# An automatic report for the dataset : Pfizer

The Relational Automatic Statistician
###### Abstract

This report was produced by the Automatic Bayesian Covariance Discovery (ABCD) algorithm.

## 1 Executive summary

The raw data and full model posterior with extrapolations are shown in figure 1. Figure 1: Raw data (left) and model posterior with extrapolation (right)

The structure search algorithm has identified five additive components in the data. The first 3 additive components explain 96.5% of the variation in the data as shown by the coefficient of determination ($R^{2}$) values in table 1. The 5 additive components explain 100.0% of the variation in the data. After the first 3 components the cross validated mean absolute error (MAE) does not decrease by more than 0.1%. This suggests that subsequent terms are modelling very short term trends, uncorrelated noise or are artefacts of the model or search procedure. Short summaries of the additive components are as follows:

• A very smooth monotonically increasing function.

• A very smooth function. This function applies from 12 Sep 2001 until 14 Sep 2001.

• A smooth function. This function applies until 12 Sep 2001 and from 14 Sep 2001 onwards.

• Uncorrelated noise.

• An approximately periodic function with a period of 4.1 weeks. This function applies until 12 Sep 2001 and from 14 Sep 2001 onwards.

Model checking statistics are summarised in table 2 in section 4. These statistics have revealed statistically significant discrepancies between the data and model in component 3.

The rest of the document is structured as follows. In section 2 the forms of the additive components are described and their posterior distributions are displayed. In section 3 the modelling assumptions of each component are discussed with reference to how this affects the extrapolations made by the model. Section 4 discusses model checking statistics, with plots showing the form of any detected discrepancies between the model and observed data.

## 2 Detailed discussion of additive components

### 2.1 Component 1 : A very smooth monotonically increasing function

This component is a very smooth and monotonically increasing function.

This component explains -1.0% of the total variance. The addition of this component reduces the cross validated MAE by 95.8% from 25.0 to 1.1. Figure 2: Pointwise posterior of component 1 (left) and the posterior of the cumulative sum of components with data (right) Figure 3: Pointwise posterior of residuals after adding component 1

### 2.2 Component 2 : A very smooth function. This function applies from 12 Sep 2001 until 14 Sep 2001

This component is a very smooth function. This component applies from 12 Sep 2001 until 14 Sep 2001.

This component explains 41.7% of the residual variance; this increases the total variance explained from -1.0% to 41.1%. The addition of this component reduces the cross validated MAE by 10.89% from 1.05 to 0.94. Figure 4: Pointwise posterior of component 2 (left) and the posterior of the cumulative sum of components with data (right) Figure 5: Pointwise posterior of residuals after adding component 2

### 2.3 Component 3 : A smooth function. This function applies until 12 Sep 2001 and from 14 Sep 2001 onwards

This component is a smooth function with a typical lengthscale of 2.8 days. This component applies until 12 Sep 2001 and from 14 Sep 2001 onwards.

This component explains 94.1% of the residual variance; this increases the total variance explained from 41.1% to 96.5%. The addition of this component reduces the cross validated MAE by 8.11% from 0.94 to 0.86. Figure 6: Pointwise posterior of component 3 (left) and the posterior of the cumulative sum of components with data (right) Figure 7: Pointwise posterior of residuals after adding component 3

### 2.4 Component 4 : Uncorrelated noise

This component models uncorrelated noise.

This component explains 26.6% of the residual variance; this increases the total variance explained from 96.5% to 97.5%. The addition of this component reduces the cross validated MAE by 0.00% from 0.86 to 0.86. This component explains residual variance but does not improve MAE which suggests that this component describes very short term patterns, uncorrelated noise or is an artefact of the model or search procedure. Figure 8: Pointwise posterior of component 4 (left) and the posterior of the cumulative sum of components with data (right) Figure 9: Pointwise posterior of residuals after adding component 4

### 2.5 Component 5 : An approximately periodic function with a period of 4.1 weeks. This function applies until 12 Sep 2001 and from 14 Sep 2001 onwards

This component is approximately periodic with a period of 4.1 weeks. Across periods the shape of this function varies smoothly with a typical lengthscale of 4.5 weeks. The shape of this function within each period has a typical lengthscale of 18.0 hours. This component applies until 12 Sep 2001 and from 14 Sep 2001 onwards.

This component explains 100.0% of the residual variance; this increases the total variance explained from 97.5% to 100.0%. The addition of this component increases the cross validated MAE by 3.17% from 0.86 to 0.89. This component explains residual variance but does not improve MAE which suggests that this component describes very short term patterns, uncorrelated noise or is an artefact of the model or search procedure. Figure 10: Pointwise posterior of component 5 (left) and the posterior of the cumulative sum of components with data (right)

## 3 Extrapolation

Summaries of the posterior distribution of the full model are shown in figure 11. The plot on the left displays the mean of the posterior together with pointwise variance. The plot on the right displays three random samples from the posterior. Figure 11: Full model posterior with extrapolation. Mean and pointwise variance (left) and three random samples (right)

Below are descriptions of the modelling assumptions associated with each additive component and how they affect the predictive posterior. Plots of the pointwise posterior and samples from the posterior are also presented, showing extrapolations from each component and the cuulative sum of components.

### 3.1 Component 1 : A very smooth monotonically increasing function

This component is assumed to continue very smoothly but is also assumed to be stationary so its distribution will eventually return to the prior. The prior distribution places mass on smooth functions with a marginal mean of zero and a typical lengthscale of 5.1 years. [This is a placeholder for a description of how quickly the posterior will start to resemble the prior]. Figure 12: Posterior of component 1 (top) and cumulative sum of components (bottom) with extrapolation. Mean and pointwise variance (left) and three random samples from the posterior distribution (right).

### 3.2 Component 2 : A very smooth function. This function applies from 12 Sep 2001 until 14 Sep 2001

This component is assumed to stop before the end of the data and will therefore be extrapolated as zero. Figure 13: Posterior of component 2 (top) and cumulative sum of components (bottom) with extrapolation. Mean and pointwise variance (left) and three random samples from the posterior distribution (right).

### 3.3 Component 3 : A smooth function. This function applies until 12 Sep 2001 and from 14 Sep 2001 onwards

This component is assumed to continue smoothly but is also assumed to be stationary so its distribution will return to the prior. The prior distribution places mass on smooth functions with a marginal mean of zero and a typical lengthscale of 2.8 days. [This is a placeholder for a description of how quickly the posterior will start to resemble the prior]. Figure 14: Posterior of component 3 (top) and cumulative sum of components (bottom) with extrapolation. Mean and pointwise variance (left) and three random samples from the posterior distribution (right).

### 3.4 Component 4 : Uncorrelated noise

This component assumes the uncorrelated noise will continue indefinitely. Figure 15: Posterior of component 4 (top) and cumulative sum of components (bottom) with extrapolation. Mean and pointwise variance (left) and three random samples from the posterior distribution (right).

### 3.5 Component 5 : An approximately periodic function with a period of 4.1 weeks. This function applies until 12 Sep 2001 and from 14 Sep 2001 onwards

This component is assumed to continue to be approximately periodic. The shape of the function is assumed to vary smoothly between periods but will return to the prior. The prior is entirely uncertain about the phase of the periodic function. Consequently the pointwise posterior will appear to lose its periodicity, but this merely reflects the uncertainty in the shape and phase of the function. [This is a placeholder for a description of how quickly the posterior will start to resemble the prior]. Figure 16: Posterior of component 5 (top) and cumulative sum of components (bottom) with extrapolation. Mean and pointwise variance (left) and three random samples from the posterior distribution (right).

## 4 Model checking

Several posterior predictive checks have been performed to assess how well the model describes the observed data. These tests take the form of comparing statistics evaluated on samples from the prior and posterior distributions for each additive component. The statistics are derived from autocorrelation function (ACF) estimates, periodograms and quantile-quantile (qq) plots.

Table 2 displays cumulative probability and $p$-value estimates for these quantities. Cumulative probabilities near 0/1 indicate that the test statistic was lower/higher under the posterior compared to the prior unexpectedly often i.e. they contain the same information as a $p$-value for a two-tailed test and they also express if the test statistic was higher or lower than expected. $p$-values near 0 indicate that the test statistic was larger in magnitude under the posterior compared to the prior unexpectedly often.

The nature of any observed discrepancies is now described and plotted and hypotheses are given for the patterns in the data that may not be captured by the model.

### 4.1 Moderately statistically significant discrepancies

#### 4.1.1 Component 3 : A smooth function. This function applies until 12 Sep 2001 and from 14 Sep 2001 onwards

The following discrepancies between the prior and posterior distributions for this component have been detected.

• The qq plot has an unexpectedly large negative deviation from equality ($x=y$). This discrepancy has an estimated $p$-value of 0.035.

• The qq plot has an unexpectedly large positive deviation from equality ($x=y$). This discrepancy has an estimated $p$-value of 0.036.

The negative deviation in the qq-plot can indicate light positive tails if it occurs at the right of the plot or heavy negative tails if it occurs as the left. The positive deviation in the qq-plot can indicate heavy positive tails if it occurs at the right of the plot or light negative tails if it occurs as the left. Figure 17: ACF (top left), periodogram (top right) and quantile-quantile (bottom left) uncertainty plots. The blue line and shading are the pointwise mean and 90% confidence interval of the plots under the prior distribution for component 3. The green line and green dashed lines are the corresponding quantities under the posterior.

### 4.2 Model checking plots for components without statistically significant discrepancies

#### 4.2.1 Component 1 : A very smooth monotonically increasing function

No discrepancies between the prior and posterior of this component have been detected Figure 18: ACF (top left), periodogram (top right) and quantile-quantile (bottom left) uncertainty plots. The blue line and shading are the pointwise mean and 90% confidence interval of the plots under the prior distribution for component 1. The green line and green dashed lines are the corresponding quantities under the posterior.

#### 4.2.2 Component 2 : A very smooth function. This function applies from 12 Sep 2001 until 14 Sep 2001

No discrepancies between the prior and posterior of this component have been detected Figure 19: ACF (top left), periodogram (top right) and quantile-quantile (bottom left) uncertainty plots. The blue line and shading are the pointwise mean and 90% confidence interval of the plots under the prior distribution for component 2. The green line and green dashed lines are the corresponding quantities under the posterior.

#### 4.2.3 Component 4 : Uncorrelated noise

No discrepancies between the prior and posterior of this component have been detected Figure 20: ACF (top left), periodogram (top right) and quantile-quantile (bottom left) uncertainty plots. The blue line and shading are the pointwise mean and 90% confidence interval of the plots under the prior distribution for component 4. The green line and green dashed lines are the corresponding quantities under the posterior.

#### 4.2.4 Component 5 : An approximately periodic function with a period of 4.1 weeks. This function applies until 12 Sep 2001 and from 14 Sep 2001 onwards

No discrepancies between the prior and posterior of this component have been detected Figure 21: ACF (top left), periodogram (top right) and quantile-quantile (bottom left) uncertainty plots. The blue line and shading are the pointwise mean and 90% confidence interval of the plots under the prior distribution for component 5. The green line and green dashed lines are the corresponding quantities under the posterior.