## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!

## #StackBounty: #time-series #autoregressive #entropy #maximum-entropy Generalization of Burg's Maximum Entropy Theorem

### Bounty: 50

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $$X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $$E X_iX_{i+k}=alpha_k$$, $$1leq kleq K$$. Here $$K$$ is fixed, and $$epsilon_i$$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series $${Y_i}_i$$ is also constrained. More precisely, the problem is to characterize the time series $$X_i$$ with maximal entropy rate satisfying the constraints $$EX_iX_{i+k}=alpha_k, 1leq kleq K_1$$ and $$EX_iY_{i+k}=beta_k, 1leq kleq K_2$$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series $${X_i}$$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?

Get this bounty!!!