How To Read Graphs For Beginners

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A Handbook For Beginner Crocheter-Learn To Read Crochet Patterns And Graphs: Learn To Read Crochet Patterns And Graphs

A lot of people choose to occupy new passions and uncover the toughest aspect is understanding the fundamentals. Standard crochet instructions can be tough to search for which is not really a craft without first understanding the fundamental crochet stitches you’re ready to simply teach-yourself. Often is couldn’t be easy to search for others who knows crochet. If you have A lot of people choose to occupy new passions and uncover the toughest aspect is understanding the fundamentals. Standard crochet instructions can be tough to search for which is not really a craft without first understanding the fundamental crochet stitches you’re ready to simply teach-yourself. Often is couldn’t be easy to search for others who knows crochet. If you have a grandmother who liked to crochet and will definitely show you you’ll typically have greater fortune. However, many who would like to analyze this modern craft simply don’t have one to show them the basics of crocheting.
Fortunately, the world wide web generally has information on any interest you’d prefer to start. Common crochet instructions aren’t any different. Should you spend a great deal of time seeking you will identify information with this attention also? While tons of people think crocheting is something just the elderly do they are not correct. Crocheting is now popular again with several young adults trying out the passion. This can be probably due to the numerous gorgeous things it’s possible to produce with a little yarn.
When there is a person just starting out in crocheting they’ll need to begin with understanding a pair of necessities. Where to start should be to discover exactly what there is along with one crochet a row.

The first step in standard crochet instructions could be to have a few basic crochet materials. You will need a crochet hook. You may usually look for a bundle of hooks offering various styles. You ought to pick as it is one of the most frequent, the one that carries a dimension G crochet hook. You can also need a yarn needle, tiny scissors, plus string.
. more

Beginners Guide to Regression Analysis and Plot Interpretations

Introduction

“The road to machine learning starts with Regression. Are you ready?”

If you are aspiring to become a data scientist, regression is the first algorithm you need to learn master. Not just to clear job interviews, but to solve real world problems. Till today, a lot of consultancy firms continue to use regression techniques at a larger scale to help their clients. No doubt, it’s one of the easiest algorithms to learn, but it requires persistent effort to get to the master level.

Running a regression model is a no-brainer. A simple model does the job. But optimizing this model for higher accuracy is a real challenge. Let’s say your model gives adjusted R² = 0.678; how will you improve it?

In this article, I’ll introduce you to crucial concepts of regression analysis with practice in R. Data is given for download below. Once you are finished reading this article, you’ll able to build, improve, and optimize regression models on your own. Regression has several types; however, in this article I’ll focus on linear and multiple regression.

Note: This article is best suited for people new to machine learning with requisite knowledge of statistics. You should have R installed in your laptops.

Table of Contents

  1. What is Regression? How does it work?
  2. What are the assumptions made in Regression?
  3. How do I know if these assumptions are violated in my data?
  4. How can I improve the accuracy of a Regression Model?
  5. How can I access the fit of a Regression Model?
  6. Practice Time – Solving a Regression Problem

What is Regression ? How does it work ?

Regression is a parametric technique used to predict continuous (dependent) variable given a set of independent variables. It is parametric in nature because it makes certain assumptions (discussed next) based on the data set. If the data set follows those assumptions, regression gives incredible results. Otherwise, it struggles to provide convincing accuracy. Don’t worry. There are several tricks (we’ll learn shortly) we can use to obtain convincing results.

Mathematically, regression uses a linear function to approximate (predict) the dependent variable given as:

where, Y – Dependent variable
X – Independent variable
βo – Intercept
β1 – Slope
∈ – Error

βo and β1 are known as coefficients. This is the equation of simple linear regression. It’s called ‘linear’ because there is just one independent variable ( X ) involved. In multiple regression, we have many independent variables ( Xs ). If you recall, the equation above is nothing but a line equation ( y = mx + c ) we studied in schools. Let’s understand what these parameters say:

Y – This is the variable we predict
X – This is the variable we use to make a prediction
βo – This is the intercept term. It is the prediction value you get when X = 0
β1 – This is the slope term. It explains the change in Y when X changes by 1 unit. ∈ – This represents the residual value, i.e. the difference between actual and predicted values.

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Error is an inevitable part of the prediction-making process. No matter how powerful the algorithm we choose, there will always remain an (∈) irreducible error which reminds us that the “future is uncertain.”

Yet, we humans have a unique ability to persevere, i.e. we know we can’t completely eliminate the (∈) error term, but we can still try to reduce it to the lowest. Right? To do this, regression uses a technique known as Ordinary Least Square (OLS).

So the next time when you say, I am using linear /multiple regression, you are actually referring to the OLS technique. Conceptually, OLS technique tries to reduce the sum of squared errors ∑[Actual(y) – Predicted(y’)]² by finding the best possible value of regression coefficients (β0, β1, etc).

Is OLS the only technique regression can use? No! There are other techniques such as Generalized Least Square, Percentage Least Square, Total Least Squares, Least absolute deviation, and many more. Then, why OLS? Let’s see.

  1. It uses squared error which has nice mathematical properties, thereby making it easier to differentiate and compute gradient descent.
  2. OLS is easy to analyze and computationally faster, i.e. it can be quickly applied to data sets having 1000s of features.
  3. Interpretation of OLS is much easier than other regression techniques.

Let’s understand OLS in detail using an example:

We are given a data set with 100 observations and 2 variables, namely Heightand Weight. We need to predict weight(y) given height(x1). The OLS equation can we written as:

When using R, Python or any computing language, you don’t need to know how these coefficients and errors are calculated. As a matter of fact, most people don’t care. But you must know, and that’s how you’ll get close to becoming a master.

The formula to calculate these coefficients is easy. Let’s say you are given the data, and you don’t have access to any statistical tool for computation. Can you still make any prediction? Yes!

The most intuitive and closest approximation of Y is mean of Y, i.e. even in the worst case scenario our predictive model should at least give higher accuracy than mean prediction. The formula to calculate coefficients goes like this:

Now you know ymean plays a crucial role in determining regression coefficients and furthermore accuracy. In OLS, the error estimates can be divided into three parts:

Residual Sum of Squares (RSS) – ∑[Actual(y) – Predicted(y)]²

Explained Sum of Squares (ESS) – ∑[Predicted(y) – Mean(ymean)]²

Total Sum of Squares (TSS) – ∑[Actual(y) – Mean(ymean)]²

The most important use of these error terms is used in the calculation of the Coefficient of Determination (R²).

R² metric tells us the amount of variance explained by the independent variables in the model. In the upcoming section, we’ll learn and see the importance of this coefficient and more metrics to compute the model’s accuracy.

What are the assumptions made in regression ?

As we discussed above, regression is a parametric technique, so it makes assumptions. Let’s look at the assumptions it makes:

  1. There exists a linear and additive relationship between dependent (DV) and independent variables (IV). By linear, it means that the change in DV by 1 unit change in IV is constant. By additive, it refers to the effect of X on Y is independent of other variables.
  2. There must be no correlation among independent variables. Presence of correlation in independent variables lead to Multicollinearity. If variables are correlated, it becomes extremely difficult for the model to determine the true effect of IVs on DV.
  3. The error terms must possess constant variance. Absence of constant variance leads to heteroskedestacity.
  4. The error terms must be uncorrelated i.e. error at ∈t must not indicate the at error at ∈t+1 . Presence of correlation in error terms is known as Autocorrelation. It drastically affects the regression coefficients and standard error values since they are based on the assumption of uncorrelated error terms.
  5. The dependent variable and the error terms must possess a normal distribution.

Presence of these assumptions make regression quite restrictive. By restrictive I meant, the performance of a regression model is conditioned on fulfillment of these assumptions.

How do I know these assumptions are violated in my data?

Once these assumptions get violated, regression makes biased, erratic predictions. I’m sure you are tempted to ask me, “How do I know these assumptions are getting violated?”

Of course, you can check performance metrics to estimate violation. But the real treasure is present in the diagnostic a.k.a residual plots. Let’s look at the important ones:

1. Residual vs. Fitted Values Plot

Ideally, this plot shouldn’t show any pattern. But if you see any shape (curve, U shape), it suggests non-linearity in the data set. In addition, if you see a funnel shape pattern, it suggests your data is suffering from heteroskedasticity, i.e. the error terms have non-constant variance.

2. Normality Q-Q Plot

As the name suggests, this plot is used to determine the normal distribution of errors. It uses standardized values of residuals. Ideally, this plot should show a straight line. If you find a curved, distorted line, then your residuals have a non-normal distribution (problematic situation).

3. Scale Location Plot

This plot is also useful to determine heteroskedasticity. Ideally, this plot shouldn’t show any pattern. Presence of a pattern determine heteroskedasticity. Don’t forget to corroborate the findings of this plot with the funnel shape in residual vs. fitted values.

If you are a non-graphical person, you can also perform quick tests / methods to check assumption violations:

  1. Durbin Watson Statistic (DW) – This test is used to check autocorrelation. Its value lies between 0 and 4. A DW=2 value shows no autocorrelation. However, a value between 0 =10 suggests high multicollinearity. Alternatively, you can also look at the tolerance (1/VIF) value to determine correlation in IVs. In addition, you can also create a correlation matrix to determine collinear variables.
  2. Breusch-Pagan / Cook Weisberg Test – This test is used to determine presence of heteroskedasticity. If you find p 0.05, we can remove that variable from model since at p> 0.05, we’ll always fail to reject null hypothesis.

How can you access the fit of regression model?

The ability to determine model fit is a tricky process. The metrics used to determine model fit can have different values based on the type of data. Hence, we need to be extremely careful while interpreting regression analysis. Following are some metrics you can use to evaluate your regression model:

  1. R Square (Coefficient of Determination) – As explained above, this metric explains the percentage of variance explained by covariates in the model. It ranges between 0 and 1. Usually, higher values are desirable but it rests on the data quality and domain. For example, if the data is noisy, you’d be happy to accept a model at low R² values. But it’s a good practice to consider adjusted R² than R² to determine model fit.
  2. Adjusted R²– The problem with R² is that it keeps on increasing as you increase the number of variables, regardless of the fact that the new variable is actually adding new information to the model. To overcome that, we use adjusted R² which doesn’t increase (stays same or decrease) unless the newly added variable is truly useful.
  3. F Statistics – It evaluates the overall significance of the model. It is the ratio of explained variance by the model by unexplained variance. It compares the full model with an intercept only (no predictors) model. Its value can range between zero and any arbitrary large number. Naturally, higher the F statistics, better the model.
  4. RMSE / MSE / MAE – Error metric is the crucial evaluation number we must check. Since all these are errors, lower the number, better the model. Let’s look at them one by one:
    • MSE – This is mean squared error. It tends to amplify the impact of outliers on the model’s accuracy. For example, suppose the actual y is 10 and predictive y is 30, the resultant MSE would be (30-10)² = 400.
    • MAE – This is mean absolute error. It is robust against the effect of outliers. Using the previous example, the resultant MAE would be (30-10) = 20
    • RMSE – This is root mean square error. It is interpreted as how far on an average, the residuals are from zero. It nullifies squared effect of MSE by square root and provides the result in original units as data. Here, the resultant RMSE would be √(30-10)² = 20. Don’t get baffled when you see the same value of MAE and RMSE. Usually, we calculate these numbers after summing overall values (actual – predicted) from the data.

Solving a Regression Problem

Let’s use our theoretical knowledge and create a model practically. As mentioned above, you should install R in your laptops. I’ve taken the data set from UCI Machine Learning repository. Originally, the data set is available in .txt file. To save you some time, I’ve converted it into .csv, and you can download it here.

Let’s load the data set and do initial data analysis:

This data has 5 independent variables and Sound_pressure_level as the dependent variable (to be predicted). In predictive modeling, we should always check missing values in data. If any data is missing, we can use methods like mean, median, and predictive modeling imputation to make up for missing data. This data set has no missing values. Good for us! Now, to avoid multicollinearity, let’s check correlation matrix. After you see carefully, you’d infer that Angle_of_Attack and Displacement show 75% correlation. It’s up to us if we should consider this correlation % as a damaging level. Usually, correlation above 80% (subjective) is considered higher. Therefore, we can forego this combination and won’t remove any variable.

In R, the base function lm is used for regression. We can run regression on this data by:

. tells lm to use all the independent variables. Let’s understand the regression output in detail:

  • Intercept – This is the βo value. It’s the prediction made by model when all the independent variables are set to zero.
  • Estimate – This represents regression coefficients for respective variables. It’s the value of slope. Let’s interpret it for Chord_Length . We can say, when Chord_Length is increased by 1 unit, holding other variables constant, Sound_pressure_level decreases by a value of -35.69.
  • Std. Error – This determines the level of variability associated with the estimates. Smaller the standard error of an estimate is, more accurate will be the predictions.
  • t value – t statistic is generally used to determine variable significance, i.e. if a variable is significantly adding information to the model. t value > 2 suggests the variable is significant. I used it as an optional value as the same information can be extracted from the p value.
  • p value – It’s the probability value of respective variables determining their significance in the model. p value linear.fit(x_train, y_train) after loading scikit learn library.

Did you find this tutorial helpful ? Let me know if there is anything you don’t understand while reading this article. I’d love to answer your questions.

R Tutorial

Obtaining R

R is available for Linux, MacOS, and Windows. Software can be downloaded from The Comprehensive R Archive Network (CRAN).

Startup

After R is downloaded and installed, simply find and launch R from your Applications folder.

Entering Commands

R is a command line driven program. The user enters commands at the prompt (> by default) and each command is executed one at a time.

The Workspace

The workspace is your current R working environment and includes any user-defined objects (vectors, matrices, data frames, lists, functions). At the end of an R session, the user can save an image of the current workspace that is automatically reloaded the next time R is started.

Graphic User Interfaces

Aside from the built in R console, RStudio is the most popular R code editor, and it interfaces with R for Windows, MacOS, and Linux platforms.

Operators in R

R’s binary and logical operators will look very familiar to programmers. Note that binary operators work on vectors and matrices as well as scalars.

Arithmetic Operators include:

Operator Description
+ addition
subtraction
* multiplication
/ division
^ or ** exponentiation

Logical Operators include:

Operator Description
> greater than
>= greater than or equal to
== exactly equal to
!= not equal to

Data Types

R has a wide variety of data types including scalars, vectors (numerical, character, logical), matrices, data frames, and lists.

Creating New Variables

Use the assignment operator # An example of computing the mean with variables

Functions

Almost everything in R is done through functions. A function is a piece of code written to carry out a specified task; it may accept arguments or parameters (or not) and it may return one or more values (or not!). In R, a function is defined with the construct:

function ( arglist )

The code in between the curly braces is the body of the function. Note that by using built-in functions, the only thing you need to worry about is how to effectively communicate the correct input arguments (arglist) and manage the return value/s (if any).

Importing Data

Importing data into R is fairly simple. R offers options to import many file types, from CSVs to databases.

For example, this is how to import a CSV into R.

# first row contains variable names, comma is separator
# assign the variable id to row names
# note the / instead of \ on mswindows systems

Descriptive Statistics

R provides a wide range of functions for obtaining summary statistics. One way to get descriptive statistics is to use the sapply( ) function with a specified summary statistic.

Below is how to get the mean with the sapply( ) function:

# get means for variables in data frame mydata
# excluding missing values
sapply(mydata, mean, na.rm=TRUE)

Possible functions used in sapply include mean, sd, var, min, max, median, range, and quantile.

Plotting in R

In R, graphs are typically created interactively. Here is an example:

# Creating a Graph
attach(mtcars)
plot(wt, mpg)
abline(lm(mpg

wt))
title(“Regression of MPG on Weight”)

The plot( ) function opens a graph window and plots weight vs. miles per gallon. The next line of code adds a regression line to this graph. The final line adds a title.

Packages

Packages are collections of R functions, data, and compiled code in a well-defined format. The directory where packages are stored is called the library. R comes with a standard set of packages. Others are available for download and installation. Once installed, they have to be loaded into the session to be used.

.libPaths() # get library location
library() # see all packages installed
search() # see packages currently loaded

Getting Help

Once R is installed, there is a comprehensive built-in help system. At the program’s command prompt you can use any of the following:

help.start() # general help
help(foo) # help about function foo
?foo # same thing
apropos(“foo“) # list all functions containing string foo
example(foo) # show an example of function foo

Going Further

If you prefer an online interactive environment to learn R, this free R tutorial by DataCamp is a great way to get started.

Graph Databases for Beginners: Graph Theory and Predictive Modeling

Connecting nodes in a graph database is fundamentally different than filling a table with data. If you’re new to graph databases: Read this.

Join the DZone community and get the full member experience.

Graphs are everywhere, from illustrating the connections between Game of Thrones characters to tracking the interactions betweens hundreds of thousands of servers in a public network.

Throughout this blog series, we’ve talked a lot about the practical details of working with graph databases. Now it’s time to discuss graph theory, with its a far more practical application to everyday life.

As a more developed field, graph theory helps us gain insight into new domains. Combined with the social sciences, there are many concepts that can be straightforwardly used to gain insight from graph data.

In last week’s post, we explained the lower-level traversal mechanisms of graph algorithms. If you haven’t read it yet, I would recommend doing so in order to best understand the higher-order analyses we are about to discuss. Now let’s take a look at some key concepts in social graph theory.

Triadic Closures

One of the most common properties of social graphs is that of triadic closures. This is the observation that if two nodes are connected via a path with a mutual third node, there is an increased likelihood of the two nodes becoming directly connected in the future.

In a social setting, a triadic closure would be a situation where two people with a mutual friend have a higher chance of meeting each other and becoming acquainted.

The triadic closure property is most likely to be upheld when a graph has a node A with a strong relationship to two other nodes, B and C. This then gives B and C a chance of a relationship, whether it be weak or strong. Although this is not a guarantee of a potential relationship, it serves as a credible predictive indicator.

Let’s take a look at this example.

Above is an organizational hierarchy where Alice manages both Bob and Charlie. This is rather strange, as it would be unlikely for Bob and Charlie to be unacquainted with one another while sharing the same manager.

As it is, there is a strong possibility they will end up working together due to the triadic closure property. This will create either a WORKS_WITH (strong) or PEER_OF (weak) relationship between the two of them, closing the triangle — hence the term triadic closure.

Structural Balance

However, another aspect to consider in the formation of stable triadic closures is the quality of the relationships involved in the graph. To illustrate the next concept, assume that the MANAGES relationship is somewhat negative while the PEER_OF and WORKS_WITH relationships are more positive.

Based off of the triadic closure property, we can assume that we can fill in the third relationship with any label, such as having everyone manage each other like in the first image below or the weird situation in the second image below.

However, you can see how uncomfortable those working situations would be in reality. In the second image, Charlie finds himself both the peer of a boss and a fellow worker. It would be difficult for Bob to figure out how to treat Charlie — as a fellow coworker or as the peer of his boss?

We have an innate preference for structural symmetry and rational layering. In graph theory, this is known as structural balance.

A structurally balanced triadic closure is made of relationships of all strong, positive sentiments (such as the first example below) or two relationships with negative sentiments and a single positive relationship (second example).

Balanced closures help with predictive modeling in graphs. The simple action of searching for chances to create balanced closures allows for the modification of the graph structure for accurate predictive analysis.

Local Bridges

We can go further and gain more valuable insight into the communications flow of our organizations by looking at local bridges. These refer to a tie between two nodes where the endpoints of the local bridge are not otherwise connected, nor do they share any common neighbors. You can think of local bridges as connections between two distinct clusters of the graph. In this case, one of the ties has to be weak.

For example, the concept of weak links is relevant in algorithms for job search. Studies have shown that the best sources of jobs come from looser acquaintances rather than close friends. This is because closer friends tend to share a similar worldview (are in the same graph component) but looser friends across a local bridge are in a different social network (and are in a different graph component).

In the image above, Davina and Alice are connected by a local bridge but belong to different graph components. If Davina were to look for a new job, she would be more likely to find a successful recommendation from Alice than from Frances.

This property of local bridges being weak links is something that is found throughout social graphs. As a result, we can make predictive analyses based on empirically derived local bridge and strong triadic closure notions.

The Final Takeaway

While graphs and our understanding of them are rooted in hundreds of years of study, we continue to find new ways to apply them to our personal, social and business lives. Technology today offers another method of understanding these principles in the form of the modern graph database.

As we have seen throughout the “Graph Databases for Beginners” blog series, we simply need to understand how to apply graph theory algorithms and analytical techniques in order to achieve our goals. Take a look back at the other posts in this series and you’ll gain the skills you need to tap into the power of graphs.

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