I used Gaussian Process Regression to predict a time series, what I have is sensor's readings that come every hour ( I have data for about 3 years) I chose a periodic kernel function which looks like this

$$ K(x,x′)=\sigma_1^2 \exp(−2\sin^2(π|x−x′|/24)/\mathcal{l_1}^2)+\sigma_2^2 \exp(−2\sin^2(π|x−x′|/8)/\mathcal{l_2}^2) $$

I got the following predictions However, the confidence interval looks very weird, (same size along the predictions, and wide even around the data), I can't understand why I am getting this answer, can anyone explain to me what's wrong in my work!?

Briefly, this is what I do :

1. I get the data (sensor readings) and prepare training ($$ y(N*1) $$) and testing data ($$ y_{*}(M*1) $$) from the readings and preparing two corresponding series of numbers $$ x=[1,2,3,....N] $$ and $$ x_*=[1,2,3,....M] $$
2. Estimate the parameters using maximum likelihood
3. Calculate $$ K(x,x), K(x_*,x)$$ and $$ K(x_*,x_*)$$
4. Calculate $$ L=K(x_*,x)*{K(x,x)}^{-1} $$
5. Calculate $$ Prediction=L*y $$ and $$ Covariance =k(x_*,x_*)-L*K(x,x_*)$$
6. Calculate the confidence interval like this:

$$ lower=Prediction+0.95*sqrt(Covariance) $$ $$ upper=Prediction-0.95*sqrt(Covariance) $$

1. Do I do anything wrong?! what is the problem?!
2. Is it right to consider the input (the time) in this way? and for the prediction mode, if I am going to predict the coming 24 hours should my input be $$ x_{pre}=[1,2,3,....,24] $$

I used Gaussian Process Regression to predict a time series, what I have is sensor's readings that come every hour ( I have data for about 3 years) I chose a periodic kernel function which looks like this

$$ K(x,x′)=\sigma_1^2 \exp(−2\sin^2(π|x−x′|/24)/\mathcal{l_1}^2)+\sigma_2^2 \exp(−2\sin^2(π|x−x′|/8)/\mathcal{l_2}^2) $$

I got the following predictions However, the confidence interval looks very weird, (same size along the predictions, and wide even around the data), I can't understand why I am getting this answer, can anyone explain to me what's wrong in my work!?

Briefly, this is what I do :

1. I get the data (sensor readings) and prepare training ($$ y(N*1) $$) and testing data ($$ y_{*}(M*1) $$) from the readings and preparing two corresponding series of numbers $$ x=[1,2,3,....N] $$ and $$ x_*=[1,2,3,....M] $$
2. Estimate the parameters using maximum likelihood
3. Calculate $$ K(x,x) +\sigma_{nois}, K(x_*,x)$$ and $$ K(x_*,x_*)$$
4. Calculate $$ L=K(x_*,x)*{K(x,x)}^{-1} $$
5. Calculate $$ Prediction=L*y $$ and $$ Covariance =k(x_*,x_*)-L*K(x,x_*)$$
6. Calculate the confidence interval like this:

$$ lower=Prediction+0.95*sqrt(Covariance) $$ $$ upper=Prediction-0.95*sqrt(Covariance) $$

1. Do I do anything wrong?! what is the problem?!
2. Is it right to consider the input (the time) in this way? and for the prediction mode, if I am going to predict the coming 24 hours should my input be $$ x_{pre}=[1,2,3,....,24] $$

I used Gaussian Process Regression to predict a time series, what I have is sensor's readings that come every hour ( I have data for about 3 years) I chose a periodic kernel function which looks like this

$$ K(x,x′)=\sigma1^2 \exp(−2\sin^2(π|x−x′|/24)/\mathcal{l1}^2)+\sigma2^2 \exp(−2\sin^2(π|x−x′|/8)/\mathcal{l2}^2) $$

I got the following predictions However, the confidence interval looks very weird, (same size along the predictions, and wide even around the data), I can't understand why I am getting this answer, can anyone explain to me what's wrong in my work!?

Briefly, this is what I do :

1. I get the data (sensor readings) and prepare training (y) and testing data (y*) from the readings and preparing two corresponding series of numbers x=[1,2,3,....N] and x*=[1,2,3,....M]
2. Estimate the parameters using maximum likelihood
3. Calculate K(x,x), K(x*,x) and K(x*,x*)
4. Calculate L=K(x*,x)*K(x,x)^-1
5. Calculate Prediction=Ly and Covariance =k(x,x*)-LK(x,x)
6. Calculate the confidence interval like this:

lower=Prediction+0.95sqrt(Covariance); upper=Prediction-0.95sqrt(Covariance)

1. Do I do anything wrong?! what is the problem?!
2. Is it right to consider the input (the time) in this way? and for the prediction mode, if I am going to predict the coming 24 hours should my input be x_pre=[1,2,3,....,24] ?

I used Gaussian Process Regression to predict a time series, what I have is sensor's readings that come every hour ( I have data for about 3 years) I chose a periodic kernel function which looks like this

$$ K(x,x′)=\sigma_1^2 \exp(−2\sin^2(π|x−x′|/24)/\mathcal{l_1}^2)+\sigma_2^2 \exp(−2\sin^2(π|x−x′|/8)/\mathcal{l_2}^2) $$

I got the following predictions However, the confidence interval looks very weird, (same size along the predictions, and wide even around the data), I can't understand why I am getting this answer, can anyone explain to me what's wrong in my work!?

Briefly, this is what I do :

1. I get the data (sensor readings) and prepare training ($$ y(N*1) $$) and testing data ($$ y_{*}(M*1) $$) from the readings and preparing two corresponding series of numbers $$ x=[1,2,3,....N] $$ and $$ x_*=[1,2,3,....M] $$
2. Estimate the parameters using maximum likelihood
3. Calculate $$ K(x,x), K(x_*,x)$$ and $$ K(x_*,x_*)$$
4. Calculate $$ L=K(x_*,x)*{K(x,x)}^{-1} $$
5. Calculate $$ Prediction=L*y $$ and $$ Covariance =k(x_*,x_*)-L*K(x,x_*)$$
6. Calculate the confidence interval like this:

$$ lower=Prediction+0.95*sqrt(Covariance) $$ $$ upper=Prediction-0.95*sqrt(Covariance) $$

1. Do I do anything wrong?! what is the problem?!
2. Is it right to consider the input (the time) in this way? and for the prediction mode, if I am going to predict the coming 24 hours should my input be $$ x_{pre}=[1,2,3,....,24] $$