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Cardiovascular disease (CVD) is number one in death rate worldwide. The latest statistics from the…

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ALS affects men more often than women, and Caucasian and non-Hispanic men are at a slightly higher risk. However, ALS does not discriminate: anyone could be at risk of developing the disease.

Experts estimate that between five and 10 percent of ALS cases are hereditary (classified as familial ALS).

When there is a hereditary element in ALS, only one parent needs to carry the gene for children to inherit it. However, the gene mutation doesn’t guarantee a grim prognosis — it may lead to ALS at some point, but also may not.

Around 90 percent of ALS cases are “sporadic” (they cannot be traced to any genetic risk), which can make it difficult to diagnose early. In these cases, family members of the patient are not at any increased risk of developing ALS.

Research on ALS has escalated in recent years (notably since the ALS Ice Bucket Challenge swept through the online community), and doctors have made gains in diagnosis and treatment. However, there is no cure for ALS; in order to slow down progression and preserve independence for longer, it’s important to find and treat the disease early.

Weakness. Loss of coordination. Spasticity.

As more voluntary muscles are affected, more drastic symptoms of nerve damage and muscle atrophy will develop: the arms and legs may become paralyzed, then the rest of the body, and eventually the deep muscles in charge of swallowing and breathing will fail. Tragically, most people with ALS die from respiratory failure within five years from the onset of symptoms.

ALS does spare some areas and processes. Intelligence and comprehension are usually not affected, and all the senses continue to operate as usual.

Most patients remain in control of their eye movements, bladder function, and bowel function. However, there is a possible link between late-stage ALS and dementia.

Diagnosing ALS means ruling out other diseases. The tests used (like X-rays, electromyography (EMG), and nerve conduction studies) seek to distinguish the specific nerve damage of ALS from other types of nerve damage.

MRI scans can help pick up underlying problems: while regular MRI scans won’t turn up evidence of ALS, they can reveal other conditions that could be causing the ALS-like symptoms, such as a herniated disk, a cyst in the spinal cord, or a tumor.

Hence, the forecasting hypothesis is the following:

H

Convex (Non-linear) Relationship.

The relative forecasting accuracy increases, as we move forward in time, compared to the benchmark’s.

H

Linear Relationship.

The relative forecasting accuracy remains constant, as we move forward in time, compared to the benchmark’s.

The full sample starts in 1880 and ends in 2017, while the forecasted period f lies in between: \( T_{window + 1} \le f \le T_{end} \) , where \( T_{window + 1} \) refers to the first period in the sample after the model’s estimation window. As mentioned before, we use a rolling window with different time-lengths in order to examine the forecasting performance of emissions when more recent values are included in the training sample.

The time-length of the variable rolling window is selected from the following set N : \( N = \left\{ {30,40,50,60,70,80,90,100} \right\} \)

Hence, the output of this iterative procedure is a series of 1 step-ahead forecasts of \( T\_dts_{f,i} \) over the remaining observations. When N equals 30, the training sample is between 1880 and 1909 leaving the rest of the period until 2017 for forecasting. On the other hand when the maximum N = 100 is used, the training sample stops in 1979, allowing for a more recent sample only to be forecasted, compared with the first window.

To evaluate the forecasting performance of the four models produced by Eqs.( Cole Haan Womens Air Lainey OT Pump Summer Khaki 7 eWMrEG
) and ( Marc New York by Andrew Marc Mens wythe Sneaker Cobble/Dark Cobble/Bone jHcWF0gFaC
), we compare it with an appropriate ARIMA model that is used as a benchmark. Since the benchmark does not include emissions in its right hand side, it enables us to cross-compare the forecast accuracy of our models.

$$ \left( {1 - \varphi_{1} {\rm B} - \varphi_{2} {\rm B}^{2} - \varphi_{3} {\rm B}^{3} } \right)T\_dts_{f} = c + \left( {1 + \theta_{1} {\rm B} + \theta_{2} {\rm B}^{2} + \theta_{3} {\rm B}^{3} + \theta_{4} {\rm B}^{4} } \right)\, \in_{f} ,\quad \forall w $$
(7)
Table2

Stationarity and estimation results for the ARIMA model of Eq.( 7 )

Regarding the evaluation of the forecasting performance, we select an in-sample evaluation using the forecast error measures of Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and the scale invariant measure of Theil’s inequality coefficient (Theil IC), and an out-of-sample evaluation using the Diebold–Mariano statistics for equal predictive accuracy with the benchmark model.

$$ g_{1} (e_{w} ) = e_{j}^{2} \;\ \;g_{2} (e_{j} ) = \left| {e_{jw}^{{}} } \right|, $$
$$ H_{0} :E\left[ {g_{k,i} (e_{j} ) - g_{k,benchmark} (e_{j} )} \right] = 0,\quad \forall w $$
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Fig.2

Rolling forecasting procedure’s flowchart

The results from all stages, i.e., all previous rolling correlation, structural breaks, rolling regression and rolling forecasting, will enable us to identify the dynamics, if any, in the relationship between emissions and global temperature anomalies, throughout time.

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