In one of the previous posts we looked at the assumptions underlying the classic linear regression model. In this post we’ll take a look at the Gauss-Markov theorem which uses these assumptions to state that the OLS estimators are the best linear unbiased estimators, in the class of linear unbiased estimators.
Gauss-Markov Theorem
To recap briefly, some of the assumptions we made were that:
On average, the stochastic error term is zero i.e. .
The data is homoscedastic i.e. the variance / standard deviation of the stochastic error term is a finite value regardless of .
The model is linear in parameters i.e. and have power 1.
There is no autocorrelation between the disturbance terms.
The Gauss-Markov theorem states the following:
Given the assumptions underlying the CLRM, the OLS estimators, in the class of linear unbiased estimators, have minimum variance. That is, they are BLUE (Best Linear Unbiased Estimators).
We’ll go over what it means for an estimator to be best, linear, and unbiased.
Linear
The dependent variable is a linear function of the variables in the model. The model must be linear in parameters. Under this assumption and are both valid models since the estimators are linear in both of them.
Unbiased
This means that on average, the value of the estimator is the same as the parameter it is estimating i.e. .
Best
This means that among the class of linear estimators, the OLS estimators will have the least variance. Consider the two normal distributions shown in the graph above for two linear, unbiased estimators (represented by orange) and (represented by blue). Also let’s assume that the true value of is zero. This means that both of these are unbiased since, on average, their value is the same as the true value of the parameter. However, the OLS estimator is better since it has lesser variance compared to the other estimator.
The calculation of the estimators and is based on sample data. As the sample drawn changes, the value of these estimators also changes. This leaves us with the question of how reliable these estimates are i.e. we’d like to determine the precision of these estimators. This can be determined by calculating the standard error or the coefficient of determination.
Standard Error
The standard error of a statistic[1] is the standard deviation of the sampling distribution of that statistic.
Suppose we have a dataset which contains incomes of people. From this dataset we start drawing samples of size n and calculating the mean. Now if we plot the distribution (the sampling distribution) of the means we calculated, we’ll get a normal distribution centered on the population mean. The standard deviation of this sampling distribution is the standard error of the statistic which in our case is mean.
Standard error is given by the formula:
Where is the standard deviation of the population. is the sample size.
Calculating the standard error shows the sampling fluctuation. Sampling fluctuation shows the extent to which a statistic takes on different values.
Notice that the standard error has an inverse relation with the sample size n. This means that the larger the sample we draw from the population, the lesser the standard error will be. This will also result in a tighter normal distribution since the standard deviation will be less.
Standard Error of OLS Estimates
The standard error of the OLS estimators and is given by:
where is the square root of true but unknown constant of homoscedastic variance .
All of the terms in the equations above except can be calculated from the sample drawn. Therefore, we will need an unbiased estimator . The denominator represents the degrees of freedom. is the residual sum of squares.
Although , its summation can be computed with an alternative formula as .
All of the terms in the above formula would have already been computed as a part of computing the estimators.
How it all ties together
As you calculate the estimators and draw the sample regression curve that passes through your data, you need some numeric measure of how good the curve fits the data i.e. a measure of “goodness of fit”. This can be given as:
This is the positive square root of the estimator of homoscedastic variance. This is the standard deviation of the values about the regerssion curve.
The standard errors of the estimators and will show you how much they fluctuate as you draw the samples. The lesser their standard error, the better.
The square of the standard error is called the mean squared error.
import pandas as pd from sklearn import datasets boston = datasets.load_boston() data, target = boston.data, boston.target df = pd.DataFrame(data=data, columns=boston.feature_names) df = df[['RM']] df['RM'] = df['RM'].apply(round) df['price'] = target
I’m choosing to use the average number of rooms as the variable to see how it affects price. I’ve rounded it off to make the calculations easier. The scatter plot of the data looks the following:
This, ofcourse, is the entire population data so let’s draw a sample.
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sample = df.sample(n=100)
Now let’s bring back our function that calculates the OLS estimates:
The plot of the regression curve on the sample data looks the following:
The sample DataFrame has our intermediate calculation and we can use that to calculate the standard error. Let’s write a function which does that.
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from math import sqrt defstandard_error(sample): return sqrt((sample['ucap'] ** 2).sum() / (len(sample) - 2))
Finally, let’s see how much standard deviation we have around the regression line.
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standard_error(sample) 7.401174774558201
Coefficient of Determination
The coefficient of determination denoted by (for bivariate regression) or (for multivariate regression) is a ratio of the explained variance to the total variance of the data. The higher the coefficient of determination, the more accurately the regressors (average number of rooms in the house) explain the regressand (the price of the house) i.e. the better your regression model is.
For bivariate linear regression, value is given by:
For the sample we’ve drawn, the value comes out to be 0.327. This means that only 32.7% of the variance in the total data is explained by the regression model.
Finito.
[1] A “statistic” is a numerical quantity such as mean, or median.
In the previous post we calculated the values of and from a given sample and in closing I mentioned that we’ll take a look at finding out whether the sample regression line we obtained is a good fit. In other words, we need to assess how close and are to and or how close is to . Recall that the population regression function (PRF) is given by . This shows that the value of depends on and . To be able to make any statistical inference about (and and ) we need to make assumptions about how the values of and were generated. We’ll discuss these assumptions in this post as they’re important for the valid interpretation of the regression estimates.
Classic Linear Regression Model (CLRM)
There are 7 assumptions that Gaussian or Classic linear regression model (CLRM) makes. These are:
1. The regression model is linear in parameters
Recall that the population regression function is given by . This is linear in parameters since and have power 1. The explanatory variables, however, do not have to be linear.
2. Values of X are fixed in repeated sampling
In this series of posts, we’ll be looking at fixed regressor models[1] where the values of X are considered to be fixed when drawing a random sample. For example, in the previous post the values of X were fixed between 80 and 260. If another sample were to be drawn, the sample values of X would be used.
3. Zero mean value of the error term
This assumption states that for every given , the error term is zero, on average. This should be intuitive. The regression line passes through the conditional mean for every . The error term are distances of the individual points from the conditional mean. Some of them lie above the regression line and some of them lie below. However, they cancel each other out; the positive cancel out the negative and therefore is zero, on average. Notationally, .
What this assumption means that all those variables that were not considered to be a part of the regression model do not systematically affect i.e. their effect on average is zero on .
This assumption also means is that there is no specification error or specification bias. This happens when we chose the wrong functional form for the regression model, exclude necessary explanatory variables or include unnecessary ones.
4. Constant variance of (homoscedasticity)
This assumption states that the variation of is the same regardless of .
With this assumption, we assume that the variance of for a given (i.e. conditional variance) is a positive constant value given by . This means that the variance for the various populations is the same. What this also means is that the variance around the regression line is the same and it neither increases nor decreases with change in values of .
5. No autocorrelation between disturbances
Given any two values, and (), the correlation between any two and is zero. In other words, the observations are independently sampled. Notationally, . This assumption states that there is no serial correlation or autocorrelation i.e. and are uncorrelated.
To build an intuition, let’s consider the population regression function
Now suppose that and are positively correlated. This means that depends not only on but also on because that affects . Autocorrelation occurs in timeseries data like stock market trends where the the observation of one day depends on the observation of the previous day. What we assume is that each observation is independent as it is in case of the income-expense example we saw.
6. The sample size must be greater than the number of parameters to estimate
This is fairly simple to understand. To be able to estimate and , we need atleast 2 pairs of observations.
7. Nature of X variables
There are a few things we assume about the value of variables. One, for a given sample they are not all the same i.e. is a positive number. Second, there are no outliers among the values.
If we have all the values of the same in a given sample, the regression line would be a horizontal line and therefore it will be impossible to estimate and thus . This will happen because in the equation , the denominator will become zero since and will be the same.
Furthermore, we assume that there are no outliers in the values. Suppose there are a few values which are far apart from the mean value, the regression line which we get will be vastly different depending on whether the sample drawn contains these outliers or not.
These are the 7 assumptions underlying OLS. In the next section, I’d like to elucidate on assumption 3 and 5 a little more.
More on Assumption 3. and 5.
In assumption 3. we state that i.e. on average the disturbance term is zero. For the expected value of a variable given another variable be zero means the two variables are uncorrelated. We assume these two variables to be uncorrelated so that we can assess the impact each has on the predicted value of . If the disturbance and the explanatory variable were correlated positively or negatively then it means that contains some factor that affects the predicted value of i.e. we’ve skipped an important explanatory variable that should’ve been included in the regression model. This means we have functionally misspecified our regression model and introduced specification bias.
Functionally misspecifying our regression model will also introduce autocorrelation in our model. Consider the two plots shown below.
In the first plot the line fits the sample points properly. In the second one, it doesn’t fit properly. The sample points seem to form a curve of some sort while we try to fit a straight line through it. What we get is runs of negative residuals followed by positive residuals which indicates autocorrelation.
Autocorrelation can also be introduced not just by functionally misspecifying the regression model but also by the nature of data itself. For example, consider timeseries data representing stock market movement. The value of depends on i.e. the two are correlated.
When there is autocorrelation, the OLS estimators are no longer the best estimators for predicting the value of .
Conclusion
In this post we looked at the assumptions underlying OLS. In the coming post we’ll look at the Gauss-Markov theorem and the BLUE property of OLS estimators.
[1] There are also stochastic regression models where the values of X are not fixed.
In the previous post we built the intuition behind linear regression. In this post we’ll dig deeper into the simplest form of linear regression which involves one dependant variable and one explanatory variable. Since we only have two variables involved, it’s called bivariate linear regression.
Ordinary Least Squares
We defined sample regression function as:
The term is called the residual. If we re-write the sample regression function, we get . The residual then represents how far off our estimated value is from the actual value .[1] A reasonable assumption to make is that we should pick and such that the sum of is the smallest. This would mean that our estimates are close to the actual values.
This assumption, however, does not work. Let’s assume that our values for are -10, 3, -3, 10. Here the sum is zero but we can see that the first and last predictions are far apart from the actual values. To mitigate this, we can add the squares of the residuals. We need to pick and such that the sum of the square of the residuals is the minimum i.e. is the least.
Again, assuming the values of to be -10, 3, -3, 10, the squares are 100, 9, 9, 100. This sums to 218. This shows us that the predicted values are far from the actual values. is called the residual sum of squares (RSS) or the squared error term.
The method of OLS provides us with estimators and such that, for a given sample set, is the least. These estimators are given by the formula:
where = , . These are the differences of individual values from their corresponding sample mean or . The estimators thus obtained are called least-squares estimators.
Now that we’ve covered significant ground, let’s solve a sum by hand.
Example
Let’s start by drawing a random sample from the original dataset. The sample I’ve drawn looks like this:
To see how this contrasts with the original dataset, here’s a plot of the original dataset and the random sample. The faint green points represent the original dataset whereas the red ones are the sample we’ve drawn. As always, our task is to use the sample dataset to come up with a regression line that’s as close as possible to the population dataset i.e. use the red dots to come up with a line that’d be similar to the line we’d get if we had all the faint green dots.
I’ll write some Python + Pandas code to come up with the intermediate calculations and the final and values. I highly recommend that you solve this by hand to get the feel for it.
Let’s plot the line we obtained as a result of this.
Interpreting the Results
Our calculations gave us the results that and . Having a slope () of means that for every unit increase in income, there’s an increase of in expenditure. The intercept is where the regression line meets the Y axis. This means that even without any income, a persion would have an expenditure of . In a lot of cases, however, the intercept term doesn’t really matter as much as the slope and how to interpret the intercept depends upon what the Y axis represents.
Conclusion
That’s it for this post on bivariate linear regression. We saw how to calculate and and worked on a sample problem. We also saw how to interpret the result of the calculations. In the coming post we’ll look at how to assess whether the regression line is a good fit.
[1] It’s important to not confuse the stochastic error term with the residual . The stochastic error term represents the inherent variability of data whereas the residual represents the difference between the predicted and actual values of .
The first topic we’ll look at is linear regression. As mentioned previously, regression is about understanding dependence of one variable on another. We’ll start by building an intuitive understanding of what it is before we dive into the math-y details of it. We’ll cover the theory, learn to solve some problems by hand, look at the assumptions supporting it, then look at what happens if these assumptions aren’t held. Once this is done we’ll look at how we can do the same by using gradient descent.
What is Regression Analysis?
Regression analysis is the study of the dependence of one variable (dependent variable, regressand) over one or more other variables (explanatory variables, regressors). The aim is to see how the dependent variable changes, on average, if there is a change in the explanatory variables. This is better explained with an example.
The plot shown here is of daily income and daily expenditure. The dependent variable here is the expenditure which depends on the explanatory variable income. As we can see from the plot, for every income level we have a range of expenditures. However, as the regression line shows, on average as income goes up, the expenditure goes up, too.
Why do we say that the income goes up on average? Let’s look at the raw data when daily income is 80 or 100.
The highest expenditure when income is 80 (75) is greater than the lowest expenditure when income level is 100 (65). If we look at the averages, the expenditure has gone up. There are a number of factors which may have caused the person with income 80 to expend 75 but as a whole, people with his daily income tend to expend less than those who earn 100 a day. The regression line passes through these averages.
That’s the basic idea behind regression analysis.
The Math behind Regression Analysis
The regression line passes through the average expenditure for every given income level. In other words, it passes through the conditional expected value — . This is read as “the expected value of expenditure (Y) given an income level ()”. Therefore, is a function of i.e. . The question now is: what is the function ?
We can start by assuming to be a straight line function which means that we’re assuming expenditure to be linearly related to income. So, . This is the slope-intercept form of a line where is the intercept and is the slope of the line. Both and are called the regression coefficients (or parameters). The function is called the population regression function.
Now given that we know how to calculate the average expenditure for a given income level, how do we calculate the expenditure of each individual at that income level? We can say that the individual’s expenditure is “off the average expenditure by a certain margin”. This can be written as . The term denotes how far off an individual’s expenditure is from the average and is called the stochastic error term.
No matter how good our regression model is, there is always going to be some inherent variability. There are factors other than income which define a person’s expenditure. Factors like gender, age, etc. affect the expenditure. All of these factors are subsumed into the stochastic error term.
When it comes to applying regression analysis, we won’t have access to the population data. Rather, we have access to sample data. Our task is to come up with sample regression coefficients such that they’ll be as close as possible to the population regression coefficients. The sample regression function is defined as:
Here , , , and are estimators of , , , and respectively. An estimator is simply a formula which tells us how to estimate the population parameter from the information provided by the sample data available to us.
What it means to be ‘Linear’
Linearity can be defined as being either linear in parameters or linear in explanatory variables. To be linear in parameters means that all your parameters in the regression function will have power 1. By that definition, is linear. In contrast, is linear in explanatory variables since has power 1. We’ll be looking at models that are linear in parameters. Henceforth, “linear” would mean “linear in parameters”.
That’s it. This is what sets the foundation for further studying linear regression.
A while ago I’d started writing a series on machine learning titled ML for Newbies which I never got around to finishing. This series of posts is a redoing of the older series with greater mathematical, and theoretical rigor. Some of the content will be ported over from that series but most of it will be new content.
The aim of the series is to summarize everything that I know about machine learning in an approachable, intuituve manner, and also be notes for myself.
What is Machine Learning?
ML is all about automated learning. It is about creating algorithms that can “learn” from the input given to them. These algorithms can then do a number of tasks such as filter out spam emails, predict the weather, etc. More formally, it’s about converting experience into expertise. The input to the algorithm become its “experience” and the task that it performs becomes its “expertise”.
Although ML is a sub-field of AI (Artificial Intelligence), they’re different. AI focuses on replicating human intelligence whereas ML focuses on complementing human intelligence by doing tasks that fall beyond human capabilities like looking for patterns in massive data sets.
How does a Machine Learn?
Like I mentioned earlier, a machine learns from the input data given to it. The way the machine learns can also be classified as supervised, unsupervised, active, or passive.
Supervised Learning
Let’s say your task is to build a spam filter. You have a large data set of emails, each of which is marked as either “spam” or “not spam”. You’ll feed this data set to your algorithm and it’ll learn from it. Later, you’ll input an email and it’ll tell you if it is spam or not. This type of learning is called supervised learning. It is called “supervised” because the algorithm was first taught what is spam and what isn’t by a data set generated manually. You had to supervise its learning process.
Unsupervised Learning
Let’s stay with the spam filter example. Let’s now suppose that the emails in the data set are not marked as “spam” or “not spam”. It’s the task of the algorithm to figure it out somehow. Such learning is called unsupervised learning.
Passive Learning
In passive learning, the algorithm only observes the data provided to it and learns from it, without influencing it in anyway. Suppose now that the spam filter starts out blank with no notion of what is and what isn’t spam. It’s up to the user to mark the incoming emails so that the algorithm can learn from it. This is passive learning.
Active Learning
In contrast, let’s say that the spam filter now looked at the email first, and if it found something suspicious, asked the user to mark it as spam or not. This is active learning.
What can ML do?
There are a number of ML algorithms, each focusing on a different task. These algorithms can do a number of things like:
Clustering
In simple terms, clustering is grouping. It’s an unsupervised form of learning in which the algorithm looks at the data and groups the similar pieces of together based on some similarity measure. In the spam filter examples above, when the algorithm, on its own, figured out which of the emails are spam, it clustered them together.
Classification
Classification is about finding out which group a piece of data belongs to. It’s a form of supervised learning because the algorithm needs to know beforehand what each of the group looks like before it can be asked to classify a piece of data. When the spam filter was given a data set with marked emails, all it had to do was to answer the question “is this email spam?”. Basically, it was asked to classify that email into a group.
Regression
Regression is about understanding relationships between variables. If one variable changes, how would it affect the other? For example, if the amount of power needed for a phone to stay on increased, how would it affect battery life?
Association Rules
Finding out association rules in data sets is also about finding out relationships between variables. Suppose you have a large data set from an online store, association rules is about finding out what items you are likely to buy based on what you’ve previously bought. For example, if you buy milk and sugar, you’re likely to buy eggs. Your rule then becomes {milk, sugar} => eggs.
For now, don’t worry about the theoretical rigor of these. We’ll get to that in subsequent posts. That’s all for now. I hope this post was motivation enough for you to start learning ML. :)
So, I am taking a break from my JVM JIT series to write this post about a table summarization algorithm that I had implemented way back in my college days. Simply put, the algorithm finds the most descriptive patterns in the table that succinctly convey the meaning of the rows contained i.e. summarize it. This post will focus on what the algorithm is and how it’s implemented in JavaScript. Take this post with a grain of salt because the implementation was a part of my college assignment and hasn’t been thoroughly checked for correctness.
Given a table where rows correspond to records and columns correspond to attributes, we want to find a small number of patterns that succinctly summarize the dataset. … TSum, a method that provides a sequence of patterns ordered by their “representativeness.” It can decide both which these patterns are, as well as how many are necessary to properly summarize the data.
An algorithm like this is useful in situations where the volume of data is so large that it’d be impossible for a human to deduce anything from it simply by looking at the data. For example, given patient records, a description like “most patients are middle-aged men with cholestrol, followed by child patients with chickenpox” provides a very human-readable summary of the data.
The algorithm views finding the best pattern as a compression problem — the best pattern will give the best compression.
Definitions
Pattern and Pattern Size
A pattern is a tuple where can can be either a specific value or a “don’t care” value represented by . The number of matching attributes in a pattern is called the size of the pattern and is denoted by . An attribute is considered matching if and only if its value doesn’t equal “don’t care” value.
Pattern List and Pattern Set
A pattern list is an ordered sequence of patterns while a pattern set is a set of patterns with no order.
Compression Saving
Compression saving of a pattern on a table , denoted as is the amount of compression it can achieve. Specifically, it is the difference of bits between he uncompressed and compressed representations of the data records covered by the pattern . Let be the number of records covered by pattern and be the number of attributes in . Then,
where
and
is the average number of bits in the ith attribute. is the number of attributes.
Residue Data Table
Given a table and a pattern collection , the residue data table contains the records that are not covered by any of the patterns in .
Pattern Marhsalling
Given a set of patterns , the pattern marshalling algorithm picks a pattern from which has the highest compression saving. After every iteration, the records in the table which have been covered by the patterns chosen so far will be removed from consideration.
Generating Patterns - Local Expansion
Local expansion is a pattern generation strategy that tries to “grow” the patterns to increase the compression savings on the data record. In this approach, the algorithm will start with single attributes first and and find a single-condition pattern that has the best compression saving, and then expand the pattern by adding other conditions until the compression cost cannot be improved. To find the next pattern, the same procedure is repeated, but only on the residue data table - the part of the table not covered by any of the patterns found so far.
Code
We’ll start with simpler code first. Let’s start by writing a function to calculate the benefit.
"use strict"; const _ = require("lodash"); /** * The benefit of using this pattern to represent the table * @param T the table containing rows * @param pattern the pattern to be applied */ functionbenefit(pattern, T){ if(pattern == null) return0;
// a function for internal use. // find out the bits needed to encode the value functionbitsFor(value) { if( typeof(value) === "string" || value instanceofString){ // JS uses UTF-16 return value.length * 16; } if( typeof(value) === "number" || value instanceofNumber){ // 64-bit for each number return64; } return0; }
let N = _.filter(T, pattern).length; let W = 0; let attributes = Object.keys(pattern);
attributes.forEach(attribute => { W += bitsFor(pattern[attribute]); });
return (N-1) * W; } module.exports = benefit;
The actual translation of formula to code happens from line #25. We start by finding N which is the number of records in the table that match the pattern. Im using lodash‘s filter function to avoid the boilerplate of having to find the matching records myself. W is the accumulator in which I will sum the number of bits each attribute in the pattern take. attributes are the attributes in the pattern. Then for each attribute, we use bitsFor and add it up with the value of W. Finally, we return the value according to the formula.
Simply put, benefit is times the total number of bits each attribute in the pattern would take.
Next, let’s write code to find
log.js
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"use strict"; /** * Calculates log*(N). * @param N number of rows in the table * @returns {number} */ functionlog(N){ return2 * Math.log2( N+2 ); } module.exports = log;
This is a fairly straightforward translation of the formula to code and needs no explanation.
We now have all the code we need to calculate overhead so let’s do that.
overhead.js
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"use strict"; const _ = require("lodash"); const log = require("./log"); /** * The overhead of using this pattern to represent this table * @param T the table containing rows * @param pattern the pattern to be applied */ functionoverhead(pattern, T){ let N = _.filter(T, pattern).length; let D = Object.keys(T[0]).length; // number of attributes return D + log(N); } module.exports = overhead;
Notice that I start by require-ing the log.js module we just saw. I am using filter to find the number of rows in the table that match the pattern. On the next line I find the number of attributes in a given record of the table. Since the algorithm assumes no null / empty values for any attribute, we can safely pick up the 0th record and see how many attributes it contains.
Now that we have benefit and overhead in place, let’s calculate saving
saving.js
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"use strict"; const benefit = require("./benefit"); const overhead = require("./overhead"); /** * The compression saving of a pattern P on a data table T, * denoted by saving(P,T) is the amount of compression it can achieve. * @param T the table containing rows * @param pattern the pattern to be applied */ functionsaving(pattern, T){ returnbenefit(pattern, T) - overhead(pattern, T); } module.exports = saving;
functionpatternMarshalling(S, T){ // chosen patterns let patterns = []; // empty = don't care for every attribute let remainingPatterns = _.cloneDeep(S);
while( remainingPatterns.length > 0 ){ // select the pattern with the top incremental compression saving let bTop = Number.NEGATIVE_INFINITY; let pTop; let residualTable = residue(patterns, T);
Given a set of patterns and a table , we start by making a copy of the patterns passed to us. The aim is to select a pattern which gives us non-negative compression. If we find such a pattern, we add it to our patterns list and remove the chosen pattern from the current list of patterns. We continue the next itreation on the residue table. We repeat this until we have no more patterns to consider.
The next piece of the puzzle is to write code for local expansion.
/** * Local expansion pattern generation strategy that directly looks * for patterns that could minimize the compression cost * @param T an array of JSON objects * @returns {Array} of patterns that best summarize the data */ functionlocalExpansion(T){ let patterns = []; // final list of patterns to return
// while we still have rows while( T.length > 0 ){ // expand from an empty pattern (Algorithm 4) let pattern = expand(T, {}); // stop if we cannot achieve more compression saving if( _.isEqual(pattern, {}) ){ break; } // found a new pattern patterns.push( pattern ); // remove all the rows that match the pattern i.e.residual table for the pattern T = _.difference(T, _.filter(T, pattern)); }
return patterns; }
module.exports = localExpansion;
As mentioned in the definitions, local expansion grows a pattern, starting with an empty pattern. The expansion is done in expand function.
/** * Expands a pattern to improve compression saving. See algorithm 4. * @paramT * @parampattern * @returns {*} */ functionexpand(T, pattern){ // find the attributes not included in the pattern let allAttributes = Object.keys(T[0]); let matchedAttributes = Object.keys(pattern); let attributes = _.difference(allAttributes, matchedAttributes);
let bestPattern; let highestCompressionSaving = 0;
attributes.forEach( attribute => { // find all the unique values for the current attribute let values = _.map(T, attribute); values = _.uniq(values);
values.forEach( value => { // an expanded pattern created by appending the current // attribute and its value to the existing pattern let newPattern = _.cloneDeep(pattern); newPattern[attribute] = value;
The final piece of code we need to look at is to calculate the coverage i.e. how much of the data is described by a given pattern.
coverage.js
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"use strict";
const _ = require("lodash"); functioncoverage(pattern,T){ let matchingRows = _.filter(T,pattern); let coverage = matchingRows.length / T.length; return coverage * 100; // % of coverage }
module.exports = coverage;
Now that we have all the machinery in place, let’s write a simple test. We’ll take the patients example given in the paper and turn it into JSON objects. Then we’ll write a simple script to run our code using this data and check whether the results make sense.
The output says that the most descriptive pattern is “adult male” which makes up 50% of the rows followed by “children with low blood pressure” which make up 30% of the rows. if we look at the sample data, out of the 10 rows, 5 are “adult male” and 3 are “children with low blood pressure”. So the output of the algorithm checks out.
In the previous post we looked at how JVM generates assembly code and how we can take a look at it. In this post we will look at yet another optimization technique called loop unrolling.
the Server VM features loop unrolling, a standard compiler optimization that enables faster loop execution. Loop unrolling increases the loop body size while simultaneously decreasing the number of iterations.
Okay so it is an optimization that makes our loops faster by increasing the body size, and reducing the iterations. This is much better explained by example.
A loop like this:
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for(inti=0; i < N; i++) { S(i); }
can be unrolled into this:
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for(inti=0; i < N; i += 4) { S(i); S(i+1); S(i+2); S(i+3); }
So the size of the loop has increased because instead of calling method S just once per iteration, we call it 4x. The iterations have reduced because we now have a stride of 4 (i += 4). This is a space-time tradeoff because you gain speed by increasing the size of the program. If this for loop were to be a part of a hot method that got compiled to assembly, more code cache would be consumed because of unrolling.
Loop unrolling, however, wins at a different front — speed. By increasing the stride of the loop, we’re reducing the number of jumps that the CPU has to make. Jumps are costly and a reduction in jumps is a huge performance boost.
for (inti=0; i < nums.length; i++) { nums[i] = 0x42; } } }
The code above is a JMH benchmark which will fork 1 JVM (@Fork), warm it for 10 iterations (@Warmup), then run our benchmark for 5 iterations (@Measurement). The generated assembly looks like this:
So the first block of code is moving 0x42 to some memory location (line #3), incrementing the counter %ebp (line #4), and jumping back to the start of the loop for the next iteration (line #6). Then there is the second block which increments the counter by 4 (line #11) and uses vmovdqu (line #10).
In the second block, vmovdqu moves %xmm0 (a 128-bit register) to some memory location (line #10) i.e. we’re writing 4 32-bit numbers in one go. This is equivalent of writing nums[0 .. 3] in one iteration.
This means that our entire loop which writes 10 array elements will not run for 10 iterations at all. It will run for 4 — 2 iterations to write 8 elements in groups of 4 each (block 2), and 2 more to write the remaining 2 elements (block 1).
What we saw above was loop unrolling to enable vectorization. Since this happened without any effort on our part, it’s called auto-vectorization.
What is vectorization?
There are certain CPU instructions which are capable of operating on multiple data elements simultaneously. Such instructions are called SIMD instructions — Single Instruction Multiple Data or vectorized instructions. In the example above, all 4 elements were packed into a single register XMM0 and then moved to their memory location with vmovdqu. This results in faster processing by saving jumps and clock cycles.
Auto-vectorization is when the compiler converts the scalar instructions (which operate on a single data element at a time) to vector instructions (which operate on multiple data elements at a time) without any effort on the part of the programmer.
Vectorizing the code enables superword level parallelism (SLP). SLP is a type of SIMD parallelism in which source and result operands are packed in a storage location.
The example shows statement packing in which the operands have been packed into registers and the scalar addition and multiplication have been replaced by their vectorized counterparts.
This is a loop containing isomorphic statements — statements that contain same operations in the same order. Such statements are easy to unroll. So the loop above can be transformed as follows:
Now it becomes easy to see how SLP can be used to speed up the execution of the loop. Loop unrolling sets the stage for SLP.
Which flags control loop unrolling and SLP?
There are a couple of flags - -XX:LoopUnrollLimit and -XX:+UseSuperWord. LoopUnrollLimit controls how many times your loop will be unrolled and UseSuperWord controls the transformation of scalar operations into vectorized operations.
Conclusion
Loop unrolling makes your code faster by doing more per iteration and increasing the stride. It’s a space-time trade off where you make your code larger so that the execution takes less time. Loop unrolling can be taken a step further to make scalar instructions into vector instructions. These instructions operate on packed data and achieve more per instruction than their scalar counterparts.
The conversion from scalar instructions to vector instructions is called vectorization. Since this happens transparently to the programmer, it is called auto-vectorization.
Recommended reading
There are a few blog posts that I recommend you read.
A fundamental introduction to x86 assembly programming This post provides a brief introduction to x86 assembly programming. Having studied assembly programming in university a long time ago, the post is quick refresher on the basic concepts of assembly programming like addresing modes, registers, etc. I keep it handy and refer to it when I have to read the assembly code generated by JVM.
JMH is a Java harness for building, running, and analysing nano/micro/milli/macro benchmarks written in Java and other languages targetting the JVM.
The post will show you how to write a simple benchmark, how to avoid common pitfalls, etc. It is a good read regardless of this series for two reasons: one, microbenchmarking is coming to Java 9 out-of-the-box and two, microbenchmarking is hard to get right.
Exploiting Superword Level Parallelism with Multimedia Instruction Sets This is the research paper on superword level parallelism. I’ve borrowed examples and diagrams from this paper. The paper will introduce you to what SLP is, and also provide an algorithm on how to find parts of code which are amenable to SLP.
In the previous post we saw how JIT inlining works. We also saw how the JVM performs OSR to replace the interpreted version of the method to the compiled version on the fly. In this post we’ll dig even deeper and see the assembly code that is generated when the method gets compiled.
Prerequisites
The flag which enables us to see assembly code is -XX:+PrintAssembly. However, viewing assembly code does not work out of the box. You’ll need to have the disassembler on your path. You’ll need to get hsdis (HotSpot Disassembler) and build it for your system. There’s a prebuilt version available for Mac and that’s the one I am going to use.
So this is the assembly code that we get when we run the program. It’s a lot to grok in one go so let’s break it down.
Line #7 and #8 are self explanatory; they show which method we’re looking at. Line #9 and #10 (and #13 to #17) are for thread synchronization. The JVM can get rid of thread synchronization if it sees that there is no need for it (lock eliding) but since we are using static methods here, it needs to add code for synchronization. It doesn’t know that we only have only one thread running.
Our actual program is on line #11 where we are moving the value 4 to %eax register. This is the register which holds, by convention, the return value for our methods. This shows that the JVM has optimized our code. Our call chain was inline1 ⟶ inline2 ⟶ inline3 and it was inline3 which returned 4. However, JVM is smart enough to see that these method calls are superfluous and decided to get rid of them. Very nifty!
Line #21 to #23 has code to handle exceptions. We know there won’t be any exceptions but the JVM doesn’t so it has to be prepared to deal with that.
And finally, there’s code to deoptimize. In addition to static optimizations, there are some optimizations that the JVM makes which are speculative. This means that the JVM generates assembly code expecting things to go a certain way after it has profiled the interpreted code. However, if the speculation is wrong, the JVM can go back to running the interpreted version.
Which flags control compilation?
-XX:CompileThreshold is the flag which controls the number of call / branch invocations after which the JVM compiles bytecodes to assembly. You can use -XX:+PrintFlagsFinal to see the value. By default it is 10000.
Compiling a method to assembly depends on two factors: the number of times that method has been invoked (method entry counter) and the number of times a loop has been executed (back-edge counter). Once the sum of the two counters is above CompileThreshold, the method will be compiled to assembly.
Maintaining the two counters separately is very useful. If the back-edge counter alone exceeds the threshold, the JVM can compile just the loop (and not the entire method) to assembly. It will perform an OSR and start using the compiled version of the loop while the loop is executing instead of waiting for the next method invocation. When the method is invoked the next time around, it’ll use the compiled version of the code.
So since compiled code is better than interpreted code, and CompileThreshold controls when a method will be compiled to assembly, reducing the CompileThreshold would mean we have a lot more assembly code.
There is one advantage to reducing the CompileThreshold - it will reduce the time taken for the branches / methods to be deemed hot i.e. reduce the JVM warmup time.
In older JDKs, there was another reason to reduce CompileThreshold. The method entry and back-edge counters would decay at every safepoint. This would mean that some methods would not compile to assembly since the counters kept decaying. These are the “lukewarm” methods that never became hot. With JDK 8+, the counters no longer decay at safepoints so there won’t be any lukewarm methods.
In addition, JDK 8+ come with tiered compilation enabled and the CompileThreshold is ignored. The idea of there being a “compile threshold”, though, does not change. I’m defering the topic of tiered compilation for the sake of simplicity.
Where is the compiled code stored?
The compiled code is stored in JVM’s code cache. As more methods become hot, the cache starts to get filled. Once the cache is filled, the JVM can no longer compile anything to assembly and will resort to purely interpreteting the bytecodes.
The size of code cache is platform dependent.
Also, JVM ensures that the access to cache is optimized. The hlt instructions in the assembly code exist for aligning the addresses. It is much more efficient for the CPU to read from even addresses than it is to read from odd addresses in memory. The hlt instructions ensure that the code is at an even address in memory.
Which flags control code cache size?
There are two flags which are important in setting the code cache size - InitialCodeCacheSize and ReservedCodeCacheSize. The first flag indicates the code cache size the JVM will start with and the latter indicates the size to which the code cache can grow. With JDK 8+, ReservedCodeCacheSize is large enough so you don’t need to set it explicitly. On my machine it is 240 MB (5x what it is for Java 7, 48 MB).
Conclusion
The JVM compiles hot code to assembly and stores it at even addresses in it’s code cache for faster access. Executing assembly code is much more efficient than interpreting the bytecodes. You don’t really need to look at the assembly code generated everyday but knowing what is generated as your code executes gives you an insight into what the JVM does to make your code run faster.
In the previous post we looked at how interpreted and compiled languages work. To recap, an interpreter works by generating assembly code for every bytecode it encounters. This is a very simple way to execute a program and also a very slow one. It ends up redoing a lot of translation from bytecode to assembly. Also, this simplistic approach means that the interpreter cannot do optimizations as it executes the bytecodes. Then there are compilers which produce assembly ahead-of-time. This overcomes having to generate assembly again and again but once the assembly is generated it cannot be changed on the fly.
JVM comes with both an interpreter and a compiler. When the execution of the code begins, the bytecodes are interpreted. For the sake of this series, I’ll be looking at Oracle HotSpot JVM which looks for “hot spots” in the code as the bytecodes get interpreted. These are the parts of the code which are most frequently executed and the performance of the application depends on these. Once the code is identified as “hot”, JVM can go from interpreting the code to compiling it to assembly i.e. the code is compiled “just-in-time”. In addition, since the code is being profiled as it is run, the compiled code is optimized.
In this post we’ll look at one such optimization: inlining.
Inlining
Inlining is an optimization where the call to a method is replaced by the body of the called method i.e. at the call site, the caller and the callee are melded together. When a method is called, the JVM has to push a stack frame so that it can resume from where it left off after the called method has finished executing. Inlining improves performance since JVM will not have to push a stack frame.
I’ll start with a simple example to demonstrate how inlining works.
Line #1 shows that inline1 was compiled to assembly. Line #2 and Line #6 show that inline2 and inline3 were also compiled to assembly. Line #3 to line #5 show inlining. We can see that inline3 was merged into inline2. Similarly, line #8 and #9 show that inline2 was merged into inline1. So basically, all the methods were inlined into inline1. This means that once a certain threshold is crossed, we’ll no longer be making methods calls at all. This gives a significant performance boost.
Which flags control inlining?
When you run a Java program, you can view the flags with which it ran using -XX:+PrintFlagsFinal. Let’s do that and look at a few flags of interest.
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java -XX:+PrintFlagsFinal Inline 10000
You’ll see a bunch of flags and their default values. The ones we are interested in are CompileThreshold, MaxInlineLevel, MaxInlineSize, and FreqInlineSize.
CompileThreshold is the number of invocations before compiling a method to native. MaxInlineLevel is a limit on how deep you’d go before you stop inlining. The default value is 9. This means if we had method calls like inline1 ⟶ inline2 … ⟶ inline20, we’d only inline upto inline10. There after, we’d invoke inline11. MaxInlineSize decides the maximum size of a method, in bytecodes, to be inlined. The default value is 35. This means that if the method to be inlined has mre than 35 bytecodes, it will not be inlined. FreqInlineSize, in contrast, decides the maximum size of a hot method, in bytecodes, to be inlined. This is a platform-dependent value and on my machine it is 325.
You can tweak these flags to change how inlining behaves for your program.
What is On Stack Replacement (OSR)?
When we make a method call, JVM pushes a stack frame. When a method is deemed hot, the JVM replaces the intrepreted version with the compiled version by replacing the old stack frame with a new one. This is done while the method is running. We saw OSR being indicated in our example. The % indicates that an OSR was made.
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66 4 % Inline::main @ -2 (28 bytes) made not entrant
Let’s write some code to see OSR in action once again.
So this is a loop that will never terminate, right? Let’s run the program and see.
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... 434062 16828 59 % OSR::main @ -2 (48 bytes) made not entrant Finished!
What just happened? When the JVM decided to perform an OSR, it saw that there was no use for the unused object and decided to set it to null, causing the WeakReference to return null and thus breaking the loop. When an OSR is performed, the method that is invoked doesn’t restart execution from the start. Rather, It continues from the “back-edge”. In our case, it would be the loop. Since the JVM saw that there was no use for the unused object after this back-edge, it was removed and the loop could terminate.
Being able to resume execution from the back-edge is very efficient. This means that once a method has been compiled to native code it can be used rightaway rather than at the next invocation of the method.
Conclusion
To recap, we saw how JVM inlines code. Fusing the caller and the callee provides for improved performance since the overhead of method dispatch is avoided. We saw the flags which control inlining and we saw how JVM performs OSR.
Inlining is a very useful optimization because it forms the basis for other optimizations like escape analysis and dead code elimination.
My day job requires me to write code in Clojure. This means the code is eventually compiled to bytecode and run on the JVM. Intrigued by how the JVM does what it does, I decided to dig a little deeper and look at how it optimizes the code on the fly. In this series of posts I will be looking at JVM JIT (Just-In-Time) compiler.
Myriad ways to run a program
Before I go into how JVM JIT works, I want to take a quick look at how interpreted and compiled languages work. For this post, I’ll take a look at the working of Python (an interpreted language) and C (a compiled language).
Python
Python, by default, ships with CPython - the original Python interpreter that runs C code for every bytecode. There’s other implementations like IronPython or PyPy. IronPython turns Python into a fully compiled language running on top of Microsoft’s .NET Common Language Runtime (CLR) whereas PyPy turns Python into a JIT compiled language. For the sake of this post, however, I will look at CPython and how it works.
I’ll start with some code which will print the bytecodes for another Python file that is passed to it.
The loop starts on line #4. For every element in the list, we’re pushing print and n onto the stack, calling the function, popping the stack, and repeating the loop. For each of the bytecodes, there’s associated C code i.e. FOR_ITER, STORE_NAME, etc. have associated C code.
This is a very simple way to run a program and also a very inefficient one. We’re repeating the stack operations and jumps over and over again. There’s no scope for optimizations like loop unrolling.
C
In contrast to Python is C. All the C code is compiled to assembly ahead-of-time. Here’s a simple C program which will print “EVEN” if a number is even.
numbers.c
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#include<stdio.h>
intmain() { for(int i = 1; i < 10000; i += 2) { if( i % 2 == 0 ) { printf("EVEN!"); } else { printf(""); } } return0; }
Next, let’s compile this code.
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gcc -S numbers.c
This will generate numbers.s. The assembly is fairly long so I’ll just cover the relevant parts.
Lines #2 - #3 show that if we’ve reached the limit of 10k, we’ll jump to LBB0_7 and the program ends. If not, on line #5 we perform a signed division (idivl) and check if it is not zero. If it is not zero, we jump to LBB0_4 and print L_.str.1 which is just a whitespace.
We will always end up making this jump because we’ll never reach the condition where we have an even number. This is the problem with ahead-of-time compilation where you cannot speculate what the data is going to be and therefore you have to be ready to handle all the possibilities.
JVM JIT
JVM JIT combines the best of both the worlds. When you execute your program the first time, the bytecodes are interpreted. As the code continues to execute, JVM collects statistics about it and the more frequently used code (“hot” code) is compiled to assembly. In addition, there are optimizations like loop unrolling. Loop unrolling looks like this:
Unrolling a loop helps avoid jumps and thus makes execution faster.
Also, since JVM collects statistics about code, it can make optimizations on the fly. For example, in the case where an even number is never reached, JVM can generate assembly code that’ll only have the else part of the branch.
Conclusion
JVM does some fairly intersting optimizations under the hood. The aim of this series of posts is to cover as much of this as possible. We’ll start simple and build upon this as we go.
One of the nice things that you’ll come across in Clojure is transducer. In this post I’ll go over what transducers are, how you can use them, how you can make one, and what transducible contexts are.
What are transducers?
In simple terms, transducers are composable transformation pipelines. A transducer does not care about where the input comes from or where the output will go; it simply cares about the transformation of the data that that flows through the pipeline. Let’s look at an example:
Here xf (external function) is our transducer which will increment every number and then will keep only the even numbers. Calling sequence functions like map, filter, etc. with single arity returns a transducer which you can then compose. The transducer doesn’t know where it will be used - will it be used with a collection or with a channel? So, a transducer captures the essence of your transformation. sequence is responsible for providing the input to the transducer. This is the context in which the transducer will run.
Here’s how the same thing can be done using threading macro:
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(->> (range 10) (map inc) (filter even?))
The difference here is that the 2-arity version of map and filter will create intermediate collections while the 1-artity versions won’t. Transducers are much more efficient than threading together sequence functions.
Let’s look at the 1-arity version of map and see what makes a transducer.
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(defn map ([f] (fn [rf] (fn ([] (rf)) ([result] (rf result)) ([result input] (rf result (f input))) ([result input & inputs] (rf result (apply f input inputs)))))) ...)
When you call 1-arity version of map you get back a transducer which, as shown above, is a function. Functions like map, filter, etc. take a collection and return a collection. Transducers, on the otherhand, take one reducing function and return another. The function returned is expected to have three arities:
0-arity (init): This kickstarts the transformation pipeline. The only thing you do here is call the reducing function `rf`.
2-arity (step): This is where you'll perform the transformation. You get the result so far and the next input. In case of `map`, you call the reducung function `rf` by applying the function `f` to the input. How the value is going to be added to the result is the job of `rf`. If you don't want to add anything to the result, just return the result as-is. You may call `rf` once, multiple times, or not at all.
1-arity (end): This is called when the transducer is terminating. Here you must call `rf` exactly once and call the 1-arity version. This results in the production of the final value.
You can use a transducer in a context. There’s four contexts which come out of the box — into, transduce, sequence, and educe.
into
The simplest way to use a transducer is to pass it to into. This will add your transformed elements to an already-existing collection after applying the transducer. In this example, we’re simply adding a range into a vector.
eduction lets you capture applying a transducer to a collection. The value returned is an iterable application of the transducer on the collection items which can then be passed to, say, reduce.
As mentioned before, transducers run in transducible contexts. The ones that come as a part of clojure.core would suffice most real-world needs and you’ll rarely see yourself writing new ones. Let’s look at transduce.
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(defn transduce ([xform f coll] (transduce xform f (f) coll)) ([xform f init coll] (let [f (xform f) ret (if (instance? clojure.lang.IReduceInit coll) (.reduce ^clojure.lang.IReduceInit coll f init) (clojure.core.protocols/coll-reduce coll f init))] (f ret))))
transduce is just like reduce. The 3-arity version expects an initial value to be supplied by calling the 0-arity version of the supplied function. The 4-arity version is slightly more involved. IReduceInit is an interface implemented by collections to let them provide an initial value. It lets a collection reduce itself. If not, the call goes to coll-reduce which is a faster way to reduce a collection than using first/next recursion.
Stateful transducers
It’s possible for transducers to maintain reduction state.
Here’s a transducer which will multiply all the incoming numbers. We maintain state by using a Volatile. Whenever we get a new input we multiply it with the product and update the state of Volatile using vswap!. Let’s see this in action:
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(into [] (multiply-xf) [1 2 3]) => [1 2 6]
Early Termination
The way the above transducer is written, it’ll process all the inputs even if one of the inputs is zero. We know that once we encounter a zero, we can safely end the reduction process and return a zero. reduced lets you return a reduced value and end the reduction. Let’s make a minor change to the above transducer and add in early termination.
In the 2-arity function, we check if the new-product is zero. If it is, we know we have a reduced value. We end the reduction by returning the result we have so far. Let’s see this in action:
Transducers can be a very useful tool in your Clojure toolkit that let you process large collections, channels, etc. effectively by letting you make composable transformation pipelines that process one element at a time. They require a little getting used-to but once you’re past the learning curve, performance galore!
