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As a first step I would ask myself whether the series looks mean-reverting. Since the series has a specific "inverted U" shape, it does not seem to be mean-reverting. Therefore I suggest it is not stationary.

However there is chance that your series could be trend-stationary. If you are able to find any function f(t) of time t that describes the trend of "inverted U", you can remove the trend and what remains could (maybe) be stationary.

One shape of such deterministic function f(t) is a linear trend that you have written in your first formula. However this is not an option for you, because clearly the trend is not linear but "inverted U" shape.

You could try to capture the deterministic trend by another f(t) function, for example a quadratic form (instead of linear) or a form based on dummy time variables.

If you are able to find such f(t) that you can fit into your regression formula and you will obtain stationary residuals, then it is a win for you, because you can apply such model.

If you are not able to find such f(t) that would lead to stationary residuals in your regression formula, then the time series is indeed non stationary. In that case you must apply differencing (of order 1 or possibly 2) on your time series.

Your series is pretty short (few observations). To build a robust model, you would do better to connect all days together into one series. Such series will have strong seasonality (morning/evening/noon) and you would be able to employ some seasonal time series model.

As a first step I would ask myself whether the series looks mean-reverting. Since the series has a specific "inverted U" shape, it does not seem to be mean-reverting. Therefore I suggest it is not stationary.

However there is chance that your series could be trend-stationary. If you are able to find any function f(t) of time t that describes the trend of "inverted U", you can remove the trend and what remains could (maybe) be stationary.

One shape of such deterministic function f(t) is a linear trend that you have written in your first formula. However this is not an option for you, because clearly the trend is not linear but "inverted U" shape.

You could try to capture the deterministic trend by another f(t) function, for example a quadratic form (instead of linear), but the assumption of a quadratic trend in the generating process appears "too artificial" to me (i.e. there is no rational expectation for the variable "intensity" to be a quadratic time process)

If you are able to find such f(t) that you can fit into your regression formula and you will obtain stationary residuals, then it is a win for you, because you can apply such model (although I have also seen doubts about this approach on a theoretical level)

If you are not able to find such f(t) that would lead to stationary residuals in your regression formula, then the time series is indeed non stationary. In that case you must apply differencing (of order 1 or possibly 2) on your time series.

However, your series is pretty short (few observations). To build a robust model, you would do better to connect all days together into one series. Such series will have strong seasonality (morning/evening/noon) and you would be able to employ some seasonal time series model. Because indeed the "inverted U" shape appears to be a seasonal pattern and nothing else.

1. As a first hint I would ask myself whether the series looks mean-reverting. With knowledge to the process (low in mornings and evenings, high during the middays) the series does not seem to be mean-reverting. Therefore I would suggest it is not stationary. You can perform statistical test (Dickey-Fuller test).

2. You can not use the model if series is not stationary. (Moreover it also makes no sense to use t as a variable since the series goes up in the morning but down in the evening. You would do better with some dummy time variables or with some quadratic time regression to capture the very specific shape)

3. You can try differencing of 1 or 2 orders in ARIMA model. Or you can perform the differencing first and then try some of the models you proposed in step 2 on the differenced time series (if it would be stationary).

4. You would build more robust model if you connect all days into one time series, you would obtain a lot more observation points. Then you would also be able to model seasonal behavior of mornings/evenings (which is more complex topic although).

As a first step I would ask myself whether the series looks mean-reverting. Since the series has a specific "inverted U" shape, it does not seem to be mean-reverting. Therefore I suggest it is not stationary.

However there is chance that your series could be trend-stationary. If you are able to find any function f(t) of time t that describes the trend of "inverted U", you can remove the trend and what remains could (maybe) be stationary.

One shape of such deterministic function f(t) is a linear trend that you have written in your first formula. However this is not an option for you, because clearly the trend is not linear but "inverted U" shape.

You could try to capture the deterministic trend by another f(t) function, for example a quadratic form (instead of linear) or a form based on dummy time variables.

If you are able to find such f(t) that you can fit into your regression formula and you will obtain stationary residuals, then it is a win for you, because you can apply such model.

If you are not able to find such f(t) that would lead to stationary residuals in your regression formula, then the time series is indeed non stationary. In that case you must apply differencing (of order 1 or possibly 2) on your time series.

Your series is pretty short (few observations). To build a robust model, you would do better to connect all days together into one series. Such series will have strong seasonality (morning/evening/noon) and you would be able to employ some seasonal time series model.

1. As a first hint I would ask myself whether the series looks mean-reverting. With knowledge to the process (low in mornings and evenings, high during the middays) the series does not seem to be mean-reverting. Therefore I would suggest it is not stationary. You can perform statistical test (Dickey-Fuller test).

2. You can not use the model if series is not stationary. (Moreover it also makes no sense to use t as a variable since the series goes up in the morning but down in the evening. You would do better with some dummy time variables)

3. You can try differencing of 1 or 2 orders in ARIMA model. Or you can perform the differencing first and then try some of the models you proposed in step 2 on the differenced time series (if it would be stationary).

4. You would build more robust model if you connect all days into one time series, you would obtain a lot more observation points. Then you would also be able to model seasonal behavior of mornings/evenings (which is more complex topic although).

1. As a first hint I would ask myself whether the series looks mean-reverting. With knowledge to the process (low in mornings and evenings, high during the middays) the series does not seem to be mean-reverting. Therefore I would suggest it is not stationary. You can perform statistical test (Dickey-Fuller test).

2. You can not use the model if series is not stationary. (Moreover it also makes no sense to use t as a variable since the series goes up in the morning but down in the evening. You would do better with some dummy time variables or with some quadratic time regression to capture the very specific shape)

3. You can try differencing of 1 or 2 orders in ARIMA model. Or you can perform the differencing first and then try some of the models you proposed in step 2 on the differenced time series (if it would be stationary).

4. You would build more robust model if you connect all days into one time series, you would obtain a lot more observation points. Then you would also be able to model seasonal behavior of mornings/evenings (which is more complex topic although).