Data Mining (and machine learning) DM Lecture 3: Basic Statistics and Coursework 1 David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Communities and Crime Here is an interesting dataset David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

-- state: US state (by number) - -- county: numeric code for county -- community: numeric code for community - -- communityname: community name – -- fold: fold number for non-random 10 fold cross validation, -- population: population for community: (numeric - decimal) -- householdsize: mean people per household (numeric - decimal) -- racepctblack: percentage of population that is african american (numeric - decimal) -- racePctWhite: percentage of population that is caucasian (numeric - decimal) -- racePctAsian: percentage of population that is of asian heritage (numeric - decimal) -- racePctHisp: percentage of population that is of hispanic heritage (numeric - decimal) -- agePct12t21: percentage of population that is 12-21 in age (numeric - decimal) -- agePct12t29: percentage of population that is 12-29 in age (numeric - decimal) -- agePct16t24: percentage of population that is 16-24 in age (numeric - decimal) -- agePct65up: percentage of population that is 65 and over in age (numeric - decimal) -- numbUrban: number of people living in areas classified as urban (numeric - decimal) -- pctUrban: percentage of people living in areas classified as urban (numeric - decimal) -- medIncome: median household income (numeric - decimal) – -- pctWWage: percentage of households with wage or salary income in 1989 (numeric - decimal) -- pctWFarmSelf: percentage of households with farm or self employment income in 1989 [etc etc etc --- 128 fields altogether] -- ViolentCrimesPerPop: total number of violent crimes per 100K popuation (numeric - decimal) GOAL attribute (to be predicted) David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Mining the C&C data Let’s do some basic preprocessing and mining of these data, to start to grasp whether we can find any patterns that will predict certain levels of violent crime David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

etc … about 2,000 instances David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

First: some sensible preprocessing The first 5 fields are (probably) not useful for prediction – they are more like “ID” fields for the record. So, let’s remove them. There are many cases of missing data here too – let’s remove any field which has any missing data in it at all. This is OK for the C&C data, still leaving 100 fields. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

First: some sensible preprocessing I downloaded the data. First I converted it to a space-separated form, rather than comma-separated, because I prefer it that way. I wrote an awk script to do this called cs2ss.awk, here: http://www.macs.hw.ac.uk/~dwcorne/Teaching/DMML/cs2ss.awk I did that with this command line on a unix machine: awk –f cs2ss.awk < communities.data > commss.txt Placing the new version in “commss.txt” Then, I wanted to remove the first 5 fields, and remove any field in which any record contained missing values. I wrote an awk script for that too: http://www.macs.hw.ac.uk/~dwcorne/Teaching/DMML/fixcommdata.awk and did this: awk –f fixcommdata.awk < comss.txt > commssfixed.txt David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Normalisation The fields in these data happen to be already min-max normalised into [0,1]; I wondered whether it would also be good to z-normalise the fields. So I wrote an awk script for z-normalisation, and produced a version that had that done http://www.macs.hw.ac.uk/~dwcorne/Teaching/DMML/znorm.awk awk –f znorm.awk < commsfixed.txt > commssfixedznorm.txt In these data, the class value is numeric, between 0 and 1, indicating (already normalized) the relative amount of violent crime in the community in question. To make it easier to find patterns and relationships, I produced new versions of each dataset where the class value was either 0 (low) or 1 (high) – 0 in the cases where it had been <= 0.4, and 1 otherwise. I used an awk script for that too, and did some renaming of files, and ended up with: commssfixed.txt two-class commssfixedznorm.txt two-class z-normlalised David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Now, I wonder: how good is 1-NN at predicting the class for these data? If only using fields 20—30 to work out the distance values, the answer is: Unchanged data (in this case, already min-max normalised to [0,1]): 81.1% Z-normalised: 81.5% But note that 81% of the data is class 0 – so if you always guess “0”, your accuracy will be 81.0%. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Now let’s look at the data in more detail; some histograms of the fields Here is the distribution of values in field 6 for class 0 – it is a “5-bin” distribution. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Let’s look at the distributions of field 6 for class 0 and class 1 together (% of pop that is Hispanic) David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Field 7 (% of pop aged 12--21) David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Field 8 (% of pop aged 12—29) David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Field 9 (% of pop aged 16—24) David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Field 10 (% of pop aged >= 65 David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Which two fields seem most useful for discriminating between classes 0 and 1? David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Fields 6 and 7 Maybe we will get better 1-NN results using only the important fields? 2 is (most often) too small a number of fields, but anyway … I produced versions of the dataset that had only fields 6, 7 and 100 (these two, and the class field): I then calculated 1-NN accuracy for these. Results: `Unchanged’ version: fields 6 and 7: 70.8% (was 81.1%) Z-normalised: fields 6 and 7: 70.9% (was 81.5%) Not very successful! But I didn’t expect that – working with several more of the important fields would quite possibly give better accuracies, but may take too much time to demonstrate, or do in your assignment. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Coursework 1 You will do what we just did, for four datasets: For each one: Download it (of course), then do some simple preparation Produce a version of the data set where each non-class field is min-max normalised (for the Communities and Crime dataset, do z-normalisation instead) Convert into a two class dataset; do this for both the original and normalised cases. Calculate the accuracy of 1-nearest-neighbour classification for your dataset; do this for both original and normalised versions Generate 5-bin histograms of the distribution of the first five fields, for each of the two classes. Write 100—200 words describing how the distributions differ between the two classes, and describing what you think are the two most important fields for discriminating between the classes. Produce a reduced dataset (two versions: original and normalised) which contains only three fields: the two you considered most important, and the class field. Repeat step 4, but this time for the reduced datasets. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

What you email me One brief paragraph that tells me how you did it / what tools you used. This will not affect the marks – I would just like to know. For step 4: A table that tells me the answers for each dataset, followed by a paragraph that attempts to explain any differences in performance between the normalised and original versions, or explains why performance is similar. For steps 5 and 6: 1 page per dataset; on each page, the 10 histograms, and the discussion (step 7). For step 8: sane as step 4. That must all be done within 6 sides of A4. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

You should know what Z-normalisation is, so here is a brief lecture on basic statistics, including that David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Fundamental Statistics Definitions A Population is the total collection of all items/individuals/events under consideration • A Sample is that part of a population which has been observed or selected for analysis E.g. all students is a population. Students at HWU is a sample; this class is a sample, etc … • A Statistic is a measure which can be computed to describe a characteristic of the sample (e.g. the sample mean) The reason for doing this is almost always to estimate (i.e. make a good guess) things about that characteristic in the population David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

E.g. This class is a sample from the population of students at HWU (it can also be considered as a sample of other populations – like what?) One statistic of this sample is your mean weight. Suppose that is 65Kg. I.e. this is the sample mean. Is 65Kg a good estimate for the mean weight of the population? Another statistic: suppose 10% of you are married. Is this a good estimate for the proportion that are married in the population? David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Some Simple Statistics The Mean (average) is the sum of the values in a sample divided by the number of values The Median is the midpoint of the values in a sample (50% above; 50% below) after they have been ordered (e.g. from the smallest to the largest) The Mode is the value that appears most frequently in a sample The Range is the difference between the smallest and largest values in a sample The Variance is a measure of the dispersion of the values in a sample – how closely the observations cluster around the mean of the sample The Standard Deviation is the square root of the variance of a sample David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Statistical moments The m-th moment about the mean (μ) of a sample is: Where n is the number of items in the sample. The first moment (m = 1) is 0! The second moment (m = 2) is the variance (and: square root of the variance is the standard deviation) The third moment can be used in tests for skewness The fourth moment can be used in tests for kurtosis David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Variation and Standard Deviation The variance of a sample is the 2nd moment Where n is the number of items in the sample. square root of the variance is the standard deviation) David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Distributions / Histograms A Normal (aka Gaussian) distribution (image from Mathworld) David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

`Normal’ or Gaussian distributions … … tend to be everywhere Given a typical numeric field in a typical dataset, it is common that most values are centred around a particular value (the mean), and the proportion with larger or smaller values tends to tail off. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

`Normal’ or Gaussian distributions … We just saw this – fields 7—10 were Normal-ish Heights, weights, times (e.g. for 100m sprint, for lengths of software projects), measurements (e.g. length of a leaf, waist measurement, coursework marks, level of protein A in a blood sample, …) all tend to be Normally distributed. Why?? David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Sometimes distributions are uniform Uniform distributions. Every possible value tends to be equally likely David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

This figure is from: http://mathworld.wolfram.com/Dice.html One die: uniform distribution of possible totals: But look what happens as soon as the value is a sum of things; The more things, the more Gaussian the distribution. Are measurements (etc.) usually the sum of many factors? David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Probability Distributions If a population (e.g. field of a dataset) is expected to match a standard probability distribution then a wealth of statistical knowledge and results can be brought to bear on its analysis Many standard statistical techniques are based on the assumption that the underlying distribution of a population is Normal (Gaussian) Usually this assumption works fine David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

A closer look at the normal distribution This is the ND with mean mu and std sigma David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

More than just a pretty bell shape Suppose mean of your sample is 1.8; and suppose std of your sample is 0.12 Theory tells us that if a population is Normal, the sample std is a fairly good guess at the population std So, we can say with some confidence, for example, that 99.7% of the population lies between 1.44 and 2.16 David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Remember, MUCH of science relies on making guesses about populations The CLT helps us make the guesses reasonable rather than crazy. Assuming normal dist, the stats of a sample tells us lots about the stats of the population And, assuming normal dist helps us detect errors and outliers – how? David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Z-normalisation (or z-score normalisation) Given any collection of numbers (e.g. the values of a particular field in a dataset) we can work out the mean and the standard deviation. Z-score normalisation means converting the numbers into units of standard deviation. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Simple z-normalisation example values 2.8 17.6 4.1 12.7 3.5 11.8 12.2 11.1 15.8 19.6 David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Simple z-normalisation example values 2.8 17.6 4.1 12.7 3.5 11.8 12.2 11.1 15.8 19.6 Mean: 11.12 STD: 5.93 David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Simple z-normalisation example values 2.8 17.6 4.1 12.7 3.5 11.8 12.2 11.1 15.8 19.6 Mean subtracted -8.32 6.48 -7.02 1.58 -7.62 0.68 1.08 -0.02 4.68 8.48 Mean: 11.12 STD: 5.93 subtract mean, so that these are centred around zero David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Simple z-normalisation example values 2.8 17.6 4.1 12.7 3.5 11.8 12.2 11.1 15.8 19.6 Mean subtracted -8.32 6.48 -7.02 1.58 -7.62 0.68 1.08 -0.02 4.68 8.48 In Z units -1.403 1.092 -1.18 0.27 -1.28 0.11 0.18 -0.003 0.79 1.43 Mean: 11.12 STD: 5.93 subtract mean, so that these are centred around zero Divide each value by the std; we now see how usual or unusual each value is David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

The take-home lesson (for those new to statistics) Your data contains 100 values for x, and you have good reason to believe that x is normally distributed. Thanks to the Central Limit Theorem, you can: Make a lot of good estimates about the statistics of the population Make justified conclusions about two distributions being different (e.g. the distribution of field X for class 1, and the distribution of field X for class 2) Maybe find outliers and spot other problems in the data David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

Next week – back to baskets -- a classic Data Mining Algorithm! David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html

The Central Limit Theorem is this: As more and more samples are taken from a population the distribution of the sample means conforms to a normal distribution • The average of the samples more and more closely approximates the average of the entire population • A very powerful and useful theorem • The normal distribution is such a common and useful distribution that additional statistics have been developed to measure how closely a population conforms to it and to test for divergence from it due to skewness and kurtosis David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail.com These slides and related resources: http://www.macs.hw.ac.uk/~dwcorne/Teaching/dmml.html