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit>

MSE(X, f) is small<latexit sha1_base64="NCrZ+qUDUC/CETSpsfdAKxmQwmg=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHaQlPKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSVmNhiKUDZcoJnjAasBBsEYkGfFdwepu/2zo1++YVDwMbmAQsZZPugH3OCWgpba56wC7h+Ty+jwtOT0CSSPdx94efhjrXGHlEyHStlm0ytYIeJrYGSmiDNW2+eV0Qhr7LAAqiFJN24qglRAJnAqWFpxYsYjQPumypqYB8ZlqJaOHUryjlQ72QqkrADxSf08kxFdq4Lu60yfQU5PeUPzPa8bgHbcSHkQxsICOF3mxwBDiYTq4wyWjIAaaECq5vhXTHpGEgs6woEOwJ1+eJrWD8knZvjosVk6zNPJoC22jErLREaqgC1RFNUTRI3pGr+jNeDJejHfjY9yaM7KZTfQHxucP7Aacuw==</latexit><latexit sha1_base64="NCrZ+qUDUC/CETSpsfdAKxmQwmg=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHaQlPKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSVmNhiKUDZcoJnjAasBBsEYkGfFdwepu/2zo1++YVDwMbmAQsZZPugH3OCWgpba56wC7h+Ty+jwtOT0CSSPdx94efhjrXGHlEyHStlm0ytYIeJrYGSmiDNW2+eV0Qhr7LAAqiFJN24qglRAJnAqWFpxYsYjQPumypqYB8ZlqJaOHUryjlQ72QqkrADxSf08kxFdq4Lu60yfQU5PeUPzPa8bgHbcSHkQxsICOF3mxwBDiYTq4wyWjIAaaECq5vhXTHpGEgs6woEOwJ1+eJrWD8knZvjosVk6zNPJoC22jErLREaqgC1RFNUTRI3pGr+jNeDJejHfjY9yaM7KZTfQHxucP7Aacuw==</latexit><latexit sha1_base64="NCrZ+qUDUC/CETSpsfdAKxmQwmg=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHaQlPKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSVmNhiKUDZcoJnjAasBBsEYkGfFdwepu/2zo1++YVDwMbmAQsZZPugH3OCWgpba56wC7h+Ty+jwtOT0CSSPdx94efhjrXGHlEyHStlm0ytYIeJrYGSmiDNW2+eV0Qhr7LAAqiFJN24qglRAJnAqWFpxYsYjQPumypqYB8ZlqJaOHUryjlQ72QqkrADxSf08kxFdq4Lu60yfQU5PeUPzPa8bgHbcSHkQxsICOF3mxwBDiYTq4wyWjIAaaECq5vhXTHpGEgs6woEOwJ1+eJrWD8knZvjosVk6zNPJoC22jErLREaqgC1RFNUTRI3pGr+jNeDJejHfjY9yaM7KZTfQHxucP7Aacuw==</latexit><latexit sha1_base64="NCrZ+qUDUC/CETSpsfdAKxmQwmg=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHaQlPKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSVmNhiKUDZcoJnjAasBBsEYkGfFdwepu/2zo1++YVDwMbmAQsZZPugH3OCWgpba56wC7h+Ty+jwtOT0CSSPdx94efhjrXGHlEyHStlm0ytYIeJrYGSmiDNW2+eV0Qhr7LAAqiFJN24qglRAJnAqWFpxYsYjQPumypqYB8ZlqJaOHUryjlQ72QqkrADxSf08kxFdq4Lu60yfQU5PeUPzPa8bgHbcSHkQxsICOF3mxwBDiYTq4wyWjIAaaECq5vhXTHpGEgs6woEOwJ1+eJrWD8knZvjosVk6zNPJoC22jErLREaqgC1RFNUTRI3pGr+jNeDJejHfjY9yaM7KZTfQHxucP7Aacuw==</latexit>

Answer 1: Time Series Imputation

Page 13: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

An Example:

“Ground Truth” of interest: as described beforef(t), t 2 R<latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit>

Observation: w.p. 0.1, observe X(t) where for some

X(t) ⇠ min

⇣C,Poisson(f(t))

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C � 1<latexit sha1_base64="yQ9CIWzx+isvrDDb9I61VxGdVfo=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG/FXjxWMLbQhrLZTtqlm026uxFK6Z/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8MBVcG9f9dgpr6xubW8Xt0s7u3v5B+fDoUSeZYuizRCSqFVKNgkv0DTcCW6lCGocCm+GwPvObT6g0T+SDGacYxLQvecQZNVZq1UmnjyPidcsVt+rOQVaJl5MK5Gh0y1+dXsKyGKVhgmrd9tzUBBOqDGcCp6VOpjGlbEj72LZU0hh1MJnfOyVnVumRKFG2pCFz9ffEhMZaj+PQdsbUDPSyNxP/89qZia6DCZdpZlCyxaIoE8QkZPY86XGFzIixJZQpbm8lbEAVZcZGVLIheMsvrxL/onpT9e4vK7XbPI0inMApnIMHV1CDO2iADwwEPMMrvDkj58V5dz4WrQUnnzmGP3A+fwAkZ47r</latexit><latexit sha1_base64="yQ9CIWzx+isvrDDb9I61VxGdVfo=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG/FXjxWMLbQhrLZTtqlm026uxFK6Z/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8MBVcG9f9dgpr6xubW8Xt0s7u3v5B+fDoUSeZYuizRCSqFVKNgkv0DTcCW6lCGocCm+GwPvObT6g0T+SDGacYxLQvecQZNVZq1UmnjyPidcsVt+rOQVaJl5MK5Gh0y1+dXsKyGKVhgmrd9tzUBBOqDGcCp6VOpjGlbEj72LZU0hh1MJnfOyVnVumRKFG2pCFz9ffEhMZaj+PQdsbUDPSyNxP/89qZia6DCZdpZlCyxaIoE8QkZPY86XGFzIixJZQpbm8lbEAVZcZGVLIheMsvrxL/onpT9e4vK7XbPI0inMApnIMHV1CDO2iADwwEPMMrvDkj58V5dz4WrQUnnzmGP3A+fwAkZ47r</latexit><latexit sha1_base64="yQ9CIWzx+isvrDDb9I61VxGdVfo=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG/FXjxWMLbQhrLZTtqlm026uxFK6Z/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8MBVcG9f9dgpr6xubW8Xt0s7u3v5B+fDoUSeZYuizRCSqFVKNgkv0DTcCW6lCGocCm+GwPvObT6g0T+SDGacYxLQvecQZNVZq1UmnjyPidcsVt+rOQVaJl5MK5Gh0y1+dXsKyGKVhgmrd9tzUBBOqDGcCp6VOpjGlbEj72LZU0hh1MJnfOyVnVumRKFG2pCFz9ffEhMZaj+PQdsbUDPSyNxP/89qZia6DCZdpZlCyxaIoE8QkZPY86XGFzIixJZQpbm8lbEAVZcZGVLIheMsvrxL/onpT9e4vK7XbPI0inMApnIMHV1CDO2iADwwEPMMrvDkj58V5dz4WrQUnnzmGP3A+fwAkZ47r</latexit><latexit sha1_base64="yQ9CIWzx+isvrDDb9I61VxGdVfo=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG/FXjxWMLbQhrLZTtqlm026uxFK6Z/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8MBVcG9f9dgpr6xubW8Xt0s7u3v5B+fDoUSeZYuizRCSqFVKNgkv0DTcCW6lCGocCm+GwPvObT6g0T+SDGacYxLQvecQZNVZq1UmnjyPidcsVt+rOQVaJl5MK5Gh0y1+dXsKyGKVhgmrd9tzUBBOqDGcCp6VOpjGlbEj72LZU0hh1MJnfOyVnVumRKFG2pCFz9ffEhMZaj+PQdsbUDPSyNxP/89qZia6DCZdpZlCyxaIoE8QkZPY86XGFzIixJZQpbm8lbEAVZcZGVLIheMsvrxL/onpT9e4vK7XbPI0inMApnIMHV1CDO2iADwwEPMMrvDkj58V5dz4WrQUnnzmGP3A+fwAkZ47r</latexit>

Goal: produce estimate so that X(t)<latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit><latexit sha1_base64="EFqmz6ajaT46cBOvKGq6FeEglTw=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSLUS0lEUG9FLx4rGFtIQ9lst+3SzSbsToQS+jO8eFDx6r/x5r9x2+agrQ8GHu/NMDMvSqUw6Lrfzsrq2vrGZmmrvL2zu7dfOTh8NEmmGfdZIhPdjqjhUijuo0DJ26nmNI4kb0Wj26nfeuLaiEQ94DjlYUwHSvQFo2iloDOkmLcnNTwj3UrVrbszkGXiFaQKBZrdylenl7As5gqZpMYEnptimFONgkk+KXcyw1PKRnTAA0sVjbkJ89nJE3JqlR7pJ9qWQjJTf0/kNDZmHEe2M6Y4NIveVPzPCzLsX4W5UGmGXLH5on4mCSZk+j/pCc0ZyrEllGlhbyVsSDVlaFMq2xC8xZeXiX9ev6579xfVxk2RRgmO4QRq4MElNOAOmuADgwSe4RXeHHRenHfnY9664hQzR/AHzucP5r6Qig==</latexit>

g(t) = E⇥min(C,Poisson(f(t))

⇤<latexit sha1_base64="Tr73D3xGWlqWRBgWJzus7TOUAQE=">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</latexit><latexit sha1_base64="Tr73D3xGWlqWRBgWJzus7TOUAQE=">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</latexit><latexit sha1_base64="Tr73D3xGWlqWRBgWJzus7TOUAQE=">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</latexit><latexit sha1_base64="Tr73D3xGWlqWRBgWJzus7TOUAQE=">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</latexit>

MSE(X, g) is small<latexit sha1_base64="jSe2aD/wXiiKcDyIh+9JUD/TbYY=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHYQlvKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSZlDQxHKuksUEzxgDnAQrB5JRnxXsJrbPxv6tTsmFQ+DGxhErOWTbsA9TgloqW3uNoHdQ3J5fZ6Wmj0CST3dx909/DDWucLKJ0KkbbNola0R8DSxM1JEGapt86vZCWnsswCoIEo1bCuCVkIkcCpYWmjGikWE9kmXNTQNiM9UKxk9lOIdrXSwF0pdAeCR+nsiIb5SA9/VnT6Bnpr0huJ/XiMG77iV8CCKgQV0vMiLBYYQD9PBHS4ZBTHQhFDJ9a2Y9ogkFHSGBR2CPfnyNHEOyidl++qwWDnN0sijLbSNSshGR6iCLlAVOYiiR/SMXtGb8WS8GO/Gx7g1Z2Qzm+gPjM8f7ZucvA==</latexit><latexit sha1_base64="jSe2aD/wXiiKcDyIh+9JUD/TbYY=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHYQlvKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSZlDQxHKuksUEzxgDnAQrB5JRnxXsJrbPxv6tTsmFQ+DGxhErOWTbsA9TgloqW3uNoHdQ3J5fZ6Wmj0CST3dx909/DDWucLKJ0KkbbNola0R8DSxM1JEGapt86vZCWnsswCoIEo1bCuCVkIkcCpYWmjGikWE9kmXNTQNiM9UKxk9lOIdrXSwF0pdAeCR+nsiIb5SA9/VnT6Bnpr0huJ/XiMG77iV8CCKgQV0vMiLBYYQD9PBHS4ZBTHQhFDJ9a2Y9ogkFHSGBR2CPfnyNHEOyidl++qwWDnN0sijLbSNSshGR6iCLlAVOYiiR/SMXtGb8WS8GO/Gx7g1Z2Qzm+gPjM8f7ZucvA==</latexit><latexit sha1_base64="jSe2aD/wXiiKcDyIh+9JUD/TbYY=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHYQlvKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSZlDQxHKuksUEzxgDnAQrB5JRnxXsJrbPxv6tTsmFQ+DGxhErOWTbsA9TgloqW3uNoHdQ3J5fZ6Wmj0CST3dx909/DDWucLKJ0KkbbNola0R8DSxM1JEGapt86vZCWnsswCoIEo1bCuCVkIkcCpYWmjGikWE9kmXNTQNiM9UKxk9lOIdrXSwF0pdAeCR+nsiIb5SA9/VnT6Bnpr0huJ/XiMG77iV8CCKgQV0vMiLBYYQD9PBHS4ZBTHQhFDJ9a2Y9ogkFHSGBR2CPfnyNHEOyidl++qwWDnN0sijLbSNSshGR6iCLlAVOYiiR/SMXtGb8WS8GO/Gx7g1Z2Qzm+gPjM8f7ZucvA==</latexit><latexit sha1_base64="jSe2aD/wXiiKcDyIh+9JUD/TbYY=">AAACD3icbVBNS8NAEN3Ur1q/oh69LBaxgpREBPVWFMGLUNHYQlvKZrtpl24+2J2IJcR/4MW/4sWDilev3vw3btsctPXBwOO9GWbmuZHgCizr28jNzM7NL+QXC0vLK6tr5vrGrQpjSZlDQxHKuksUEzxgDnAQrB5JRnxXsJrbPxv6tTsmFQ+DGxhErOWTbsA9TgloqW3uNoHdQ3J5fZ6Wmj0CST3dx909/DDWucLKJ0KkbbNola0R8DSxM1JEGapt86vZCWnsswCoIEo1bCuCVkIkcCpYWmjGikWE9kmXNTQNiM9UKxk9lOIdrXSwF0pdAeCR+nsiIb5SA9/VnT6Bnpr0huJ/XiMG77iV8CCKgQV0vMiLBYYQD9PBHS4ZBTHQhFDJ9a2Y9ogkFHSGBR2CPfnyNHEOyidl++qwWDnN0sijLbSNSshGR6iCLlAVOYiiR/SMXtGb8WS8GO/Gx7g1Z2Qzm+gPjM8f7ZucvA==</latexit>

Answer 1: Time Series Imputation

Page 14: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Algorithm:Transform to Matrix, Do Matrix Estimation, Undo Transformation

X(1) X(2) … X(L) X(L+1) … X(2L) … … … X(T-L+1) … X(T)

X(1)

X(2)

X(L)

X(L+1)

X(L+2)

X(2L)

… … …

X(T-L+1)

X(T)

X(T-L+2)X 1 ⌘

<latexit sha1_base64="rZtUOnTMiDbkJkSzT0PkVaiNM74=">AAAB+XicbVBNS8NAEN34WetXqkcvi0XwVBIR1FvRi8cKxhaaWDbbSbt0s4m7m0qJ/SlePKh49Z9489+4bXPQ1gcDj/dmmJkXppwp7Tjf1tLyyuraemmjvLm1vbNrV/buVJJJCh5NeCJbIVHAmQBPM82hlUogccihGQ6uJn5zCFKxRNzqUQpBTHqCRYwSbaSOXcl9Sjhuje9d7MNDxoYdu+rUnCnwInELUkUFGh37y+8mNItBaMqJUm3XSXWQE6kZ5TAu+5mClNAB6UHbUEFiUEE+PX2Mj4zSxVEiTQmNp+rviZzESo3i0HTGRPfVvDcR//PamY7Og5yJNNMg6GxRlHGsEzzJAXeZBKr5yBBCJTO3YtonklBt0iqbENz5lxeJd1K7qLk3p9X6ZZFGCR2gQ3SMXHSG6ugaNZCHKHpEz+gVvVlP1ov1bn3MWpesYmYf/YH1+QPzoJNa</latexit><latexit sha1_base64="rZtUOnTMiDbkJkSzT0PkVaiNM74=">AAAB+XicbVBNS8NAEN34WetXqkcvi0XwVBIR1FvRi8cKxhaaWDbbSbt0s4m7m0qJ/SlePKh49Z9489+4bXPQ1gcDj/dmmJkXppwp7Tjf1tLyyuraemmjvLm1vbNrV/buVJJJCh5NeCJbIVHAmQBPM82hlUogccihGQ6uJn5zCFKxRNzqUQpBTHqCRYwSbaSOXcl9Sjhuje9d7MNDxoYdu+rUnCnwInELUkUFGh37y+8mNItBaMqJUm3XSXWQE6kZ5TAu+5mClNAB6UHbUEFiUEE+PX2Mj4zSxVEiTQmNp+rviZzESo3i0HTGRPfVvDcR//PamY7Og5yJNNMg6GxRlHGsEzzJAXeZBKr5yBBCJTO3YtonklBt0iqbENz5lxeJd1K7qLk3p9X6ZZFGCR2gQ3SMXHSG6ugaNZCHKHpEz+gVvVlP1ov1bn3MWpesYmYf/YH1+QPzoJNa</latexit><latexit sha1_base64="rZtUOnTMiDbkJkSzT0PkVaiNM74=">AAAB+XicbVBNS8NAEN34WetXqkcvi0XwVBIR1FvRi8cKxhaaWDbbSbt0s4m7m0qJ/SlePKh49Z9489+4bXPQ1gcDj/dmmJkXppwp7Tjf1tLyyuraemmjvLm1vbNrV/buVJJJCh5NeCJbIVHAmQBPM82hlUogccihGQ6uJn5zCFKxRNzqUQpBTHqCRYwSbaSOXcl9Sjhuje9d7MNDxoYdu+rUnCnwInELUkUFGh37y+8mNItBaMqJUm3XSXWQE6kZ5TAu+5mClNAB6UHbUEFiUEE+PX2Mj4zSxVEiTQmNp+rviZzESo3i0HTGRPfVvDcR//PamY7Og5yJNNMg6GxRlHGsEzzJAXeZBKr5yBBCJTO3YtonklBt0iqbENz5lxeJd1K7qLk3p9X6ZZFGCR2gQ3SMXHSG6ugaNZCHKHpEz+gVvVlP1ov1bn3MWpesYmYf/YH1+QPzoJNa</latexit><latexit sha1_base64="rZtUOnTMiDbkJkSzT0PkVaiNM74=">AAAB+XicbVBNS8NAEN34WetXqkcvi0XwVBIR1FvRi8cKxhaaWDbbSbt0s4m7m0qJ/SlePKh49Z9489+4bXPQ1gcDj/dmmJkXppwp7Tjf1tLyyuraemmjvLm1vbNrV/buVJJJCh5NeCJbIVHAmQBPM82hlUogccihGQ6uJn5zCFKxRNzqUQpBTHqCRYwSbaSOXcl9Sjhuje9d7MNDxoYdu+rUnCnwInELUkUFGh37y+8mNItBaMqJUm3XSXWQE6kZ5TAu+5mClNAB6UHbUEFiUEE+PX2Mj4zSxVEiTQmNp+rviZzESo3i0HTGRPfVvDcR//PamY7Og5yJNNMg6GxRlHGsEzzJAXeZBKr5yBBCJTO3YtonklBt0iqbENz5lxeJd1K7qLk3p9X6ZZFGCR2gQ3SMXHSG6ugaNZCHKHpEz+gVvVlP1ov1bn3MWpesYmYf/YH1+QPzoJNa</latexit>

Answer 1: Time Series Imputation

Page 15: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Theorem (Informal):

For L large enough (depending upon model params) with

The matrix satisfies Latent Variable Model with Lipschitz function.

L2 ⌧ T<latexit sha1_base64="NDpwDXpQw97tG5dPpraK0zrXd1Y=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe6CoHZBGwuLCDkTSc6wt9lLluzuHbt7QjjyK2wsVGz9O3b+GzfJFZr4YODx3gwz88KEM21c99sprKyurW8UN0tb2zu7e+X9g3sdp4pQn8Q8Vu0Qa8qZpL5hhtN2oigWIaetcHQ99VtPVGkWy6YZJzQQeCBZxAg2Vnq4fayhLueo2StX3Ko7A1omXk4qkKPRK391+zFJBZWGcKx1x3MTE2RYGUY4nZS6qaYJJiM8oB1LJRZUB9ns4Ak6sUofRbGyJQ2aqb8nMiy0HovQdgpshnrRm4r/eZ3URBdBxmSSGirJfFGUcmRiNP0e9ZmixPCxJZgoZm9FZIgVJsZmVLIheIsvLxO/Vr2sendnlfpVnkYRjuAYTsGDc6jDDTTABwICnuEV3hzlvDjvzse8teDkM4fwB87nD88Zj0w=</latexit><latexit sha1_base64="NDpwDXpQw97tG5dPpraK0zrXd1Y=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe6CoHZBGwuLCDkTSc6wt9lLluzuHbt7QjjyK2wsVGz9O3b+GzfJFZr4YODx3gwz88KEM21c99sprKyurW8UN0tb2zu7e+X9g3sdp4pQn8Q8Vu0Qa8qZpL5hhtN2oigWIaetcHQ99VtPVGkWy6YZJzQQeCBZxAg2Vnq4fayhLueo2StX3Ko7A1omXk4qkKPRK391+zFJBZWGcKx1x3MTE2RYGUY4nZS6qaYJJiM8oB1LJRZUB9ns4Ak6sUofRbGyJQ2aqb8nMiy0HovQdgpshnrRm4r/eZ3URBdBxmSSGirJfFGUcmRiNP0e9ZmixPCxJZgoZm9FZIgVJsZmVLIheIsvLxO/Vr2sendnlfpVnkYRjuAYTsGDc6jDDTTABwICnuEV3hzlvDjvzse8teDkM4fwB87nD88Zj0w=</latexit><latexit sha1_base64="NDpwDXpQw97tG5dPpraK0zrXd1Y=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe6CoHZBGwuLCDkTSc6wt9lLluzuHbt7QjjyK2wsVGz9O3b+GzfJFZr4YODx3gwz88KEM21c99sprKyurW8UN0tb2zu7e+X9g3sdp4pQn8Q8Vu0Qa8qZpL5hhtN2oigWIaetcHQ99VtPVGkWy6YZJzQQeCBZxAg2Vnq4fayhLueo2StX3Ko7A1omXk4qkKPRK391+zFJBZWGcKx1x3MTE2RYGUY4nZS6qaYJJiM8oB1LJRZUB9ns4Ak6sUofRbGyJQ2aqb8nMiy0HovQdgpshnrRm4r/eZ3URBdBxmSSGirJfFGUcmRiNP0e9ZmixPCxJZgoZm9FZIgVJsZmVLIheIsvLxO/Vr2sendnlfpVnkYRjuAYTsGDc6jDDTTABwICnuEV3hzlvDjvzse8teDkM4fwB87nD88Zj0w=</latexit><latexit sha1_base64="NDpwDXpQw97tG5dPpraK0zrXd1Y=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe6CoHZBGwuLCDkTSc6wt9lLluzuHbt7QjjyK2wsVGz9O3b+GzfJFZr4YODx3gwz88KEM21c99sprKyurW8UN0tb2zu7e+X9g3sdp4pQn8Q8Vu0Qa8qZpL5hhtN2oigWIaetcHQ99VtPVGkWy6YZJzQQeCBZxAg2Vnq4fayhLueo2StX3Ko7A1omXk4qkKPRK391+zFJBZWGcKx1x3MTE2RYGUY4nZS6qaYJJiM8oB1LJRZUB9ns4Ak6sUofRbGyJQ2aqb8nMiy0HovQdgpshnrRm4r/eZ3URBdBxmSSGirJfFGUcmRiNP0e9ZmixPCxJZgoZm9FZIgVJsZmVLIheIsvLxO/Vr2sendnlfpVnkYRjuAYTsGDc6jDDTTABwICnuEV3hzlvDjvzse8teDkM4fwB87nD88Zj0w=</latexit>

the resulting estimator is consistent as long as the fraction of observed

data, p, is large enough (+ good Matrix Estimation Black-Box).

For example:

Sum of harmonics with period in {1,…,n}, is sufficientT = !(n2)<latexit sha1_base64="jWhnrV2pw3tYNCLcidHViwfhYro=">AAAB93icbVDLSsNAFJ34rPXRqEs3g0Wom5IUQV0IRTcuKzS20MYymU7aofMIMxOhhn6JGxcqbv0Vd/6N0zYLbT1w4XDOvdx7T5Qwqo3nfTsrq2vrG5uFreL2zu5eyd0/uNcyVZgEWDKp2hHShFFBAkMNI+1EEcQjRlrR6Gbqtx6J0lSKphknJORoIGhMMTJW6rmlJryCXcnJAFXEQ+2055a9qjcDXCZ+TsogR6PnfnX7EqecCIMZ0rrje4kJM6QMxYxMit1UkwThERqQjqUCcaLDbHb4BJ5YpQ9jqWwJA2fq74kMca3HPLKdHJmhXvSm4n9eJzXxRZhRkaSGCDxfFKcMGgmnKcA+VQQbNrYEYUXtrRAPkULY2KyKNgR/8eVlEtSql1X/7qxcv87TKIAjcAwqwAfnoA5uQQMEAIMUPINX8OY8OS/Ou/Mxb11x8plD8AfO5w8QyJGa</latexit><latexit sha1_base64="jWhnrV2pw3tYNCLcidHViwfhYro=">AAAB93icbVDLSsNAFJ34rPXRqEs3g0Wom5IUQV0IRTcuKzS20MYymU7aofMIMxOhhn6JGxcqbv0Vd/6N0zYLbT1w4XDOvdx7T5Qwqo3nfTsrq2vrG5uFreL2zu5eyd0/uNcyVZgEWDKp2hHShFFBAkMNI+1EEcQjRlrR6Gbqtx6J0lSKphknJORoIGhMMTJW6rmlJryCXcnJAFXEQ+2055a9qjcDXCZ+TsogR6PnfnX7EqecCIMZ0rrje4kJM6QMxYxMit1UkwThERqQjqUCcaLDbHb4BJ5YpQ9jqWwJA2fq74kMca3HPLKdHJmhXvSm4n9eJzXxRZhRkaSGCDxfFKcMGgmnKcA+VQQbNrYEYUXtrRAPkULY2KyKNgR/8eVlEtSql1X/7qxcv87TKIAjcAwqwAfnoA5uQQMEAIMUPINX8OY8OS/Ou/Mxb11x8plD8AfO5w8QyJGa</latexit><latexit sha1_base64="jWhnrV2pw3tYNCLcidHViwfhYro=">AAAB93icbVDLSsNAFJ34rPXRqEs3g0Wom5IUQV0IRTcuKzS20MYymU7aofMIMxOhhn6JGxcqbv0Vd/6N0zYLbT1w4XDOvdx7T5Qwqo3nfTsrq2vrG5uFreL2zu5eyd0/uNcyVZgEWDKp2hHShFFBAkMNI+1EEcQjRlrR6Gbqtx6J0lSKphknJORoIGhMMTJW6rmlJryCXcnJAFXEQ+2055a9qjcDXCZ+TsogR6PnfnX7EqecCIMZ0rrje4kJM6QMxYxMit1UkwThERqQjqUCcaLDbHb4BJ5YpQ9jqWwJA2fq74kMca3HPLKdHJmhXvSm4n9eJzXxRZhRkaSGCDxfFKcMGgmnKcA+VQQbNrYEYUXtrRAPkULY2KyKNgR/8eVlEtSql1X/7qxcv87TKIAjcAwqwAfnoA5uQQMEAIMUPINX8OY8OS/Ou/Mxb11x8plD8AfO5w8QyJGa</latexit><latexit sha1_base64="jWhnrV2pw3tYNCLcidHViwfhYro=">AAAB93icbVDLSsNAFJ34rPXRqEs3g0Wom5IUQV0IRTcuKzS20MYymU7aofMIMxOhhn6JGxcqbv0Vd/6N0zYLbT1w4XDOvdx7T5Qwqo3nfTsrq2vrG5uFreL2zu5eyd0/uNcyVZgEWDKp2hHShFFBAkMNI+1EEcQjRlrR6Gbqtx6J0lSKphknJORoIGhMMTJW6rmlJryCXcnJAFXEQ+2055a9qjcDXCZ+TsogR6PnfnX7EqecCIMZ0rrje4kJM6QMxYxMit1UkwThERqQjqUCcaLDbHb4BJ5YpQ9jqWwJA2fq74kMca3HPLKdHJmhXvSm4n9eJzXxRZhRkaSGCDxfFKcMGgmnKcA+VQQbNrYEYUXtrRAPkULY2KyKNgR/8eVlEtSql1X/7qxcv87TKIAjcAwqwAfnoA5uQQMEAIMUPINX8OY8OS/Ou/Mxb11x8plD8AfO5w8QyJGa</latexit>

Contrast with in the best case for additive noiseT = !(n)<latexit sha1_base64="sgZ11mZr3SSY3UcIqS4EzWjAzk4=">AAAB83icbVBNSwMxEJ31s9avqkcvwSLUS9kVQT0IRS8eK3RtoV1KNs22odlkTbKFsvR3ePGg4tU/481/Y9ruQVsfDDzem2FmXphwpo3rfjsrq2vrG5uFreL2zu7efung8FHLVBHqE8mlaoVYU84E9Q0znLYSRXEcctoMh3dTvzmiSjMpGmac0CDGfcEiRrCxUtBAN6gjY9rHFXHWLZXdqjsDWiZeTsqQo94tfXV6kqQxFYZwrHXbcxMTZFgZRjidFDuppgkmQ9ynbUsFjqkOstnRE3RqlR6KpLIlDJqpvycyHGs9jkPbGWMz0IveVPzPa6cmugoyJpLUUEHmi6KUIyPRNAHUY4oSw8eWYKKYvRWRAVaYGJtT0YbgLb68TPzz6nXVe7go127zNApwDCdQAQ8uoQb3UAcfCDzBM7zCmzNyXpx352PeuuLkM0fwB87nD2zmkMU=</latexit><latexit sha1_base64="sgZ11mZr3SSY3UcIqS4EzWjAzk4=">AAAB83icbVBNSwMxEJ31s9avqkcvwSLUS9kVQT0IRS8eK3RtoV1KNs22odlkTbKFsvR3ePGg4tU/481/Y9ruQVsfDDzem2FmXphwpo3rfjsrq2vrG5uFreL2zu7efung8FHLVBHqE8mlaoVYU84E9Q0znLYSRXEcctoMh3dTvzmiSjMpGmac0CDGfcEiRrCxUtBAN6gjY9rHFXHWLZXdqjsDWiZeTsqQo94tfXV6kqQxFYZwrHXbcxMTZFgZRjidFDuppgkmQ9ynbUsFjqkOstnRE3RqlR6KpLIlDJqpvycyHGs9jkPbGWMz0IveVPzPa6cmugoyJpLUUEHmi6KUIyPRNAHUY4oSw8eWYKKYvRWRAVaYGJtT0YbgLb68TPzz6nXVe7go127zNApwDCdQAQ8uoQb3UAcfCDzBM7zCmzNyXpx352PeuuLkM0fwB87nD2zmkMU=</latexit><latexit sha1_base64="sgZ11mZr3SSY3UcIqS4EzWjAzk4=">AAAB83icbVBNSwMxEJ31s9avqkcvwSLUS9kVQT0IRS8eK3RtoV1KNs22odlkTbKFsvR3ePGg4tU/481/Y9ruQVsfDDzem2FmXphwpo3rfjsrq2vrG5uFreL2zu7efung8FHLVBHqE8mlaoVYU84E9Q0znLYSRXEcctoMh3dTvzmiSjMpGmac0CDGfcEiRrCxUtBAN6gjY9rHFXHWLZXdqjsDWiZeTsqQo94tfXV6kqQxFYZwrHXbcxMTZFgZRjidFDuppgkmQ9ynbUsFjqkOstnRE3RqlR6KpLIlDJqpvycyHGs9jkPbGWMz0IveVPzPa6cmugoyJpLUUEHmi6KUIyPRNAHUY4oSw8eWYKKYvRWRAVaYGJtT0YbgLb68TPzz6nXVe7go127zNApwDCdQAQ8uoQb3UAcfCDzBM7zCmzNyXpx352PeuuLkM0fwB87nD2zmkMU=</latexit><latexit sha1_base64="sgZ11mZr3SSY3UcIqS4EzWjAzk4=">AAAB83icbVBNSwMxEJ31s9avqkcvwSLUS9kVQT0IRS8eK3RtoV1KNs22odlkTbKFsvR3ePGg4tU/481/Y9ruQVsfDDzem2FmXphwpo3rfjsrq2vrG5uFreL2zu7efung8FHLVBHqE8mlaoVYU84E9Q0znLYSRXEcctoMh3dTvzmiSjMpGmac0CDGfcEiRrCxUtBAN6gjY9rHFXHWLZXdqjsDWiZeTsqQo94tfXV6kqQxFYZwrHXbcxMTZFgZRjidFDuppgkmQ9ynbUsFjqkOstnRE3RqlR6KpLIlDJqpvycyHGs9jkPbGWMz0IveVPzPa6cmugoyJpLUUEHmi6KUIyPRNAHUY4oSw8eWYKKYvRWRAVaYGJtT0YbgLb68TPzz6nXVe7go127zNApwDCdQAQ8uoQb3UAcfCDzBM7zCmzNyXpx352PeuuLkM0fwB87nD2zmkMU=</latexit>

“Quadratic” loss may be min’l cost of “university” w.r.t. model/noise (?!)

Answer 1: Time Series Imputation

Page 16: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

mixture of periodic, trend and auto-regressivewith additive zero-mean noise and randomly missing values

Answer 1: Time Series Imputation

Page 17: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

mixture of periodic, trend and auto-regressivewith Poisson “noise” and randomly missing values

Answer 1: Time Series Imputation

Page 18: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

mixture of periodic, trend and auto-regressivewith Poisson “noise” and randomly missing values

Answer 1: Time Series Imputation

Page 19: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Answer 2: Time Series Forecasting

“Ground Truth” of interest: , for examplef(t), t 2 R<latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit><latexit sha1_base64="THq1vN8lh7PIVhCPVCBEPiNJzPQ=">AAAB/3icbVBNS8NAEN3Ur1q/qh48eFksQgUpiQjqrejFYxVrC00om+2mXbrZhN2JUEI9+Fe8eFDx6t/w5r9x0+agrQ8GHu/NMDPPjwXXYNvfVmFhcWl5pbhaWlvf2Nwqb+/c6yhRlDVpJCLV9olmgkvWBA6CtWPFSOgL1vKHV5nfemBK80jewShmXkj6kgecEjBSt7wXVOHoGD8CdrnEbkhg4Pvp7bhbrtg1ewI8T5ycVFCORrf85fYimoRMAhVE645jx+ClRAGngo1LbqJZTOiQ9FnHUElCpr108sAYHxqlh4NImZKAJ+rviZSEWo9C33RmF+pZLxP/8zoJBOdeymWcAJN0uihIBIYIZ2ngHleMghgZQqji5lZMB0QRCiazkgnBmX15njRPahc15+a0Ur/M0yiifXSAqshBZ6iOrlEDNRFFY/SMXtGb9WS9WO/Wx7S1YOUzu+gPrM8flSSVVQ==</latexit>

f(t) =KX

k=1

↵k sin(!kt) + �k cos(!kt)<latexit sha1_base64="JMQZ33PW5zdOv88wLf+LVSQ9+Fs=">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</latexit><latexit sha1_base64="JMQZ33PW5zdOv88wLf+LVSQ9+Fs=">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</latexit><latexit sha1_base64="JMQZ33PW5zdOv88wLf+LVSQ9+Fs=">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</latexit><latexit sha1_base64="JMQZ33PW5zdOv88wLf+LVSQ9+Fs=">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</latexit>

f(t) =KX

k=1

↵kt�k

<latexit sha1_base64="d+XGi/F38Dd4v4ljmf5IQX7cdBI=">AAACEHicbZC7SgNBFIZn4y3G26qlzWAQYhN2RVCLQNBGsIngmkA2WWYns8mQ2QszZ4Ww7CvY+Co2Fiq2lna+jZNLoYk/DHz85xzOnN9PBFdgWd9GYWl5ZXWtuF7a2Nza3jF39+5VnErKHBqLWLZ8opjgEXOAg2CtRDIS+oI1/eHVuN58YFLxOLqDUcI6IelHPOCUgLY8sxJU4BjXsKvS0MuGNTvv3mCXiGRAvCGGbub6DDTmnlm2qtZEeBHsGZTRTA3P/HJ7MU1DFgEVRKm2bSXQyYgETgXLS26qWELokPRZW2NEQqY62eSiHB9pp4eDWOoXAZ64vycyEio1Cn3dGRIYqPna2Pyv1k4hOO9kPEpSYBGdLgpSgSHG43hwj0tGQYw0ECq5/iumAyIJBR1iSYdgz5+8CM5J9aJq356W65ezNIroAB2iCrLRGaqja9RADqLoET2jV/RmPBkvxrvxMW0tGLOZffRHxucPKy2cOQ==</latexit><latexit sha1_base64="d+XGi/F38Dd4v4ljmf5IQX7cdBI=">AAACEHicbZC7SgNBFIZn4y3G26qlzWAQYhN2RVCLQNBGsIngmkA2WWYns8mQ2QszZ4Ww7CvY+Co2Fiq2lna+jZNLoYk/DHz85xzOnN9PBFdgWd9GYWl5ZXWtuF7a2Nza3jF39+5VnErKHBqLWLZ8opjgEXOAg2CtRDIS+oI1/eHVuN58YFLxOLqDUcI6IelHPOCUgLY8sxJU4BjXsKvS0MuGNTvv3mCXiGRAvCGGbub6DDTmnlm2qtZEeBHsGZTRTA3P/HJ7MU1DFgEVRKm2bSXQyYgETgXLS26qWELokPRZW2NEQqY62eSiHB9pp4eDWOoXAZ64vycyEio1Cn3dGRIYqPna2Pyv1k4hOO9kPEpSYBGdLgpSgSHG43hwj0tGQYw0ECq5/iumAyIJBR1iSYdgz5+8CM5J9aJq356W65ezNIroAB2iCrLRGaqja9RADqLoET2jV/RmPBkvxrvxMW0tGLOZffRHxucPKy2cOQ==</latexit><latexit sha1_base64="d+XGi/F38Dd4v4ljmf5IQX7cdBI=">AAACEHicbZC7SgNBFIZn4y3G26qlzWAQYhN2RVCLQNBGsIngmkA2WWYns8mQ2QszZ4Ww7CvY+Co2Fiq2lna+jZNLoYk/DHz85xzOnN9PBFdgWd9GYWl5ZXWtuF7a2Nza3jF39+5VnErKHBqLWLZ8opjgEXOAg2CtRDIS+oI1/eHVuN58YFLxOLqDUcI6IelHPOCUgLY8sxJU4BjXsKvS0MuGNTvv3mCXiGRAvCGGbub6DDTmnlm2qtZEeBHsGZTRTA3P/HJ7MU1DFgEVRKm2bSXQyYgETgXLS26qWELokPRZW2NEQqY62eSiHB9pp4eDWOoXAZ64vycyEio1Cn3dGRIYqPna2Pyv1k4hOO9kPEpSYBGdLgpSgSHG43hwj0tGQYw0ECq5/iumAyIJBR1iSYdgz5+8CM5J9aJq356W65ezNIroAB2iCrLRGaqja9RADqLoET2jV/RmPBkvxrvxMW0tGLOZffRHxucPKy2cOQ==</latexit><latexit sha1_base64="d+XGi/F38Dd4v4ljmf5IQX7cdBI=">AAACEHicbZC7SgNBFIZn4y3G26qlzWAQYhN2RVCLQNBGsIngmkA2WWYns8mQ2QszZ4Ww7CvY+Co2Fiq2lna+jZNLoYk/DHz85xzOnN9PBFdgWd9GYWl5ZXWtuF7a2Nza3jF39+5VnErKHBqLWLZ8opjgEXOAg2CtRDIS+oI1/eHVuN58YFLxOLqDUcI6IelHPOCUgLY8sxJU4BjXsKvS0MuGNTvv3mCXiGRAvCGGbub6DDTmnlm2qtZEeBHsGZTRTA3P/HJ7MU1DFgEVRKm2bSXQyYgETgXLS26qWELokPRZW2NEQqY62eSiHB9pp4eDWOoXAZ64vycyEio1Cn3dGRIYqPna2Pyv1k4hOO9kPEpSYBGdLgpSgSHG43hwj0tGQYw0ECq5/iumAyIJBR1iSYdgz5+8CM5J9aJq356W65ezNIroAB2iCrLRGaqja9RADqLoET2jV/RmPBkvxrvxMW0tGLOZffRHxucPKy2cOQ==</latexit>

f(t) =KX

k=1

↵kf(t� k)<latexit sha1_base64="EOwrRrym2h8aWJBvFq6Oag9DSYE=">AAACDHicbVDLSsNAFJ3UV62vqEs3g1VoF5ZEBHVRKLoR3FSwttDEMJlO2iEzSZiZCCX0B9z4K25cqLj1A9z5N07bLLT1wIXDOfdy7z1+wqhUlvVtFBYWl5ZXiqultfWNzS1ze+dOxqnApIVjFouOjyRhNCItRRUjnUQQxH1G2n54OfbbD0RIGke3apgQl6N+RAOKkdKSZx4EFVWFdejIlHtZWLdH99fQQSwZIC+E2jwKq9Azy1bNmgDOEzsnZZCj6ZlfTi/GKSeRwgxJ2bWtRLkZEopiRkYlJ5UkQThEfdLVNEKcSDebfDOCh1rpwSAWuiIFJ+rviQxxKYfc150cqYGc9cbif143VcGZm9EoSRWJ8HRRkDKoYjiOBvaoIFixoSYIC6pvhXiABMJKB1jSIdizL8+T1nHtvGbfnJQbF3kaRbAH9kEF2OAUNMAVaIIWwOARPINX8GY8GS/Gu/ExbS0Y+cwu+APj8wdZsplo</latexit><latexit sha1_base64="EOwrRrym2h8aWJBvFq6Oag9DSYE=">AAACDHicbVDLSsNAFJ3UV62vqEs3g1VoF5ZEBHVRKLoR3FSwttDEMJlO2iEzSZiZCCX0B9z4K25cqLj1A9z5N07bLLT1wIXDOfdy7z1+wqhUlvVtFBYWl5ZXiqultfWNzS1ze+dOxqnApIVjFouOjyRhNCItRRUjnUQQxH1G2n54OfbbD0RIGke3apgQl6N+RAOKkdKSZx4EFVWFdejIlHtZWLdH99fQQSwZIC+E2jwKq9Azy1bNmgDOEzsnZZCj6ZlfTi/GKSeRwgxJ2bWtRLkZEopiRkYlJ5UkQThEfdLVNEKcSDebfDOCh1rpwSAWuiIFJ+rviQxxKYfc150cqYGc9cbif143VcGZm9EoSRWJ8HRRkDKoYjiOBvaoIFixoSYIC6pvhXiABMJKB1jSIdizL8+T1nHtvGbfnJQbF3kaRbAH9kEF2OAUNMAVaIIWwOARPINX8GY8GS/Gu/ExbS0Y+cwu+APj8wdZsplo</latexit><latexit sha1_base64="EOwrRrym2h8aWJBvFq6Oag9DSYE=">AAACDHicbVDLSsNAFJ3UV62vqEs3g1VoF5ZEBHVRKLoR3FSwttDEMJlO2iEzSZiZCCX0B9z4K25cqLj1A9z5N07bLLT1wIXDOfdy7z1+wqhUlvVtFBYWl5ZXiqultfWNzS1ze+dOxqnApIVjFouOjyRhNCItRRUjnUQQxH1G2n54OfbbD0RIGke3apgQl6N+RAOKkdKSZx4EFVWFdejIlHtZWLdH99fQQSwZIC+E2jwKq9Azy1bNmgDOEzsnZZCj6ZlfTi/GKSeRwgxJ2bWtRLkZEopiRkYlJ5UkQThEfdLVNEKcSDebfDOCh1rpwSAWuiIFJ+rviQxxKYfc150cqYGc9cbif143VcGZm9EoSRWJ8HRRkDKoYjiOBvaoIFixoSYIC6pvhXiABMJKB1jSIdizL8+T1nHtvGbfnJQbF3kaRbAH9kEF2OAUNMAVaIIWwOARPINX8GY8GS/Gu/ExbS0Y+cwu+APj8wdZsplo</latexit><latexit sha1_base64="EOwrRrym2h8aWJBvFq6Oag9DSYE=">AAACDHicbVDLSsNAFJ3UV62vqEs3g1VoF5ZEBHVRKLoR3FSwttDEMJlO2iEzSZiZCCX0B9z4K25cqLj1A9z5N07bLLT1wIXDOfdy7z1+wqhUlvVtFBYWl5ZXiqultfWNzS1ze+dOxqnApIVjFouOjyRhNCItRRUjnUQQxH1G2n54OfbbD0RIGke3apgQl6N+RAOKkdKSZx4EFVWFdejIlHtZWLdH99fQQSwZIC+E2jwKq9Azy1bNmgDOEzsnZZCj6ZlfTi/GKSeRwgxJ2bWtRLkZEopiRkYlJ5UkQThEfdLVNEKcSDebfDOCh1rpwSAWuiIFJ+rviQxxKYfc150cqYGc9cbif143VcGZm9EoSRWJ8HRRkDKoYjiOBvaoIFixoSYIC6pvhXiABMJKB1jSIdizL8+T1nHtvGbfnJQbF3kaRbAH9kEF2OAUNMAVaIIWwOARPINX8GY8GS/Gu/ExbS0Y+cwu+APj8wdZsplo</latexit>

or or, their combination….

Observation: s. t. if observed (w.p. p)X(t), t 2 {0, 1, . . . , T}<latexit sha1_base64="yohLOkCg8/GZ2+mfaDnBKHv4v4w=">AAACCHicbVBNS8NAFNz4WetX1KOXxSJUKCURQb0VvXis0NhCE8pmu22XbjZh90UooR69+Fe8eFDx6k/w5r9x2+agrQMLw8wb3r4JE8E1OM63tbS8srq2Xtgobm5t7+zae/t3Ok4VZR6NRaxaIdFMcMk84CBYK1GMRKFgzXB4PfGb90xpHssGjBIWRKQveY9TAkbq2LhVhpMKfgDsc4n9zKlgt4L9bgy6ghv+GHfsklN1psCLxM1JCeWod+wvk6ZpxCRQQbRuu04CQUYUcCrYuOinmiWEDkmftQ2VJGI6yKaXjPGxUbq4FyvzJOCp+juRkUjrURSayYjAQM97E/E/r51C7yLIuExSYJLOFvVSgSHGk1pwlytGQYwMIVRx81dMB0QRCqa8oinBnT95kXin1cuqe3tWql3lbRTQITpCZeSic1RDN6iOPETRI3pGr+jNerJerHfrYza6ZOWZA/QH1ucPMsmXDA==</latexit><latexit sha1_base64="yohLOkCg8/GZ2+mfaDnBKHv4v4w=">AAACCHicbVBNS8NAFNz4WetX1KOXxSJUKCURQb0VvXis0NhCE8pmu22XbjZh90UooR69+Fe8eFDx6k/w5r9x2+agrQMLw8wb3r4JE8E1OM63tbS8srq2Xtgobm5t7+zae/t3Ok4VZR6NRaxaIdFMcMk84CBYK1GMRKFgzXB4PfGb90xpHssGjBIWRKQveY9TAkbq2LhVhpMKfgDsc4n9zKlgt4L9bgy6ghv+GHfsklN1psCLxM1JCeWod+wvk6ZpxCRQQbRuu04CQUYUcCrYuOinmiWEDkmftQ2VJGI6yKaXjPGxUbq4FyvzJOCp+juRkUjrURSayYjAQM97E/E/r51C7yLIuExSYJLOFvVSgSHGk1pwlytGQYwMIVRx81dMB0QRCqa8oinBnT95kXin1cuqe3tWql3lbRTQITpCZeSic1RDN6iOPETRI3pGr+jNerJerHfrYza6ZOWZA/QH1ucPMsmXDA==</latexit><latexit sha1_base64="yohLOkCg8/GZ2+mfaDnBKHv4v4w=">AAACCHicbVBNS8NAFNz4WetX1KOXxSJUKCURQb0VvXis0NhCE8pmu22XbjZh90UooR69+Fe8eFDx6k/w5r9x2+agrQMLw8wb3r4JE8E1OM63tbS8srq2Xtgobm5t7+zae/t3Ok4VZR6NRaxaIdFMcMk84CBYK1GMRKFgzXB4PfGb90xpHssGjBIWRKQveY9TAkbq2LhVhpMKfgDsc4n9zKlgt4L9bgy6ghv+GHfsklN1psCLxM1JCeWod+wvk6ZpxCRQQbRuu04CQUYUcCrYuOinmiWEDkmftQ2VJGI6yKaXjPGxUbq4FyvzJOCp+juRkUjrURSayYjAQM97E/E/r51C7yLIuExSYJLOFvVSgSHGk1pwlytGQYwMIVRx81dMB0QRCqa8oinBnT95kXin1cuqe3tWql3lbRTQITpCZeSic1RDN6iOPETRI3pGr+jNerJerHfrYza6ZOWZA/QH1ucPMsmXDA==</latexit><latexit sha1_base64="yohLOkCg8/GZ2+mfaDnBKHv4v4w=">AAACCHicbVBNS8NAFNz4WetX1KOXxSJUKCURQb0VvXis0NhCE8pmu22XbjZh90UooR69+Fe8eFDx6k/w5r9x2+agrQMLw8wb3r4JE8E1OM63tbS8srq2Xtgobm5t7+zae/t3Ok4VZR6NRaxaIdFMcMk84CBYK1GMRKFgzXB4PfGb90xpHssGjBIWRKQveY9TAkbq2LhVhpMKfgDsc4n9zKlgt4L9bgy6ghv+GHfsklN1psCLxM1JCeWod+wvk6ZpxCRQQbRuu04CQUYUcCrYuOinmiWEDkmftQ2VJGI6yKaXjPGxUbq4FyvzJOCp+juRkUjrURSayYjAQM97E/E/r51C7yLIuExSYJLOFvVSgSHGk1pwlytGQYwMIVRx81dMB0QRCqa8oinBnT95kXin1cuqe3tWql3lbRTQITpCZeSic1RDN6iOPETRI3pGr+jNerJerHfrYza6ZOWZA/QH1ucPMsmXDA==</latexit>

E[X(t)] = f(t)<latexit sha1_base64="957RoC9KjYPGJnd/idpIfrEtmAw=">AAACAHicbVDLSsNAFL2pr1pfUTeCm8Ei1E1JRFAXQlEElxWMLaShTKaTdujkwcxEKKFu/BU3LlTc+hnu/BsnbRbaemCYwzn3cu89fsKZVJb1bZQWFpeWV8qrlbX1jc0tc3vnXsapINQhMY9F28eSchZRRzHFaTsRFIc+py1/eJX7rQcqJIujOzVKqBfifsQCRrDSUtfc64RYDXw/ux677Zo68tAFCvRf6ZpVq25NgOaJXZAqFGh2za9OLyZpSCNFOJbSta1EeRkWihFOx5VOKmmCyRD3qatphEMqvWxywRgdaqWHgljoFyk0UX93ZDiUchT6ujLfV856ufif56YqOPMyFiWpohGZDgpSjlSM8jhQjwlKFB9pgolgeldEBlhgonRoeQj27MnzxDmun9ft25Nq47JIowz7cAA1sOEUGnADTXCAwCM8wyu8GU/Gi/FufExLS0bRswt/YHz+ADqLlP0=</latexit><latexit sha1_base64="957RoC9KjYPGJnd/idpIfrEtmAw=">AAACAHicbVDLSsNAFL2pr1pfUTeCm8Ei1E1JRFAXQlEElxWMLaShTKaTdujkwcxEKKFu/BU3LlTc+hnu/BsnbRbaemCYwzn3cu89fsKZVJb1bZQWFpeWV8qrlbX1jc0tc3vnXsapINQhMY9F28eSchZRRzHFaTsRFIc+py1/eJX7rQcqJIujOzVKqBfifsQCRrDSUtfc64RYDXw/ux677Zo68tAFCvRf6ZpVq25NgOaJXZAqFGh2za9OLyZpSCNFOJbSta1EeRkWihFOx5VOKmmCyRD3qatphEMqvWxywRgdaqWHgljoFyk0UX93ZDiUchT6ujLfV856ufif56YqOPMyFiWpohGZDgpSjlSM8jhQjwlKFB9pgolgeldEBlhgonRoeQj27MnzxDmun9ft25Nq47JIowz7cAA1sOEUGnADTXCAwCM8wyu8GU/Gi/FufExLS0bRswt/YHz+ADqLlP0=</latexit><latexit sha1_base64="957RoC9KjYPGJnd/idpIfrEtmAw=">AAACAHicbVDLSsNAFL2pr1pfUTeCm8Ei1E1JRFAXQlEElxWMLaShTKaTdujkwcxEKKFu/BU3LlTc+hnu/BsnbRbaemCYwzn3cu89fsKZVJb1bZQWFpeWV8qrlbX1jc0tc3vnXsapINQhMY9F28eSchZRRzHFaTsRFIc+py1/eJX7rQcqJIujOzVKqBfifsQCRrDSUtfc64RYDXw/ux677Zo68tAFCvRf6ZpVq25NgOaJXZAqFGh2za9OLyZpSCNFOJbSta1EeRkWihFOx5VOKmmCyRD3qatphEMqvWxywRgdaqWHgljoFyk0UX93ZDiUchT6ujLfV856ufif56YqOPMyFiWpohGZDgpSjlSM8jhQjwlKFB9pgolgeldEBlhgonRoeQj27MnzxDmun9ft25Nq47JIowz7cAA1sOEUGnADTXCAwCM8wyu8GU/Gi/FufExLS0bRswt/YHz+ADqLlP0=</latexit><latexit sha1_base64="957RoC9KjYPGJnd/idpIfrEtmAw=">AAACAHicbVDLSsNAFL2pr1pfUTeCm8Ei1E1JRFAXQlEElxWMLaShTKaTdujkwcxEKKFu/BU3LlTc+hnu/BsnbRbaemCYwzn3cu89fsKZVJb1bZQWFpeWV8qrlbX1jc0tc3vnXsapINQhMY9F28eSchZRRzHFaTsRFIc+py1/eJX7rQcqJIujOzVKqBfifsQCRrDSUtfc64RYDXw/ux677Zo68tAFCvRf6ZpVq25NgOaJXZAqFGh2za9OLyZpSCNFOJbSta1EeRkWihFOx5VOKmmCyRD3qatphEMqvWxywRgdaqWHgljoFyk0UX93ZDiUchT6ujLfV856ufif56YqOPMyFiWpohGZDgpSjlSM8jhQjwlKFB9pgolgeldEBlhgonRoeQj27MnzxDmun9ft25Nq47JIowz7cAA1sOEUGnADTXCAwCM8wyu8GU/Gi/FufExLS0bRswt/YHz+ADqLlP0=</latexit>

(+ independence, conditions on “noise”)

Goal: produce estimate so that X(T + 1)<latexit sha1_base64="Q3RyyilO7LmaIuCiBmFKNX/KjVA=">AAAB8nicbVBNS8NAEJ34WetX1aOXYBEqQklEUG9FLx4rNLbQhLLZbtqlm03YnQgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZemAqu0XG+rZXVtfWNzdJWeXtnd2+/cnD4qJNMUebRRCSqExLNBJfMQ46CdVLFSBwK1g5Hd1O//cSU5ols4ThlQUwGkkecEjSS7w8J5p1JrXXunvUqVafuzGAvE7cgVSjQ7FW+/H5Cs5hJpIJo3XWdFIOcKORUsEnZzzRLCR2RAesaKknMdJDPbp7Yp0bp21GiTEm0Z+rviZzEWo/j0HTGBId60ZuK/3ndDKPrIOcyzZBJOl8UZcLGxJ4GYPe5YhTF2BBCFTe32nRIFKFoYiqbENzFl5eJd1G/qbsPl9XGbZFGCY7hBGrgwhU04B6a4AGFFJ7hFd6szHqx3q2PeeuKVcwcwR9Ynz85m5Cw</latexit><latexit sha1_base64="Q3RyyilO7LmaIuCiBmFKNX/KjVA=">AAAB8nicbVBNS8NAEJ34WetX1aOXYBEqQklEUG9FLx4rNLbQhLLZbtqlm03YnQgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZemAqu0XG+rZXVtfWNzdJWeXtnd2+/cnD4qJNMUebRRCSqExLNBJfMQ46CdVLFSBwK1g5Hd1O//cSU5ols4ThlQUwGkkecEjSS7w8J5p1JrXXunvUqVafuzGAvE7cgVSjQ7FW+/H5Cs5hJpIJo3XWdFIOcKORUsEnZzzRLCR2RAesaKknMdJDPbp7Yp0bp21GiTEm0Z+rviZzEWo/j0HTGBId60ZuK/3ndDKPrIOcyzZBJOl8UZcLGxJ4GYPe5YhTF2BBCFTe32nRIFKFoYiqbENzFl5eJd1G/qbsPl9XGbZFGCY7hBGrgwhU04B6a4AGFFJ7hFd6szHqx3q2PeeuKVcwcwR9Ynz85m5Cw</latexit><latexit sha1_base64="Q3RyyilO7LmaIuCiBmFKNX/KjVA=">AAAB8nicbVBNS8NAEJ34WetX1aOXYBEqQklEUG9FLx4rNLbQhLLZbtqlm03YnQgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZemAqu0XG+rZXVtfWNzdJWeXtnd2+/cnD4qJNMUebRRCSqExLNBJfMQ46CdVLFSBwK1g5Hd1O//cSU5ols4ThlQUwGkkecEjSS7w8J5p1JrXXunvUqVafuzGAvE7cgVSjQ7FW+/H5Cs5hJpIJo3XWdFIOcKORUsEnZzzRLCR2RAesaKknMdJDPbp7Yp0bp21GiTEm0Z+rviZzEWo/j0HTGBId60ZuK/3ndDKPrIOcyzZBJOl8UZcLGxJ4GYPe5YhTF2BBCFTe32nRIFKFoYiqbENzFl5eJd1G/qbsPl9XGbZFGCY7hBGrgwhU04B6a4AGFFJ7hFd6szHqx3q2PeeuKVcwcwR9Ynz85m5Cw</latexit><latexit sha1_base64="Q3RyyilO7LmaIuCiBmFKNX/KjVA=">AAAB8nicbVBNS8NAEJ34WetX1aOXYBEqQklEUG9FLx4rNLbQhLLZbtqlm03YnQgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZemAqu0XG+rZXVtfWNzdJWeXtnd2+/cnD4qJNMUebRRCSqExLNBJfMQ46CdVLFSBwK1g5Hd1O//cSU5ols4ThlQUwGkkecEjSS7w8J5p1JrXXunvUqVafuzGAvE7cgVSjQ7FW+/H5Cs5hJpIJo3XWdFIOcKORUsEnZzzRLCR2RAesaKknMdJDPbp7Yp0bp21GiTEm0Z+rviZzEWo/j0HTGBId60ZuK/3ndDKPrIOcyzZBJOl8UZcLGxJ4GYPe5YhTF2BBCFTe32nRIFKFoYiqbENzFl5eJd1G/qbsPl9XGbZFGCY7hBGrgwhU04B6a4AGFFJ7hFd6szHqx3q2PeeuKVcwcwR9Ynz85m5Cw</latexit>

Eh�X(T + 1)�X(T + 1)

�2iis small

<latexit sha1_base64="zbDdxxoJCrcQmU7udqggp6rXaYg=">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</latexit><latexit sha1_base64="zbDdxxoJCrcQmU7udqggp6rXaYg=">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</latexit><latexit sha1_base64="zbDdxxoJCrcQmU7udqggp6rXaYg=">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</latexit><latexit sha1_base64="zbDdxxoJCrcQmU7udqggp6rXaYg=">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</latexit>

Page 20: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Algorithm:Transform to Matrix, Do Matrix Estimation, Regression, Prediction

X(1) X(2) … X(L) X(L+1) … X(2L) … … …

X(1)

X(2)

X(L)

X(L+1)

X(L+2)

X(2L)

… … …X 1 ⌘<latexit sha1_base64="rZtUOnTMiDbkJkSzT0PkVaiNM74=">AAAB+XicbVBNS8NAEN34WetXqkcvi0XwVBIR1FvRi8cKxhaaWDbbSbt0s4m7m0qJ/SlePKh49Z9489+4bXPQ1gcDj/dmmJkXppwp7Tjf1tLyyuraemmjvLm1vbNrV/buVJJJCh5NeCJbIVHAmQBPM82hlUogccihGQ6uJn5zCFKxRNzqUQpBTHqCRYwSbaSOXcl9Sjhuje9d7MNDxoYdu+rUnCnwInELUkUFGh37y+8mNItBaMqJUm3XSXWQE6kZ5TAu+5mClNAB6UHbUEFiUEE+PX2Mj4zSxVEiTQmNp+rviZzESo3i0HTGRPfVvDcR//PamY7Og5yJNNMg6GxRlHGsEzzJAXeZBKr5yBBCJTO3YtonklBt0iqbENz5lxeJd1K7qLk3p9X6ZZFGCR2gQ3SMXHSG6ugaNZCHKHpEz+gVvVlP1ov1bn3MWpesYmYf/YH1+QPzoJNa</latexit><latexit sha1_base64="rZtUOnTMiDbkJkSzT0PkVaiNM74=">AAAB+XicbVBNS8NAEN34WetXqkcvi0XwVBIR1FvRi8cKxhaaWDbbSbt0s4m7m0qJ/SlePKh49Z9489+4bXPQ1gcDj/dmmJkXppwp7Tjf1tLyyuraemmjvLm1vbNrV/buVJJJCh5NeCJbIVHAmQBPM82hlUogccihGQ6uJn5zCFKxRNzqUQpBTHqCRYwSbaSOXcl9Sjhuje9d7MNDxoYdu+rUnCnwInELUkUFGh37y+8mNItBaMqJUm3XSXWQE6kZ5TAu+5mClNAB6UHbUEFiUEE+PX2Mj4zSxVEiTQmNp+rviZzESo3i0HTGRPfVvDcR//PamY7Og5yJNNMg6GxRlHGsEzzJAXeZBKr5yBBCJTO3YtonklBt0iqbENz5lxeJd1K7qLk3p9X6ZZFGCR2gQ3SMXHSG6ugaNZCHKHpEz+gVvVlP1ov1bn3MWpesYmYf/YH1+QPzoJNa</latexit><latexit sha1_base64="rZtUOnTMiDbkJkSzT0PkVaiNM74=">AAAB+XicbVBNS8NAEN34WetXqkcvi0XwVBIR1FvRi8cKxhaaWDbbSbt0s4m7m0qJ/SlePKh49Z9489+4bXPQ1gcDj/dmmJkXppwp7Tjf1tLyyuraemmjvLm1vbNrV/buVJJJCh5NeCJbIVHAmQBPM82hlUogccihGQ6uJn5zCFKxRNzqUQpBTHqCRYwSbaSOXcl9Sjhuje9d7MNDxoYdu+rUnCnwInELUkUFGh37y+8mNItBaMqJUm3XSXWQE6kZ5TAu+5mClNAB6UHbUEFiUEE+PX2Mj4zSxVEiTQmNp+rviZzESo3i0HTGRPfVvDcR//PamY7Og5yJNNMg6GxRlHGsEzzJAXeZBKr5yBBCJTO3YtonklBt0iqbENz5lxeJd1K7qLk3p9X6ZZFGCR2gQ3SMXHSG6ugaNZCHKHpEz+gVvVlP1ov1bn3MWpesYmYf/YH1+QPzoJNa</latexit><latexit sha1_base64="rZtUOnTMiDbkJkSzT0PkVaiNM74=">AAAB+XicbVBNS8NAEN34WetXqkcvi0XwVBIR1FvRi8cKxhaaWDbbSbt0s4m7m0qJ/SlePKh49Z9489+4bXPQ1gcDj/dmmJkXppwp7Tjf1tLyyuraemmjvLm1vbNrV/buVJJJCh5NeCJbIVHAmQBPM82hlUogccihGQ6uJn5zCFKxRNzqUQpBTHqCRYwSbaSOXcl9Sjhuje9d7MNDxoYdu+rUnCnwInELUkUFGh37y+8mNItBaMqJUm3XSXWQE6kZ5TAu+5mClNAB6UHbUEFiUEE+PX2Mj4zSxVEiTQmNp+rviZzESo3i0HTGRPfVvDcR//PamY7Og5yJNNMg6GxRlHGsEzzJAXeZBKr5yBBCJTO3YtonklBt0iqbENz5lxeJd1K7qLk3p9X6ZZFGCR2gQ3SMXHSG6ugaNZCHKHpEz+gVvVlP1ov1bn3MWpesYmYf/YH1+QPzoJNa</latexit>

Answer 2: Time Series Forecasting

Page 21: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Algorithm:Transform to Matrix, Do Matrix Estimation, Regression, Prediction

X(k) … X(L+k-1) X(L+k)…X(2L+k-1) … … …

X(k)

X(k+1)

X(L+k-1)

X(L+k)

X(L+k+1)

X(2L+k-1)

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Answer 2: Time Series Forecasting

Page 22: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Algorithm:Transform to Matrix, Do Matrix Estimation, Regression, Prediction

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BB-ME

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Target

FeaturesLinear

Regression

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Transform to Matrix, Do Matrix Estimation, Regression, Prediction

Answer 2: Time Series Forecasting

Page 23: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Algorithm:Transform to Matrix, Do Matrix Estimation, Regression, Prediction

[X(T-L+2) … X(T)]

Project on Coln Space of Mk

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Inner product with �k

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X(T + 1)<latexit sha1_base64="Q3RyyilO7LmaIuCiBmFKNX/KjVA=">AAAB8nicbVBNS8NAEJ34WetX1aOXYBEqQklEUG9FLx4rNLbQhLLZbtqlm03YnQgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZemAqu0XG+rZXVtfWNzdJWeXtnd2+/cnD4qJNMUebRRCSqExLNBJfMQ46CdVLFSBwK1g5Hd1O//cSU5ols4ThlQUwGkkecEjSS7w8J5p1JrXXunvUqVafuzGAvE7cgVSjQ7FW+/H5Cs5hJpIJo3XWdFIOcKORUsEnZzzRLCR2RAesaKknMdJDPbp7Yp0bp21GiTEm0Z+rviZzEWo/j0HTGBId60ZuK/3ndDKPrIOcyzZBJOl8UZcLGxJ4GYPe5YhTF2BBCFTe32nRIFKFoYiqbENzFl5eJd1G/qbsPl9XGbZFGCY7hBGrgwhU04B6a4AGFFJ7hFd6szHqx3q2PeeuKVcwcwR9Ynz85m5Cw</latexit><latexit sha1_base64="Q3RyyilO7LmaIuCiBmFKNX/KjVA=">AAAB8nicbVBNS8NAEJ34WetX1aOXYBEqQklEUG9FLx4rNLbQhLLZbtqlm03YnQgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZemAqu0XG+rZXVtfWNzdJWeXtnd2+/cnD4qJNMUebRRCSqExLNBJfMQ46CdVLFSBwK1g5Hd1O//cSU5ols4ThlQUwGkkecEjSS7w8J5p1JrXXunvUqVafuzGAvE7cgVSjQ7FW+/H5Cs5hJpIJo3XWdFIOcKORUsEnZzzRLCR2RAesaKknMdJDPbp7Yp0bp21GiTEm0Z+rviZzEWo/j0HTGBId60ZuK/3ndDKPrIOcyzZBJOl8UZcLGxJ4GYPe5YhTF2BBCFTe32nRIFKFoYiqbENzFl5eJd1G/qbsPl9XGbZFGCY7hBGrgwhU04B6a4AGFFJ7hFd6szHqx3q2PeeuKVcwcwR9Ynz85m5Cw</latexit><latexit sha1_base64="Q3RyyilO7LmaIuCiBmFKNX/KjVA=">AAAB8nicbVBNS8NAEJ34WetX1aOXYBEqQklEUG9FLx4rNLbQhLLZbtqlm03YnQgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZemAqu0XG+rZXVtfWNzdJWeXtnd2+/cnD4qJNMUebRRCSqExLNBJfMQ46CdVLFSBwK1g5Hd1O//cSU5ols4ThlQUwGkkecEjSS7w8J5p1JrXXunvUqVafuzGAvE7cgVSjQ7FW+/H5Cs5hJpIJo3XWdFIOcKORUsEnZzzRLCR2RAesaKknMdJDPbp7Yp0bp21GiTEm0Z+rviZzEWo/j0HTGBId60ZuK/3ndDKPrIOcyzZBJOl8UZcLGxJ4GYPe5YhTF2BBCFTe32nRIFKFoYiqbENzFl5eJd1G/qbsPl9XGbZFGCY7hBGrgwhU04B6a4AGFFJ7hFd6szHqx3q2PeeuKVcwcwR9Ynz85m5Cw</latexit><latexit sha1_base64="Q3RyyilO7LmaIuCiBmFKNX/KjVA=">AAAB8nicbVBNS8NAEJ34WetX1aOXYBEqQklEUG9FLx4rNLbQhLLZbtqlm03YnQgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZemAqu0XG+rZXVtfWNzdJWeXtnd2+/cnD4qJNMUebRRCSqExLNBJfMQ46CdVLFSBwK1g5Hd1O//cSU5ols4ThlQUwGkkecEjSS7w8J5p1JrXXunvUqVafuzGAvE7cgVSjQ7FW+/H5Cs5hJpIJo3XWdFIOcKORUsEnZzzRLCR2RAesaKknMdJDPbp7Yp0bp21GiTEm0Z+rviZzEWo/j0HTGBId60ZuK/3ndDKPrIOcyzZBJOl8UZcLGxJ4GYPe5YhTF2BBCFTe32nRIFKFoYiqbENzFl5eJd1G/qbsPl9XGbZFGCY7hBGrgwhU04B6a4AGFFJ7hFd6szHqx3q2PeeuKVcwcwR9Ynz85m5Cw</latexit>

k =

�T + 1 mod L

�+ 1

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where

Answer 2: Time Series Forecasting

Page 24: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Theorem (Informal):

For L large enough (depending upon model params) with

the expectation of the transformed matrix obeys approx “linear regression”

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structure in addition to Latent Variable Model. For additive symmetric

noise model with good Matrix Estimation Black-Box

Eh(X(T + 1)�X(T + 1))2

i C�2/p

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where is noise variance, C is a universal constant�2<latexit sha1_base64="DRe8q9G8lOKJt3WChWHyv0OBjjM=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWMLbQxrLZbtqlu5u4uxFK6J/w4kHFq7/Hm//GTZuDtj4YeLw3w8y8MOFMG9f9dkorq2vrG+XNytb2zu5edf/gXsepItQnMY9VJ8Saciapb5jhtJMoikXIaTscX+d++4kqzWJ5ZyYJDQQeShYxgo2VOj3NhgI/NPrVmlt3Z0DLxCtIDQq0+tWv3iAmqaDSEI617npuYoIMK8MIp9NKL9U0wWSMh7RrqcSC6iCb3TtFJ1YZoChWtqRBM/X3RIaF1hMR2k6BzUgvern4n9dNTXQRZEwmqaGSzBdFKUcmRvnzaMAUJYZPLMFEMXsrIiOsMDE2oooNwVt8eZn4jfpl3bs9qzWvijTKcATHcAoenEMTbqAFPhDg8Ayv8OY8Oi/Ou/Mxby05xcwh/IHz+QMxno+b</latexit><latexit sha1_base64="DRe8q9G8lOKJt3WChWHyv0OBjjM=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWMLbQxrLZbtqlu5u4uxFK6J/w4kHFq7/Hm//GTZuDtj4YeLw3w8y8MOFMG9f9dkorq2vrG+XNytb2zu5edf/gXsepItQnMY9VJ8Saciapb5jhtJMoikXIaTscX+d++4kqzWJ5ZyYJDQQeShYxgo2VOj3NhgI/NPrVmlt3Z0DLxCtIDQq0+tWv3iAmqaDSEI617npuYoIMK8MIp9NKL9U0wWSMh7RrqcSC6iCb3TtFJ1YZoChWtqRBM/X3RIaF1hMR2k6BzUgvern4n9dNTXQRZEwmqaGSzBdFKUcmRvnzaMAUJYZPLMFEMXsrIiOsMDE2oooNwVt8eZn4jfpl3bs9qzWvijTKcATHcAoenEMTbqAFPhDg8Ayv8OY8Oi/Ou/Mxby05xcwh/IHz+QMxno+b</latexit><latexit sha1_base64="DRe8q9G8lOKJt3WChWHyv0OBjjM=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWMLbQxrLZbtqlu5u4uxFK6J/w4kHFq7/Hm//GTZuDtj4YeLw3w8y8MOFMG9f9dkorq2vrG+XNytb2zu5edf/gXsepItQnMY9VJ8Saciapb5jhtJMoikXIaTscX+d++4kqzWJ5ZyYJDQQeShYxgo2VOj3NhgI/NPrVmlt3Z0DLxCtIDQq0+tWv3iAmqaDSEI617npuYoIMK8MIp9NKL9U0wWSMh7RrqcSC6iCb3TtFJ1YZoChWtqRBM/X3RIaF1hMR2k6BzUgvern4n9dNTXQRZEwmqaGSzBdFKUcmRvnzaMAUJYZPLMFEMXsrIiOsMDE2oooNwVt8eZn4jfpl3bs9qzWvijTKcATHcAoenEMTbqAFPhDg8Ayv8OY8Oi/Ou/Mxby05xcwh/IHz+QMxno+b</latexit><latexit sha1_base64="DRe8q9G8lOKJt3WChWHyv0OBjjM=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWMLbQxrLZbtqlu5u4uxFK6J/w4kHFq7/Hm//GTZuDtj4YeLw3w8y8MOFMG9f9dkorq2vrG+XNytb2zu5edf/gXsepItQnMY9VJ8Saciapb5jhtJMoikXIaTscX+d++4kqzWJ5ZyYJDQQeShYxgo2VOj3NhgI/NPrVmlt3Z0DLxCtIDQq0+tWv3iAmqaDSEI617npuYoIMK8MIp9NKL9U0wWSMh7RrqcSC6iCb3TtFJ1YZoChWtqRBM/X3RIaF1hMR2k6BzUgvern4n9dNTXQRZEwmqaGSzBdFKUcmRvnzaMAUJYZPLMFEMXsrIiOsMDE2oooNwVt8eZn4jfpl3bs9qzWvijTKcATHcAoenEMTbqAFPhDg8Ayv8OY8Oi/Ou/Mxby05xcwh/IHz+QMxno+b</latexit>

Answer 2: Time Series Forecasting

Page 25: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

mixture of periodic, trend and auto-regressivewith additive zero-mean noise and randomly missing values

Answer 2: Time Series Forecasting

Page 26: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Google Flu Trend in Peru

Answer 2: Time Series Forecasting

Page 27: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Answer 3: Synthetic Control

time 1 time time TT0<latexit sha1_base64="f77q8BAinuZuDRiGeuJfEYQz3hQ=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWPFxhbaUDbbSbt0swm7G6GE/gQvHlS8+o+8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6iRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE9k04xSDmA4kjzijxkoPzZ7bq1TdmjsDWSZeQapQoNGrfHX7CctilIYJqnXHc1MT5FQZzgROyt1MY0rZiA6wY6mkMeogn506IadW6ZMoUbakITP190ROY63HcWg7Y2qGetGbiv95ncxEV0HOZZoZlGy+KMoEMQmZ/k36XCEzYmwJZYrbWwkbUkWZsemUbQje4svLxD+vXde8+4tq/aZIowTHcAJn4MEl1OEOGuADgwE8wyu8OcJ5cd6dj3nrilPMHMEfOJ8/QT6NTw==</latexit><latexit sha1_base64="f77q8BAinuZuDRiGeuJfEYQz3hQ=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWPFxhbaUDbbSbt0swm7G6GE/gQvHlS8+o+8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6iRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE9k04xSDmA4kjzijxkoPzZ7bq1TdmjsDWSZeQapQoNGrfHX7CctilIYJqnXHc1MT5FQZzgROyt1MY0rZiA6wY6mkMeogn506IadW6ZMoUbakITP190ROY63HcWg7Y2qGetGbiv95ncxEV0HOZZoZlGy+KMoEMQmZ/k36XCEzYmwJZYrbWwkbUkWZsemUbQje4svLxD+vXde8+4tq/aZIowTHcAJn4MEl1OEOGuADgwE8wyu8OcJ5cd6dj3nrilPMHMEfOJ8/QT6NTw==</latexit><latexit sha1_base64="f77q8BAinuZuDRiGeuJfEYQz3hQ=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWPFxhbaUDbbSbt0swm7G6GE/gQvHlS8+o+8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6iRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE9k04xSDmA4kjzijxkoPzZ7bq1TdmjsDWSZeQapQoNGrfHX7CctilIYJqnXHc1MT5FQZzgROyt1MY0rZiA6wY6mkMeogn506IadW6ZMoUbakITP190ROY63HcWg7Y2qGetGbiv95ncxEV0HOZZoZlGy+KMoEMQmZ/k36XCEzYmwJZYrbWwkbUkWZsemUbQje4svLxD+vXde8+4tq/aZIowTHcAJn4MEl1OEOGuADgwE8wyu8OcJ5cd6dj3nrilPMHMEfOJ8/QT6NTw==</latexit><latexit sha1_base64="f77q8BAinuZuDRiGeuJfEYQz3hQ=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWPFxhbaUDbbSbt0swm7G6GE/gQvHlS8+o+8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6iRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE9k04xSDmA4kjzijxkoPzZ7bq1TdmjsDWSZeQapQoNGrfHX7CctilIYJqnXHc1MT5FQZzgROyt1MY0rZiA6wY6mkMeogn506IadW6ZMoUbakITP190ROY63HcWg7Y2qGetGbiv95ncxEV0HOZZoZlGy+KMoEMQmZ/k36XCEzYmwJZYrbWwkbUkWZsemUbQje4svLxD+vXde8+4tq/aZIowTHcAJn4MEl1OEOGuADgwE8wyu8OcJ5cd6dj3nrilPMHMEfOJ8/QT6NTw==</latexit>

donor 1

donor i

target

3 4

1

1 4

3

2 3

51

3 2

2

3

6

pre-intervention post-intervention

what would it have been without intervention?

Page 28: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Answer 3: Synthetic Control

pre-intervention post-intervention

Algorithm:

Grey Matrix through BB-ME

Regression

Target: Blue

Features: Denoised Grey

Restriction: pre-intervention

Prediction (Synthetic control)

Predict Blue post-intervention

using post-intervention denoised Grey

3 4

1

1 4

3

2 3

51

3 2

2

3

6

Page 29: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Theorem (Informal):

If the matrix satisfies Latent Variable Model

then the synthetic control estimates true non-intervention outcome

such that the mean-squared error decays as

Answer 3: Synthetic Control

Popular factor model from Econometrics literature (cf. Abadie et al):

eO�T

�1/20 /p

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where p is the fraction of observed data

satisfy low-rank (rank 2) structure and a (very) special instance of

Latent Variable Model.

Page 30: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Answer 3: Synthetic Control

Did Terrorism have impact on Economy (of Basque Country)?

Page 31: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Answer 3: Synthetic Control

Did Terrorism have impact on Economy (of Basque Country)?

Page 32: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Answer 3: Synthetic Control

Did Terrorism have impact on Economy (of Basque Country)?

Page 33: Time Series Analysis via Matrix EstimationQuestions from Retail and Time Series Analysis Question 1: demand rate estimation estimating latent state of a time-series with missing values

Summary

Matrix Estimation

A remarkable method with applications beyond obvious

Time Series Analysis

Matrix Estimation provides “universal” solution

Going forward

“Time Series Prediction DataBase” using such algorithm