Programming assignment #4

    • Chemical composition of pottery

You work for a pottery manufacturing company that produces two di erent types of products A and B. Product A uses raw material from Llanedyrn and Product B uses raw material from Isle Thorns and Ashley Rails1. Your company is informed that Llanedyrn will be closing soon for maintenance and your entire production of product A is at risk.

You received a potential new source of raw material from site Caldicot and you analyzed two samples to compare them to your existing samples from the other three sites. As the new data scientist of the company you are asked to look into the data and give your recommendation regarding the suitability of raw material from Caldicot as a replacement for Llanedyrn.

The ask "look into the data" normally calls for an unsupervised learning exercise, since there is no clear output you are asked to predict. You will investigate the multivariate chem-ical composition of the raw material from four di erent sources using Principal Component Analysis.


1.1    Data exploration with PCA

    1. Import and view the data. How many columns do you have? Which columns will you use in your PCA?

    2. Pre-process the data and perform PCA with 3 PCs.

    3. Plot the cumulative explained variance graph. What percent of the variance do the rst 2 and 3 components describe?

    4. Plot the scores-loadings graph for PC1-PC2. Visualize the di erent sites with a di er-ent colour or symbol.

    5. How does the map of scores-loadings explains the reason that your company uses the raw material from Isle Thorns and Ashley Rails to manufacture Product B?


1These are sites in Great Britain (or Game of Thrones if you prefer). The data come from ancient pottery ndings and the problem is ctitious.

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    6. Is the raw material from Caldicot a good replacement for Llanedyrn? Yes or no and why?

    7. What are the biggest di erences in the two big clusters? How are the two samples from the candidate Caldicot di erent than the Llanedyrn samples?

    8. Con rm the answers by producing the boxplot of the 5 variables grouped by the site of the raw material shown below.











































Final note: In this problem, we reduced the number of variables from 5 to 2 in order to visualize the characteristics captured in the 5 variables. With more than 5 variables you realize that it becomes di cult to visualize and compare the di erent samples. Dimension reduction methods like PCA are crucial to understand multivariate data. Conventional statistical analysis like the boxplot shown here do not show the correlations between the variables which are simply captured in the PCA plots you created.

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    • Batch data analysis

In this problem, we will look into batch data; dynamic time-series of a nite duration 2. Batch manufacturing processes are very common in chemical, pharma, bioengineering and semi-conductor industries such as baker’s yeast production, beer brewing and vaccines production.

In theory, a reactor is designed with temperature, pressure, level, pH control and multiple sensors that measure these variables among others. A perfect batch (again in theory) is one that is tightly controlled to the speci cations and as a result the productivity and quality of the nal product is optimized.

In real life, a typical batch is run from a few hours up to a week or two and a lot of things can go wrong during this period. There is always variability either because the process is very sensitive to minor uctuations in some variables or the control of some variables failed for a period of time.

In a company that implements Data Analytics or Multivariate Statistical Process Control (MSPC) monitoring is typically implemented with the following steps:

    1. Identify a number of reference, perfect historical batches (15-20), both in terms of high productivity/quality and minimum anomalies or uctuations around the setpoints.

    2. Create a PCA model of the perfect batches identi ed. This is your model.

    3. Every time your site is running a new batch, t your data online or as soon as your data infrastructure allows you to do so. Fitting will tell you whether your batch is similar to the perfect batches or it is deviating from the reference behaviour.

Next, you will follow these steps to build a Batch Statistical Process Control and implement it to monitor a new batch (we will assume that you got the data at the end of the batch and t them to the model). The dataset is from a baker’s yeast production facility in Solna, Sweden capturing the last step of the fermentation.


2.1    Build a Batch Statistical Process Control model

    1. Import the data from ’bakers yeast reference batches.xlsx’. Identify how many batches are in the data. What is the duration of each batch and how many data points are there per batch? How many variables are measured (including time)?

    2. Plot the variables time-pro les in a 2x4 subplot. Inspect the graphs (don’t just plot them). Look for potential outliers. Which variables have the largest variability? Which variables are tightly controlled?


2The data are taken from Chapter 16 of Multi- and Megavariate Data Analysis from Umetrics Academy. The problem in this assignment though is reformulated and is not the same as the one described in the book.


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    3. Select the features (including the Time column), pre-process the data and perform PCA with 5 principal components. Extract the scores and loadings.

    4. In order to plot the scores-loading plot, you need to pivot the scores BatchID with index ’Time’ (use pandas pivot table).

    5. Plot the scores-loadings plot with one line per batch (this is why the pivot in the previous step was needed). The output should look similar to the plot below. You may choose a di erent scaling, but the trend should be the same as this graph.























    6. Explain this graph. In which quarter do the batches start and end? What happens at the kink where the direction of the lines changes? Can you tell from this graph which variables do not change in the rst phase and which in the second phase?

    7. Plot the cumulative explained variance. How much variance do the rst two principal components capture?


2.2    Use the model to monitor running batches

The goal of building an unsupervised model is to monitor the running batches. Your site runs two reactors in parallel and here you will t the data from these two reactors to the model previously built and identify potential problems and outliers3.

    1. Load the data from the le ’todays batches.xlsx’ and repeat the same procedure as in the steps 3-4 of the previous section with the exception of the PCA modeling. Here, instead of t the data to the model and transform, you will only transform them with the model object you created in the previous section.


    • Ideally, in most industries you have the data available online and you get a new data point every minute or so. Then you t every coming point to the model and overlay it with the graph from the previous section.

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    2. Plot the same scores-loadings plot for the data in the batches you used to develop the model with solid lines. Overlay the new incoming data from the two current batches with dashed lines and two di erent colours to distinguish them. Also, add a legend for the two batches so that the viewer can distinguish them.

    3. Do the batches show behaviour similar to that of the reference ones or there are outliers indicating potential problems?





















































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