NAMP Module 17: “Introduction to Multivariate Analysis” Tier 2, Rev.: 5 Program for North American Mobility in Higher Education Introducing Process Integration

NAMP Module 17: “Introduction to Multivariate Analysis” Tier 2, Rev.: 5

Program for North American Mobility in Higher Education

Introducing Process Integration for Environmental Control in Engineering Curricula

MODULE 17: “Introduction to Multivariate Analysis”

Created at:Ecole Polytechnique de Montreal &

North Carolina State University, 2003.

NC STATEUNIVERSITY


TIER 2:

Worked Examples


Tier 2: Statement of intent:

The goal of Tier 2 is to demonstrate the various MVA concepts using real examples. At the end of Tier 2, the student should be able to grasp the following:

• How to read the basic MVA outputs• How to deal with raw, messy data• How to deal with a large number of variables• How to deal with shorter timespans

The purpose is to teach the concepts behind MVA, and not merely how to run the software itself, which could be gleaned from any user’s manual. The biggest danger of this technique is using the software blindly, without understanding what’s inside the ‘black box’.

Tier 2: Statement of Intent


Tier 2 is broken down into four sections:

2.1 Where are the data coming from?

2.2 Example 1: PCA on Raw, Messy Data

2.3 Example 2: Using Fewer Variables

2.4 Example 3: Using Shorter Timescales

At the end of Tier 2 there is a short multiple-answer quiz.

Tier 2: Contents


2.1: Where are the data coming from?


Where are data coming from?

A standard joke is that teenagers think milk comes from a refrigerator.

Similarly, we could wrongly say that process data come from the plant’s data historian. They are, of course, generated somewhere else. We must fully understand each data “tag” if we are to make sense of the final MVA results.


Types of Data “Tags”A “tag” is a label or address for a certain measurement. For instance, the

tag “TempRT01” might refer to the temperature measured by a thermocouple in the top of reactor 1, in degrees Celsius, updated every 5 seconds. There are five major categories of tags, shown in descending order of “immediacy”:

• Immediate, on-lineThese are instantaneous readings, like those provided by a pressure

gauge. Even if the instrument operates continuously, there will be a sampling frequency which we must know and understand.

• Delayed, on-line These are delayed readings, like those from an on-line water quality

analyser. Not only must we understand the sampling frequency, but also the lag between the time the sample is taken and the time the values are logged.


Types of Data “Tags” (cont’d.)

• Delayed, off-lineThis category is even further removed, in that samples are

taken manually to an automatic analyser. Here the lag between sampling and logging of analytical results may be different from sample to sample.

• Manual, off-lineThese are laboratory measurements which are logged by

hand, often literally typed into the system on a keyboard by a human being.

• CalculationsThese are values calculated from other tags.


Timescales

Each value in the database will also have a timescale associated with it.

Discrete values are taken only at the precise instant in question. For example: the main steam header pressure at exactly 10:00 a.m., zero seconds. If no reading was taken at that precise moment, the discrete value is 0 (or 999 or blank or N/A…)

Average values are the mean or median over some designated timespan, for instance the average main header steam pressure between 9:59 a.m. and 10:00 a.m.

Frequency of measurement and of data-logging is extremely important. Some values may be updated every few seconds, while others only twice a day.


Process LagsIf you are using daily averages for your MVA, then a ten-minute residence time in a reactor or vessel will not impact your results. However, if you are comparing one-minute averages, then obviously such a process lag must be taken into account.

Estimating these lags is not obvious, since they can change with time (e.g., fluctuating tank levels).


Preparing the SpreadsheetGenerally, the data are downloaded into a standard spreadsheet,

which then serves as the input to the MVA software.

This offers several advantages:

1) Rows and columns can be set up appropriately, with tag numbers, long variable names, short variable names (to show on plots), observation numbers, time stamps and so forth. This greatly facilitates the use of the MVA software.

2) Additional calculations can be done, if required, for instance taking the log of certain variables for use in the MVA analysis.

3) Time lags can be incorporated right from the start, by shifting data from certain tags forward or backward in time. For instance, input variables for a process with a 30-minute residence time can be shifted to the same row as product quality variables measured 30 minutes later.


2.2: Example (1)

PCA of Raw, Messy Data


Process Example: TMP Refining line

All the examples in Tier 2 are based on the thermo-mechanical pulping (TMP) process, used to convert wood chips into pulp. This is a straightforward process, with well known underlying physical characteristics.

A generic flowsheet for the TMP process is shown on the next page. The wood chips are about 3 cm x 4 cm x 0,5 cm. They are pre-heated and pass through two refiners where huge spinning disks cut them down into individual cellulose fibre strands. The resulting pulp, a cellulose-water slurry, resembles the stuffing in a disposable diaper. This pulp is held for 45 minutes in the latency chest, to allow the cellulose strands to disentangle themselves, before being sent to the papermaking section of the plant.

TMP is used to make newsprint

Example 1


Thermomechanical Pulping (TMP)Generic Flowsheet

Y’s

X’s

45-minute residence time

No expertise on the TMP process is required to understand the examples.

Example 1


Dozens of Variables Measured

The many dozens of variables that are measured on a TMP line fall into two categories, those which impact the process (X’s) and those which are impacted by the process (Y’s). Note that for some variables, this categorisation is not obvious.

Unit operation #1

Unit operation #2

Raw material quality

X Y

Final product quality

What about intermediate product quality?

X or Y?Example 1


The Actual Data Used

The data used in this example came from a real TMP mill in North America. The data have been modified to ensure that no confidential information is revealed.

About 130 tags were selected, corresponding to the X and Y list on the next page. It is not necessary for the student to understand all these, just to be aware that it is complicated and involves many different measurements.

Remember the terminology:

• “Variables”: These are the types of measurements or tags (e.g., refiner body temperature). Variables are shown on the Loadings plot.

• “Observations”: These are the individual measurements, separated in time (March 19, 2000). Observations are shown on the Score plot.

Example 1


The X and Y Variables

The X variables for the TMP process are:• Incoming chips: size distribution, bulk density, humidity.• Refiner operating data: throughput; specific energy imparted to the chips; energy split between the primary and secondary refiner; vertical and conical plate distances; dilution rates; levels, pressures and temperatures in various units immediately connected to the refiners; voltage at chip screw conveyors; specific hydrosulphite consumption; refiner body temperature.• Season, represented by the average monthly temperature measured at a nearby meteorological station.

The Y variables are:• Steam generation rate (an indicator of waste heat generated by friction inside the refiners)• Pulp quality data after the latency chest (automated, on-line analysis of grab samples): standard industry parameters including fibre length distribution, freeness, consistency, and brightness.

Example 1


Pretreatment of data

For this first example, daily averages were obtained for all 130 tags over a 34-month period, corresponding to 1044 observations. Note that the data historian can provide averages over many different time periods, from seconds to months.

The purpose of this exercise was simply to determine which variables trended together over this multi-year period. The spreadsheet contained over 100,000 values (130 variables x 1044 observations), obviously far too much for manual analysis.

Because these are daily averages, the 45-minute residence time in the latency was ignored.

Daily averages

Example 1


PCA of All the DataAs a first step, all the data were put into the MVA program to look for outliers. No distinction was made between X’s and Y’s (everything lumped together).

The software immediately rejected four variables for having “zero or close to zero variance”. This means that they did not vary enough to be of use to the MVA exercise (remember, this is not a planned experiment). The rest of the variables were accepted.

The score plot for this initial PCA exercise is shown on the next page.

Some variables did not change enough to be accepted by the MVA software tool

Example 1


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t[2]

t[1]

32-months version 2.M1 (PCA-X), Untitledt[1]/t[2]

1999-11-231999-11-241999-11-251999-11-261999-11-271999-11-281999-11-291999-11-30

1999-12-011999-12-021999-12-031999-12-041999-12-051999-12-06

1999-12-07

1999-12-081999-12-09

1999-12-101999-12-111999-12-121999-12-131999-12-141999-12-15 1999-12-161999-12-171999-12-181999-12-191999-12-201999-12-211999-12-221999-12-231999-12-241999-12-251999-12-261999-12-271999-12-281999-12-291999-12-302000-01-002000-01-012000-01-022000-01-03

2000-01-042000-01-052000-01-06 2000-01-07

2000-01-082000-01-092000-01-102000-01-112000-01-122000-01-132000-01-142000-01-152000-01-16

2000-01-172000-01-182000-01-192000-01-202000-01-212000-01-22

2000-01-232000-01-242000-01-252000-01-262000-01-272000-01-282000-01-292000-01-302000-01-31

2000-02-01 2000-02-022000-02-032000-02-04

2000-02-052000-02-062000-02-072000-02-082000-02-092000-02-102000-02-112000-02-122000-02-13

2000-02-142000-02-152000-02-162000-02-172000-02-18

2000-02-192000-02-202000-02-21 2000-02-222000-02-232000-02-242000-02-252000-02-262000-02-272000-02-282000-02-292000-03-012000-03-02

2000-03-03 2000-03-042000-03-052000-03-062000-03-072000-03-082000-03-09 2000-03-102000-03-112000-03-122000-03-132000-03-14 2000-03-15 2000-03-162000-03-172000-03-182000-03-192000-03-202000-03-212000-03-22

2000-03-232000-03-242000-03-252000-03-262000-03-272000-03-282000-03-292000-03-302000-03-31 2000-04-012000-04-022000-04-032000-04-042000-04-052000-04-062000-04-07

2000-04-08

2000-04-092000-04-10 2000-04-112000-04-122000-04-132000-04-14 2000-04-152000-04-16

2000-04-172000-04-182000-04-192000-04-202000-04-21 2000-04-222000-04-232000-04-242000-04-252000-04-262000-04-27

2000-04-282000-04-292000-04-30

2000-05-01

2000-05-022000-05-032000-05-042000-05-05

2000-05-062000-05-072000-05-082000-05-092000-05-102000-05-112000-05-12

2000-05-132000-05-142000-05-152000-05-162000-05-17

2000-05-182000-05-192000-05-202000-05-212000-05-222000-05-23 2000-05-242000-05-252000-05-262000-05-272000-05-282000-05-292000-05-302000-05-312000-06-012000-06-022000-06-03

2000-06-042000-06-052000-06-062000-06-072000-06-082000-06-092000-06-102000-06-112000-06-122000-06-132000-06-142000-06-152000-06-162000-06-172000-06-182000-06-192000-06-202000-06-212000-06-222000-06-232000-06-242000-06-252000-06-262000-06-272000-06-282000-06-292000-06-302000-07-012000-07-022000-07-032000-07-042000-07-052000-07-062000-07-07

2000-07-082000-07-092000-07-102000-07-112000-07-122000-07-132000-07-142000-07-152000-07-16

2000-07-172000-07-182000-07-19

2000-07-202000-07-212000-07-222000-07-232000-07-242000-07-252000-07-262000-07-272000-07-282000-07-292000-07-302000-07-312000-08-012000-08-022000-08-032000-08-042000-08-052000-08-062000-08-072000-08-082000-08-092000-08-102000-08-112000-08-122000-08-132000-08-14

2000-08-152000-08-162000-08-172000-08-182000-08-192000-08-20

2000-08-212000-08-222000-08-232000-08-242000-08-25 2000-08-262000-08-272000-08-282000-08-292000-08-302000-08-31 2000-09-012000-09-022000-09-032000-09-042000-09-052000-09-062000-09-072000-09-082000-09-092000-09-102000-09-112000-09-122000-09-132000-09-142000-09-152000-09-162000-09-172000-09-18

2000-09-192000-09-202000-09-212000-09-222000-09-232000-09-242000-09-252000-09-262000-09-272000-09-282000-09-292000-09-302000-10-01

2000-10-022000-10-032000-10-042000-10-052000-10-062000-10-072000-10-082000-10-092000-10-102000-10-11

2000-10-122000-10-132000-10-142000-10-152000-10-162000-10-172000-10-182000-10-192000-10-202000-10-212000-10-222000-10-232000-10-242000-10-252000-10-262000-10-272000-10-282000-10-292000-10-302000-10-312000-11-01

2000-11-022000-11-032000-11-04

2000-11-052000-11-06

2000-11-07

2000-11-08

2000-11-092000-11-10 2000-11-112000-11-122000-11-13

2000-11-14

2000-11-152000-11-162000-11-172000-11-182000-11-192000-11-202000-11-212000-11-222000-11-232000-11-242000-11-252000-11-262000-11-272000-11-282000-11-292000-11-302000-12-012000-12-022000-12-032000-12-042000-12-052000-12-062000-12-072000-12-082000-12-092000-12-102000-12-112000-12-122000-12-13

2000-12-14

2000-12-152000-12-162000-12-172000-12-182000-12-192000-12-202000-12-212000-12-222000-12-23

2000-12-24

2000-12-25

2000-12-26

2000-12-272000-12-282000-12-292000-12-302000-12-312001-01-012001-01-022001-01-032001-01-042001-01-052001-01-062001-01-072001-01-082001-01-09 2001-01-102001-01-112001-01-122001-01-132001-01-142001-01-152001-01-162001-01-172001-01-182001-01-192001-01-202001-01-21

2001-01-222001-01-232001-01-24

2001-01-25

2001-01-262001-01-27

2001-01-282001-01-292001-01-302001-01-312001-02-012001-02-022001-02-03

2001-02-042001-02-052001-02-062001-02-072001-02-082001-02-092001-02-102001-02-112001-02-122001-02-132001-02-142001-02-152001-02-162001-02-172001-02-182001-02-192001-02-20

2001-02-212001-02-22 2001-02-232001-02-242001-02-252001-02-262001-02-272001-02-28

2001-03-01

2001-03-022001-03-03

2001-03-042001-03-05

2001-03-062001-03-072001-03-082001-03-09 2001-03-102001-03-112001-03-12

2001-03-132001-03-14

2001-03-152001-03-162001-03-172001-03-182001-03-192001-03-202001-03-212001-03-22 2001-03-232001-03-242001-03-252001-03-262001-03-272001-03-282001-03-292001-03-302001-03-312001-04-012001-04-022001-04-032001-04-04

2001-04-052001-04-062001-04-072001-04-082001-04-09 2001-04-102001-04-112001-04-122001-04-132001-04-142001-04-152001-04-162001-04-172001-04-182001-04-192001-04-202001-04-212001-04-222001-04-232001-04-242001-04-25 2001-04-262001-04-272001-04-282001-04-292001-04-30

2001-05-012001-05-022001-05-032001-05-042001-05-052001-05-062001-05-072001-05-082001-05-092001-05-102001-05-112001-05-122001-05-132001-05-142001-05-152001-05-162001-05-172001-05-18 2001-05-192001-05-202001-05-212001-05-222001-05-232001-05-242001-05-252001-05-262001-05-272001-05-28

2001-05-292001-05-302001-05-312001-06-012001-06-022001-06-032001-06-042001-06-052001-06-062001-06-072001-06-082001-06-09

2001-06-102001-06-112001-06-122001-06-132001-06-142001-06-152001-06-162001-06-172001-06-18 2001-06-192001-06-202001-06-212001-06-222001-06-23

2001-06-24

2001-06-25

2001-06-26

2001-06-272001-06-282001-06-292001-06-302001-07-012001-07-022001-07-032001-07-04 2001-07-052001-07-062001-07-072001-07-082001-07-092001-07-102001-07-112001-07-12

2001-07-132001-07-142001-07-152001-07-162001-07-172001-07-182001-07-192001-07-20 2001-07-212001-07-222001-07-23 2001-07-242001-07-252001-07-262001-07-272001-07-282001-07-292001-07-302001-07-312001-08-012001-08-022001-08-032001-08-042001-08-052001-08-06

2001-08-072001-08-082001-08-092001-08-102001-08-112001-08-122001-08-13

2001-08-142001-08-152001-08-16

2001-08-172001-08-182001-08-192001-08-202001-08-212001-08-222001-08-232001-08-242001-08-252001-08-26

2001-08-272001-08-282001-08-292001-08-302001-08-312001-09-012001-09-022001-09-032001-09-042001-09-05 2001-09-062001-09-072001-09-08 2001-09-092001-09-102001-09-112001-09-122001-09-132001-09-142001-09-152001-09-162001-09-172001-09-182001-09-192001-09-202001-09-21

2001-09-222001-09-232001-09-242001-09-252001-09-262001-09-272001-09-282001-09-292001-09-30

2001-10-012001-10-022001-10-032001-10-042001-10-052001-10-062001-10-07

2001-10-082001-10-092001-10-10 2001-10-112001-10-122001-10-132001-10-14

2001-10-152001-10-162001-10-172001-10-182001-10-192001-10-20

2001-10-212001-10-222001-10-23 2001-10-24 2001-10-252001-10-262001-10-272001-10-282001-10-292001-10-302001-10-312001-11-01 2001-11-022001-11-032001-11-042001-11-052001-11-062001-11-072001-11-082001-11-092001-11-102001-11-112001-11-12 2001-11-132001-11-142001-11-152001-11-162001-11-172001-11-182001-11-192001-11-202001-11-212001-11-22 2001-11-232001-11-242001-11-252001-11-262001-11-272001-11-282001-11-292001-11-30

2001-12-012001-12-02

2001-12-032001-12-042001-12-052001-12-062001-12-072001-12-082001-12-09

2001-12-10

2001-12-11

2001-12-122001-12-13 2001-12-14

2001-12-152001-12-162001-12-172001-12-182001-12-192001-12-202001-12-212001-12-222001-12-23

2001-12-242001-12-25

2001-12-26

2001-12-272001-12-282001-12-292001-12-30

2001-12-31 2002-01-012002-01-022002-01-032002-01-042002-01-052002-01-062002-01-072002-01-082002-01-092002-01-102002-01-112002-01-122002-01-132002-01-142002-01-152002-01-162002-01-172002-01-18

2002-01-192002-01-20

2002-01-212002-01-22 2002-01-232002-01-242002-01-252002-01-262002-01-272002-01-28

2002-01-29

2002-01-302002-01-312002-02-012002-02-022002-02-032002-02-042002-02-05

2002-02-062002-02-072002-02-082002-02-092002-02-102002-02-112002-02-12 2002-02-132002-02-14 2002-02-152002-02-162002-02-172002-02-182002-02-19

2002-02-202002-02-21

2002-02-222002-02-232002-02-242002-02-252002-02-262002-02-272002-02-282002-03-012002-03-022002-03-032002-03-042002-03-052002-03-062002-03-07

2002-03-082002-03-092002-03-102002-03-112002-03-122002-03-132002-03-142002-03-152002-03-162002-03-172002-03-182002-03-19

2002-03-202002-03-212002-03-222002-03-232002-03-242002-03-252002-03-26

2002-03-27

2002-03-282002-03-292002-03-302002-03-312002-04-012002-04-022002-04-032002-04-042002-04-052002-04-062002-04-072002-04-082002-04-09

2002-04-102002-04-11

2002-04-12 2002-04-132002-04-142002-04-152002-04-162002-04-172002-04-182002-04-192002-04-202002-04-212002-04-222002-04-232002-04-242002-04-25 2002-04-262002-04-272002-04-282002-04-292002-04-302002-05-012002-05-022002-05-032002-05-042002-05-052002-05-062002-05-072002-05-082002-05-092002-05-102002-05-112002-05-122002-05-132002-05-142002-05-152002-05-162002-05-172002-05-182002-05-192002-05-20

2002-05-212002-05-222002-05-232002-05-242002-05-252002-05-262002-05-272002-05-282002-05-292002-05-302002-05-312002-06-012002-06-022002-06-032002-06-042002-06-052002-06-062002-06-072002-06-082002-06-092002-06-10

2002-06-112002-06-12 2002-06-132002-06-142002-06-152002-06-162002-06-172002-06-182002-06-192002-06-202002-06-212002-06-222002-06-232002-06-242002-06-252002-06-262002-06-272002-06-282002-06-29

2002-06-302002-07-012002-07-022002-07-03

2002-07-042002-07-05

2002-07-062002-07-072002-07-082002-07-092002-07-102002-07-112002-07-12 2002-07-132002-07-142002-07-152002-07-162002-07-172002-07-182002-07-192002-07-202002-07-212002-07-22

2002-07-232002-07-24

2002-07-252002-07-262002-07-272002-07-282002-07-292002-07-302002-07-31

2002-08-01

Initial PCA: Score Plot

Already something looks suspicious. Note how a small number of observations dominate the rest. MVA is extremely sensitive to outliers.

What do you notice about the dates?

Example 1


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t[1]


1999-11-231999-11-241999-11-251999-11-261999-11-271999-11-281999-11-291999-11-30

1999-12-011999-12-021999-12-031999-12-041999-12-051999-12-06

1999-12-07

1999-12-081999-12-09

1999-12-101999-12-111999-12-121999-12-131999-12-141999-12-15 1999-12-161999-12-171999-12-181999-12-191999-12-201999-12-211999-12-221999-12-231999-12-241999-12-251999-12-261999-12-271999-12-281999-12-291999-12-302000-01-002000-01-012000-01-022000-01-03

2000-01-042000-01-052000-01-06 2000-01-07

2000-01-082000-01-092000-01-102000-01-112000-01-122000-01-132000-01-142000-01-152000-01-16

2000-01-172000-01-182000-01-192000-01-202000-01-212000-01-22

2000-01-232000-01-242000-01-252000-01-262000-01-272000-01-282000-01-292000-01-302000-01-31

2000-02-01 2000-02-022000-02-032000-02-04

2000-02-052000-02-062000-02-072000-02-082000-02-092000-02-102000-02-112000-02-122000-02-13

2000-02-142000-02-152000-02-162000-02-172000-02-18

2000-02-192000-02-202000-02-21 2000-02-222000-02-232000-02-242000-02-252000-02-262000-02-272000-02-282000-02-292000-03-012000-03-02

2000-03-03 2000-03-042000-03-052000-03-062000-03-072000-03-082000-03-09 2000-03-102000-03-112000-03-122000-03-132000-03-14 2000-03-15 2000-03-162000-03-172000-03-182000-03-192000-03-202000-03-212000-03-22

2000-03-232000-03-242000-03-252000-03-262000-03-272000-03-282000-03-292000-03-302000-03-31 2000-04-012000-04-022000-04-032000-04-042000-04-052000-04-062000-04-07

2000-04-08

2000-04-092000-04-10 2000-04-112000-04-122000-04-132000-04-14 2000-04-152000-04-16

2000-04-172000-04-182000-04-192000-04-202000-04-21 2000-04-222000-04-232000-04-242000-04-252000-04-262000-04-27

2000-04-282000-04-292000-04-30

2000-05-01

2000-05-022000-05-032000-05-042000-05-05

2000-05-062000-05-072000-05-082000-05-092000-05-102000-05-112000-05-12

2000-05-132000-05-142000-05-152000-05-162000-05-17

2000-05-182000-05-192000-05-202000-05-212000-05-222000-05-23 2000-05-242000-05-252000-05-262000-05-272000-05-282000-05-292000-05-302000-05-312000-06-012000-06-022000-06-03

2000-06-042000-06-052000-06-062000-06-072000-06-082000-06-092000-06-102000-06-112000-06-122000-06-132000-06-142000-06-152000-06-162000-06-172000-06-182000-06-192000-06-202000-06-212000-06-222000-06-232000-06-242000-06-252000-06-262000-06-272000-06-282000-06-292000-06-302000-07-012000-07-022000-07-032000-07-042000-07-052000-07-062000-07-07

2000-07-082000-07-092000-07-102000-07-112000-07-122000-07-132000-07-142000-07-152000-07-16

2000-07-172000-07-182000-07-19

2000-07-202000-07-212000-07-222000-07-232000-07-242000-07-252000-07-262000-07-272000-07-282000-07-292000-07-302000-07-312000-08-012000-08-022000-08-032000-08-042000-08-052000-08-062000-08-072000-08-082000-08-092000-08-102000-08-112000-08-122000-08-132000-08-14

2000-08-152000-08-162000-08-172000-08-182000-08-192000-08-20

2000-08-212000-08-222000-08-232000-08-242000-08-25 2000-08-262000-08-272000-08-282000-08-292000-08-302000-08-31 2000-09-012000-09-022000-09-032000-09-042000-09-052000-09-062000-09-072000-09-082000-09-092000-09-102000-09-112000-09-122000-09-132000-09-142000-09-152000-09-162000-09-172000-09-18

2000-09-192000-09-202000-09-212000-09-222000-09-232000-09-242000-09-252000-09-262000-09-272000-09-282000-09-292000-09-302000-10-01

2000-10-022000-10-032000-10-042000-10-052000-10-062000-10-072000-10-082000-10-092000-10-102000-10-11

2000-10-122000-10-132000-10-142000-10-152000-10-162000-10-172000-10-182000-10-192000-10-202000-10-212000-10-222000-10-232000-10-242000-10-252000-10-262000-10-272000-10-282000-10-292000-10-302000-10-312000-11-01

2000-11-022000-11-032000-11-04

2000-11-052000-11-06

2000-11-07

2000-11-08

2000-11-092000-11-10 2000-11-112000-11-122000-11-13

2000-11-14

2000-11-152000-11-162000-11-172000-11-182000-11-192000-11-202000-11-212000-11-222000-11-232000-11-242000-11-252000-11-262000-11-272000-11-282000-11-292000-11-302000-12-012000-12-022000-12-032000-12-042000-12-052000-12-062000-12-072000-12-082000-12-092000-12-102000-12-112000-12-122000-12-13

2000-12-14

2000-12-152000-12-162000-12-172000-12-182000-12-192000-12-202000-12-212000-12-222000-12-23

2000-12-24

2000-12-25

2000-12-26

2000-12-272000-12-282000-12-292000-12-302000-12-312001-01-012001-01-022001-01-032001-01-042001-01-052001-01-062001-01-072001-01-082001-01-09 2001-01-102001-01-112001-01-122001-01-132001-01-142001-01-152001-01-162001-01-172001-01-182001-01-192001-01-202001-01-21

2001-01-222001-01-232001-01-24

2001-01-25

2001-01-262001-01-27

2001-01-282001-01-292001-01-302001-01-312001-02-012001-02-022001-02-03

2001-02-042001-02-052001-02-062001-02-072001-02-082001-02-092001-02-102001-02-112001-02-122001-02-132001-02-142001-02-152001-02-162001-02-172001-02-182001-02-192001-02-20

2001-02-212001-02-22 2001-02-232001-02-242001-02-252001-02-262001-02-272001-02-28

2001-03-01

2001-03-022001-03-03

2001-03-042001-03-05

2001-03-062001-03-072001-03-082001-03-09 2001-03-102001-03-112001-03-12

2001-03-132001-03-14

2001-03-152001-03-162001-03-172001-03-182001-03-192001-03-202001-03-212001-03-22 2001-03-232001-03-242001-03-252001-03-262001-03-272001-03-282001-03-292001-03-302001-03-312001-04-012001-04-022001-04-032001-04-04

2001-04-052001-04-062001-04-072001-04-082001-04-09 2001-04-102001-04-112001-04-122001-04-132001-04-142001-04-152001-04-162001-04-172001-04-182001-04-192001-04-202001-04-212001-04-222001-04-232001-04-242001-04-25 2001-04-262001-04-272001-04-282001-04-292001-04-30

2001-05-012001-05-022001-05-032001-05-042001-05-052001-05-062001-05-072001-05-082001-05-092001-05-102001-05-112001-05-122001-05-132001-05-142001-05-152001-05-162001-05-172001-05-18 2001-05-192001-05-202001-05-212001-05-222001-05-232001-05-242001-05-252001-05-262001-05-272001-05-28

2001-05-292001-05-302001-05-312001-06-012001-06-022001-06-032001-06-042001-06-052001-06-062001-06-072001-06-082001-06-09

2001-06-102001-06-112001-06-122001-06-132001-06-142001-06-152001-06-162001-06-172001-06-18 2001-06-192001-06-202001-06-212001-06-222001-06-23

2001-06-24

2001-06-25

2001-06-26

2001-06-272001-06-282001-06-292001-06-302001-07-012001-07-022001-07-032001-07-04 2001-07-052001-07-062001-07-072001-07-082001-07-092001-07-102001-07-112001-07-12

2001-07-132001-07-142001-07-152001-07-162001-07-172001-07-182001-07-192001-07-20 2001-07-212001-07-222001-07-23 2001-07-242001-07-252001-07-262001-07-272001-07-282001-07-292001-07-302001-07-312001-08-012001-08-022001-08-032001-08-042001-08-052001-08-06

2001-08-072001-08-082001-08-092001-08-102001-08-112001-08-122001-08-13

2001-08-142001-08-152001-08-16

2001-08-172001-08-182001-08-192001-08-202001-08-212001-08-222001-08-232001-08-242001-08-252001-08-26

2001-08-272001-08-282001-08-292001-08-302001-08-312001-09-012001-09-022001-09-032001-09-042001-09-05 2001-09-062001-09-072001-09-08 2001-09-092001-09-102001-09-112001-09-122001-09-132001-09-142001-09-152001-09-162001-09-172001-09-182001-09-192001-09-202001-09-21

2001-09-222001-09-232001-09-242001-09-252001-09-262001-09-272001-09-282001-09-292001-09-30

2001-10-012001-10-022001-10-032001-10-042001-10-052001-10-062001-10-07

2001-10-082001-10-092001-10-10 2001-10-112001-10-122001-10-132001-10-14

2001-10-152001-10-162001-10-172001-10-182001-10-192001-10-20

2001-10-212001-10-222001-10-23 2001-10-24 2001-10-252001-10-262001-10-272001-10-282001-10-292001-10-302001-10-312001-11-01 2001-11-022001-11-032001-11-042001-11-052001-11-062001-11-072001-11-082001-11-092001-11-102001-11-112001-11-12 2001-11-132001-11-142001-11-152001-11-162001-11-172001-11-182001-11-192001-11-202001-11-212001-11-22 2001-11-232001-11-242001-11-252001-11-262001-11-272001-11-282001-11-292001-11-30

2001-12-012001-12-02

2001-12-032001-12-042001-12-052001-12-062001-12-072001-12-082001-12-09

2001-12-10

2001-12-11

2001-12-122001-12-13 2001-12-14

2001-12-152001-12-162001-12-172001-12-182001-12-192001-12-202001-12-212001-12-222001-12-23

2001-12-242001-12-25

2001-12-26

2001-12-272001-12-282001-12-292001-12-30

2001-12-31 2002-01-012002-01-022002-01-032002-01-042002-01-052002-01-062002-01-072002-01-082002-01-092002-01-102002-01-112002-01-122002-01-132002-01-142002-01-152002-01-162002-01-172002-01-18

2002-01-192002-01-20

2002-01-212002-01-22 2002-01-232002-01-242002-01-252002-01-262002-01-272002-01-28

2002-01-29

2002-01-302002-01-312002-02-012002-02-022002-02-032002-02-042002-02-05

2002-02-062002-02-072002-02-082002-02-092002-02-102002-02-112002-02-12 2002-02-132002-02-14 2002-02-152002-02-162002-02-172002-02-182002-02-19

2002-02-202002-02-21

2002-02-222002-02-232002-02-242002-02-252002-02-262002-02-272002-02-282002-03-012002-03-022002-03-032002-03-042002-03-052002-03-062002-03-07

2002-03-082002-03-092002-03-102002-03-112002-03-122002-03-132002-03-142002-03-152002-03-162002-03-172002-03-182002-03-19

2002-03-202002-03-212002-03-222002-03-232002-03-242002-03-252002-03-26

2002-03-27

2002-03-282002-03-292002-03-302002-03-312002-04-012002-04-022002-04-032002-04-042002-04-052002-04-062002-04-072002-04-082002-04-09

2002-04-102002-04-11

2002-04-12 2002-04-132002-04-142002-04-152002-04-162002-04-172002-04-182002-04-192002-04-202002-04-212002-04-222002-04-232002-04-242002-04-25 2002-04-262002-04-272002-04-282002-04-292002-04-302002-05-012002-05-022002-05-032002-05-042002-05-052002-05-062002-05-072002-05-082002-05-092002-05-102002-05-112002-05-122002-05-132002-05-142002-05-152002-05-162002-05-172002-05-182002-05-192002-05-20

2002-05-212002-05-222002-05-232002-05-242002-05-252002-05-262002-05-272002-05-282002-05-292002-05-302002-05-312002-06-012002-06-022002-06-032002-06-042002-06-052002-06-062002-06-072002-06-082002-06-092002-06-10

2002-06-112002-06-12 2002-06-132002-06-142002-06-152002-06-162002-06-172002-06-182002-06-192002-06-202002-06-212002-06-222002-06-232002-06-242002-06-252002-06-262002-06-272002-06-282002-06-29

2002-06-302002-07-012002-07-022002-07-03

2002-07-042002-07-05

2002-07-062002-07-072002-07-082002-07-092002-07-102002-07-112002-07-12 2002-07-132002-07-142002-07-152002-07-162002-07-172002-07-182002-07-192002-07-202002-07-212002-07-22

2002-07-232002-07-24

2002-07-252002-07-262002-07-272002-07-282002-07-292002-07-302002-07-31

2002-08-01

Extreme Outliers

Some of these strange dates fall on Christmas Eve and Christmas Day! These holidays are radically different somehow. An obvious guess is that production was lower on those days. To confirm this we check the original data.

Example 1


-30

-20

-10

0

-20 -10 0 10

t[2]

t[1]


1999-11-231999-11-241999-11-251999-11-261999-11-271999-11-281999-11-291999-11-30

1999-12-011999-12-021999-12-031999-12-041999-12-051999-12-06

1999-12-07

1999-12-081999-12-09

1999-12-101999-12-111999-12-121999-12-131999-12-141999-12-15 1999-12-161999-12-171999-12-181999-12-191999-12-201999-12-211999-12-221999-12-231999-12-241999-12-251999-12-261999-12-271999-12-281999-12-291999-12-302000-01-002000-01-012000-01-022000-01-03

2000-01-042000-01-052000-01-06 2000-01-07

2000-01-082000-01-092000-01-102000-01-112000-01-122000-01-132000-01-142000-01-152000-01-16

2000-01-172000-01-182000-01-192000-01-202000-01-212000-01-22

2000-01-232000-01-242000-01-252000-01-262000-01-272000-01-282000-01-292000-01-302000-01-31

2000-02-01 2000-02-022000-02-032000-02-04

2000-02-052000-02-062000-02-072000-02-082000-02-092000-02-102000-02-112000-02-122000-02-13

2000-02-142000-02-152000-02-162000-02-172000-02-18

2000-02-192000-02-202000-02-21 2000-02-222000-02-232000-02-242000-02-252000-02-262000-02-272000-02-282000-02-292000-03-012000-03-02

2000-03-03 2000-03-042000-03-052000-03-062000-03-072000-03-082000-03-09 2000-03-102000-03-112000-03-122000-03-132000-03-14 2000-03-15 2000-03-162000-03-172000-03-182000-03-192000-03-202000-03-212000-03-22

2000-03-232000-03-242000-03-252000-03-262000-03-272000-03-282000-03-292000-03-302000-03-31 2000-04-012000-04-022000-04-032000-04-042000-04-052000-04-062000-04-07

2000-04-08

2000-04-092000-04-10 2000-04-112000-04-122000-04-132000-04-14 2000-04-152000-04-16

2000-04-172000-04-182000-04-192000-04-202000-04-21 2000-04-222000-04-232000-04-242000-04-252000-04-262000-04-27

2000-04-282000-04-292000-04-30

2000-05-01

2000-05-022000-05-032000-05-042000-05-05

2000-05-062000-05-072000-05-082000-05-092000-05-102000-05-112000-05-12

2000-05-132000-05-142000-05-152000-05-162000-05-17

2000-05-182000-05-192000-05-202000-05-212000-05-222000-05-23 2000-05-242000-05-252000-05-262000-05-272000-05-282000-05-292000-05-302000-05-312000-06-012000-06-022000-06-03

2000-06-042000-06-052000-06-062000-06-072000-06-082000-06-092000-06-102000-06-112000-06-122000-06-132000-06-142000-06-152000-06-162000-06-172000-06-182000-06-192000-06-202000-06-212000-06-222000-06-232000-06-242000-06-252000-06-262000-06-272000-06-282000-06-292000-06-302000-07-012000-07-022000-07-032000-07-042000-07-052000-07-062000-07-07

2000-07-082000-07-092000-07-102000-07-112000-07-122000-07-132000-07-142000-07-152000-07-16

2000-07-172000-07-182000-07-19

2000-07-202000-07-212000-07-222000-07-232000-07-242000-07-252000-07-262000-07-272000-07-282000-07-292000-07-302000-07-312000-08-012000-08-022000-08-032000-08-042000-08-052000-08-062000-08-072000-08-082000-08-092000-08-102000-08-112000-08-122000-08-132000-08-14

2000-08-152000-08-162000-08-172000-08-182000-08-192000-08-20

2000-08-212000-08-222000-08-232000-08-242000-08-25 2000-08-262000-08-272000-08-282000-08-292000-08-302000-08-31 2000-09-012000-09-022000-09-032000-09-042000-09-052000-09-062000-09-072000-09-082000-09-092000-09-102000-09-112000-09-122000-09-132000-09-142000-09-152000-09-162000-09-172000-09-18

2000-09-192000-09-202000-09-212000-09-222000-09-232000-09-242000-09-252000-09-262000-09-272000-09-282000-09-292000-09-302000-10-01

2000-10-022000-10-032000-10-042000-10-052000-10-062000-10-072000-10-082000-10-092000-10-102000-10-11

2000-10-122000-10-132000-10-142000-10-152000-10-162000-10-172000-10-182000-10-192000-10-202000-10-212000-10-222000-10-232000-10-242000-10-252000-10-262000-10-272000-10-282000-10-292000-10-302000-10-312000-11-01

2000-11-022000-11-032000-11-04

2000-11-052000-11-06

2000-11-07

2000-11-08

2000-11-092000-11-10 2000-11-112000-11-122000-11-13

2000-11-14

2000-11-152000-11-162000-11-172000-11-182000-11-192000-11-202000-11-212000-11-222000-11-232000-11-242000-11-252000-11-262000-11-272000-11-282000-11-292000-11-302000-12-012000-12-022000-12-032000-12-042000-12-052000-12-062000-12-072000-12-082000-12-092000-12-102000-12-112000-12-122000-12-13

2000-12-14

2000-12-152000-12-162000-12-172000-12-182000-12-192000-12-202000-12-212000-12-222000-12-23

2000-12-24

2000-12-25

2000-12-26

2000-12-272000-12-282000-12-292000-12-302000-12-312001-01-012001-01-022001-01-032001-01-042001-01-052001-01-062001-01-072001-01-082001-01-09 2001-01-102001-01-112001-01-122001-01-132001-01-142001-01-152001-01-162001-01-172001-01-182001-01-192001-01-202001-01-21

2001-01-222001-01-232001-01-24

2001-01-25

2001-01-262001-01-27

2001-01-282001-01-292001-01-302001-01-312001-02-012001-02-022001-02-03

2001-02-042001-02-052001-02-062001-02-072001-02-082001-02-092001-02-102001-02-112001-02-122001-02-132001-02-142001-02-152001-02-162001-02-172001-02-182001-02-192001-02-20

2001-02-212001-02-22 2001-02-232001-02-242001-02-252001-02-262001-02-272001-02-28

2001-03-01

2001-03-022001-03-03

2001-03-042001-03-05

2001-03-062001-03-072001-03-082001-03-09 2001-03-102001-03-112001-03-12

2001-03-132001-03-14

2001-03-152001-03-162001-03-172001-03-182001-03-192001-03-202001-03-212001-03-22 2001-03-232001-03-242001-03-252001-03-262001-03-272001-03-282001-03-292001-03-302001-03-312001-04-012001-04-022001-04-032001-04-04

2001-04-052001-04-062001-04-072001-04-082001-04-09 2001-04-102001-04-112001-04-122001-04-132001-04-142001-04-152001-04-162001-04-172001-04-182001-04-192001-04-202001-04-212001-04-222001-04-232001-04-242001-04-25 2001-04-262001-04-272001-04-282001-04-292001-04-30

2001-05-012001-05-022001-05-032001-05-042001-05-052001-05-062001-05-072001-05-082001-05-092001-05-102001-05-112001-05-122001-05-132001-05-142001-05-152001-05-162001-05-172001-05-18 2001-05-192001-05-202001-05-212001-05-222001-05-232001-05-242001-05-252001-05-262001-05-272001-05-28

2001-05-292001-05-302001-05-312001-06-012001-06-022001-06-032001-06-042001-06-052001-06-062001-06-072001-06-082001-06-09

2001-06-102001-06-112001-06-122001-06-132001-06-142001-06-152001-06-162001-06-172001-06-18 2001-06-192001-06-202001-06-212001-06-222001-06-23

2001-06-24

2001-06-25

2001-06-26

2001-06-272001-06-282001-06-292001-06-302001-07-012001-07-022001-07-032001-07-04 2001-07-052001-07-062001-07-072001-07-082001-07-092001-07-102001-07-112001-07-12

2001-07-132001-07-142001-07-152001-07-162001-07-172001-07-182001-07-192001-07-20 2001-07-212001-07-222001-07-23 2001-07-242001-07-252001-07-262001-07-272001-07-282001-07-292001-07-302001-07-312001-08-012001-08-022001-08-032001-08-042001-08-052001-08-06

2001-08-072001-08-082001-08-092001-08-102001-08-112001-08-122001-08-13

2001-08-142001-08-152001-08-16

2001-08-172001-08-182001-08-192001-08-202001-08-212001-08-222001-08-232001-08-242001-08-252001-08-26

2001-08-272001-08-282001-08-292001-08-302001-08-312001-09-012001-09-022001-09-032001-09-042001-09-05 2001-09-062001-09-072001-09-08 2001-09-092001-09-102001-09-112001-09-122001-09-132001-09-142001-09-152001-09-162001-09-172001-09-182001-09-192001-09-202001-09-21

2001-09-222001-09-232001-09-242001-09-252001-09-262001-09-272001-09-282001-09-292001-09-30

2001-10-012001-10-022001-10-032001-10-042001-10-052001-10-062001-10-07

2001-10-082001-10-092001-10-10 2001-10-112001-10-122001-10-132001-10-14

2001-10-152001-10-162001-10-172001-10-182001-10-192001-10-20

2001-10-212001-10-222001-10-23 2001-10-24 2001-10-252001-10-262001-10-272001-10-282001-10-292001-10-302001-10-312001-11-01 2001-11-022001-11-032001-11-042001-11-052001-11-062001-11-072001-11-082001-11-092001-11-102001-11-112001-11-12 2001-11-132001-11-142001-11-152001-11-162001-11-172001-11-182001-11-192001-11-202001-11-212001-11-22 2001-11-232001-11-242001-11-252001-11-262001-11-272001-11-282001-11-292001-11-30

2001-12-012001-12-02

2001-12-032001-12-042001-12-052001-12-062001-12-072001-12-082001-12-09

2001-12-10

2001-12-11

2001-12-122001-12-13 2001-12-14

2001-12-152001-12-162001-12-172001-12-182001-12-192001-12-202001-12-212001-12-222001-12-23

2001-12-242001-12-25

2001-12-26

2001-12-272001-12-282001-12-292001-12-30

2001-12-31 2002-01-012002-01-022002-01-032002-01-042002-01-052002-01-062002-01-072002-01-082002-01-092002-01-102002-01-112002-01-122002-01-132002-01-142002-01-152002-01-162002-01-172002-01-18

2002-01-192002-01-20

2002-01-212002-01-22 2002-01-232002-01-242002-01-252002-01-262002-01-272002-01-28

2002-01-29

2002-01-302002-01-312002-02-012002-02-022002-02-032002-02-042002-02-05

2002-02-062002-02-072002-02-082002-02-092002-02-102002-02-112002-02-12 2002-02-132002-02-14 2002-02-152002-02-162002-02-172002-02-182002-02-19

2002-02-202002-02-21

2002-02-222002-02-232002-02-242002-02-252002-02-262002-02-272002-02-282002-03-012002-03-022002-03-032002-03-042002-03-052002-03-062002-03-07

2002-03-082002-03-092002-03-102002-03-112002-03-122002-03-132002-03-142002-03-152002-03-162002-03-172002-03-182002-03-19

2002-03-202002-03-212002-03-222002-03-232002-03-242002-03-252002-03-26

2002-03-27

2002-03-282002-03-292002-03-302002-03-312002-04-012002-04-022002-04-032002-04-042002-04-052002-04-062002-04-072002-04-082002-04-09

2002-04-102002-04-11

2002-04-12 2002-04-132002-04-142002-04-152002-04-162002-04-172002-04-182002-04-192002-04-202002-04-212002-04-222002-04-232002-04-242002-04-25 2002-04-262002-04-272002-04-282002-04-292002-04-302002-05-012002-05-022002-05-032002-05-042002-05-052002-05-062002-05-072002-05-082002-05-092002-05-102002-05-112002-05-122002-05-132002-05-142002-05-152002-05-162002-05-172002-05-182002-05-192002-05-20

2002-05-212002-05-222002-05-232002-05-242002-05-252002-05-262002-05-272002-05-282002-05-292002-05-302002-05-312002-06-012002-06-022002-06-032002-06-042002-06-052002-06-062002-06-072002-06-082002-06-092002-06-10

2002-06-112002-06-12 2002-06-132002-06-142002-06-152002-06-162002-06-172002-06-182002-06-192002-06-202002-06-212002-06-222002-06-232002-06-242002-06-252002-06-262002-06-272002-06-282002-06-29

2002-06-302002-07-012002-07-022002-07-03

2002-07-042002-07-05

2002-07-062002-07-072002-07-082002-07-092002-07-102002-07-112002-07-12 2002-07-132002-07-142002-07-152002-07-162002-07-172002-07-182002-07-192002-07-202002-07-212002-07-22

2002-07-232002-07-24

2002-07-252002-07-262002-07-272002-07-282002-07-292002-07-302002-07-31

2002-08-01

Low Production Days!

Days with production < 50 t/d

Days with production < 100 t/d

Our suspicions are confirmed. A quick check of the original dataset shows that all these dates correspond to lower production.

Example 1


Decision to remove outliers

Now that we know why these dates are outliers, we can remove them with confidence.

It is generally a bad idea to remove outliers without determining why they are different. It may be that these are not outliers at all, but actually interesting and important shifts in the process – the very thing we would like to know about.

Determining the cause of outliers is usually more difficult than this “Christmas holiday” example. We will see other techniques in the examples that follow.

Chopping the outliers

Example 1


PCA with extreme outliers removed

-5

0

5

-20 -10 0 10

t[2]

t[1]

32-months version 2.M2 (PCA-X), Extreme outliers removedt[1]/t[2]

1999-11-231999-11-241999-11-251999-11-261999-11-271999-11-28

1999-11-291999-11-301999-12-011999-12-02

1999-12-031999-12-041999-12-05

1999-12-061999-12-081999-12-09

1999-12-101999-12-111999-12-121999-12-131999-12-141999-12-15 1999-12-161999-12-171999-12-181999-12-191999-12-201999-12-211999-12-221999-12-231999-12-241999-12-251999-12-261999-12-271999-12-28

1999-12-29 1999-12-302000-01-002000-01-012000-01-022000-01-032000-01-042000-01-052000-01-06 2000-01-072000-01-08

2000-01-092000-01-102000-01-112000-01-122000-01-132000-01-142000-01-152000-01-162000-01-17

2000-01-182000-01-192000-01-202000-01-212000-01-22

2000-01-232000-01-24

2000-01-25

2000-01-262000-01-272000-01-282000-01-292000-01-302000-01-31

2000-02-01 2000-02-022000-02-032000-02-04

2000-02-052000-02-062000-02-072000-02-082000-02-092000-02-102000-02-11

2000-02-122000-02-132000-02-142000-02-152000-02-16

2000-02-172000-02-18 2000-02-192000-02-202000-02-212000-02-222000-02-232000-02-242000-02-252000-02-262000-02-272000-02-282000-02-292000-03-012000-03-02

2000-03-032000-03-04 2000-03-052000-03-062000-03-072000-03-082000-03-09 2000-03-102000-03-112000-03-122000-03-13

2000-03-142000-03-15 2000-03-162000-03-172000-03-182000-03-192000-03-20

2000-03-212000-03-222000-03-23 2000-03-242000-03-252000-03-262000-03-272000-03-282000-03-29

2000-03-302000-03-31

2000-04-012000-04-022000-04-03

2000-04-042000-04-052000-04-062000-04-07

2000-04-082000-04-09

2000-04-10 2000-04-112000-04-122000-04-132000-04-14 2000-04-15

2000-04-162000-04-172000-04-182000-04-192000-04-20

2000-04-21 2000-04-222000-04-232000-04-242000-04-252000-04-262000-04-27

2000-04-282000-04-292000-04-302000-05-01

2000-05-02

2000-05-032000-05-042000-05-05

2000-05-062000-05-072000-05-082000-05-09

2000-05-10

2000-05-112000-05-122000-05-13

2000-05-142000-05-152000-05-162000-05-172000-05-182000-05-192000-05-202000-05-212000-05-22

2000-05-23 2000-05-242000-05-252000-05-262000-05-272000-05-282000-05-292000-05-302000-05-312000-06-012000-06-02 2000-06-032000-06-04

2000-06-052000-06-062000-06-072000-06-082000-06-092000-06-102000-06-112000-06-12

2000-06-132000-06-142000-06-152000-06-162000-06-172000-06-18

2000-06-192000-06-20

2000-06-212000-06-222000-06-23 2000-06-242000-06-252000-06-262000-06-272000-06-282000-06-292000-06-302000-07-01

2000-07-022000-07-032000-07-042000-07-052000-07-062000-07-07 2000-07-082000-07-092000-07-102000-07-11 2000-07-122000-07-132000-07-142000-07-152000-07-162000-07-172000-07-18

2000-07-192000-07-20

2000-07-212000-07-222000-07-232000-07-24

2000-07-252000-07-262000-07-272000-07-282000-07-292000-07-302000-07-312000-08-012000-08-022000-08-032000-08-042000-08-052000-08-062000-08-072000-08-08

2000-08-092000-08-102000-08-112000-08-122000-08-132000-08-142000-08-152000-08-16

2000-08-172000-08-182000-08-192000-08-202000-08-212000-08-222000-08-232000-08-242000-08-25 2000-08-262000-08-272000-08-282000-08-29

2000-08-302000-08-31 2000-09-012000-09-022000-09-032000-09-042000-09-052000-09-062000-09-072000-09-082000-09-092000-09-10

2000-09-112000-09-122000-09-132000-09-142000-09-15

2000-09-162000-09-172000-09-18

2000-09-192000-09-20

2000-09-212000-09-222000-09-232000-09-242000-09-252000-09-262000-09-272000-09-282000-09-292000-09-302000-10-01

2000-10-022000-10-032000-10-042000-10-05 2000-10-06

2000-10-072000-10-082000-10-092000-10-102000-10-112000-10-122000-10-132000-10-142000-10-152000-10-162000-10-172000-10-182000-10-192000-10-202000-10-212000-10-222000-10-232000-10-242000-10-252000-10-262000-10-27

2000-10-282000-10-292000-10-302000-10-312000-11-012000-11-022000-11-03

2000-11-042000-11-05

2000-11-06

2000-11-092000-11-10 2000-11-112000-11-122000-11-13

2000-11-14

2000-11-152000-11-162000-11-172000-11-182000-11-192000-11-202000-11-212000-11-222000-11-232000-11-242000-11-252000-11-262000-11-272000-11-282000-11-292000-11-30

2000-12-012000-12-022000-12-032000-12-042000-12-05

2000-12-062000-12-072000-12-082000-12-092000-12-102000-12-112000-12-12

2000-12-132000-12-152000-12-16

2000-12-172000-12-182000-12-192000-12-202000-12-21

2000-12-222000-12-232000-12-27

2000-12-282000-12-292000-12-302000-12-312001-01-012001-01-022001-01-032001-01-042001-01-052001-01-062001-01-072001-01-082001-01-09 2001-01-10

2001-01-112001-01-122001-01-132001-01-142001-01-152001-01-162001-01-172001-01-182001-01-192001-01-202001-01-212001-01-222001-01-232001-01-24

2001-01-262001-01-27

2001-01-282001-01-292001-01-302001-01-312001-02-01

2001-02-022001-02-03

2001-02-042001-02-052001-02-062001-02-072001-02-08

2001-02-092001-02-102001-02-112001-02-122001-02-132001-02-142001-02-15

2001-02-162001-02-172001-02-182001-02-19

2001-02-202001-02-212001-02-22 2001-02-232001-02-24

2001-02-25

2001-02-262001-02-27

2001-02-282001-03-01

2001-03-022001-03-032001-03-04 2001-03-05

2001-03-062001-03-07

2001-03-082001-03-09

2001-03-102001-03-112001-03-12

2001-03-132001-03-14

2001-03-152001-03-162001-03-172001-03-182001-03-192001-03-202001-03-212001-03-22 2001-03-232001-03-242001-03-252001-03-262001-03-272001-03-282001-03-292001-03-302001-03-312001-04-01

2001-04-022001-04-032001-04-04

2001-04-052001-04-062001-04-072001-04-082001-04-092001-04-102001-04-11

2001-04-12

2001-04-132001-04-14

2001-04-152001-04-162001-04-172001-04-182001-04-192001-04-202001-04-212001-04-222001-04-232001-04-24

2001-04-252001-04-262001-04-272001-04-282001-04-29

2001-04-30 2001-05-012001-05-022001-05-032001-05-042001-05-052001-05-062001-05-07

2001-05-082001-05-092001-05-102001-05-112001-05-122001-05-132001-05-142001-05-152001-05-16

2001-05-172001-05-182001-05-192001-05-202001-05-212001-05-222001-05-232001-05-242001-05-252001-05-262001-05-27

2001-05-282001-05-292001-05-302001-05-312001-06-01 2001-06-022001-06-032001-06-042001-06-052001-06-062001-06-072001-06-08

2001-06-092001-06-102001-06-112001-06-122001-06-132001-06-142001-06-15

2001-06-162001-06-172001-06-18

2001-06-192001-06-202001-06-212001-06-222001-06-23

2001-06-26

2001-06-272001-06-282001-06-292001-06-302001-07-012001-07-022001-07-032001-07-04

2001-07-052001-07-062001-07-072001-07-08

2001-07-092001-07-102001-07-11

2001-07-122001-07-132001-07-142001-07-152001-07-16

2001-07-172001-07-182001-07-192001-07-202001-07-21

2001-07-222001-07-23 2001-07-242001-07-252001-07-26

2001-07-272001-07-282001-07-292001-07-302001-07-312001-08-012001-08-022001-08-032001-08-042001-08-05

2001-08-06

2001-08-07

2001-08-082001-08-092001-08-102001-08-112001-08-12

2001-08-132001-08-14

2001-08-15

2001-08-162001-08-17

2001-08-182001-08-192001-08-202001-08-212001-08-222001-08-232001-08-242001-08-252001-08-26

2001-08-272001-08-282001-08-292001-08-302001-08-312001-09-012001-09-022001-09-032001-09-04

2001-09-05 2001-09-062001-09-072001-09-08 2001-09-092001-09-102001-09-112001-09-122001-09-132001-09-142001-09-152001-09-162001-09-172001-09-182001-09-192001-09-202001-09-21

2001-09-222001-09-232001-09-242001-09-252001-09-262001-09-272001-09-282001-09-292001-09-302001-10-012001-10-022001-10-03

2001-10-042001-10-052001-10-062001-10-072001-10-082001-10-09

2001-10-10

2001-10-112001-10-122001-10-132001-10-14

2001-10-152001-10-16

2001-10-172001-10-182001-10-192001-10-20

2001-10-212001-10-22

2001-10-23 2001-10-24 2001-10-252001-10-262001-10-272001-10-282001-10-29

2001-10-302001-10-312001-11-01 2001-11-022001-11-032001-11-042001-11-052001-11-062001-11-072001-11-082001-11-092001-11-102001-11-112001-11-12

2001-11-132001-11-142001-11-152001-11-162001-11-172001-11-182001-11-192001-11-202001-11-212001-11-22

2001-11-23 2001-11-242001-11-252001-11-262001-11-272001-11-282001-11-292001-11-30

2001-12-012001-12-02

2001-12-032001-12-042001-12-05

2001-12-062001-12-072001-12-082001-12-09

2001-12-10

2001-12-122001-12-13 2001-12-14

2001-12-152001-12-162001-12-172001-12-182001-12-192001-12-20

2001-12-212001-12-222001-12-232001-12-272001-12-282001-12-292001-12-30

2001-12-312002-01-012002-01-022002-01-032002-01-042002-01-052002-01-062002-01-072002-01-082002-01-092002-01-10

2002-01-112002-01-122002-01-13

2002-01-142002-01-152002-01-162002-01-172002-01-18

2002-01-192002-01-202002-01-212002-01-22 2002-01-232002-01-242002-01-252002-01-262002-01-272002-01-282002-01-302002-01-31

2002-02-012002-02-022002-02-032002-02-042002-02-05

2002-02-062002-02-07

2002-02-082002-02-092002-02-102002-02-112002-02-12 2002-02-132002-02-14 2002-02-152002-02-162002-02-172002-02-182002-02-19

2002-02-202002-02-21

2002-02-222002-02-232002-02-242002-02-25

2002-02-262002-02-272002-02-282002-03-012002-03-022002-03-032002-03-042002-03-05

2002-03-062002-03-072002-03-08

2002-03-092002-03-102002-03-112002-03-122002-03-132002-03-142002-03-152002-03-162002-03-172002-03-182002-03-19 2002-03-202002-03-212002-03-222002-03-232002-03-242002-03-25

2002-03-262002-03-272002-03-282002-03-29 2002-03-302002-03-312002-04-012002-04-02

2002-04-032002-04-042002-04-052002-04-062002-04-072002-04-082002-04-09

2002-04-102002-04-112002-04-12

2002-04-132002-04-142002-04-152002-04-162002-04-172002-04-182002-04-192002-04-202002-04-212002-04-222002-04-232002-04-242002-04-252002-04-262002-04-272002-04-282002-04-292002-04-302002-05-012002-05-022002-05-032002-05-042002-05-05

2002-05-062002-05-07

2002-05-08 2002-05-092002-05-102002-05-112002-05-12

2002-05-132002-05-142002-05-152002-05-162002-05-172002-05-182002-05-19

2002-05-202002-05-21

2002-05-222002-05-232002-05-242002-05-252002-05-26

2002-05-272002-05-282002-05-29

2002-05-302002-05-312002-06-012002-06-022002-06-032002-06-042002-06-052002-06-062002-06-072002-06-082002-06-092002-06-10

2002-06-112002-06-122002-06-132002-06-142002-06-152002-06-162002-06-172002-06-182002-06-192002-06-20

2002-06-212002-06-222002-06-232002-06-242002-06-252002-06-262002-06-272002-06-282002-06-29

2002-06-302002-07-01

2002-07-022002-07-03

2002-07-042002-07-05

2002-07-062002-07-072002-07-082002-07-092002-07-102002-07-11

2002-07-12 2002-07-132002-07-142002-07-152002-07-162002-07-172002-07-182002-07-192002-07-202002-07-212002-07-22

2002-07-232002-07-24

2002-07-252002-07-262002-07-272002-07-282002-07-292002-07-302002-07-31

2002-08-01

Much better (on average, 5% of observations are supposed to be outside the ellipse)

Much better (on average, 5% of observations are supposed to be outside the ellipse)

Here is the new score plot, with low production days removed. It hardly resembles the initial one – proof of the extreme effect of outliers.

FIRST COMPONENT ALONG THIS AXIS

SE

CO

ND

CO

MP

ON

EN

T A

LO

NG

TH

IS A

XIS

Example 1


0.00

0.20

0.40

0.60

0.80

1.00

Com

p[1]

Com

p[2]

Com

p[3]

Comp No.

32-months version 2.M2 (PCA-X), Extreme outliers removed R2X(cum)Q2(cum)

R2 and Q2 for PCA Model

This is the R2 and Q2 plot for this same model. The R2 values tell us that the first component explains 32% of the variability in the original data, the second another 7% and the third another 6%.

The Q2 values are lower, as always. This means that the predictive power of the model is around 40% when using all three components. This may seem low, but is normal for real process data.

Example 1


“Moderate” outliers in residuals

1

2

31999-1

1-2

31999-1

1-2

41999-1

1-2

51999-1

1-2

61999-1

1-2

71999-1

1-2

81999-1

1-2

91999-1

1-3

01999-1

2-0

11999-1

2-0

21999-1

2-0

31999-1

2-0

41999-1

2-0

51999-1

2-0

61999-1

2-0

81999-1

2-0

91999-1

2-1

01999-1

2-1

11999-1

2-1

21999-1

2-1

31999-1

2-1

41999-1

2-1

51999-1

2-1

61999-1

2-1

71999-1

2-1

81999-1

2-1

91999-1

2-2

01999-1

2-2

11999-1

2-2

21999-1

2-2

31999-1

2-2

41999-1

2-2

51999-1

2-2

61999-1

2-2

71999-1

2-2

81999-1

2-2

91999-1

2-3

02000-0

1-0

02000-0

1-0

12000-0

1-0

22000-0

1-0

32000-0

1-0

42000-0

1-0

52000-0

1-0

62000-0

1-0

72000-0

1-0

82000-0

1-0

92000-0

1-1

02000-0

1-1

12000-0

1-1

22000-0

1-1

32000-0

1-1

42000-0

1-1

52000-0

1-1

62000-0

1-1

72000-0

1-1

82000-0

1-1

92000-0

1-2

02000-0

1-2

12000-0

1-2

22000-0

1-2

32000-0

1-2

42000-0

1-2

52000-0

1-2

62000-0

1-2

72000-0

1-2

82000-0

1-2

92000-0

1-3

02000-0

1-3

12000-0

2-0

12000-0

2-0

22000-0

2-0

32000-0

2-0

42000-0

2-0

52000-0

2-0

62000-0

2-0

72000-0

2-0

82000-0

2-0

92000-0

2-1

02000-0

2-1

12000-0

2-1

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1DM

odX

[3](

Norm

)

Obs ID (TIME)

32-months version 2.M2 (PCA-X), Extreme outliers removedDModX[3](Norm)

M2-D-Crit[3] = 1.141

D-Crit(0.05)

This is the “Distance to Model” or residual plot for this model. It shows the distance, in multi-dimensional space, between each real observation (date) in the initial dataset and the predicted value based on the model. Clearly there are some “moderate” outliers that need investigating, different to the “extreme” outliers we saw on the score plot. This can be done by looking at the original data, or using other techniques

EACH POINT IS AN INDIVIDUAL DAY (DATES NOT LEGIBLE)

Moderate outliers

Example 1


Looking at the Results

So what do these results mean?

Obviously the score plot showing the dates is totally illegible. We will therefore remove the date label. However, in order not to lost the seasonal information, we will colour-code each day to show which time of year it occurred in. It is very easy to modify the graphical outputs in this way.

Let’s have a look at the result.

Example 1


Score plot of first 2 components

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0

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t[2]

t[1]

34-months of 1 day rev. 2 (incl. chip data) no. 2.M4 (PCA-X), Untitledt[1]/t[2]Colored according to classes in M4

No ClassClass 1Class 2Class 3Class 4

Variation in this direction Variation in this direction appears to occur WITHIN a appears to occur WITHIN a given season (given season ( Component Component 1)1)

Variation in this direction Variation in this direction appears to occur WITHIN a appears to occur WITHIN a given season (given season ( Component Component 1)1)

Variation in this direction Variation in this direction appears to occur appears to occur BETWEEN individual BETWEEN individual seasons seasons (( Component 2) Component 2)

Variation in this direction Variation in this direction appears to occur appears to occur BETWEEN individual BETWEEN individual seasons seasons (( Component 2) Component 2)

Note that all days < 100 t/d were systematically removed, plus major outliers. In all, only a few dozen observations were removed (out of 1044).

AutumnWinter Spring Summer


Same plot as before, only backwards Same plot as before, only backwards (mathematically identical)(mathematically identical)Same plot as before, only backwards Same plot as before, only backwards (mathematically identical)(mathematically identical)

Example 1


34-months of 1 day rev. 2 (incl. chip data) no. 2.M4 (PCA-X), Bad residuals removedt[1]/t[2]/t[3]Colored according to classes in M4


First 3 components…

2000

2001

2002

To show the first 3 components, we need a 3-D plot of course. To show the first 3 components, we need a 3-D plot of course. The third component is on the vertical axis. If the points were to The third component is on the vertical axis. If the points were to “drop” onto the bottom surface, you would just get the previous “drop” onto the bottom surface, you would just get the previous image.image.

To show the first 3 components, we need a 3-D plot of course. To show the first 3 components, we need a 3-D plot of course. The third component is on the vertical axis. If the points were to The third component is on the vertical axis. If the points were to “drop” onto the bottom surface, you would just get the previous “drop” onto the bottom surface, you would just get the previous image.image.



By looking at the original data, it By looking at the original data, it became clear that the three became clear that the three years were separated in the 2nd years were separated in the 2nd component component

By looking at the original data, it By looking at the original data, it became clear that the three became clear that the three years were separated in the 2nd years were separated in the 2nd component component

Each point represents Each point represents an INDIVIDUAL DAYan INDIVIDUAL DAYEach point represents Each point represents an INDIVIDUAL DAYan INDIVIDUAL DAY


Loadings PlotThe MVA software generates a set of new axes called “components” that are statistically significant. However, the software does not tell us what these new components actually mean.

To figure out how the original variables relate to the newly created MVA components, we must look at the Loadings Plot. For this example, the 1st / 2nd component loadings plot is shown on the next slide. It looks somewhat daunting, because the tag numbers are shown.

It is not necessary for the purposes of this exercise to understand what all the tag numbers mean. The important point is that similar tags trend together, as indicated by the text box. In this case, many variables related to the throughput tend to increase and decrease together, as shown by their clustering. Also, they are clearly related to the first component, on the negative side (positive and negative are totally arbitrary in MVA component space).

Example 1


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34-months of 1 day rev. 2 (incl. chip data) no. 2.M4 (PCA-X), Bad residuals removedp[1]/p[2]

X

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PCA: Loadings Plot (p1/p2)

Pulp throughputRefining energyDilution flowsSteam generation

Pulp throughputRefining energyDilution flowsSteam generation

See-saw principle

ORIGIN

Example 1


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Conclusions: p1

INTERPRETATIONINTERPRETATIONComponent 1: ThroughputComponent 1: ThroughputINTERPRETATIONINTERPRETATIONComponent 1: ThroughputComponent 1: Throughput

+ -Example 1


Interpretation of 1st component

Our conclusion is that the first component corresponds to throughput. This is logical, for two reasons:

1) many process variables are related either directly or indirectly to throughput;

2) The extreme outliers we removed at the beginning, which dominated the model, were also related to throughput (low production days)

Now we are ready to look at the score plot again. Remember we said that the 1st component was something that varied within an individual season? Now we know what it is: throughput.

So what have we accomplished? We’ve reduced the dimensionality by going from dozens of variables to a single latent variable.

Example 1


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2nd componentSame plot as before

Bleach consumptionBleach consumption

Pulp brightnessSeason

Pulp brightnessSeason


Interpretation of 2nd component

If you recall, we said that the 2nd component explains only 7% of the total variability. It is therefore “messier” than the first component, and will be less easy to interpret.

We also noted that the three years were separated with respect to this second component.

A major clue occurs in the prominence of two important and related tags: bleach consumption and pulp brightness. This would suggest that perhaps the brightness of the incoming wood chips was different from year to year, requiring more bleaching to get a less white pulp.

Note also that “Season” is prominent. We alreadyknew this, by the obvious separation of the seasonson the score plot. This suggests that winter chipsare less bright than summer chips.

Example 1


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Conclusions: p1 & p2

INTERPRETATIONINTERPRETATIONComponent 2: Component 2: Brightness of Brightness of incoming wood chipsincoming wood chips

INTERPRETATIONINTERPRETATIONComponent 2: Component 2: Brightness of Brightness of incoming wood chipsincoming wood chips

+

-

Example 1


Looking at 3rd component

To look at the 3rd component, we must generate a new plot showing the 1st component vs. the 3rd. In other words, we ignore the 2nd component. This 3rd component is orthogonal, and thus statistically independent, to the first two components.

We said that the 3rd component explains only 6% of the total variability. It is therefore even “messier” than the 2nd component.

Let’s have a look at this new score plot. Note that this is exactly the image you would get if all the points on the 3-D score plot were projected onto the back wall.

1st vs. 3rd

Example 1


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t[3]

t[1]

34-months of 1 day rev. 2 (incl. chip data) no. 2.M4 (PCA-X), Bad residuals removedt[1]/t[3]Colored according to classes in M4


PCA: t1 & t3

No segregation No segregation by year:by year:SUMMERS VS. SUMMERS VS. WINTERS!WINTERS!

No segregation No segregation by year:by year:SUMMERS VS. SUMMERS VS. WINTERS!WINTERS! AutumnW

inter Spring Summer


Note: Orthogonal

Note: Orthogonal


Looking at 3rd component

One very interesting results is that the three years are not separated on this plot. All the winters line up, and all the summers line up.

This suggests that the 3rd component is related to the time of year, pure and simple. This is confirmed by the the corresponding loadings plot, which shows “SEASON” to be the single most prominent variable. A reasonable interpretation would be that summer chips differ from winter chips in some way other than brightness, which was already covered by the second component. This could be, for instance, the ease with which the wood fibres can be separated from each other.

Example 1

Documents

NAMP Module 17: “Introduction to Multivariate Analysis” Tier 2, Rev.: 5 Program for North American Mobility in Higher Education Introducing Process Integration