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Index

Here is an index of posts by topic:

Python basics

Introduction, and installing python for healthcare modelling

Lists

Nested Lists

Tuples

Sets

Dictionaries

math module

Variable Types

Random numbers and sequences

if, else, elif, while, and logical operators; else after while

loops and iterating

List comprehensions – one line loops

try …. except (where code might fail)

Decimal places in output

Read from and write to files

Functions

Automatically passing unpacked lists or tuples to a function (or why do you see * before lists and tuples)

Lambda functions (one line functions), and map/filter/reduce

Accessing date and time, and timing code

NumPy and Pandas

NumPy and Pandas

NumPy basics: building an array from lists, basic statistics, converting to booleans, referencing the array, and taking slices

Pandas basics: building a dataframe from lists, and retrieving data from the dataframe using row and column index references

Pandas: basic statistics

Converting between NumPy and Pandas

Array maths in NumPy

Reading and writing CSV files using NumPy and Pandas

Applying user-defined functions to NumPy and Pandas

Adding more data to NumPy arrays and Pandas dataframes

Using Pandas to merge or lookup data

Sorting and ranking with Pandas

Using masks to filter data, and perform search and replace, in NumPy and Pandas

Summarising data by groups in Pandas using pivot_tables and groupby

Reshaping Pandas data with stack, unstack, pivot and melt

Subgrouping data in Pandas with groupby

Iterating through columns and rows in NumPy and Pandas

Removing duplicate data in NumPy and Pandas

Setting width and number of decimal places in NumPy print output

Using NumPy to generate random numbers, or shuffle arrays

Matplotlib for plotting charts

Simple xy line charts, and simple save to file

Scatter plot, and adding titles to axes

Bar charts

Pie charts, and adding a title

Histograms (and obtaining histogram data with NumPy)

Boxplots

Violin plots

3D wireframe and surface plots

Common modifications to charts

A simple heatmap

Adding contour lines to a heatmap

Creating a grid of subplots

Adding error bars to charts

Adding shaded areas to charts

Statistics

Linear regression with scipy.stats

Linear regression with scikit learn

One sample t-test and Wilcoxon signed rank test

t-tests for testing the difference between two groups of data

Mann Whitney U-test

Analysis of variance (ANOVA)

Multi-comparison with Tukey’s test and the Holm-Bonferroni method

Multiple comparison of non-normally distributed data with the Kruskal-Wallace test

Confidence Interval for a single proportion

Chi-squared test

Fisher’s exact test

Distribution fitting to data

Clinical pathway simulation with SimPy

A simple bed occupancy model

An emergency department model in SimPy, with patient prioritisation and capacity limited by doctor availability

Machine Learning with Scikit Learn

The iris data set

Splitting data into training and test sets

Feature Scaling

Using logistic regression to diagnose breast cancer

Adding standard diagnostic performance metrics to a ml diagnosis model

How do you know if you have gathered enough data? By using learning rates.

Working with ordinal and categorical data

Support Vector machines

Random Forests

Neural networks

Choosing between models with stratified k-fold validation

Visualising accuracy and error in a classification model with a confusion matrix

Changing sensitivity of machine learning algorithms and performing a receiver-operator characteristic curve

Reducing data complexity, and eliminating covariance, with principal component analysis

Grouping unlabelled data with k-means clustering

Linear regression with scikit learn

Using free text for classification – ‘Bag of Words’

Some common (and hopefully useful) algorithms

The travelling community nurse problem (aka the Travelling Salesman Problem)

Miscellaneous Python

Function decorators

Resources

Download zipped files (Jupyter Notebooks, PDFs, & py files)

Open data travel times from all UK LSOA to all acute hospitals

 

 

86. Linear regression and multiple linear regression

In linear regression we seek to predict the value of a continuous variable based on either a single variable, or a set of variables.

The example we will look at below seeks to predict life span based on weight, height, physical activity, BMI, gender, and whether the person has a history of smoking.

With linear regression we assume that the output variable (lifespan in this example) is linearly related to the features we have (we will look at non-linear models in the next module).

This example uses a synthetic data set.

Load data

We’ll load the data from the web…

import pandas as pd

filename = 'https://gitlab.com/michaelallen1966/1804_python_healthcare_wordpress/raw/master/jupyter_notebooks/life_expectancy.csv'
df = pd.read_csv(filename)
df.head()
weight smoker physical_activity_scale BMI height male life_expectancy
0 51 1 6 22 152 1 57
1 83 1 5 34 156 1 36
2 78 1 10 18 208 0 78
3 106 1 3 28 194 0 49
4 92 1 7 23 200 0 67

Exploratory data analysis

import matplotlib.pyplot as plt
import seaborn as sns
%matplotlib inline

sns.set(style = 'whitegrid', context = 'notebook')
sns.pairplot(df, size = 2.5)
plt.show()
lr1

We can show the correlation matrix with np.corrcoef (np.cov would show the non-standardised covariance matrix; a correlation matrix has the same values as a covariance matrix on standardised data). Note that we need to transpose our data so that each feature is in a row rather than a column.

When building a linear regression model we are most interested in those features which have the strongest correlation with our outcome. If there are high degrees of covariance between features we may wish to consider using principal component analysis to reduce the data set.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
%matplotlib inline

np.set_printoptions(precision=3)
corr_mat = np.corrcoef(df.values.T)
print ('Correlation matrix:\n')
print (corr_mat)
print ()

# Plot correlation matrix
plt.imshow(corr_mat, interpolation='nearest')
plt.colorbar()
plt.xlabel('Feature')
plt.ylabel('Feature')
plt.title('Correlation between life expectancy features')
plt.show()
Correlation matrix:

[[ 1.    -0.012 -0.03  -0.011  0.879 -0.004 -0.009]
 [-0.012  1.     0.034 -0.027  0.006  0.018 -0.518]
 [-0.03   0.034  1.    -0.028 -0.009 -0.007  0.366]
 [-0.011 -0.027 -0.028  1.    -0.477 -0.019 -0.619]
 [ 0.879  0.006 -0.009 -0.477  1.     0.006  0.278]
 [-0.004  0.018 -0.007 -0.019  0.006  1.    -0.299]
 [-0.009 -0.518  0.366 -0.619  0.278 -0.299  1.   ]]

lr2

Fitting a linear regression model using a single feature

To illustrate linear regression, we’ll start with a single feature. We’ll pick BMI.

X = df['BMI'].values.reshape(-1, 1) 
X = X.astype('float')
y = df['life_expectancy'].values.reshape(-1, 1)
y = y.astype('float')

# Standardise X and y
# Though this may often not be necessary it may help when features are on
# very different scales. We won't use the standardised data here,
# but here is how it would be done
from sklearn.preprocessing import StandardScaler
sc_X = StandardScaler()
sc_y = StandardScaler()
X_std = sc_X.fit_transform(X)
Y_std = sc_y.fit_transform(X)

# Create linear regression model
from sklearn.linear_model import LinearRegression
slr = LinearRegression()
slr.fit(X, y)

# Print model coefficients
print ('Slope =', slr.coef_[0])
print ('Intercept =', slr.intercept_)
Slope = [-1.739]
Intercept = [ 107.874]

Predicting values

We can simply use the predict method of the linear regression model to predict values for any given X. (We use ‘flatten’ below to change from a column array toa row array).

y_pred = slr.predict(X)

print ('Actual = ', y[0:5].flatten())
print ('Predicted = ', y_pred[0:5].flatten())
Actual =  [ 57.  36.  78.  49.  67.]
Predicted =  [ 69.616  48.748  76.572  59.182  67.877]

Obtaining metrics of observed vs predicted

The metrics module from sklearn contains simple methods of reporting metrics given observed and predicted values.

from sklearn import metrics  
print('Mean Absolute Error:', metrics.mean_absolute_error(y, y_pred))  
print('Mean Squared Error:', metrics.mean_squared_error(y, y_pred))  
print('Root Mean Squared Error:', np.sqrt(metrics.mean_squared_error(y, y_pred)))
print('R-square:',metrics.r2_score(y, y_pred))
Mean Absolute Error: 5.5394378529
Mean Squared Error: 45.9654230242
Root Mean Squared Error: 6.77978045545
R-square: 0.382648884752

Plotting observed values and fitted line

plt.scatter (X, y, c = 'blue')
plt.plot (X, slr.predict(X), color = 'red')
plt.xlabel('X')
plt.ylabel('y')
plt.show()
lr3

Plotting observed vs. predicted values

Plotting observed vs. predicted can give a good sense of the accuracy of the model, and is also suitable when there are multiple X features.

plt.scatter (y, slr.predict(X), c = 'blue')
plt.xlabel('Observed')
plt.ylabel('Predicted')
plt.show()
lr4

Fitting a model to multiple X features

The method described above works with any number of X features. Generally we may wish to pick those features with the highest correlation to the outcome value, but here we will use them all.

X = df.values[:, :-1]
y = df.values[:, -1]
# Create linear regression model
from sklearn.linear_model import LinearRegression
slr = LinearRegression()
slr.fit(X, y)

# Print model coefficients
print ('Slope =', slr.coef_)
print ('Intercept =', slr.intercept_)
Slope = [  0.145 -10.12    1.104  -2.225  -0.135  -5.148]
Intercept = 132.191010159

Show metrics (notice the improvement)

y_pred = slr.predict(X)
from sklearn import metrics  
print('Mean Absolute Error:', metrics.mean_absolute_error(y, y_pred))  
print('Mean Squared Error:', metrics.mean_squared_error(y, y_pred))  
print('Root Mean Squared Error:', np.sqrt(metrics.mean_squared_error(y, y_pred)))
print('R-square:',metrics.r2_score(y, y_pred))
Mean Absolute Error: 2.4055613151
Mean Squared Error: 7.96456424623
Root Mean Squared Error: 2.82215595711
R-square: 0.89302975375

Plot observed vs. predicted:

plt.scatter (y, slr.predict(X), c = 'blue')
plt.xlabel('Observed')
plt.ylabel('Predicted')
plt.show()
lr5

Plotting residuals

Residuals are simply the difference between an observed value and its predicted value. We can plot the relationship between observed values and residuals. Ideally we like to see that there is no clear relationship between predicted value and residual – residuals should be randomly distributed. Residual plotting may also be used to look to see if there are any outliers which might be having an effect on our model (in which case we may decide that it better to remove the outliers and re-fit).

residuals = slr.predict(X) - y # predicted - observed
plt.scatter (y, residuals, c = 'blue')
plt.xlabel('Observed')
plt.ylabel('Residual')
plt.show()
lr6

 

 

85. Using free text for classification – ‘Bag of Words’

There may be times in healthcare where we would like to classify patients based on free text data we have for them. Maybe, for example, we would like to predict likely outcome based on free text clinical notes.

Using free text requires methods known as ‘Natural Language Processing’.

Here we start with one of the simplest techniques – ‘bag of words’.

In a ‘bag of words’ free text is reduced to a vector (a series of numbers) that represent the number of times a word is used in the text we are given. It is also posible to look at series of two, three or more words in case use of two or more words together helps to classify a patient.

A classic ‘toy problem’ used to help teach or develop methos is to try to judge whether people rates a film as ‘like’ or ‘did not like’ based on the free text they entered into a widely used internet film review database (www.imdb.com).

Here will will use 50,000 records from IMDb to convert each review into a ‘bag of words’, which we will then use in a simple logistic regression machine learning model.

We can use raw word counts, but in this case we’ll add an extra transformation called tf-idf (frequency–inverse document frequency) which adjusts values according to the number fo reviews that use the word. Words that occur across many reviews may be less discriminatory than words that occur more rarely, so tf-idf reduces the value of those words used frequently across reviews.

This code will take us through the following steps:

1) Load data from internet, and split into training and test sets.

2) Clean data – remove non-text, convert to lower case, reduce words to their ‘stems’ (see below for details), and reduce common ‘stop-words’ (such as ‘as’, ‘the’, ‘of’).

3) Convert cleaned reviews in word vectors (‘bag of words’), and apply the tf-idf transform.

4) Train a logistic regression model on the tr-idf transformed word vectors.

5) Apply the logistic regression model to our previously unseen test cases, and calculate accuracy of our model.

Load data

This will load the IMDb data from the web. It is loaded into a Pandas DataFrame.

import pandas as pd
import numpy as np

file_location = 'https://raw.githubusercontent.com/MichaelAllen1966/1805_nlp_basics/master/data/IMDb.csv'
data = pd.read_csv(file_location)

Show the size of the data set (rows, columns).

data.shape
Out:
(50000, 2)

Show the data fields.

list(data)
Out:
['review', 'sentiment']

Show the first record review and recorded sentiment (which will be 0 for not liked, or 1 for liked)

In [11
print ('Review:')
print (data['review'].iloc[0])
print ('\nSentiment (label):')
print (data['sentiment'].iloc[0])
  Out:
Review:
I have no read the novel on which "The Kite Runner" is based. My wife and daughter, who did, thought the movie fell a long way short of the book, and I'm prepared to take their word for it. But, on its own, the movie is good -- not great but good. How accurately does it portray the havoc created by the Soviet invasion of Afghanistan? How convincingly does it show the intolerant Taliban regime that followed? I'd rate it C+ on the first and B+ on the second. The human story, the Afghan-American who returned to the country to rescue the son of his childhood playmate, is well done but it is on this count particularly that I'm told the book was far more convincing than the movie. The most exciting part of the film, however -- the kite contests in Kabul and, later, a mini-contest in California -- cannot have been equaled by the book. I'd wager money on that.

Sentiment (label):
1

Splitting the data into training and test sets

Split the data into 70% training data and 30% test data. The model will be trained using the training data, and accuracy will be tested using the independent test data.

from sklearn.model_selection import train_test_split
X = list(data['review'])
y = list(data['sentiment'])
X_train, X_test, y_train, y_test = train_test_split(
    X,y, test_size = 0.3, random_state = 0)

Defining a function to clean the text

This function will:

1) Remove ant HTML commands in the text

2) Remove non-letters (e.g. punctuation)

3) Convert all words to lower case

4) Remove stop words (stop words are commonly used works like ‘and’ and ‘the’ which have little value in nag of words. If stop words are not already installed then open a python terminal and type the two following lines of code (these instructions will also be given when running this code if the stopwords have not already been downloaded onto the computer running this code).

import nltk

nltk.download(“stopwords”)

5) Reduce words to stem of words (e.g. ‘runner’, ‘running’, and ‘runs’ will all be converted to ‘run’)

6) Join words back up into a single string

def clean_text(raw_review):
    # Function to convert a raw review to a string of words
    
    # Import modules
    from bs4 import BeautifulSoup
    import re

    # Remove HTML
    review_text = BeautifulSoup(raw_review, 'lxml').get_text() 

    # Remove non-letters        
    letters_only = re.sub("[^a-zA-Z]", " ", review_text) 
    
    # Convert to lower case, split into individual words
    words = letters_only.lower().split()   

    # Remove stop words (use of sets makes this faster)
    from nltk.corpus import stopwords
    stops = set(stopwords.words("english"))                  
    meaningful_words = [w for w in words if not w in stops]                             

    # Reduce word to stem of word
    from nltk.stem.porter import PorterStemmer
    porter = PorterStemmer()
    stemmed_words = [porter.stem(w) for w in meaningful_words]

    # Join the words back into one string separated by space
    joined_words = ( " ".join( stemmed_words ))
    return joined_words 

Now will will define a function that will apply the cleaning function to a series of records (the clean text function works on one string of text at a time).

def apply_cleaning_function_to_series(X):
    print('Cleaning data')
    cleaned_X = []
    for element in X:
        cleaned_X.append(clean_text(element))
    print ('Finished')
    return cleaned_X

We will call the cleaning functions to clean the text of both the training and the test data. This may take a little time.

X_train_clean = apply_cleaning_function_to_series(X_train)
X_test_clean = apply_cleaning_function_to_series(X_test)
  Out:
Cleaning data
Finished
Cleaning data
Finished

Create ‘bag of words’

The ‘bag of words’ is the word vector for each review. This may be a simple word count for each review where each position of the vector represnts a word (returned in the ‘vocab’ list) and the value of that position represents the number fo times that word is used in the review.

The function below also returns a tf-idf (frequency–inverse document frequency) which adjusts values according to the number fo reviews that use the word. Words that occur across many reviews may be less discriminatory than words that occur more rarely. The tf-idf transorm reduces the value of a given word in proportion to the number of documents that it appears in.

The function returns the following:

1) vectorizer – this may be applied to any new reviews to convert the revier into the same word vector as the training set.

2) vocab – the list of words that the word vectors refer to.

3) train_data_features – raw word count vectors for each review

4) tfidf_features – tf-idf transformed word vectors

5) tfidf – the tf-idf transformation that may be applied to new reviews to convert the raw word counts into the transformed word counts in the same way as the training data.

Our vectorizer has an argument called ‘ngram_range’. A simple bag of words divides reviews into single words. If we have an ngram_range of (1,2) it means that the review is divided into single words and also pairs of consecutiev words. This may be useful if pairs of words are useful, such as ‘very good’. The max_features argument limits the size of the word vector, in this case to a maximum of 10,000 words (or 10,000 ngrams of words if an ngram may be mor than one word).

def create_bag_of_words(X):
    from sklearn.feature_extraction.text import CountVectorizer
    
    print ('Creating bag of words...')
    # Initialize the "CountVectorizer" object, which is scikit-learn's
    # bag of words tool.  
    
    # In this example features may be single words or two consecutive words
    vectorizer = CountVectorizer(analyzer = "word",   \
                                 tokenizer = None,    \
                                 preprocessor = None, \
                                 stop_words = None,   \
                                 ngram_range = (1,2), \
                                 max_features = 10000) 

    # fit_transform() does two functions: First, it fits the model
    # and learns the vocabulary; second, it transforms our training data
    # into feature vectors. The input to fit_transform should be a list of 
    # strings. The output is a sparse array
    train_data_features = vectorizer.fit_transform(X)
    
    # Convert to a NumPy array for easy of handling
    train_data_features = train_data_features.toarray()
    
    # tfidf transform
    from sklearn.feature_extraction.text import TfidfTransformer
    tfidf = TfidfTransformer()
    tfidf_features = tfidf.fit_transform(train_data_features).toarray()

    # Take a look at the words in the vocabulary
    vocab = vectorizer.get_feature_names()
   
    return vectorizer, vocab, train_data_features, tfidf_features, tfidf

We will apply our bag_of_words function to our training set. Again this might take a little time.

vectorizer, vocab, train_data_features, tfidf_features, tfidf  = (
        create_bag_of_words(X_train_clean))
  Out:
Creating bag of words...

Let’s look at the some items from the vocab list (positions 40-44). Some of the words may seem odd. That is because of the stemming.

vocab[40:45]
Out:
['accomplish', 'accord', 'account', 'accur', 'accuraci']

And we can see the raw word count represented in train_data_features.

train_data_features[0][40:45]
Out:
array([0, 0, 1, 0, 0], dtype=int64)

If we look at the tf-idf transform we can see the value reduced (words occuring in many documents will have their value reduced the most)

tfidf_features[0][40:45]
Out:
array([0.        , 0.        , 0.06988648, 0.        , 0.        ])

Training a machine learning model on the bag of words

Now we have transformed our free text reviews in vectors of numebrs (representing words) we can apply many different machine learning techniques. Here will will use a relatively simple one, logistic regression.

We’ll set up a function to train a logistic regression model.

def train_logistic_regression(features, label):
    print ("Training the logistic regression model...")
    from sklearn.linear_model import LogisticRegression
    ml_model = LogisticRegression(C = 100,random_state = 0)
    ml_model.fit(features, label)
    print ('Finished')
    return ml_model

Now we will use the tf-idf tranformed word vectors to train the model (we could use the plain word counts contained in ‘train_data_features’ (rather than using’tfidf_features’). We pass both the features and the known label corresponding to the review (the sentiment, either 0 or 1 depending on whether a person likes the film or not.

ml_model = train_logistic_regression(tfidf_features, y_train)
  Out:
Training the logistic regression model...
Finished

Applying the bag of words model to test reviews

We will now apply the bag of words model to test reviews, and assess the accuracy.

We’ll first apply our vectorizer to create a word vector for review in the test data set.

test_data_features = vectorizer.transform(X_test_clean)
# Convert to numpy array
test_data_features = test_data_features.toarray()

As we are using the tf-idf transform, we’ll apply the tfid transformer so that word vectors are transformed in the same way as the training data set.

test_data_tfidf_features = tfidf.fit_transform(test_data_features)
# Convert to numpy array
test_data_tfidf_features = test_data_tfidf_features.toarray()

Now the bit that we really want to do – we’ll predict the sentiment of the all test reviews (and it’s just a single line of code!). Did they like the film or not?

predicted_y = ml_model.predict(test_data_tfidf_features)

Now we’ll compare the predicted sentiment to the actual sentiment, and show the overall accuracy of this model.

correctly_identified_y = predicted_y == y_test
accuracy = np.mean(correctly_identified_y) * 100
print ('Accuracy = %.0f%%' %accuracy)
 Out:
Accuracy = 87%

87% accuracy. That’s not bad for a simple Natural Language Processing model, using logistic regression.

84. Function decorators

Decorators (identified by @ in line above function definition) allow code to be run before and after the function (hence ‘decorating’ it). An example of use of a decorator is shown below when a decorator function is used to time two different functions. This removes the need for duplicating code in different functions, and also keeps the functions focussed on their primary objective. Continue reading “84. Function decorators”

83. Automatically passing unpacked lists or tuples to a function (or why do you see * before lists and tuples)

Imagine we have a very simple function that adds three numebrs together:

In [1]:
def add_three_numbers (a, b, c):
    return a + b + c

We would normally pass separate numebrs to the function, e.g.

add_three_numbers (10, 20, 35)
65

But what if our numbers are in a list or a tuple. If we pass that as a list then we are passing only a single argument, and the function declares an error:

In [3]:
my_list = [10, 20, 35]

add_three_numbers (my_list)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-3-900947bff326> in <module>()
      1 my_list = [10, 20, 35]
      2 
----> 3 add_three_numbers (my_list)

TypeError: add_three_numbers() missing 2 required positional arguments: 'b' and 'c'

Python allows us to pass the list or tuple with the instruction to unpack it for input into the function. We instruct Python to unpack the list/tuple with an asterix before the list/tuple:

add_three_numbers (*my_list)
65

 

 

81. Distribution fitting to data

SciPy has over 80 distributions that may be used to either generate data or test for fitting of existing data. In this example we will test for fit against ten distributions and plot the best three fits. For a full list of distributions see:

https://docs.scipy.org/doc/scipy/reference/stats.html

In this example we’ll take the first feature (column) from the Wisconsin Breast Cancer data set and identify a statistical distribution that can approximate the observed distribution.

As usual we will start by loading general modules used, and load our data (selecting the first column for our ‘y’, the data to be fitted).

import pandas as pd
import numpy as np
import scipy
from sklearn.preprocessing import StandardScaler
import scipy.stats
import matplotlib.pyplot as plt
%matplotlib inline
# Load data and select first column

from sklearn import datasets
data_set = datasets.load_breast_cancer()
y=data_set.data[:,0]

# Create an index array (x) for data

x = np.arange(len(y))
size = len(y)

Visualise the data, and show descriptive summary

Before we do any fitting of distributions, it’s always good to do a simple visualisation of the data, and show descriptive statistics. We may, for example, decide to perform some outlier removal if we think that necessary (if there appear to be data points that don’t belong to the rest of the population).

plt.hist(y)
plt.show()
dist_fit_1

If we put the data in a Pandas Dataframe we can use the Pandas describe method to show a summary.

y_df = pd.DataFrame(y, columns=['Data'])
y_df.describe()
Out[3]:
Data
count 569.000000
mean 14.127292
std 3.524049
min 6.981000
25% 11.700000
50% 13.370000
75% 15.780000
max 28.110000

Fitting a range of distribution and test for goodness of fit

This method will fit a number of distributions to our data, compare goodness of fit with a chi-squared value, and test for significant difference between observed and fitted distribution with a Kolmogorov-Smirnov test.

The chi-squared value bins data into 50 bins (this could be reduced for smaller data sets) based on percentiles so that each bin contains approximately an equal number of values. For each fitted distribution the expected count of values in each bin is predicted from the distribution. The chi-squared value is the the sum of the relative squared error for each bin, such that:

chi-squared = sum ((observed – predicted) ** 2) / predicted)

For the observed and predicted we will use the cumulative sum of observed and predicted frequency across the bin range used.

The lower the chi-squared value the better the fit.

The Kolmogorov-Smirnov test assumes that data has been standardised: that is the mean is subtracted from all data (so the data becomes centred around zero), and that the results values are divided by the standard deviation (so all data becomes expressed as the number of standard deviations above or below the mean). A value of greater than 0.05 means that the fitted distribution is not significantly different to the observed distribution of the data.

It is worth noting that statistical distributions are theoretical models of real-world data. Statistical distributions offer a good way of approximating data (and simplifying huge amounts of data into a few parameters). But when you have a large set of real-world data it is not surprising to find that no theoretical distribution fits the data perfectly. Having he Kolmogorov-Smirnov tests for all distributions produce results of P<0.05 (fitted distribution is statistically different to the observed data distribution) is not unusual for large data sets. In that case, in modelling we are generally happy to continue with a fit that looks ‘reasonable’, being aware this is one of the simplifications present in any model.

Let’s first standardise the data using sklearn’s StandardScaler:

sc=StandardScaler() 
yy = y.reshape (-1,1)
sc.fit(yy)
y_std =sc.transform(yy)
y_std = y_std.flatten()
y_std
del yy

Now we will fit 10 different distributions, rank them by the approximate chi-squared goodness of fit, and report the Kolmogorov-Smirnov (KS) P value results. Remember that we want chi-squared to be as low as possible, and ideally we want the KS P-value to be >0.05.

Python may report warnings while running the distributions.

# Set list of distributions to test
# See https://docs.scipy.org/doc/scipy/reference/stats.html for more

# Turn off code warnings (this is not recommended for routine use)
import warnings
warnings.filterwarnings("ignore")

# Set up list of candidate distributions to use
# See https://docs.scipy.org/doc/scipy/reference/stats.html for more

dist_names = ['beta',
              'expon',
              'gamma',
              'lognorm',
              'norm',
              'pearson3',
              'triang',
              'uniform',
              'weibull_min', 
              'weibull_max']

# Set up empty lists to stroe results
chi_square = []
p_values = []

# Set up 50 bins for chi-square test
# Observed data will be approximately evenly distrubuted aross all bins
percentile_bins = np.linspace(0,100,51)
percentile_cutoffs = np.percentile(y_std, percentile_bins)
observed_frequency, bins = (np.histogram(y_std, bins=percentile_cutoffs))
cum_observed_frequency = np.cumsum(observed_frequency)

# Loop through candidate distributions

for distribution in dist_names:
    # Set up distribution and get fitted distribution parameters
    dist = getattr(scipy.stats, distribution)
    param = dist.fit(y_std)
    
    # Obtain the KS test P statistic, round it to 5 decimal places
    p = scipy.stats.kstest(y_std, distribution, args=param)[1]
    p = np.around(p, 5)
    p_values.append(p)    
    
    # Get expected counts in percentile bins
    # This is based on a 'cumulative distrubution function' (cdf)
    cdf_fitted = dist.cdf(percentile_cutoffs, *param[:-2], loc=param[-2], 
                          scale=param[-1])
    expected_frequency = []
    for bin in range(len(percentile_bins)-1):
        expected_cdf_area = cdf_fitted[bin+1] - cdf_fitted[bin]
        expected_frequency.append(expected_cdf_area)
    
    # calculate chi-squared
    expected_frequency = np.array(expected_frequency) * size
    cum_expected_frequency = np.cumsum(expected_frequency)
    ss = sum (((cum_expected_frequency - cum_observed_frequency) ** 2) / cum_observed_frequency)
    chi_square.append(ss)
        
# Collate results and sort by goodness of fit (best at top)

results = pd.DataFrame()
results['Distribution'] = dist_names
results['chi_square'] = chi_square
results['p_value'] = p_values
results.sort_values(['chi_square'], inplace=True)
    
# Report results

print ('\nDistributions sorted by goodness of fit:')
print ('----------------------------------------')
print (results)
Distributions sorted by goodness of fit:
----------------------------------------
  Distribution    chi_square  p_value
3      lognorm     30.426685  0.17957
2        gamma     44.960532  0.06151
5     pearson3     44.961716  0.06152
0         beta     48.102181  0.06558
4         norm    292.430764  0.00000
6       triang    532.742597  0.00000
7      uniform   2150.560693  0.00000
1        expon   5701.366858  0.00000
9  weibull_max  10452.188968  0.00000
8  weibull_min  12002.386769  0.00000

We will now take the top three fits, plot the fit and return the sklearn parameters. This time we will fit to the raw data rather than the standardised data.

# Divide the observed data into 100 bins for plotting (this can be changed)
number_of_bins = 100
bin_cutoffs = np.linspace(np.percentile(y,0), np.percentile(y,99),number_of_bins)

# Create the plot
h = plt.hist(y, bins = bin_cutoffs, color='0.75')

# Get the top three distributions from the previous phase
number_distributions_to_plot = 3
dist_names = results['Distribution'].iloc[0:number_distributions_to_plot]

# Create an empty list to stroe fitted distribution parameters
parameters = []

# Loop through the distributions ot get line fit and paraemters

for dist_name in dist_names:
    # Set up distribution and store distribution paraemters
    dist = getattr(scipy.stats, dist_name)
    param = dist.fit(y)
    parameters.append(param)
    
    # Get line for each distribution (and scale to match observed data)
    pdf_fitted = dist.pdf(x, *param[:-2], loc=param[-2], scale=param[-1])
    scale_pdf = np.trapz (h[0], h[1][:-1]) / np.trapz (pdf_fitted, x)
    pdf_fitted *= scale_pdf
    
    # Add the line to the plot
    plt.plot(pdf_fitted, label=dist_name)
    
    # Set the plot x axis to contain 99% of the data
    # This can be removed, but sometimes outlier data makes the plot less clear
    plt.xlim(0,np.percentile(y,99))

# Add legend and display plot

plt.legend()
plt.show()

# Store distribution paraemters in a dataframe (this could also be saved)
dist_parameters = pd.DataFrame()
dist_parameters['Distribution'] = (
        results['Distribution'].iloc[0:number_distributions_to_plot])
dist_parameters['Distribution parameters'] = parameters

# Print parameter results
print ('\nDistribution parameters:')
print ('------------------------')

for index, row in dist_parameters.iterrows():
    print ('\nDistribution:', row[0])
    print ('Parameters:', row[1] )
dist_fit_2
Distribution parameters:
------------------------

Distribution: lognorm
Parameters: (0.3411670333611477, 4.067737189292493, 9.490709944326486)

Distribution: gamma
Parameters: (5.252232022713325, 6.175162625863668, 1.5140473798580563)

Distribution: pearson3
Parameters: (0.8726704680754525, 14.127306804909308, 3.4698385545042782)

qq and pp plots

qq and pp plots are two ways of showing how well a distribution fits data, other than plotting the distribution on top of a histogram of values (as used above).

Our intention here is not to describe the basis of the plots, but to show how to plot them in Python. In very basic terms they both describe how observed data is distributed compared with a theoretical distribution. If the fit is perfect then the data will appear as a perfect diagonal line where the x and y values are the same (for standardised data, as we use here).

For more info on the plots see:

https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot
https://en.wikipedia.org/wiki/P%E2%80%93P_plot

Here is the code for the plots.

## qq and pp plots
    
data = y_std.copy()
data.sort()

# Loop through selected distributions (as previously selected)

for distribution in dist_names:
    # Set up distribution
    dist = getattr(scipy.stats, distribution)
    param = dist.fit(y_std)
    
    # Get random numbers from distribution
    norm = dist.rvs(*param[0:-2],loc=param[-2], scale=param[-1],size = size)
    norm.sort()
    
    # Create figure
    fig = plt.figure(figsize=(8,5)) 
    
    # qq plot
    ax1 = fig.add_subplot(121) # Grid of 2x2, this is suplot 1
    ax1.plot(norm,data,"o")
    min_value = np.floor(min(min(norm),min(data)))
    max_value = np.ceil(max(max(norm),max(data)))
    ax1.plot([min_value,max_value],[min_value,max_value],'r--')
    ax1.set_xlim(min_value,max_value)
    ax1.set_xlabel('Theoretical quantiles')
    ax1.set_ylabel('Observed quantiles')
    title = 'qq plot for ' + distribution +' distribution'
    ax1.set_title(title)
    
    # pp plot
    ax2 = fig.add_subplot(122)
    
    # Calculate cumulative distributions
    bins = np.percentile(norm,range(0,101))
    data_counts, bins = np.histogram(data,bins)
    norm_counts, bins = np.histogram(norm,bins)
    cum_data = np.cumsum(data_counts)
    cum_norm = np.cumsum(norm_counts)
    cum_data = cum_data / max(cum_data)
    cum_norm = cum_norm / max(cum_norm)
    
    # plot
    ax2.plot(cum_norm,cum_data,"o")
    min_value = np.floor(min(min(cum_norm),min(cum_data)))
    max_value = np.ceil(max(max(cum_norm),max(cum_data)))
    ax2.plot([min_value,max_value],[min_value,max_value],'r--')
    ax2.set_xlim(min_value,max_value)
    ax2.set_xlabel('Theoretical cumulative distribution')
    ax2.set_ylabel('Observed cumulative distribution')
    title = 'pp plot for ' + distribution +' distribution'
    ax2.set_title(title)
    
    # Display plot    
    plt.tight_layout(pad=4)
    plt.show()
 ppqq1

ppqq3

ppqq3

80. Grouping unlabelled data with k-means clustering

k-means clustering is a form of ‘unsupervised learning’. This is useful for grouping unlabelled data. For example, in the Wisconsin breast cancer data set, what if we did did not know whether the patients had cancer or not at the time the data was collected? Could we find a way of finding groups of similar patients? These groups may then form the basis of some further investigation.

k-means clustering groups data in such a way that the average distance of all points to their closest cluster centre is minimised.

Let’s start by importing the required modules, and loading our data. In order better visualise what k-means is doing we will limit our data to two features (though k-means clustering works with any number of features).

In [11]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from sklearn.preprocessing import StandardScaler
from sklearn.cluster import KMeans
  
# Load data
from sklearn import datasets
data_set = datasets.load_breast_cancer()
X=data_set.data[:,0:2] #  restrict patient data to two features.

We will normalise our data:

sc=StandardScaler() 
sc.fit(X)
X_std=sc.transform(X)

And now we will group patients into three clusters (k=3), and plot the centre of each cluster (each cluster centre has two co-ordinates, representing the normalised features we are using). If we used data with more than two features we would have a co-ordinate point for each feature.

In [13]:
kmeans = KMeans(n_clusters=3)  
kmeans.fit(X_std)  
print('\nCluster centres:')
print(kmeans.cluster_centers_)
Cluster centres:
[[-0.30445203  0.92354217]
 [ 1.52122912  0.58146379]
 [-0.50592385 -0.73299233]]

We can identify which cluster a sample has been put into with ‘kmeans.labels_’. Using that we may plot our clustering:

In [14]:
labels = kmeans.labels_

plt.scatter(X_std[:,0],X_std[:,1],
            c=labels, cmap=plt.cm.rainbow)
plt.xlabel('Normalised feature 1')
plt.ylabel('Normalised feature 2')
plt.show() 
kmeans1

How many clusters should be used?

Sometimes we may have prior knowledge that we want to group the data into a given number of clusters. Other times we may wish to investigate what may be a good number of clusters.

In the example below we look at changing the number of clusters between 1 and 100 and measure the average distance points are from their closest cluster centre (kmeans.transform gives us the distance for each sample to each cluster centre). Looking at the results we may decide that up to about 10 clusters may be useful, but after that there are diminishing returns of adding further clusters.

In [16]:
distance_to_closter_cluster_centre = []
for k in range(1,100):
    kmeans = KMeans(n_clusters=k)  
    kmeans.fit(X_std)
    distance = np.min(kmeans.transform(X_std),axis=1)
    average_distance = np.mean(distance)
    distance_to_closter_cluster_centre.append(average_distance)

clusters = np.arange(len(distance_to_closter_cluster_centre))+1
plt.plot(clusters, distance_to_closter_cluster_centre)
plt.xlabel('Number of clusters (k)')
plt.ylabel('Average distance to closest cluster centroid')
plt.ylim(0,1.5)
plt.show()
kmeans2

Continue reading “80. Grouping unlabelled data with k-means clustering”