#regression analysis

Waves by Country 

Thai BL is growing exponentially.

In 2020 they increase their output from 2019 by 1/3, which in turn was a 1/4 more than 2018, which was 1/4 more than 2017, and so forth. If they continue at the 2021 pace for the rest of this year they’ll have increased by 1/3 again. If you chart starting in 2014 it’s a darn near perfect exponential growth curve. 

Now THAT’S a wave. 

Korea looks like they’re ramping up production. 

Discounting microfilms, they only produced 4 BLs between 2006-2019. Then in 2020 they did 3 and now they’ve already done 4 this year and it’s only first quarter. 

But it is kinda out of nowhere. So maybe more an ice-bucket dumping than a wave? 

The Philippines & Vietnam are the newest to the scene, only entering the market in the last few years, increasing steady in output but too young to predict. 

The Philippines is growing its production much faster than Vietnam, but losing quality. They are tending towards longer series of lower production values. Gameboys and My Day might have been their best offerings. 

It’s a wave, but is it surf-able? 

Vietnam is clearly ramping up production, but also production values, and leaning towards shorter runs, like Korea. I think this is a smarter choice for a nascent film industry. IMHO Vietnamese BL has also been improving steadily. 

Not great yet, but consistently not awful, either. 

Baby wave. 

Not Waves 

Taiwan has not increased output at all. 

They spiked in 2017 and have been holding steady since then at 2-3 BLs a year (ranging in length). Don’t mistake the popularity of WBL for a wave, although if we are really lucky it might cause one. 

(*updated they just announced 2 more this year, so maybe we are getting lucky!) 

Japan is confusing. 

They had a consistent pattern since 2007 of 2-4 a year. Then last year they suddenly did 7, yet this year ... crickets ... just Absolute BL and that doesn’t really count. Also no announcements or rumors of manga adaptations. It’s very odd. 

Ripples, maybe?

The number of Chinese BLs has gone steadily down since 2016. 

*this is my shocked face*

I’d put in pretty graphs but I’m too darn lazy to battle excel. 

*final wave* 

This is based on data from 2020. I intend to do end of year trend wrap ups including this data going forward. 

Linear Regression

The relation between a dependent and an independent variable can be seen or predicted using linear regression models. When two or more independent variables are employed in a regression analysis, the model is referred to as a multiple regression model rather than a linear model.

Simple linear regression is a technique for predicting the value of one variable from the value of another. With linear regression, the relationship between the two variables is represented as a straight line.

In multiple regression, a dependent variable has a linear relationship with two or more independent variables. The dependent and independent variables may not follow a straight line if the relationship is non-linear.

When two or more variables are used to track a response, linear and non-linear regression are used. The non-linear regression is based on trial-and-error assumptions and is comparatively difficult to implement.

 Multiple Linear Regression

Multiple linear regression is a statistical analysis technique that uses two or more factors to predict a variable's outcome. It is also known as multiple regression and is an extension of linear regression. The dependent variable is the one that has to be predicted, and the factors that are used to forecast the value of the dependent variable are called independent or explanatory variables.

 Analysts can use multiple linear regression to determine the model's variance and the relative contribution of each independent variable. There are two types of multiple regression: linear and non-linear regression.

 Multiple Regression Equation

The equation for multiple regression with three predictor variables (x) predicting variable y is as follows:

 Y = B0 + B1 * X1 + B2 * X2 +B3 * X3

The beta coefficients are represented by the "B" values, which are the regression weights. They are the correlations between the predictor and the result variables.

Yi is predictable variable or dependent variable

B0 is the Y Intercept

B1 and B2 the regression coefficients represent the change in y as a function of a one-unit change in x1 and x2, respectively.

 Multiple Linear Regression Assumptions

        I.            The Independent Variables are not Much Correlated

Multicollinearity, which occurs when the independent variables are highly correlated, should not be present in the data. This will make finding the specific variable that contributes to the variance in the dependent variable difficult.

    II.            Relationship Between Dependent and Independent Variables

Each independent variable has a linear relationship with the dependent variable. A scatterplot is constructed and checked for linearity to check the linear relationships. If the scatterplot relationship is non-linear, the data is transferred using statistical software or a non-linear regression is done.

 III.            Observation Independence

The observations should be of each other, and the residual values should be independent. The Durbin Watson statistic works best for this.

 Multiple Linear Regression in R

Analyzing the linear relationship between Stock Index Prices and Unemployment rate in the Economy.

Multiple linear regression can be done in a variety of methods, although statistical software is the most popular. R, a free, powerful, and easily accessible piece of software, is one of the most widely used. We'll start by learning how to perform regression with R, then look at an example to make sure we understand everything.

 Steps to Apply Multiple Linear Regression in R

Step 1: Data Collection

The data needed for the forecast is gathered and collected. The purpose is to use two independents

·         Unemployment Rate

·         Interest Rate

to forecast the stock index price (the dependent variable) of a fictional economy.

Step 2: Capturing the Data in R

Data Capturing using the code in R and Importing Excel file from save folder.

Step 3: Checking Data Linearity with R

It is critical to ensure that the dependent and independent variables have a linear relationship. Scatter plots or R code can be used to do this. Scatter plots are a quick technique to check for linearity. We need to make sure that various assumptions are met before using linear regression models. Most importantly, you must ensure that the dependent variable and the independent variable(s) have a linear relationship.

In this we'll check the Linear Relationship exist between:

·         Stock Index Price (Dependent Variable) and Interest Rate (Independent Variable)

·         Stock Index Price (Dependent Variable) and Unemployment Rate (Independent Variable)

Below is the code that is used in R to plot the relations between the dependent variable that is Stock Index Price and Interest Rate.

From the Graph we can notice that there is Indeed a Linear relationship exist between the dependent variable Stock Index Price and Independent variable Interest Rate.

In can be specifically noted that When interest rates rise, the price of the stock index rises as well.

In the second scenario, we can plot the link between the Stock Index Price and the Unemployment Rate using the code below:

As you can see, the Stock Index Price and the Unemployment Rate have a linear relationship: when unemployment rates rise, the stock index price drops. we still have a linear correlation, although with a negative slope.

Step 4: Perform Multiple Linear Regression In R

To generate a set of coefficients, use code to conduct multiple linear regression in R. Template to perform Multiple Linear Regression in R is as below:

M1 <- lm (Dependent Variable ~ First Independent Variable + Second Independent Variable, Data= X)

Summary (M1)

Using the Template, the Code follows:

We will get the following summary if you execute the code in R:

The model's residuals ('Residuals'). The model fulfils heteroscedasticity assumptions if the residuals are roughly centered around zero and have similar spread on both sides (median -6.248, and min and max -158.2 and 118.8). The model's regression coefficients ('Coefficients').

To construct the multiple linear regression equation, utilize the coefficients from the summary as follows:

Stock Index Price = (Intercept) + (Interest Rate) X1* (Unemployment Rate) X2

Once you've entered the numbers from the summary: 

Stock Index Price = (1798.4) + (345.5) X1 * (-250.1) X2

Adjusted R-squared: Measures the model's fit, with a higher number indicating a better fit.

The p-value is Pr(>|t|):  Statistical significance is defined as a p-value of less than 0.05.

Step 5: Make Predictions

 To predict the Stock Index Price from the collected Data it is noted that,

X1 <= Interest Rate = 1.5

X2 <= Unemployment Rate = 5.8

And when this data in equated into the regression Equation we obtain:

Stock Index Price = (1798.4) + (345.5) * (1.5) + (-250.1) * (5.8)

Stock Index Price = 866.066666

 The Final Predicted data for the stock Index Price using Multiple Linear Regression is 866.07.

Conclusion

The stock market and our economy's relationship frequently converge and diverges from one another. The gross domestic product, unemployment, inflation, and a slew of other indicators all represent the state of the economy. These trends are expected to show the economy and markets moving in lockstep in the long run.

When the unemployment rate is high, the Monetary Policy lowers the interest rate, which causes stock market prices to rise.

Welcome back! If you have not read the earlier parts of this series, you can find them here:

Part 1 (Intro to Markov Chains)

Part 2 (Setting Up The Model)

This will be the last part of this series and will primarily involve ~actually~ using the model. Finally, amirite?

Things I’m Going To Touch On:

More About I(t)

Fitting I(t) To Data

Predicting Probabilities

For the Future

Here we go!

More About I(t)

First, let’s recall the equations we obtained in the last post. 

We can figure out the long term behavior of this I(t) by taking the limit as t approaches $\infty$. Let’s note that since $\alpha$, $\beta$ and $\gamma$ are less than 1, the exponential terms will go to zero when taking the limit. So, 

What does this mean in context?

This $\frac{\gamma}{\beta+\gamma}$ tells us the probability of someone being in the infected state in the long term (that is, after the initial “outbreak” of the meme). We can also verify this from our steady state distribution, but I’ll leave that as an exercise. 

Additionally, I think it’s interesting to note that this long term value does not depend whatsoever on $\alpha$. This means that the probability for moving from susceptible to infected has no effect on the long term probability of being in the infected state. 

Fitting I(t) to Data

I obtained my data from Google Trends. Here’s what I did:

First, I looked up a particular meme as a search term on Google Trends and got a spreadsheet of popularity values. Google plots the data so that the search term’s peak popularity is given a value of 100 and then the rest is scaled in relation to that value. Since I’m concerned with probabilities, I can’t use the values given by Google. So, I scaled the values down so that they were between 0 and 1.

Now that I have my adjusted values, I want to try and fit my I(t) equation to them. I did this by performing regression analysis using SageMath to find the $\alpha$, $\beta$, and $\gamma$ values that best fit I(t) to the data (Will provide my code upon request!). 

Below is the Google Trends graph for “dat boi”. I do want to note that each data point obtained from this graph is for a one week time span. 

And here’s a graph of the best fit I(t) (*drumroll*):

This is awesome! Look how closely I(t) matches the actual data. In this case, the $\alpha$, $\beta$, and $\gamma$ that best fit I(t) are:

$\alpha \approx$ 0.5354

$\beta \approx$ = 0.3308

$\gamma \approx$ = 0.03285

We can also see that $\frac{\gamma}{\beta+\gamma} = 0.09$.

I performed this analysis with 10 different memes/search terms. I’ll present the results in the table below, just listing the $\alpha$, $\beta$, and $\gamma$ values. 

Quick note: those terms with an asterisk had data points that were taken daily, those with a cross were monthly, and those with nothing were taken weekly.

I included “Flint, Michigan” because although it isn’t a meme, it does follow the same viral pattern that the memes do. This is due to the increased media coverage due to the Flint Water Crisis. I do want to stress that I am in no way trying to diminish the significance of the Flint Water Crisis or the struggle that those involved are facing, but rather I want to show that this model can be applied to more things than just memes. 

It’s neat that we can find these probabilities with memes that have already died down from their “outbreaks,” but is there a way we can try and predict the $\alpha$, $\beta$, and $\gamma$ values for up and coming memes? This brings us to the next section of this post. 

Predicting Probabilities

I want to try to predict the $\alpha$, $\beta$, and $\gamma$ values for a particular meme given an initial set of data points. Let’s assume that we have the data points up to and including the peak popularity. We’ll talk about why we need to assume this in a little bit. 

Let’s start by attempting to predict $\alpha$. Briefly recall that $\alpha$ is the probability of moving from the susceptible state to the infected state. I know that $\alpha$ has to do with how quickly the meme rises in popularity. Because of this, I approximate $\alpha$ using slopes in the following process:

Take the set of points up to and including the peak point.

Find all of the slopes between consecutive pairs of points.

Take the maximum over all these slopes. This is the estimated $\alpha$. 

This process is why we have to assume the peak is included in our data set. If we didn’t have the peak, we wouldn’t know if there was a slope that would be higher than the maximum we found over the points we have. 

Using this method over all 10 examples gives the following results for $\alpha$.

This seems promising at first, but with a closer look, the differences for some of the $\alpha$ values are pretty large. Is there a way to make it an even better approximation?

Spoiler alert: yes. 

Using SPSS, I performed various regressions with this estimated $\alpha$ as the independent variable and the actual $\alpha$ value obtained from the best fit I(t) as the dependent variable. See the table below:

Let’s quickly note that for every regression performed, the p-value is < 0.05, and so the relationship between $\alpha$ and our approximation of $\alpha$ is statistically significant (via F-Test of Overall Significance). We want to pick the equation that gives us the highest R-Square value. The quadratic and the cubic equations have the same R-Square value, so I’ll pick the quadratic just so I have one less term to deal with. 

Thus,

where $\alpha_{est}$ is that largest slope value. 

Using this new way to estimate $\alpha$, we find our new approximate $\alpha$ values by plugging in the maximum slope the equation above:

We can see that the differences between the estimated and the actual $\alpha$ values are much smaller than in the previous estimate! The bold values are those that have decreased significantly.

Now we have a pretty nice approximation for $\alpha$!

Can we do the same for $\beta$?

Let’s try! Since $\beta$ is the probability of moving from the infected state to the recovered state, we can’t really use the slopes as an approximation. Instead, we’ll perform a regression analysis with $\alpha$ as the independent variable and $\beta$ as the dependent variable to see if there’s a connection. 

In this case, we will pick the cubic equation. Thus,

Let’s look at how this approximation compares to the actual $\beta$ values obtained from this model. 

The average difference between the expected and the actual $\beta$ with this approximation is 0.06214. I believe it’s safe to say that provided we have an accurate $\alpha$ value, we can obtain an estimated $\beta$ value that is somewhat close to the actual $\beta$. 

Of course, there is probably a method that will better predict $\beta$, but for now, this is the best we’ve got.

What about $\gamma$?

Well, here’s where we have problems. Unfortunately, there is no significant relationship between $\alpha$ and $\gamma$ or $\beta$ and $\gamma$, as determined by performing various regressions. Because of this, we can’t use the same methods of approximation that we used for $\alpha$ and $\beta$. This brings me to the final section of this post. 

For The Future

We’ve done a lot with this already, but there is always more to be done! There are several things that my model does not account for:

If a meme has two peaks instead of just one (i.e. it died and then became popular again)

Extremely sharp increase followed by an extremely sharp decrease (say, in the matter of a couple days)

If $\beta + \gamma = \alpha$ or if $\beta + \gamma > 1$. This is because of issues with our I(t) equation. $\beta + \gamma  = \alpha$ gives a denominator of zero and $\beta + \gamma > 1$ makes $1-\beta-\gamma$ negative, which causes issues when raising it to t. 

Problem 3 is an issue because initially our Markov chain has no restrictions on what $\alpha$, $\beta$, or $\gamma$ could be, yet when I come up with the I(t) equation, we somehow have these problems. It’d be nice to have a way to circumvent this issue! 

Additionally, we still cannot predict what $\gamma$ will be, and until we can do so, we can’t create an equation given just an initial data set. Finding a way to $\gamma$ is the first thing on the list for future work. 

These are all things to consider more in the future, and I’d (of course) love to hear any ideas on how to combat these issues. 

Thanks for reading!

As always, if there are any questions/comments/concerns or even suggestions for another post, please feel free to send an ask!

Introduction

The Internal Assessment is an important component of IB Mathematics because it contributes significantly to the final grade. Many students seek support from an IB mathematics tutor to complete their IA successfully. Tutors help students choose suitable topics, apply mathematical concepts correctly, and structure reports professionally.

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Interesting and practical topic

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Scope for analysis and interpretation

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Accurate data collection is essential for meaningful mathematical analysis.

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Statistical analysis

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Probability calculations

Tutors guide students in organizing and interpreting data effectively.

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Mathematical modeling helps students apply mathematical concepts to real-world situations.

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An IB mathematics tutor explains how to use formulas, graphs, and equations to create meaningful mathematical models.

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Introduction and objectives

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This article was written as a practice exercise with reference to the information provided in the COURSERA course, specifically the Mars Crater Study.

=========================================

In this analysis, we used data on Martian craters to explore the impact of crater diameter, latitude, and longitude on their depth through multiple regression analysis. Our model shows that these variables significantly affect the variation in crater depth, with an overall R-squared of 0.345, indicating that the model explains about 34.5% of the variation in depth.

Relationship Between Explanatory Variables and Response Variable

1. Crater Diameter (DIAM_CIRCLE_IMAGE)

Beta Coefficient: 0.0151

p-value: 0.000

Relationship: Significantly positive, indicating that for each unit increase in crater diameter, the depth increases by an average of 0.0151 units.

2. Latitude (LATITUDE_CIRCLE_IMAGE)

Beta Coefficient: -6.507e-05

p-value: 0.000

Relationship: Significantly negative, suggesting that for each unit increase in latitude, the depth decreases by an average of 6.507e-05 units.

3. Longitude (LONGITUDE_CIRCLE_IMAGE)

Beta Coefficient: 3.384e-05

p-value: 0.000

Relationship: Significantly positive, indicating that for each unit increase in longitude, the depth increases by an average of 3.384e-05 units.

Hypothesis Testing

Our results support the initial hypothesis that crater diameter has a significant positive impact on crater depth. This indicates that larger craters tend to have greater depths. Additionally, geographical location (latitude and longitude) also has a significant effect on depth.

Confounding Factors Analysis

After incrementally adding latitude and longitude as control variables, we found no significant confounding effect on the relationship between diameter and depth. This means that while geographical location affects depth, it does not significantly alter the relationship between diameter and depth.

Regression Diagnostic Plots

a) Q-Q Plot

The Q-Q plot shows the normality of the residuals. The skewness of the residuals indicates some deviation from normal distribution, which may require further model adjustments.

b) Standardized Residuals Plot

The standardized residuals plot helps identify outliers and the model's fit. Some points deviate from the center line, indicating that the model may not fit well for certain observations.

c) Leverage Plot

The leverage plot shows influential observations and potential outliers. There are several points with high leverage values, which may have a significant impact on the model's fit.

Conclusion

Research Made Simple: Regression Analysis #FOANed #nursing #nursingresearch #regressionanalysis #research

Undertaking research is difficult, especially the statistical analysis part if your main role is clinial focused and research is just one aspect of your role. There is a difference between a research nurse and a nurse researcher. These Research Made Simple posts aim to help provide introductory or refresher resources for nurses undertaking clinical research projects. What is regression analysis…