Data Science Tip Day #8 - Dealing with Skewness for Error histograms after Linear Regression

In previous posts we have seen histogram based validation for errors. When a left / right skew based distribution is observed some transformation techniques to apply are

• Right Skewed - Apply Log
• Slightly Right - Square root
• Left Skewed - Exponential
• Slightly Left - Square root, Cube

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Data Science Tip Day#7 - Interaction Variables

This post is using interaction variables while performing linear regression

For illustration purpose lets construct some datasets with a three vectors (y,x,z)



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Data Science Tip Day#6 - Linear Regression, Polynomial Regression

We will be renaming the R tips to Data Science Tips moving on. Today we will look at Linear Regression, Polynomial & Variable Interactions. Today's stats class was very useful and interesting. Any topic of interest needs practise to master the fundamentals.

Linear Regression
Assume a Linear Equation y = mx+b

• Here m is slope, b is intercept value at x = 0
• Similarly, we use the equation to express relationship between dependent and independent variables
• In this equation y = mx+b, y is the dependent variable. x is the independent variable

For illustration purpose let's construct some datasets with a two vectors (y,x)

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R Day #5 Tip of the Day - Linear Regression

I have taken up Udemy Data Science Classes. Below notes from Linear Regression Classes

Linear Regression 

• Analyze relationship between two or multiple variables
• Goal is preparing equation (Outcome y, Predictors x)
• Estimate value of dependent and independent variables using relationship equations
• Used for Continuous variables that have some correlation between them
• Goodness of fit test to validate model

Linear Equations

• Explains relationship between two variables
•  X (Independent- Predictor), Y (Dependent)
•  Y = AX + B
•  A (Slope) = (Y/X)
•  B - Intercept (Value of Y when X =0)
•  Equation becomes predictor of Y

Fitting Line

• Sum of squares of vertical distances minimal
• Best Line = least residual
• Difference between model and actual values are called as residuals

Goodness of Fit

• R square measure 
• Sum of squares of distances (Sum of squares of vertical distances minimal)
• Uses residual values
• Higher R square value better the fit (close to 1 higher fit)
• Higher Correlation means better fit (R square will also be high) 

Multiple Regression

• Multiple predictors involved (X Values)
• More than one independent variable used to predict dependent variable
• Y = A1X1 + A2X2 + A3X3 +ApXp + B

11 Important Model Evaluation Techniques Everyone Should Know

Homoscedasticity -  all random variables in the sequence or vector have the same finite variance

heteroscedasticity -  variability of a variable is unequal across the range of values of a second variable that predicts it

Pearson's correlation coefficient (r) is a measure of the strength of the association between the two variables

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R Day #4 - Tip for the Day - Learning Resources

I came across this site NYU Stats R Site

Please bookmark it for a good walkthru and learning on R Core topics

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Good Reads - Data Science

Read Joaquin Quiñonero Candela's answer to In applied Machine Learning what is more important: data, infrastructure, or algorithms? on Quora

Read Joaquin Quiñonero Candela's answer to What do you look for when hiring someone for your team? on Quora

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R Day #3 - Tip for the Day

This is based on reading from notes from link 

Logistic Regression

• Applied when response is binary
• (0/1, yes/No etc..), Also known as dichotomous outcome variable 

Binomial probability model

• consists of (i) n independent trials where 
• (ii) each trial results in one of two possible outcomes (Yes/No, 1/0)
• (iii) the probability p of a success stays the same for each trial

Maximum likelihood - Find the value of the parameter(s) (in this case p) which makes the observed data most likely to have occurred

Poisson Regression
Applied for below situations

• The occurrences of the event of interest in non-overlapping “time” intervals are independent
• The probability two or more events in a small time interval is small, and
• The probability that an event occurs in a short interval of time is proportional to the length of the time interval
• Heteroscedasticity - means unequal error variances

Negative Binomial Model

• The Poisson model does not always provide a good fit to a count response. 
• An alternative model is the negative binomial distribution

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Day #2 - Multivariate Linear Regression - R

• More than one predictor involved in this case

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R Day #1 - Simple Linear Regression - Slope

UCLA Notes were very useful.

Linear Regression Model - Representing mean of response variable as function using slope and intercept parameters. Can be used for predictions. I have earlier used moving average algorithm for forecasting.

• Simple Linear Regression - Explanatory variable is 1 (Dependent variable is 1)
• Multivariate Linear Regression - Number of Explanatory variables more than 1

Good Summary of Data Quality Issues were summarized

• Data-entry errors
• Missing values
• Outliers
• Unusual (e.g. asymmetric) distributions
• Unexpected patterns

R Cookbook had good step by step examples to try out - link

Basics Maths Again

Slope - lines rate of change in the vertical direction

y = mx + b

• y = dependent variable as y depends on x
• x = independent variable
• m , b = characteristics of line
• b = y intercept where line crosses y axis

Ref - Link

Slope     = Rise / Run

              = Change in y / Change in X

Equation y = x

1 = 1

2 = 2

Slope = y/x = 2/2 = 1

Slope = y2-y1 / x2-x1

Slope > 1 tilt upwards towards y axis

Slope < 1 tilt downwards towards x axis

Ref - Link

Ref - Link

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Regression Basics

This post is on basics of Regression and Steps Involved. Linear Regression defines relationships between variables involved. We use it to identify relationships between variables.

Steps Involved

• Plot line between Independent Variable in X Axis, Dependent Variable Y Axis
• Identify if their positive or negative relationship (When X increases with respective to Y it is positive)
• Plot a line that minimizes errors between estimates / actuals

Y = B0 + B1X (B0, B1 Derived Mathematically)
where B0 is Y Intercept, B1 is Slope

R Squared 

R Squared Verification 

• How well regression line predicts actual values
• Take Actual values (compute mean of them). Distance between actual value of mean will sum up to zero
• Perfect fit R square equals 1

Standard Error of Estimates

• Compare estimated values vs Actual Values
• Distance between estimated and actual values

Correlation Coefficient

• Fit the line
• Remember slope +ve or -ve
• Scatter along Y and X Axis
• High Correlation means good fit

In next post we will look @ R Examples

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