Factors in R
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Factors in R
Yearly Archives: 2017
Logistic Polynomial Regression in R
Polynomial regression is used when you want to develop a regression model that is not linear. It is common to use this method when performing traditional least squares regression. However, it is also possible to use polynomial regression when the dependent variable is categorical. As such, in this post, we will go through an example of logistic polynomial regression.
Specifically, we will use the “Clothing” dataset from the “Ecdat” package. We will divide the “tsales” dependent variable into two categories to run the analysis. Below is the code to get started.
library(Ecdat)data(Clothing)There is little preparation for this example. Below is the code for the model
fitglm<-glm(I(tsales>900000)~poly(inv2,4),data=Clothing,family = binomial)Here is what we did
1. We created an object called “fitglm” to save our results
2. We used the “glm” function to process the model
3. We used the “I” function. This told R to process the information inside the parentheses as is. As such, we did not have to make a new variable in which we split the “tsales” variable. Simply, if sales were greater than 900000 it was code 1 and 0 if less than this amount.
4. Next, we set the information for the independent variable. We used the “poly” function. Inside this function, we placed the “inv2” variable and the highest order polynomial we want to explore.
5. We set the data to “Clothing”
6. Lastly, we set the “family” argument to “binomial” which is needed for logistic regression
Below is the results
summary(fitglm)##
## Call:
## glm(formula = I(tsales > 9e+05) ~ poly(inv2, 4), family = binomial,
## data = Clothing)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5025 -0.8778 -0.8458 1.4534 1.5681
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 3.074 2.685 1.145 0.2523
## poly(inv2, 4)1 641.710 459.327 1.397 0.1624
## poly(inv2, 4)2 585.975 421.723 1.389 0.1647
## poly(inv2, 4)3 259.700 178.081 1.458 0.1448
## poly(inv2, 4)4 73.425 44.206 1.661 0.0967 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 521.57 on 399 degrees of freedom
## Residual deviance: 493.51 on 395 degrees of freedom
## AIC: 503.51
##
## Number of Fisher Scoring iterations: 13It appears that only the 4th-degree polynomial is significant and barely at that. We will now find the range of our independent variable “inv2” and make a grid from this information. Doing this will allow us to run our model using the full range of possible values for our independent variable.
inv2lims<-range(Clothing$inv2)
inv2.grid<-seq(from=inv2lims[1],to=inv2lims[2])The “inv2lims” object has two values. The lowest value in “inv2” and the highest value. These values serve as the highest and lowest values in our “inv2.grid” object. This means that we have values started at 350 and going to 400000 by 1 in a grid to be used as values for “inv2” in our prediction model. Below is our prediction model.
predsglm<-predict(fitglm,newdata=list(inv2=inv2.grid),se=T,type="response")Next, we need to calculate the probabilities that a given value of “inv2” predicts a store has “tsales” greater than 900000. The equation is as follows.
pfit<-exp(predsglm$fit)/(1+exp(predsglm$fit))Graphing this leads to interesting insights. Below is the code
plot(pfit)
You can see the curves in the line from the polynomial expression. As it appears. As inv2 increase the probability increase until the values fall between 125000 and 200000. This is interesting, to say the least.
We now need to plot the actual model. First, we need to calculate the confidence intervals. This is done with the code below.
se.bandsglm.logit<-cbind(predsglm$fit+2*predsglm$se.fit,predsglm$fit-2*predsglm$se.fit)
se.bandsglm<-exp(se.bandsglm.logit)/(1+exp(se.bandsglm.logit))The ’se.bandsglm” object contains the log odds of each example and the “se.bandsglm” has the probabilities. Now we plot the results
plot(Clothing$inv2,I(Clothing$tsales>900000),xlim=inv2lims,type='n')
points(jitter(Clothing$inv2),I((Clothing$tsales>900000)),cex=2,pch='|',col='darkgrey')
lines(inv2.grid,pfit,lwd=4)
matlines(inv2.grid,se.bandsglm,col="green",lty=6,lwd=6)
In the code above we did the following.
1. We plotted our dependent and independent variables. However, we set the argument “type” to n which means nothing. This was done so we can add the information step-by-step.
2. We added the points. This was done using the “points” function. The “jitter” function just helps to spread the information out. The other arguments (cex, pch, col) our for aesthetics and our optional.
3. We add our logistic polynomial line based on our independent variable grid and the “pfit” object which has all of the predicted probabilities.
4. Last, we add the confidence intervals using the “matlines” function. Which includes the grid object as well as the “se.bandsglm” information.
You can see that these results are similar to when we only graphed the “pfit” information. However, we also add the confidence intervals. You can see the same dip around 125000-200000 were there is also a larger confidence interval. if you look at the plot you can see that there are fewer data points in this range which may be what is making the intervals wider.
Conclusion
Logistic polynomial regression allows the regression line to have more curves to it if it is necessary. This is useful for fitting data that is non-linear in nature.
Teaching HandWriting to Young Children
Learning to write takes a lifetime. Any author will share with you how they have matured and grown over time in the craft of writing. However, there are some basic fundamentals that need to be mastered before the process of growing as a writer can begin.
This post will provide an approach to teaching writing to young children that includes the following steps.
- Learning to write the letters
- Learning to write sentences
- Learning to write paragraphs
Learning the Letters
The first step in this process is learning to write letters. The challenge is normally developing the fine motor skills for creating letters. If you have ever seen the writing of a 5-year-old you have some idea of what I am talking about.
It is difficult for children to actually write letters. Normally this is taught through having the students trace the letters on a piece of paper. This drill and kill style eventual works as the child masters the art of tracing. An analogy would be the use of training wheels on a bicycle.
Generally, straight lines are easier to write than curves. As such, easy letters to learn first are t, i, and l. Curves with straight lines are often easier than slanted lines so the next stage of letters might include b, d, f, h, j, p, r, u, and y. Lastly, slanted lines and full circle letters are the hardest in my experience. As such, a, c, e, g, k, m, n, o, s, v, w, x, and z are the last to learn.
Learning to Write Sentences
It is discouraging to have the child learn the entire alphabet before writing something. It’s better to learn a few letters and begin making sentences immediately. This heightens relevance and it is motivating to the child to be able to read their own writing. For now, the sentences do not really need to make sense. Just have them write using a handful of letters with support.
Simple three-word sentences are enough at this moment. Many worksheets will provide blanks lines with space at the top for drawing and coloring which provides a visual of the sentence.
It is critical to provide support for the development of the sentence. You have to help the child develop the thought that they want to put on paper. This is difficult for many children. You may also be taxed with proving spelling support. Although for now, I would not worry too much about spelling. Students need to create first and follow rules of creating later.
Writing Paragraphs
The typical child will probably not be able to write paragraphs until the 3rd or 4th grade at the earliest. paragraph writing takes an extensive amount of planning for a small child as they now must have a beginning, middle, and end or a main idea with supporting details.
At this stage, the best way to learn to write is to read a lot. This provides a structure and vocabulary on which the child can develop their own ideas in writing. In addition, rules of writing can be taught such as grammar and other components of language.
Conclusion
Writing can be an enjoyable experience if children are guided initially in learning this craft. Over time, a child can provide many insightful ideas and comments through developing the ability to communicate through the use of text.
Polynomial Regression in R
Polynomial regression is one of the easiest ways to fit a non-linear line to a data set. This is done through the use of higher order polynomials such as cubic, quadratic, etc to one or more predictor variables in a model.
Generally, polynomial regression is used for one predictor and one outcome variable. When there are several predictor variables it is more common to use generalized additive modeling/ In this post, we will use the “Clothing” dataset from the “Ecdat” package to predict total sales with the use of polynomial regression. Below is some initial code.
library(Ecdat)data(Clothing) str(Clothing)
## 'data.frame': 400 obs. of 13 variables:
## $ tsales : int 750000 1926395 1250000 694227 750000 400000 1300000 495340 1200000 495340 ...
## $ sales : num 4412 4281 4167 2670 15000 ...
## $ margin : num 41 39 40 40 44 41 39 28 41 37 ...
## $ nown : num 1 2 1 1 2 ...
## $ nfull : num 1 2 2 1 1.96 ...
## $ npart : num 1 3 2.22 1.28 1.28 ...
## $ naux : num 1.54 1.54 1.41 1.37 1.37 ...
## $ hoursw : int 76 192 114 100 104 72 161 80 158 87 ...
## $ hourspw: num 16.8 22.5 17.2 21.5 15.7 ...
## $ inv1 : num 17167 17167 292857 22207 22207 ...
## $ inv2 : num 27177 27177 71571 15000 10000 ...
## $ ssize : int 170 450 300 260 50 90 400 100 450 75 ...
## $ start : num 41 39 40 40 44 41 39 28 41 37 ...We are going to use the “inv2” variable as our predictor. This variable measures the investment in automation by a particular store. We will now run our polynomial regression model.
fit<-lm(tsales~poly(inv2,5),data = Clothing)
summary(fit)##
## Call:
## lm(formula = tsales ~ poly(inv2, 5), data = Clothing)
##
## Residuals:
## Min 1Q Median 3Q Max
## -946668 -336447 -96763 184927 3599267
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 833584 28489 29.259 < 2e-16 ***
## poly(inv2, 5)1 2391309 569789 4.197 3.35e-05 ***
## poly(inv2, 5)2 -665063 569789 -1.167 0.2438
## poly(inv2, 5)3 49793 569789 0.087 0.9304
## poly(inv2, 5)4 1279190 569789 2.245 0.0253 *
## poly(inv2, 5)5 -341189 569789 -0.599 0.5497
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 569800 on 394 degrees of freedom
## Multiple R-squared: 0.05828, Adjusted R-squared: 0.04633
## F-statistic: 4.876 on 5 and 394 DF, p-value: 0.0002428The code above should be mostly familiar. We use the “lm” function as normal for regression. However, we then used the “poly” function on the “inv2” variable. What this does is runs our model 5 times (5 is the number next to “inv2”). Each time a different polynomial is used from 1 (no polynomial) to 5 (5th order polynomial). The results indicate that the 4th-degree polynomial is significant.
We now will prepare a visual of the results but first, there are several things we need to prepare. First, we want to find what the range of our predictor variable “inv2” is and we will save this information in a grade. The code is below.
inv2lims<-range(Clothing$inv2)Second, we need to create a grid that has all the possible values of “inv2” from the lowest to the highest broken up in intervals of one. Below is the code.
inv2.grid<-seq(from=inv2lims[1],to=inv2lims[2])We now have a dataset with almost 400000 data points in the “inv2.grid” object through this approach. We will now use these values to predict “tsales.” We also want the standard errors so we se “se” to TRUE
preds<-predict(fit,newdata=list(inv2=inv2.grid),se=TRUE)We now need to find the confidence interval for our regression line. This is done by making a dataframe that takes the predicted fit adds or subtracts 2 and multiples this number by the standard error as shown below.
se.bands<-cbind(preds$fit+2*preds$se.fit,preds$fit-2*preds$se.fit)With these steps completed, we are ready to create our civilization.
To make our visual, we use the “plot” function on the predictor and outcome. Doing this gives us a plot without a regression line. We then use the “lines” function to add the polynomial regression line, however, this line is based on the “inv2.grid” object (40,000 observations) and our predictions. Lastly, we use the “matlines” function to add the confidence intervals we found and stored in the “se.bands” object.
plot(Clothing$inv2,Clothing$tsales)
lines(inv2.grid,preds$fit,lwd=4,col='blue')
matlines(inv2.grid,se.bands,lwd = 4,col = "yellow",lty=4)
Conclusion
You can clearly see the curvature of the line. Which helped to improve model fit. Now any of you can tell that we are fitting this line to mostly outliers. This is one reason we the standard error gets wider and wider it is because there are fewer and fewer observations on which to base it. However, for demonstration purposes, this is a clear example of the power of polynomial regression.
Search Text within Vector in R VIDEO
Searching text within an object in R
Partial Least Squares Regression in R
Partial least squares regression is a form of regression that involves the development of components of the original variables in a supervised way. What 
In this post, we will use predict “income” in the “Mroz” dataset using pls. Below is some initial code.
library(pls);library(Ecdat)data("Mroz")
str(Mroz)## 'data.frame': 753 obs. of 18 variables:
## $ work : Factor w/ 2 levels "yes","no": 2 2 2 2 2 2 2 2 2 2 ...
## $ hoursw : int 1610 1656 1980 456 1568 2032 1440 1020 1458 1600 ...
## $ child6 : int 1 0 1 0 1 0 0 0 0 0 ...
## $ child618 : int 0 2 3 3 2 0 2 0 2 2 ...
## $ agew : int 32 30 35 34 31 54 37 54 48 39 ...
## $ educw : int 12 12 12 12 14 12 16 12 12 12 ...
## $ hearnw : num 3.35 1.39 4.55 1.1 4.59 ...
## $ wagew : num 2.65 2.65 4.04 3.25 3.6 4.7 5.95 9.98 0 4.15 ...
## $ hoursh : int 2708 2310 3072 1920 2000 1040 2670 4120 1995 2100 ...
## $ ageh : int 34 30 40 53 32 57 37 53 52 43 ...
## $ educh : int 12 9 12 10 12 11 12 8 4 12 ...
## $ wageh : num 4.03 8.44 3.58 3.54 10 ...
## $ income : int 16310 21800 21040 7300 27300 19495 21152 18900 20405 20425 ...
## $ educwm : int 12 7 12 7 12 14 14 3 7 7 ...
## $ educwf : int 7 7 7 7 14 7 7 3 7 7 ...
## $ unemprate : num 5 11 5 5 9.5 7.5 5 5 3 5 ...
## $ city : Factor w/ 2 levels "no","yes": 1 2 1 1 2 2 1 1 1 1 ...
## $ experience: int 14 5 15 6 7 33 11 35 24 21 ...First, we must prepare our data by dividing it into a training and test set. We will do this by doing a 50/50 split of the data.
set.seed(777)
train<-sample(c(T,F),nrow(Mroz),rep=T) #50/50 train/test split
test<-(!train)In the code above we set the “set.seed function in order to assure reduplication. Then we created the “train” object and used the “sample” function to make a vector with ‘T’ and ‘F’ based on the number of rows in “Mroz”. Lastly, we created the “test”” object base don everything that is not in the “train” object as that is what the exclamation point is for.
Now we create our model using the “plsr” function from the “pls” package and we will examine the results using the “summary” function. We will also scale the data since this the scale affects the development of the components and use cross-validation. Below is the code.
set.seed(777)
pls.fit<-plsr(income~.,data=Mroz,subset=train,scale=T,validation="CV")
summary(pls.fit)## Data: X dimension: 392 17
## Y dimension: 392 1
## Fit method: kernelpls
## Number of components considered: 17
##
## VALIDATION: RMSEP
## Cross-validated using 10 random segments.
## (Intercept) 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps
## CV 11218 8121 6701 6127 5952 5886 5857
## adjCV 11218 8114 6683 6108 5941 5872 5842
## 7 comps 8 comps 9 comps 10 comps 11 comps 12 comps 13 comps
## CV 5853 5849 5854 5853 5853 5852 5852
## adjCV 5837 5833 5837 5836 5836 5835 5835
## 14 comps 15 comps 16 comps 17 comps
## CV 5852 5852 5852 5852
## adjCV 5835 5835 5835 5835
##
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps 7 comps
## X 17.04 26.64 37.18 49.16 59.63 64.63 69.13
## income 49.26 66.63 72.75 74.16 74.87 75.25 75.44
## 8 comps 9 comps 10 comps 11 comps 12 comps 13 comps 14 comps
## X 72.82 76.06 78.59 81.79 85.52 89.55 92.14
## income 75.49 75.51 75.51 75.52 75.52 75.52 75.52
## 15 comps 16 comps 17 comps
## X 94.88 97.62 100.00
## income 75.52 75.52 75.52The printout includes the root mean squared error for each of the components in the VALIDATION section as well as the variance explained in the TRAINING section. There are 17 components because there are 17 independent variables. You can see that after component 3 or 4 there is little improvement in the variance explained in the dependent variable. Below is the code for the plot of these results. It requires the use of the “validationplot” function with the “val.type” argument set to “MSEP” Below is the code
validationplot(pls.fit,val.type = "MSEP")
We will do the predictions with our model. We use the “predict” function, use our “Mroz” dataset but only those index in the “test” vector and set the components to three based on our previous plot. Below is the code.
set.seed(777)
pls.pred<-predict(pls.fit,Mroz[test,],ncomp=3)After this, we will calculate the mean squared error. This is done by subtracting the results of our predicted model from the dependent variable of the test set. We then square this information and calculate the mean. Below is the code
mean((pls.pred-Mroz$income[test])^2)## [1] 63386682As you know, this information is only useful when compared to something else. Therefore, we will run the data with a tradition least squares regression model and compare the results.
set.seed(777)
lm.fit<-lm(income~.,data=Mroz,subset=train)
lm.pred<-predict(lm.fit,Mroz[test,])
mean((lm.pred-Mroz$income[test])^2)## [1] 59432814The least squares model is slightly better then our partial least squares model but if we look at the model we see several variables that are not significant. We will remove these see what the results are
summary(lm.fit)##
## Call:
## lm(formula = income ~ ., data = Mroz, subset = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -20131 -2923 -1065 1670 36246
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.946e+04 3.224e+03 -6.036 3.81e-09 ***
## workno -4.823e+03 1.037e+03 -4.651 4.59e-06 ***
## hoursw 4.255e+00 5.517e-01 7.712 1.14e-13 ***
## child6 -6.313e+02 6.694e+02 -0.943 0.346258
## child618 4.847e+02 2.362e+02 2.052 0.040841 *
## agew 2.782e+02 8.124e+01 3.424 0.000686 ***
## educw 1.268e+02 1.889e+02 0.671 0.502513
## hearnw 6.401e+02 1.420e+02 4.507 8.79e-06 ***
## wagew 1.945e+02 1.818e+02 1.070 0.285187
## hoursh 6.030e+00 5.342e-01 11.288 < 2e-16 ***
## ageh -9.433e+01 7.720e+01 -1.222 0.222488
## educh 1.784e+02 1.369e+02 1.303 0.193437
## wageh 2.202e+03 8.714e+01 25.264 < 2e-16 ***
## educwm -4.394e+01 1.128e+02 -0.390 0.697024
## educwf 1.392e+02 1.053e+02 1.322 0.186873
## unemprate -1.657e+02 9.780e+01 -1.694 0.091055 .
## cityyes -3.475e+02 6.686e+02 -0.520 0.603496
## experience -1.229e+02 4.490e+01 -2.737 0.006488 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5668 on 374 degrees of freedom
## Multiple R-squared: 0.7552, Adjusted R-squared: 0.744
## F-statistic: 67.85 on 17 and 374 DF, p-value: < 2.2e-16set.seed(777)
lm.fit<-lm(income~work+hoursw+child618+agew+hearnw+hoursh+wageh+experience,data=Mroz,subset=train)
lm.pred<-predict(lm.fit,Mroz[test,])
mean((lm.pred-Mroz$income[test])^2)## [1] 57839715As you can see the error decreased even more which indicates that the least squares regression model is superior to the partial least squares model. In addition, the partial least squares model is much more difficult to explain because of the use of components. As such, the least squares model is the favored one.
Story Grammar Components
When people tell a story, whether orally or in a movie, there are certain characteristics that seem to appears in stories as determined by culture which children attempt to imitate when they tell a story. These traits are called story grammar components and include the following
- Setting statement
- Initiating event
- Internal response
- Internal plan
- Attempt
- Direct Consequence
- Reaction
This post will explore each of these characteristics of a story.
Setting Statement
The setting statement introduces the character of the story and often identifies 
Initiating Event
The initiating event is the catalyst to get the characters to do something. For example, in the “Three Little Pigs” the pigs need shelter. In other words, the initiating event introduces the problem that the characters need to overcome during the story.
Internal Response
The internal response is the characters reaction to the initiating event. The response can talk many forms such as emotional. For example, the pigs get excited when they see they need shelter. Generally, the internal response provides motivation to do something.
Internal Plan
The internal plan is what the characters will do to overcome the initiating event problem. For the pigs, the plan was to each build a house to prepare for the wolf.
Attempt
The attempt is the action that helps the characters to reach their goal. This is the step in which the internal plan is put into action. Therefore, for the pigs, it is the actual construction of their houses.
Direct Consequence
At this step, the story indicates if the attempt was successful or not. For the pigs, this is where things are complicated. Of the three pigs, two were unsuccessful and only one was successful. Success is determined by who is the protagonist and the antagonist. As such, if the wolf is the protagonist the success would be two and the failure one.
Reaction
The reaction is the character’s response to the direct consequence. For the two unsuccessful pigs, there was no reaction because they were eaten by the wolf. However, for the last pig, he was able to live safely after his home protected him.
Conclusion
Even small children will have several of these components in their storytelling. However, it is important to remember that the components are not required in a story nor do they have to follow the order specified here. Instead, this is a broad generalize way of how people communicate through storytelling.
Principal Component Regression in R
This post will explain and provide an example of principal component regression (PCR). 
Since PCR is based on principal component analysis it is an unsupervised method, which means the dependent variable has no influence on the development of the components. As such, there are times when the components that are developed may not be beneficial for explaining the dependent variable.
Our example will use the “Mroz” dataset from the “Ecdat” package. Our goal will be to predict “income” based on the variables in the dataset. Below is the initial code
library(pls);library(Ecdat)data(Mroz)
str(Mroz)## 'data.frame': 753 obs. of 18 variables:
## $ work : Factor w/ 2 levels "yes","no": 2 2 2 2 2 2 2 2 2 2 ...
## $ hoursw : int 1610 1656 1980 456 1568 2032 1440 1020 1458 1600 ...
## $ child6 : int 1 0 1 0 1 0 0 0 0 0 ...
## $ child618 : int 0 2 3 3 2 0 2 0 2 2 ...
## $ agew : int 32 30 35 34 31 54 37 54 48 39 ...
## $ educw : int 12 12 12 12 14 12 16 12 12 12 ...
## $ hearnw : num 3.35 1.39 4.55 1.1 4.59 ...
## $ wagew : num 2.65 2.65 4.04 3.25 3.6 4.7 5.95 9.98 0 4.15 ...
## $ hoursh : int 2708 2310 3072 1920 2000 1040 2670 4120 1995 2100 ...
## $ ageh : int 34 30 40 53 32 57 37 53 52 43 ...
## $ educh : int 12 9 12 10 12 11 12 8 4 12 ...
## $ wageh : num 4.03 8.44 3.58 3.54 10 ...
## $ income : int 16310 21800 21040 7300 27300 19495 21152 18900 20405 20425 ...
## $ educwm : int 12 7 12 7 12 14 14 3 7 7 ...
## $ educwf : int 7 7 7 7 14 7 7 3 7 7 ...
## $ unemprate : num 5 11 5 5 9.5 7.5 5 5 3 5 ...
## $ city : Factor w/ 2 levels "no","yes": 1 2 1 1 2 2 1 1 1 1 ...
## $ experience: int 14 5 15 6 7 33 11 35 24 21 ...Our first step is to divide our dataset into a train and test set. We will do a simple 50/50 split for this demonstration.
train<-sample(c(T,F),nrow(Mroz),rep=T) #50/50 train/test split
test<-(!train)In the code above we use the “sample” function to create a “train” index based on the number of rows in the “Mroz” dataset. Basically, R is making a vector that randomly assigns different rows in the “Mroz” dataset to be marked as True or False. Next, we use the “train” vector and we assign everything or every number that is not in the “train” vector to the test vector by using the exclamation mark.
We are now ready to develop our model. Below is the code
set.seed(777)
pcr.fit<-pcr(income~.,data=Mroz,subset=train,scale=T,validation="CV")To make our model we use the “pcr” function from the “pls” package. The “subset” argument tells r to use the “train” vector to select examples from the “Mroz” dataset. The “scale” argument makes sure everything is measured the same way. This is important when using a component analysis tool as variables with different scale have a different influence on the components. Lastly, the “validation” argument enables cross-validation. This will help us to determine the number of components to use for prediction. Below is the results of the model using the “summary” function.
summary(pcr.fit)## Data: X dimension: 381 17
## Y dimension: 381 1
## Fit method: svdpc
## Number of components considered: 17
##
## VALIDATION: RMSEP
## Cross-validated using 10 random segments.
## (Intercept) 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps
## CV 12102 11533 11017 9863 9884 9524 9563
## adjCV 12102 11534 11011 9855 9878 9502 9596
## 7 comps 8 comps 9 comps 10 comps 11 comps 12 comps 13 comps
## CV 9149 9133 8811 8527 7265 7234 7120
## adjCV 9126 9123 8798 8877 7199 7172 7100
## 14 comps 15 comps 16 comps 17 comps
## CV 7118 7141 6972 6992
## adjCV 7100 7123 6951 6969
##
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps 7 comps
## X 21.359 38.71 51.99 59.67 65.66 71.20 76.28
## income 9.927 19.50 35.41 35.63 41.28 41.28 46.75
## 8 comps 9 comps 10 comps 11 comps 12 comps 13 comps 14 comps
## X 80.70 84.39 87.32 90.15 92.65 95.02 96.95
## income 47.08 50.98 51.73 68.17 68.29 68.31 68.34
## 15 comps 16 comps 17 comps
## X 98.47 99.38 100.00
## income 68.48 70.29 70.39There is a lot of information here.The VALIDATION: RMSEP section gives you the root mean squared error of the model broken down by component. The TRAINING section is similar the printout of any PCA but it shows the amount of cumulative variance of the components, as well as the variance, explained for the dependent variable “income.” In this model, we are able to explain up to 70% of the variance if we use all 17 components.
We can graph the MSE using the “validationplot” function with the argument “val.type” set to “MSEP”. The code is below.
validationplot(pcr.fit,val.type = "MSEP")
How many components to pick is subjective, however, there is almost no improvement beyond 13 so we will use 13 components in our prediction model and we will calculate the means squared error.
set.seed(777)
pcr.pred<-predict(pcr.fit,Mroz[test,],ncomp=13)
mean((pcr.pred-Mroz$income[test])^2)## [1] 48958982MSE is what you would use to compare this model to other models that you developed. Below is the performance of a least squares regression model
set.seed(777)
lm.fit<-lm(income~.,data=Mroz,subset=train)
lm.pred<-predict(lm.fit,Mroz[test,])
mean((lm.pred-Mroz$income[test])^2)## [1] 47794472If you compare the MSE the least squares model performs slightly better than the PCR one. However, there are a lot of non-significant features in the model as shown below.
summary(lm.fit)##
## Call:
## lm(formula = income ~ ., data = Mroz, subset = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27646 -3337 -1387 1860 48371
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.215e+04 3.987e+03 -5.556 5.35e-08 ***
## workno -3.828e+03 1.316e+03 -2.909 0.00385 **
## hoursw 3.955e+00 7.085e-01 5.582 4.65e-08 ***
## child6 5.370e+02 8.241e+02 0.652 0.51512
## child618 4.250e+02 2.850e+02 1.491 0.13673
## agew 1.962e+02 9.849e+01 1.992 0.04709 *
## educw 1.097e+02 2.276e+02 0.482 0.63013
## hearnw 9.835e+02 2.303e+02 4.270 2.50e-05 ***
## wagew 2.292e+02 2.423e+02 0.946 0.34484
## hoursh 6.386e+00 6.144e-01 10.394 < 2e-16 ***
## ageh -1.284e+01 9.762e+01 -0.132 0.89542
## educh 1.460e+02 1.592e+02 0.917 0.35982
## wageh 2.083e+03 9.930e+01 20.978 < 2e-16 ***
## educwm 1.354e+02 1.335e+02 1.014 0.31115
## educwf 1.653e+02 1.257e+02 1.315 0.18920
## unemprate -1.213e+02 1.148e+02 -1.057 0.29140
## cityyes -2.064e+02 7.905e+02 -0.261 0.79421
## experience -1.165e+02 5.393e+01 -2.159 0.03147 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6729 on 363 degrees of freedom
## Multiple R-squared: 0.7039, Adjusted R-squared: 0.69
## F-statistic: 50.76 on 17 and 363 DF, p-value: < 2.2e-16Removing these and the MSE is almost the same for the PCR and least square models
set.seed(777)
lm.fit2<-lm(income~work+hoursw+hearnw+hoursh+wageh,data=Mroz,subset=train)
lm.pred2<-predict(lm.fit2,Mroz[test,])
mean((lm.pred2-Mroz$income[test])^2)## [1] 47968996Conclusion
Since the least squares model is simpler it is probably the superior model. PCR is strongest when there are a lot of variables involve and if there are issues with multicollinearity.
Combining and Splitting Strings in R VIDEO
Combining and Splitting Strings in R
Accommodation Theory
Accommodation theory attempts to explain how people adjust the way they talk depending on who the audience is. Generally, there are two ways in which a person can adjust their speech. The two ways are convergence and divergence. In this post, we will look at these two ways of accommodating.
Speech Convergence
Converging is when you change the way you talk about sound more like the person you are talking to. This is seen as polite in many cultures and signals that you are accepting the person who is talking.
There are many different ways in which convergence can take place. The speaker may begin to use similar vocabulary. Another way is to imitate the pronunciation of the person you are talking to. Another common way os to translate technical jargon into simpler English.
Speech Divergence
Speech divergence is often seen as the opposite of speech convergence. Speech divergence is deliberately selecting a style of language different from the speaker. This often communicates dissatisfaction with the person you are speaking with. For example, most teenagers deliberately speak differently from their parents. This serves a role in their identifying with peers and to distances from their parents.
However, a slight divergence is expected of non-native speakers. Many people enjoy the accents of athletes and actresses. To have perfect control of two languages is at times seen negatively in some parts of the world.
A famous example of speech divergence is the speaking of former Federal Reserve Chairman Alan Greenspan and his ‘Fedspeak.’ Fedspeak was used whenever Greenspan appears before Congress or made announcements about changing the Federal Reserve interest rate. The goal of this form of communication was to sound as divergent and incoherent as possible below is an example.
The members of the Board of Governors and the Reserve Bank presidents foresee an implicit strengthening of activity after the current rebalancing is over, although the central tendency of their individual forecasts for real GDP still shows a substantial slowdown, on balance, for the year as a whole.
Make little sense unless you have an MBA in finance. It sounds like he sees no change in the growth of the economy
The reason behind this mysterious form of communication was that people place a strong emphasis on whatever the Federal Reserve and Alan Greenspan said. This led to swings in the stock market. To prevent this, Greenspan diverged his language to make it as confusing as possible to avoid massive changes in the stock market
Conclusion
When communicating we can choose to adapt ourselves are deliberately diverge. Which choice we choose depends a great deal on the context that we find ourselves end
Ways Language Change Spreads
All languages change if there is any doubt just pick up a book that is over 100 years old and regardless of the language it will at a minimum sound slightly different from current language use or radically different.
In this post, we will look at three common ways in which language change is spread. These three ways are from…
- From one group to another
- From one style to another
- Lexical diffusion
Changes from Group to Group
This view of language change is that changes in a language move from one group. A group can be any sort of social or work circle. Examples can include family, colleagues, church affiliation, etc.
The change of language in a group is often facilitated by “gatekeepers.” Gatekeepers are people who are members of different groups. Most people are members of many different groups at the same time.
What happens is that a person picks up language in one group and shares this style of communication in another. An example would be a child learning slang at school and using it at home. Naturally, language change moves at different speeds in different groups depending on the acceptability of the change.
Changes from Style to Style
A style is a way of communicating. It simple terms a person’s style can be formal and informal with varying shades of grey in-between. These two extremes can also be viewed as prestigious vs not prestigious.
Normally, formal/prestigious styles of language move down into informal styles of language. For example, a movie star or some other celebrity speaks a certain way and thus style is transferred downward among those who are not so famous.
There are times where informal and un-prestigious language change spreads upward. Normally, this is much slower than change moving downward. In addition, it also involves words and styles that are so old that what really happened was the young people who used these words become part of the “establishment” in their middle age and continued to use the style. For example, the word “cool” used to be slang but is commonly used among some of the most elite leaders of the world now. Therefore, it wasn’t the language that changed as much as the people who used it with the passing of one generation to another.
Lexical Diffusion
Lexical diffusion is the change of how a word is pronounced. This is often an exceedingly slow process and can take centuries. The English language is full of words that have strange pronunciations when considering the spelling. This is due to English being thoroughly influenced by other languages such as French and Latin.
Conclusion
These three theories are just some of the ways langauge change can spread. In addition, it may not be practical to thinnk of them each happening independently from the others. Rather, these three theories can often be seen as working at the same time to slowly change a language over time.
Example of Best Subset Regression in R
This post will provide an example of best subset regression. This is a topic 
library(leaps);library(Ecdat)data(HI)
str(HI)## 'data.frame': 22272 obs. of 13 variables:
## $ whrswk : int 0 50 40 40 0 40 40 25 45 30 ...
## $ hhi : Factor w/ 2 levels "no","yes": 1 1 2 1 2 2 2 1 1 1 ...
## $ whi : Factor w/ 2 levels "no","yes": 1 2 1 2 1 2 1 1 2 1 ...
## $ hhi2 : Factor w/ 2 levels "no","yes": 1 1 2 2 2 2 2 1 1 2 ...
## $ education : Ord.factor w/ 6 levels "<9years"<"9-11years"<..: 4 4 3 4 2 3 5 3 5 4 ...
## $ race : Factor w/ 3 levels "white","black",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ hispanic : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...
## $ experience: num 13 24 43 17 44.5 32 14 1 4 7 ...
## $ kidslt6 : int 2 0 0 0 0 0 0 1 0 1 ...
## $ kids618 : int 1 1 0 1 0 0 0 0 0 0 ...
## $ husby : num 12 1.2 31.3 9 0 ...
## $ region : Factor w/ 4 levels "other","northcentral",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ wght : int 214986 210119 219955 210317 219955 208148 213615 181960 214874 214874 ...To develop a model we use the “regsubset” function from the “leap” package. Most of the coding is the same as linear regression. The only difference is the “nvmax” argument which is set to 13. The default setting for “nvmax” is 8. This is good if you only have 8 variables. However, the results from the “str” function indicate that we have 13 functions. Therefore, we need to set the “nvmax” argument to 13 instead of the default value of 8 in order to be sure to include all variables. Below is the code
regfit.full<-regsubsets(whrswk~.,HI, nvmax = 13)We can look at the results with the “summary” function. For space reasons, the code is shown but the results will not be shown here.
summary(regfit.full)If you run the code above in your computer you will 13 columns that are named after the variables created. A star in a column means that that variable is included in the model. To the left is the numbers 1-13 which. One means one variable in the model two means two variables in the model etc.
Our next step is to determine which of these models is the best. First, we need to decide what our criteria for inclusion will be. Below is a list of available fit indices.
names(summary(regfit.full))## [1] "which" "rsq" "rss" "adjr2" "cp" "bic" "outmat" "obj"For our purposes, we will use “rsq” (r-square) and “bic” “Bayesian Information Criteria.” In the code below we are going to save the values for these two fit indices in their own objects.
rsq<-summary(regfit.full)$rsq
bic<-summary(regfit.full)$bicNow let’s plot them
plot(rsq,type='l',main="R-Square",xlab="Number of Variables")
plot(bic,type='l',main="BIC",xlab="Number of Variables")
You can see that for r-square the values increase and for BIC the values decrease. We will now make both of these plots again but we will have r tell the optimal number of variables when considering each model index. For we use the “which” function to determine the max r-square and the minimum BIC
which.max(rsq)## [1] 13which.min(bic)## [1] 12The model with the best r-square is the one with 13 variables. This makes sense as r-square always improves as you add variables. Since this is a demonstration we will not correct for this. For BIC the lowest values was for 12 variables. We will now plot this information and highlight the best model in the plot using the “points” function, which allows you to emphasis one point in a graph
plot(rsq,type='l',main="R-Square with Best Model Highlighted",xlab="Number of Variables")
points(13,(rsq[13]),col="blue",cex=7,pch=20)
plot(bic,type='l',main="BIC with Best Model Highlighted",xlab="Number of Variables")
points(12,(bic[12]),col="blue",cex=7,pch=20)
Since BIC calls for only 12 variables it is simpler than the r-square recommendation of 13. Therefore, we will fit our final model using the BIC recommendation of 12. Below is the code.
coef(regfit.full,12)## (Intercept) hhiyes whiyes
## 30.31321796 1.16940604 18.25380263
## education.L education^4 education^5
## 6.63847641 1.54324869 -0.77783663
## raceblack hispanicyes experience
## 3.06580207 -1.33731802 -0.41883100
## kidslt6 kids618 husby
## -6.02251640 -0.82955827 -0.02129349
## regionnorthcentral
## 0.94042820So here is our final model. This is what we would use for our test set.
Conclusion
Best subset regression provides the researcher with insights into every possible model as well as clues as to which model is at least statistically superior. This knowledge can be used for developing models for data science applications.
Characters Vectors in R VIDEO
Understanding character vectors in R
Koines
There are many different ways that languages or a language can interact with each other. One way is how different dialects of a language interact. A koine is 
In this post, we will discuss the follow
- Koine vs pidgins and creoles
- Development of koines
Koine vs Pidgins and Creoles
Koine is a lesser known term to the general public in comparison to pidgin and creole. The word “koine” comes from the same Greek word that means “common.” In other words, koine is what two or more dialects have in common. For those familiar with biblical languages you would n=know that the term “Koine” Greek is the language of the New Testament, which means the New Testament was written in common Greek.
As mentioned in the introduction a koine is the interaction of two or more dialects to create new common dialect. In contrast, a pidgin is the interaction of two languages to make a new third language. A pidgin can eventually mature into a Creole which is a pidgin that is spoken as the native language of people. There seems to be no term for a koine that is spoken natively.
Developing a Koine
The process of developing a Koine is known as koineization. There are both linguistic and social factors that contribute to the development of a koine. For linguistic factors, they include leveling and simplification and for the social factor, it involves accommodation, prestige.
Levelling is the elimination from the koine distinct sounds from the colliding dialects. An example of this is the loss of the post-vocalic [r] in parts of England.
Simplification is the process of the simplest characteristics of the dialects being included in the new koine. The dialect with the fewer rules and exceptions almost always emerges as having more influence on the koine.
The social factor of accommodations means that people will copy something they believe is prestigious or “cool.” If a certain dialect is considered unacceptable it will not lead to a koine because of people’s dislike of it. As such accommodation and prestige are interrelated concepts.
Conclusion
Koine and koineization are two words that help to explain where dialects come from. As such, for those who have an interest in linguistics, these are terms to be familiar with
High Dimensionality Regression
There are times when least squares regression is not able to provide accurate predictions or explanation in an object. One example in which least scares
This post will explain the consequences of what happens when high dimensions is a problem and also how to address the problem.
Inaccurate measurements
One problem with high dimensions in regression is that the results for the various metrics are overfitted to the data. Below is an example using the “attitude” dataset. There are 2 variables and 3 examples for developing a model. This is not strictly high dimensions but it is an example of a small sample size.
data("attitude")
reg1 <- lm(complaints[1:3]~rating[1:3],data=attitude[1:3])
summary(reg1)##
## Call:
## lm(formula = complaints[1:3] ~ rating[1:3], data = attitude[1:3])
##
## Residuals:
## 1 2 3
## 0.1026 -0.3590 0.2564
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 21.95513 1.33598 16.43 0.0387 *
## rating[1:3] 0.67308 0.02221 30.31 0.0210 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4529 on 1 degrees of freedom
## Multiple R-squared: 0.9989, Adjusted R-squared: 0.9978
## F-statistic: 918.7 on 1 and 1 DF, p-value: 0.021With only 3 data points the fit is perfect. You can also examine the mean squared error of the model. Below is a function for this followed by the results
mse <- function(sm){
mean(sm$residuals^2)}
mse(reg1)## [1] 0.06837607Almost no error. Lastly, let’s look at a visual of the model
with(attitude[1:3],plot(complaints[1:3]~ rating[1:3]))
title(main = "Sample Size 3")
abline(lm(complaints[1:3]~rating[1:3],data = attitude))
You can see that the regression line goes almost perfectly through each data point. If we tried to use this model on the test set in a real data science problem there would be a huge amount of bias. Now we will rerun the analysis this time with the full sample.
reg2<- lm(complaints~rating,data=attitude)
summary(reg2)##
## Call:
## lm(formula = complaints ~ rating, data = attitude)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.3880 -6.4553 -0.2997 6.1462 13.3603
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.2445 7.6706 1.075 0.292
## rating 0.9029 0.1167 7.737 1.99e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.65 on 28 degrees of freedom
## Multiple R-squared: 0.6813, Adjusted R-squared: 0.6699
## F-statistic: 59.86 on 1 and 28 DF, p-value: 1.988e-08You can clearly see a huge reduction in the r-square from .99 to .68. Next, is the mean-square error
mse(reg2)## [1] 54.61425The error has increased a great deal. Lastly, we fit the regression line
with(attitude,plot(complaints~ rating))
title(main = "Full Sample Size")
abline(lm(complaints~rating,data = attitude))
Naturally, the second model is more likely to perform better with a test set. The problem is that least squares regression is too flexible when the number of features is greater than or equal to the number of examples in a dataset.
What to Do?
If least squares regression must be used. One solution to overcoming high dimensionality is to use some form of regularization regression such as ridge, lasso, or elastic net. Any of these regularization approaches will help to reduce the number of variables or dimensions in the final model through the use of shrinkage.
However, keep in mind that no matter what you do as the number of dimensions increases so does the r-square even if the variable is useless. This is known as the curse of dimensionality. Again, regularization can help with this.
Remember that with a large number of dimensions there are normally several equally acceptable models. To determine which is most useful depends on understanding the problem and context of the study.
Conclusion
With the ability to collect huge amounts of data has led to the growing problem of high dimensionality. One there are more features than examples it can lead to statistical errors. However, regularization is one tool for dealing with this problem.
Regression with Shrinkage Methods
One problem with least squares regression is determining what variables to keep in a model. One solution to this problem is the use of shrinkage methods.
In this post, we will look at two common forms of regularization and these are.
Ridge
Ridge regression includes a tuning parameter called lambda that can be used to reduce to weak coefficients almost to zero. This shrinkage penalty helps with the bias-variance trade-off. Lambda can be set to any value from 0 to infinity. A lambda set to 0 is the same as least square regression while a lambda set to infinity will produce a null model. The technical term for lambda when ridge is used is the “l2 norm”
Finding the right value of lambda is the primary goal when using this algorithm,. Finding it involves running models with several values of lambda and seeing which returns the best results on predetermined metrics.
The primary problem with ridge regression is that it does not actually remove any variables from the model. As such, the prediction might be excellent but explanatory power is not improve if there are a large number of variables.
Lasso
Lasso regression has the same characteristics as Ridge with one exception. The one exception is the Lasso algorithm can actually remove variables by setting them to zero. This means that lasso regression models are usually superior in terms of the ability to interpret and explain them. The technical term for lambda when lasso is used is the “l1 norm.”
It is not clear when lasso or ridge is superior. Normally, if the goal is explanatory lasso is often stronger. However, if the goal is prediction, ridge may be an improvement but not always.
Conclusion
Shrinkage methods are not limited to regression. Many other forms of analysis can employ shrinkage such as artificial neural networks. Most machine learning models can accommodate shrinkage.
Generally, ridge and lasso regression is employed when you have a huge number of predictors as well as a larger dataset. The primary goal is the simplification of an overly complex model. Therefore, the shrinkage methods mentioned here are additional ways to use statistical models in regression.
Logical Vectors in R VIDEO
Logical vectors in r studio
Subset Selection Regression
There are many different ways in which the variables of a regression model can be selected. In this post, we look at several common ways in which to select
- Best subset regression
- Stepwise selection
Best Subset Regression
Best subset regression fits a regression model for every possible combination of variables. The “best” model can be selected based on such criteria as the adjusted r-square, BIC (Bayesian Information Criteria), etc.
The primary drawback to best subset regression is that it becomes impossible to compute the results when you have a large number of variables. Generally, when the number of variables exceeds 40 best subset regression becomes too difficult to calculate.
Stepwise Selection
Stepwise selection involves adding or taking away one variable at a time from a regression model. There are two forms of stepwise selection and they are forward and backward selection.
In forward selection, the computer starts with a null model ( a model that calculates the mean) and adds one variable at a time to the model. The variable chosen is the one the provides the best improvement to the model fit. This process reduces greatly the number of models that need to be fitted in comparison to best subset regression.
Backward selection starts the full model and removes one variable at a time based on which variable improves the model fit the most. The main problem with either forward or backward selection is that the best model may not always be selected in this process. In addition, backward selection must have a sample size that is larger than the number of variables.
Deciding Which to Choose
Best subset regression is perhaps most appropriate when you have a small number of variables to develop a model with, such as less than 40. When the number of variables grows forward or backward selection are appropriate. If the sample size is small forward selection may be a better choice. However, if the sample size is large as in the number of examples is greater than the number of variables it is now possible to use backward selection.
Conclusion
The examples here are some of the most basic ways to develop a regression model. However, these are not the only ways in which this can be done. What these examples provide is an introduction to regression model development. In addition, these models provide some sort of criteria for the addition or removal of a variable based on statistics rather than intuition.
Leave One Out Cross Validation in R
Leave one out cross validation. (LOOCV) is a variation of the validation approach in that instead of splitting the dataset in half, LOOCV uses one
In this post, we will use the “Hedonic” dataset from the “Ecdat” package to assess several different models that predict the taxes of homes In order to do this, we will also need to use the “boot” package. Below is the code.
library(Ecdat);library(boot)data(Hedonic) str(Hedonic)
## 'data.frame': 506 obs. of 15 variables:
## $ mv : num 10.09 9.98 10.45 10.42 10.5 ...
## $ crim : num 0.00632 0.02731 0.0273 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 ...
## $ chas : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...
## $ nox : num 28.9 22 22 21 21 ...
## $ rm : num 43.2 41.2 51.6 49 51.1 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 ...
## $ dis : num 1.41 1.6 1.6 1.8 1.8 ...
## $ rad : num 0 0.693 0.693 1.099 1.099 ...
## $ tax : int 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 ...
## $ blacks : num 0.397 0.397 0.393 0.395 0.397 ...
## $ lstat : num -3 -2.39 -3.21 -3.53 -2.93 ...
## $ townid : int 1 2 2 3 3 3 4 4 4 4 ...First, we need to develop our basic least squares regression model. We will do this with the “glm” function. This is because the “cv.glm” function (more on this later) only works when models are developed with the “glm” function. Below is the code.
tax.glm<-glm(tax ~ mv+crim+zn+indus+chas+nox+rm+age+dis+rad+ptratio+blacks+lstat, data = Hedonic)We now need to calculate the MSE. To do this we will use the “cv.glm” function. Below is the code.
cv.error<-cv.glm(Hedonic,tax.glm)
cv.error$delta## [1] 4536.345 4536.075cv.error$delta contains two numbers. The first is the MSE for the training set and the second is the error for the LOOCV. As you can see the numbers are almost identical.
We will now repeat this process but with the inclusion of different polynomial models. The code for this is a little more complicated and is below.
cv.error=rep(0,5)
for (i in 1:5){
tax.loocv<-glm(tax ~ mv+poly(crim,i)+zn+indus+chas+nox+rm+poly(age,i)+dis+rad+ptratio+blacks+lstat, data = Hedonic)
cv.error[i]=cv.glm(Hedonic,tax.loocv)$delta[1]
}
cv.error## [1] 4536.345 4515.464 4710.878 7047.097 9814.748Here is what happen.
- First, we created an empty object called “cv.error” with five empty spots, which we will use to store information later.
- Next, we created a for loop that repeats 5 times
- Inside the for loop, we create the same regression model except we added the “poly” function in front of “age”” and also “crim”. These are the variables we want to try polynomials 1-5 one to see if it reduces the error.
- The results of the polynomial models are stored in the “cv.error” object and we specifically request the results of “delta” Finally, we printed “cv.error” to the console.
From the results, you can see that the error decreases at a second order polynomial but then increases after that. This means that high order polynomials are not beneficial generally.
Conclusion
LOOCV is another option in assessing different models and determining which is most appropriate. As such, this is a tool that is used by many data scientist.
Validation Set for Regression in R
Estimating error and looking for ways to reduce it is a key component of machine learning. In this post, we will look at a simple way of addressing this problem
The validation set method is a standard approach in model development. To put it simply, you divide your dataset into a training and a hold-out set. The model is developed on the training set and then the hold-out set is used for prediction purposes. The error rate of the hold-out set is assumed to be reflective of the test error rate.
In the example below, we will use the “Carseats” dataset from the “ISLR” package. Our goal is to predict the competitors’ price for a carseat based on the other available variables. Below is some initial code
library(ISLR)
data("Carseats")
str(Carseats)## 'data.frame': 400 obs. of 11 variables:
## $ Sales : num 9.5 11.22 10.06 7.4 4.15 ...
## $ CompPrice : num 138 111 113 117 141 124 115 136 132 132 ...
## $ Income : num 73 48 35 100 64 113 105 81 110 113 ...
## $ Advertising: num 11 16 10 4 3 13 0 15 0 0 ...
## $ Population : num 276 260 269 466 340 501 45 425 108 131 ...
## $ Price : num 120 83 80 97 128 72 108 120 124 124 ...
## $ ShelveLoc : Factor w/ 3 levels "Bad","Good","Medium": 1 2 3 3 1 1 3 2 3 3 ...
## $ Age : num 42 65 59 55 38 78 71 67 76 76 ...
## $ Education : num 17 10 12 14 13 16 15 10 10 17 ...
## $ Urban : Factor w/ 2 levels "No","Yes": 2 2 2 2 2 1 2 2 1 1 ...
## $ US : Factor w/ 2 levels "No","Yes": 2 2 2 2 1 2 1 2 1 2 ...We need to divide our dataset into two part. One will be the training set and the other the hold-out set. Below is the code.
set.seed(7)
train<-sample(x=400,size=200)Now, for those who are familiar with R you know that we haven’t actually made our training set. We are going to use the “train” object to index items from the “Carseat” dataset. What we did was set the seed so that the results can be replicated. Then we used the “sample” function using two arguments “x” and “size”. X represents the number of examples in the “Carseat” dataset. Size represents how big we want the sample to be. In other words, we want a sample size of 200 of the 400 examples to be in the training set. Therefore, R will randomly select 200 numbers from 400.
We will now fit our initial model
car.lm<-lm(CompPrice ~ Income+Sales+Advertising+Population+Price+ShelveLoc+Age+Education+Urban, data = Carseats,subset = train)The code above should not be new. However, one unique twist is the use of the “subset” argument. What this argument does is tell R to only use rows that are in the “train” index. Next, we calculate the mean squared error.
mean((Carseats$CompPrice-predict(car.lm,Carseats))[-train]^2)## [1] 77.13932Here is what the code above means
- We took the “CompPrice” results and subtracted them from the prediction made by the “car.lm” model we developed.
- Used the test set which here is identified as “-train” minus means everything that is not in the “train”” index
- the results were squared.
The results here are the baseline comparison. We will now make two more models each with a polynomial in one of the variables. First, we will square the “income” variable
car.lm2<-lm(CompPrice ~ Income+Sales+Advertising+Population+I(Income^2)+Price+ShelveLoc+Age+Education+Urban, data = Carseats,subset = train)
mean((Carseats$CompPrice-predict(car.lm2,Carseats))[-train]^2)## [1] 75.68999You can see that there is a small decrease in the MSE. Also, notice the use of the “I” function which allows us to square “income”. Now, let’s try a cubic model
car.lm3<-lm(CompPrice ~ Income+Sales+Advertising+Population+I(Income^3)+Price+ShelveLoc+Age+Education+Urban, data = Carseats,subset = train)
mean((Carseats$CompPrice-predict(car.lm3,Carseats))[-train]^2)## [1] 75.84575This time there was an increase when compared to the second model. As such, higher order polynomials will probably not improve the model.
Conclusion
This post provided a simple example of assessing several different models use the validation approach. However, in practice, this approach is not used as frequently as there are so many more ways to do this now. Yet, it is still good to be familiar with a standard approach such as this.
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Understanding boolean logic in r programming
Teaching Small Children to Write
Teaching a child to write is an interesting experience. In this post, I will share some basics ideas on one way this can be done.
To Read or not to Read
Often writing is taught after the child has learned to read. A major exception 
Generally, I teach young children how to read first. This is because I want the child to know the letters before trying to write them.
The Beginning
If the child is already familiar with the basics of reading writing is probably more about hand-eye coordination than anything else. The first few letters are quite the experience. This is affected by age as well. Smaller children will have much more difficulty with writing than older children.
A common strategy to motivate a child to write is to have them first learn to spell their name. This can work depending on how hard the child’s name is to spell. A kid named “Dan” will master writing his name quickly. However, a kid with a longer name or a transliterated name from another language is going to have a tough time. I knew one student who misspelled their name for almost a year and a half because it was so hard to write in English.
A common way to teach actually writing is to allow the child to trace the words on dot paper. By doing this they develop the muscle memory for writing. Once this is successful the child will then attempt to write the letters with the tracing paper. This process can easily take a year.
Sentences and Paragraphs
After, they learn to write letters and words it is time to begin writing sentences. A six-year-old, with good penmanship, will probably not be able to write a sentence with support. Writing and spelling and different skills initially and it is the adult’s job to provide support for the spelling aspect as the child explains what they want to write about.
With help, children can create short little stories that may be one to two paragraphs in length. Yet they will still need a lot of support to do this.
By eight years of age, a child can probably write a paragraph on their own about simple concepts or stories. This is when the teaching and learning can really get interesting as the child can now write to learn instead of focusing on learning to write.
Conclusion
Writing is a skill that is hard to find these days. With so many other forms of communication, writing is not a skill that children want to focus on. Nevertheless, learning to write by basic literacy is an excellent way to develop communication skills and interact with people in situations where face-to-face contact is not possible.
Homeschooling Concerns
Parents frequently have questions about homeschooling. In this post, we look at three common questions related to homeschooling.
- How do you know if your child has learned
- What do you do about socializing
- What about college
How do You know if they Learned
One definition of learning is a change in observable behavior. In other words, 
A lot of the more advanced forms of assessment including standardized test was created in order to assess the progress of a huge number of students. In the context of homeschooling with only a few students, such rigorous measures are unnecessary. governments need sophisticated measures of achievement because of the huge populations that they serve which would be inappropriate when dealing with one or two elementary students.
Another way to know what your child has learned is to look at what they are studying right now. For example, if my child is reading I know that they have probably mastered the alphabet. Otherwise, how could the read? I also know that they probably have mastered the most of the phonics. In other words, current struggles are an indication of what was mastered before.
What about Socializing
The answer to this question really depends on your position on socializing. Many parents want their child to act like other children. For example, if my child is 7 I want him to act like other 7-year-olds.
Other parents want their child to learn how to act like an adult. For them, they want their 7-year-old child to imitate the behavior of them (the parents) rather than the behavior of other 7-year-olds. A child will only rise to the expectations of those around them. Being around children encourages childish behavior because that’s the example. Again for many parents, this is what they want, however, others see this differently.
The reality is that until middle-age most of the people we interact with are older than us. As such, it is beneficial for a child to spend a large amount of time around people who are older than them and understand the importance of setting an example that can be imitated.
All socializing is not the same. Adult-to-child socializing provides a child with an example of how to be an adult rather than how to be a child. Besides, most small children would love to be around their parents all day. They only grow to love friends so much because those are the people who give them the most attention.
What about College
This question is the hardest to answer as it depends on context a great deal. Concerns with college can be alleviated by having the child take the GED in the US or local college entrance examinations in other countries.
It is also important to keep careful records of what the child studies during high school. Most colleges do not care about K-8 learning but really want to know what happens during grades 9-12. Keep records of the courses the child took as well as the grades. It will also be necessary to take the SAT or ACT in most countries as well.
Conclusion
Homeschooling is an option for people who want to spend the maximum amount of time possible with their children. Concerns about learning, socializing, and college are unnecessary if the parents are willing to thoroughly dedicate themselves and provide their children with a learning environment that develops their children wholistically.
What it Takes to Homeschool
Some may be wondering what does it take to homeschool. Below are some characteristics of the homeschool.
Time management
Being able to adhere to a schedule is a prerequisite for homeschooling. It is 
This is not to say there should be no flexibility. Rather, the schedule should not be cheated because of laziness. There must be a set schedule for studying for the sake of behavior management of the children. If the child doesn’t know what to expect they may challenge you when you flippantly decide they need to study. Consistency is a foundational principle of homeschooling.
Discipline
Discipline means being able to do something even when you do not feel like doing it. In homeschooling, you have to teach whether you want to or not. Remember, sometimes we had to work at our jobs when we didn’t feel like it and the same with teaching in the home. If you’re tired you still have to teach, if you’re a little sick you still have to teach, if you’re angry you still have to teach.
The child is relying on you to provide them with the academic skills needed to compete in the world. This cannot be neglected for trivial reasons. Lesson plans are key. Either buy them or make them. Keep track of completed assignment and note the progress of the student.
Toughness
As a homeschooling parent, you are the only authority in the child’s life. This means all discipline falls under your jurisdiction. One reasons parents enjoy sending their kids to school is to burden the public school teachers with their own child’s poor behavior. “Let the school deal with him” is a common comment I have heard when I was a k12 teacher. However, when you teach as a homeschool parent only you have the pleasure of disciplining your child.
Discipline is not only about taking away privileges and causing general suffering for unacceptable behavior. Discipline also includes communicating clearly with your child to prevent poor behavior, have clear rules that are always enforced, as well as providing a stable environment in which to study.
Patience
Homeschooling also requires patience. For example, you are teaching a basic first-grade math concept to your child that takes several weeks for them to learn. Naturally, you start to get angry with the child and yourself for the lack of progress. You may even begin to question if you have what it takes to do this. However, after waiting for what seems an eternity they child finally gets it.
This is the reality of homeschooling. No matter how bad you think you are the child will eventually get it when they are ready. This requires patience in the parent and some confidence in their own ability to help their child to grow.
Conclusion
There are many more ideas I could share. However, this is sufficient for now. In general, I would not recommend homeschooling for the typical family as the above traits are usually missing in the parents. Many parents want to homeschool for emotional reasons. The problem with this is that when they feel bad they will not want to continue the experience. Homeschooling can involve love but it must transcend emotions in order to endure for several years.
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Working with vectors in R
Teaching Math in the Homeschool
Teaching a child to count and do simple math is much more challenge then many would believe. Below is a simple process that I accidentally developed from working with kindergarten home-school student for two years. Keep in mind that often these steps overlapped.
- Number recognition
- Counting
- Counting with manipulatives
- Flashcards with larger numbers
- Writing numbers
- Adding with manipulatives
- Subtraction with manipulatives
- Visual math
1. Number Recognition
Number recognition simple involved the use of flashcards with the child. I would 
2. Counting
With the numbers memorized, the next step was to actually learn to count. I did this by holding up the same flashcards. After the child identify what number it was I would then flip the flashcard over and have them count the number of objects on the card. My goal was to have them make a connection between the abstract number and the actual amount that could be seen and counted.
Again it took about six months for the four and half-year-old student to master this from numbers 1-20. It was a really stressful six months.
3. Counting with Manipulatives
The next few steps happen concurrently for the most part. I started to have the student count with manipulatives. I would show or say a number and expect the student to count the correct number using the manipulatives. This was done with numbers 1-20 only.
4. Flashcards with Larger Numbers
At the same time, I worked with the student to learn numbers beyond 20. This was strictly for memorization purposes. This continued from 4.5 to 6 years of age. Eventually, the child could identify numbers 1-999. However, the never discovered the pattern of counting. By pattern, I mean how the 0-9 cycle repeats in the tens, how the 1-9 cycle repeats for the tens when moving to 100s, etc. The child only knew the numbers through brute memorization.
5. Writing Numbers
Writing numbers was used as preparation for doing addition. It was as simple as giving the student some numbers to trace on paper. It took about 8 months for the student to write numbers with any kind of consistency.
6. Adding with Manipulatives
This involved me writing a math problem and having the student solve the problem use manipulatives. For example, 2 + 2 would be solved by having the student count two manipulatives and then count two more and then count the total.
My biggest concern was having the child understand the + and = sign. The plus sign was easy but the equal sign was mysterious for a long time. However, the learning rate was picking up and the kid learn this in about 3 months
7. Subtraction with Manipulatives
Same as above but only took one month to learn
8. Visual Math
At this stage, the child was doing worksheets on their own. Manipulatives were allowed as a crutch to get through the problems. However, the child was now being encouraged to use their fingers for counting purposes. This was a disaster for several weeks as the lack the coordination to open and close the fingers independent of each other.
Conclusion
This entire process took two years to complete from ages 4-6 working with the child one-on-one. By the age of six, the child could add and subtract anything from 1-30 and was ready for 1st grade.
I would recommend waiting longer to start math with a child. Being 4 was probably too young for this particular child. Better to wait untili 5 or 6 to learn numbers and counting. There more danger in starting early then there is in starting late.
Confusing Words for Small Children
In this post, we will look at some commonly used words that can bring a great deal of frustration to adults when communicating with small children. The terms 
- Deictic terms
- Interrogatives
- Locational terms
- Temporal terms
Deictic Terms
Deictic terms fall under the umbrella of pragmatic development or understanding of the context in which words are used. Examples of deictic terms include such words as this, that, these, those, here, there, etc. What makes these words confusing for young children and even ESL speakers is that the meaning of these words depends on the context. Below is a clear way to communicate followed by a way that is unclear using a deixis term
Clear communication: Take the book
Unclear communication: Take that
The first sentence makes it clear what to take which in this example is the book. However, for a child or ESL speaker, the second sentences can be mysterious. What does “that” mean. It takes pragmatic or contextual knowledge to determine what “that” is referring to in the sentence. Children usually cannot figure this out while an ESL speaker will watch the body language (nonlinguistic cues) of the speaker to figure this out.
Interrogatives
A unique challenge for children is understanding interrogatives. These are such words as who, what, where, when, and why. The challenge with these questions is they involve explaining the cause, time, and or reasons. Many parents have asked the following question without receiving an adequate answer
Why did you take the book?
The typical 3-year old is going to wonder what the word “why” means. Off course, you can combine a deictic term with an interrogative and completely lose a child
Why did you do that?
Locational Terms
Locational terms are prepositions words such as in, under, above, behind etc. These words can be challenging for young children because they have to understand the perspective of the person speaking. Below is an example.
Put the book under the table.
Naturally, the child is trying to understand what “under” means. We can also completely confuse a child by using terms from all the categories we have discussed so far.
Why did you put that under the table?
This sentence would probably be unclear to many native speakers. The ambiguity is high especially with the term “that” included.
Temporal Terms
Temporal terms are about time. Commonly used words include before, after, while, etc. These terms are difficult for children because young children do not quite grasp the concept of time. Below is an example of a sentence with a temporal term.
Before, dinner, grab the book
The child is probably wondering when they are supposed to get the book. Naturally, we can combine all of our terms to make a truly nightmarish sentence.
Why did you put that under the table after dinner?
Conclusion
The different terms mentioned here are terms that can cause frustration when trying to communicate. To alleviate, these problems parents and teachers should avoid these terms when possible by using nouns. In addition, using body language to indicate position or pointing to whatever you are talking about can help young children to infer the meaning
Additive Assumption and Multiple Regression
In regression, one of the assumptions is the additive assumption. This 
In this post, we will look at how to address violations of the additive assumption through the use of interactions in a regression model.
An interaction effect is when you have two predictor variables whose effect on the dependent variable is not the same. As such, their effect must be considered simultaneously rather than separately. This is done through the use of an interaction term. An interaction term is the product of the two predictor variables.
Let’s begin by making a regular regression model with an interaction. To do this we will use the “Carseats” data from the “ISLR” package to predict “Sales”. Below is the code.
library(ISLR);library(ggplot2)
data(Carseats)
saleslm<-lm(Sales~.,Carseats)
summary(saleslm)##
## Call:
## lm(formula = Sales ~ ., data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8692 -0.6908 0.0211 0.6636 3.4115
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.6606231 0.6034487 9.380 < 2e-16 ***
## CompPrice 0.0928153 0.0041477 22.378 < 2e-16 ***
## Income 0.0158028 0.0018451 8.565 2.58e-16 ***
## Advertising 0.1230951 0.0111237 11.066 < 2e-16 ***
## Population 0.0002079 0.0003705 0.561 0.575
## Price -0.0953579 0.0026711 -35.700 < 2e-16 ***
## ShelveLocGood 4.8501827 0.1531100 31.678 < 2e-16 ***
## ShelveLocMedium 1.9567148 0.1261056 15.516 < 2e-16 ***
## Age -0.0460452 0.0031817 -14.472 < 2e-16 ***
## Education -0.0211018 0.0197205 -1.070 0.285
## UrbanYes 0.1228864 0.1129761 1.088 0.277
## USYes -0.1840928 0.1498423 -1.229 0.220
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.019 on 388 degrees of freedom
## Multiple R-squared: 0.8734, Adjusted R-squared: 0.8698
## F-statistic: 243.4 on 11 and 388 DF, p-value: < 2.2e-16The results are rather excellent for the social sciences. The model explains 87.3% of the variance in “Sales”. The current results that we have are known as main effects. These are effects that directly influence the dependent variable. Most regression models only include main effects.
We will now examine an interaction effect between two continuous variables. Let’s see if there is an interaction between “Population” and “Income”.
saleslm1<-lm(Sales~CompPrice+Income+Advertising+Population+Price+Age+Education+US+
Urban+ShelveLoc+Population*Income, Carseats)
summary(saleslm1)##
## Call:
## lm(formula = Sales ~ CompPrice + Income + Advertising + Population +
## Price + Age + Education + US + Urban + ShelveLoc + Population *
## Income, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8699 -0.7624 0.0139 0.6763 3.4344
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.195e+00 6.436e-01 9.625 <2e-16 ***
## CompPrice 9.262e-02 4.126e-03 22.449 <2e-16 ***
## Income 7.973e-03 3.869e-03 2.061 0.0400 *
## Advertising 1.237e-01 1.107e-02 11.181 <2e-16 ***
## Population -1.811e-03 9.524e-04 -1.901 0.0580 .
## Price -9.511e-02 2.659e-03 -35.773 <2e-16 ***
## Age -4.566e-02 3.169e-03 -14.409 <2e-16 ***
## Education -2.157e-02 1.961e-02 -1.100 0.2722
## USYes -2.160e-01 1.497e-01 -1.443 0.1498
## UrbanYes 1.330e-01 1.124e-01 1.183 0.2375
## ShelveLocGood 4.859e+00 1.523e-01 31.901 <2e-16 ***
## ShelveLocMedium 1.964e+00 1.255e-01 15.654 <2e-16 ***
## Income:Population 2.879e-05 1.253e-05 2.298 0.0221 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.013 on 387 degrees of freedom
## Multiple R-squared: 0.8751, Adjusted R-squared: 0.8712
## F-statistic: 226 on 12 and 387 DF, p-value: < 2.2e-16The new contribution is at the bottom of the coefficient table and is the “Income:Population” coefficient. What this means is “the increase of Sales given a one unit increase in Income and Population simultaneously” In other words the “Income:Population” coefficient looks at their combined simultaneous effect on Sales rather than just their independent effect on Sales.
This makes practical sense as well. The larger the population the more available income and vice versa. However, for our current model, the improvement in the r-squared is relatively small. The actual effect is a small increase in sales. Below is a graph of income and population by sales. Notice how the lines cross. This is a visual of what an interaction looks like. The lines are not parallel by any means.
ggplot(data=Carseats, aes(x=Income, y=Sales, group=1)) +geom_smooth(method=lm,se=F)+
geom_smooth(aes(Population,Sales), method=lm, se=F,color="black")+xlab("Income and Population")+labs(
title="Income in Blue Population in Black")We will now repeat this process but this time using a categorical variable and a continuous variable. We will look at the interaction between “US” location (categorical) and “Advertising” (continuous).
saleslm2<-lm(Sales~CompPrice+Income+Advertising+Population+Price+Age+Education+US+
Urban+ShelveLoc+US*Advertising, Carseats)
summary(saleslm2)##
## Call:
## lm(formula = Sales ~ CompPrice + Income + Advertising + Population +
## Price + Age + Education + US + Urban + ShelveLoc + US * Advertising,
## data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8531 -0.7140 0.0266 0.6735 3.3773
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.6995305 0.6023074 9.463 < 2e-16 ***
## CompPrice 0.0926214 0.0041384 22.381 < 2e-16 ***
## Income 0.0159111 0.0018414 8.641 < 2e-16 ***
## Advertising 0.2130932 0.0530297 4.018 7.04e-05 ***
## Population 0.0001540 0.0003708 0.415 0.6782
## Price -0.0954623 0.0026649 -35.823 < 2e-16 ***
## Age -0.0463674 0.0031789 -14.586 < 2e-16 ***
## Education -0.0233500 0.0197122 -1.185 0.2369
## USYes -0.1057320 0.1561265 -0.677 0.4987
## UrbanYes 0.1191653 0.1127047 1.057 0.2910
## ShelveLocGood 4.8726025 0.1532599 31.793 < 2e-16 ***
## ShelveLocMedium 1.9665296 0.1259070 15.619 < 2e-16 ***
## Advertising:USYes -0.0933384 0.0537807 -1.736 0.0834 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.016 on 387 degrees of freedom
## Multiple R-squared: 0.8744, Adjusted R-squared: 0.8705
## F-statistic: 224.5 on 12 and 387 DF, p-value: < 2.2e-16Again, you can see that when the store is in the US you have to also consider the advertising budget as well. When these two variables are considered there is a slight decline in sales. What this means in practice is that advertising in the US is not as beneficial as advertising outside the US.
Below you can again see a visual of the interaction effect when the lines for US yes and no cross each other in the plot below.
ggplot(data=Carseats, aes(x=Advertising, y=Sales, group = US, colour = US)) + geom_smooth(method=lm,se=F)+scale_x_continuous(limits = c(0, 25))+scale_y_continuous(limits = c(0, 25))
Lastly, we will look at an interaction effect for two categorical variables.
saleslm3<-lm(Sales~CompPrice+Income+Advertising+Population+Price+Age+Education+US+
Urban+ShelveLoc+ShelveLoc*US, Carseats)
summary(saleslm3)##
## Call:
## lm(formula = Sales ~ CompPrice + Income + Advertising + Population +
## Price + Age + Education + US + Urban + ShelveLoc + ShelveLoc *
## US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8271 -0.6839 0.0213 0.6407 3.4537
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.8120748 0.6089695 9.544 <2e-16 ***
## CompPrice 0.0929370 0.0041283 22.512 <2e-16 ***
## Income 0.0158793 0.0018378 8.640 <2e-16 ***
## Advertising 0.1223281 0.0111143 11.006 <2e-16 ***
## Population 0.0001899 0.0003721 0.510 0.6100
## Price -0.0952439 0.0026585 -35.826 <2e-16 ***
## Age -0.0459380 0.0031830 -14.433 <2e-16 ***
## Education -0.0267021 0.0197807 -1.350 0.1778
## USYes -0.3683074 0.2379400 -1.548 0.1225
## UrbanYes 0.1438775 0.1128171 1.275 0.2030
## ShelveLocGood 4.3491643 0.2734344 15.906 <2e-16 ***
## ShelveLocMedium 1.8967193 0.2084496 9.099 <2e-16 ***
## USYes:ShelveLocGood 0.7184116 0.3320759 2.163 0.0311 *
## USYes:ShelveLocMedium 0.0907743 0.2631490 0.345 0.7303
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.014 on 386 degrees of freedom
## Multiple R-squared: 0.8753, Adjusted R-squared: 0.8711
## F-statistic: 208.4 on 13 and 386 DF, p-value: < 2.2e-16In this case, we can see that when the store is in the US and the shelf location is good it has an effect on Sales when compared to a bad location. The plot below is a visual of this. However, it is harder to see this because the x-axis has only two categories
ggplot(data=Carseats, aes(x=US, y=Sales, group = ShelveLoc, colour = ShelveLoc)) +
geom_smooth(method=lm,se=F)Conclusion
Interactions effects are a great way to fine-tune a model, especially for explanatory purposes. Often, the change in r-square is not strong enough for prediction but can be used for nuanced understanding of the relationships among the variables.
Vectors and Functions in R VIDEO
Vectors and functions in R
Concepts to Consider for Model Development
When assessing how to conduct quantitative data analysis there are several factors to consider. In this post, we look at common either-or choices in data analysis. The concepts are as follows
- Parametric vs Non-Parametric
- Bias vs Variance
- Accuracy vs Interpretability
- Supervised vs Unsupervised
- Regression vs Classifications
Parametric vs Non-Parametric
Parametric models make assumptions about the shape of a function. Often, the assumption is that the function is a linear in nature such as in linear 
Non-parametric models do not make any assumptions about the shape of the function. This allows the function to take any shape possible. Examples of non-parametric models include decision trees, support vector machines, and artificial neural networks.
The main concern with non-parametric models is they require a huge dataset when compared to parametric models.
Bias vs Variance Tradeoff
In relation to parametric and non-parametric models is the bias-variance tradeoff. A bias model is simply a model that does not listen to the data when it is tested on the test dataset. The function does what it wants as if it was not trained on the data. Parametric models tend to suffer from high bias.
Variance is the change in the function if it was estimated using new data. Variance is usually much higher in non-parametric models as they are more sensitive to the unique nature of each dataset.
Accuracy vs Interpretability
It is also important to determine what you want to know. If your goal is accuracy then a complicated model may be more appropriate. However, if you want to infer and make conclusions about your model then it is preferred to make a simpler model that can be explained.
A model can be made simpler or more complex through the inclusion of more features or the use of a more complex algorithm. For example, regression is much easier to interpret than artificial neural networks.
Supervised vs Unsupervised
Supervised learning is learning that involves a dependent variable. Examples of supervised learning include regression, support vector machines, K nearest neighbor, and random forest.
Unsupervised learning involves a dataset that does not have a dependent variable. In this situation, you are looking for patterns in the data. Examples of this include kmeans, principal component analysis, and cluster analysis.
Regression vs Classifications
Lastly, a problem can call for regression or classification. A regression problem involves a numeric dependent variable. A classification problem has a categorical dependent variable. Almost all models that are used for supervised machine learning can address regression or classification.
For example, regression includes numeric regression and logistic regression for classification. K nearest neighbor can do both as can random forest, support vector machines, and artificial neural networks.
Ways to Comprehend Academic Texts
In this post, we will look at some practical ways to better understand an academic text. The tips are broken down into three sections which are, what to do before, during, and after reading.
Before Reading
Read the Preface. The preface lays out the entire scope of the book. It provides the framework in which you can place the details of the chapters. This is critical in order to put the pieces together to make use of 
Read the Chapter titles. The chapter titles give you an idea of what the chapter is about. Again it helps you to zoom down one level to understand the subject of the book from one aspect of it. Again, most students fly past this when the chapter title provides clear clues as to what to expect in the text
Read the Objectives. The objectives tell you what you are going to visit in the chapter. They serve as signposts of what to expect and provide a framework for placing the details of the text.
Read the Chapter Headings. An academic text is broken down into chapters which are broken down into headings. Examining the headings provides more information about the chapter and the book. I also should mention that often the objectives and the headings of a book are the same with slight rewording. This seems lazy but is actually much clearer than when they two are not similar.
Look at the visuals (tables, graphs, pictures). Visuals summarize critical information. It is easy for anybody to become overwhelmed when reading a text. Therefore, visuals are created to summarize the most important information. Just like figure 1.2 above.
If you do these things you now know what to expect when you read. You are also beginning to develop an idea of what you did not know about the given subject. This leads to the next major point.
Ask Questions. After this inspection of the text, you should do the following.
- Decide what you already know about this topic
- Decide what you want to know about this topic and make questions
This two-step process prepares you for connecting your current level of understanding with the new knowledge within the text. You know what is a review for you and you focus on finding answers to the concepts and idea that are new for you.
During Reading
After all of this preparatory work, it is now time to read. Having done all this you already know the following
- Title of the chapter
- Major headings/objectives of the chapter
- What you already know about this subject
- What you do not know about this subject
Now you read the text and answer your questions. You also can highlight key ideas as well as write in the margins of the text. Highlighting should generally be limited to main ideas in order to reduce the clutter of highlighting everything. Writing in the margins allows you to make quick notes to yourself about key points and or to summarize a dense concept. Doing either of these is a way to wrestle with a text in an active manner which is important for comprehension.
After Reading
After you have read and answered your questions in a text. There are several things left to do.
Determine what did you learn. Write briefly a few notes to yourself about what exactly you learned. This is for you and helps to make sense of all the details in your mind at the moment.
Look at the Resources at the back of the chapter. Many textbooks have several study tools at the back of the chapter. Example this includes an outline of the chapter which is a great summary, discussion questions which help in developing critical thinking skills, and often vocabulary words are here as well. When preparing for an exam this is an excellent resource.
Conclusion
This process is not as much work as it seems. With practice, it can become natural. In addition, you need to modify this so that you can be successful as a reader. The ideas here provide a framework in which you can develop your own style.
Insights into Reading Academic Text
In my experience as a teacher for several years at university, I have noticed how students consistently struggle with reading an academic text. It seemed as those they were able to “read” the words but always lack the ability to understand what the text was about. I’ve thought about this challenge and have been lead to the following conclusions.
- Many Students believe reading and understanding are the same thing
- Many Students believe they have no responsibility to think about what they have read
- Many Students believe there is no reason to connect what they are reading to anything they currently know
- Many Students see no point to determine how to use or apply what they have read
- Many Students do not understand how academic writers structure their writing
None of these points apply to everybody. However, it is common for me to ask my students if they read something and they usually that they yes the did read it. However, as I begin to ask questions and to explore the text with them it quickly becomes clear they did not understand anything that they read. This is 
In other words, reading is not the problem, rather it is what to do with what they have read. The purpose of studying is to use what you have learned. Few of us have the time, to simply learn for fun. Often, we learn to do something for monetary reasons. In other words, some sort of immediate application is critical to reading success.
Another important aspect of reading comprehension is understanding the structure of academic writing. Textbooks have different subjects but they all have a surprisingly similar structure which often starts with the big picture and zooms down to the details. If students can see the structure it can greatly improve their ability to understand what they are reading.
The Tour Guide Analogy
The analogy that I like to use is that of a tour guide. A tour guide’s job is to show you around a particular place. It could be an entire city or a single tourist attraction it all depends on the level of detail that he or she wants to provide you. Often, at the beginning of a tour, the tour guide will explain the itinerary of the tour. This provides the big picture purpose of the tour group as well as what to expect during the journey.
If the trip is especially detail you may visit several different places. At each place, there will be several places to see at each place that the tour guide will mention. For example, If I go to Thailand for vacation and visit Bangkok there will be several locations within Bangkok that I would visit such as Malls and maybe a museum. It is the tour guide’s job to guide me in the learning experience.
The author of an academic text is like a tour guide. Their job is to show you around the subject they are an expert in. The tour guide has an itinerary while the academic author has a preface/introduction. In the preface, the author explains the purpose of the book, as well as the major themes or “places” they will show you on the tour. The preface also explains who the book is for.
Each chapter in an academic text is one specific place the author wants to show you on the tour. Just as a tour guide may show you a museum in Bangkok so an author will show you one aspect of a subject in a chapter. Furthermore, every chapter has several headings within it. This is the same as me seeing the dinosaur exhibit at the museum or the ancient Thai instruments exhibit. These are the places within the place that you visited.
| Tour guide | Writer |
| Expert in their area | Exepert in their area |
| Shows you around the tourist attraction | Shows you around a subject area |
| Explains what you will see today | Explains what they will share in a book/chapter |
| Provides details about the different sights | Provides details for the main ideas |
We can break this down further about subheadings and more but I think the point is clear. The layout for an academic text is not mysterious but rather highly consistent. Having said this here are some critical ideas to remember when you read.
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Statistical Learning
Statistical learning is a discipline that focuses on understanding data. Understanding data can happen through classifying or making a numeric prediction which is called supervised learning or finding patterns in data which is called unsupervised learning,
In this post, we will examine the following
- History of statistical learning
- The purpose of statistical learning
- Statistical learning vs Machine learning
History Of Statistical Learning
The early pioneers of statistical learning focused exclusively on supervised learning. Linear regression was developed in the 19th century by Legendre and 
The developments of the late 19th century to the mid 20th century were limited due to the lack of computational power. However, by the 1970’s things began to change and new algorithms emerged, specifically ones that can handle non-linear relationships
In the 1980’s Friedman and Stone developed classification and regression trees. The term generalized additive models were first used by Hastie and Tibshirani for non-linear generalized models.
Purpose of Statistical Learning
The primary goal of statistical learning is to develop a model of data you currently have to make decisions about the future. In terms of supervised learning with a numeric dependent variable, a teacher may have data on their students and want to predict future academic performance. For a categorical variable, a doctor may use data he has to predict whether someone has cancer or not. In both situations, the goal is to use what one knows to predict what one does not know.
A unique characteristic of supervised learning is that the purpose can be to predict future values or to explain the relationship between the dependent variable and another independent variable(s). Generally, data science is much more focused on prediction while the social sciences seem more concerned with explanations.
For unsupervised learning, there is no dependent variable. In terms of a practical example, a company may want to use the data they have to determine several unique categories of customers they have. Understanding large groups of customer behavior can allow the company to adjust their marketing strategy to cater to the different needs of their vast clientele.
Statistical Learning vs Machine Learning
The difference between statistical learning and machine learning is so small that for the average person it makes little difference. Generally, although some may disagree, these two terms mean essentially the same thing. Often statisticians speak of statistical learning while computer scientist speak of machine learning
Machine learning is the more popular term as it is easier to conceive of a machine learning rather than statistics learning.
Conclusion
Statistical or machine learning is a major force in the world today. With some much data and so much computing power, the possibilities are endless in terms of what kind of beneficial information can be gleaned. However, all this began with people creating a simple linear model in the 19th century.
Conversational Analysis: Request and Response
Within conversational analysis (CA) it is common to analysis peoples request as well as people’s response to a request in the context of a conversation. In this 
Request
Requests are a specific type of question in conversational analysis. Request almost always involve some sort of action. Either the person asking the request wants to do something or the speaker wants the listener to do something. As such there are only two categories in which request can be classified and they are…
- Action request
- Permission request
Action Request
An action request is made when the speaker wants the listener to do something.
A: Can you turn off the light?
Permission Request
A permission request is made when the speaker wants to perform an action and is seeking approval from the listener.
A: You mind if I turn off the light?
Response to Request
The response to a request can be positive or negative. However, when a response is negative it is often indirect. As such, there are three categories in which a response to a request can be placed.
- Accept
- Reject
- Evade
Accept
Accepting is to grant permission either for the speaker to do something or that the listener will perform the request.
A: Could you turn the light off?
B: No problem
Reject
Rejecting means that a person states directly that they cannot do something
A: Can you turn the light off?
B: No, I can’t
Evading
Evading is the art of saying “no” indirectly to a request. This is done through giving a reason why something cannot be done rather than directly responding
A: Can you turn the light off?
B: I’m busy with the baby
In the example above, person B never says no. Rather, they provide an excuse for not completing the task.
Conclusion
We all have used these various ways of requesting and responding to request. The benefit of CA is being able to breakdown these conversational pairs and understand what is happening beyond the surface level.
Conversational Analysis: Questions & Responses
Conversational analysis (CA) is the study of social interactions in everyday 
Questions
In CA, there are generally three types of questions and they are as follows…
- Identification question
- Polarity question
- Confirmation question
Identification Question
Identification questions are questions that employees one of the five W’s (who, what, where, when, why). The response can be opened or closed-ended. An example is below
Where are the keys?
Polarity Question
A polarity question is a question the calls for a yes/no response.
Can you come to work tomorrow?
Confirmation Question
Similiar to the polarity question, a confirmation question is a question that is seeking to gather support for something the speaker already said.
Didn’t Sam go to the store already?
This question is seeking an affirmative yes.
Responses
There are also several ways in which people respond to a question. Below is a list of common ways.
- Comply
- Supply
- Imply
- Evade
- Disclaim
Comply
Complying means give a clear direct answer to a question. Below is an example
A: What time is it?
B: 6:30pm
Supply
Supplying is the act of giving a partial response, that is often irrelevant and fails to answer the question.
A: Is this your dog?
B: Well…I do feed it once in awhile
In the example above, person A asks a clear question. However, person B states what they do for the dog (feed it) rather than indicate if the dog belongs to them. Feeding the dog is irrelevant to ownership.
Imply
Implying is providing information indirectly to answer a question.
A: What time do you want to leave?
B: Not too late
The response from person B does not indicate any sort of specific time to leave. This leaves it up to person A to determine what is meant by “too late.”
Disclaim
Disclaiming is the person stating they do not n]know the answer.
A: Where are the keys?
B: I don’t know
Evade
Evading is the act of answering with really answering the question
A: Where is the car
B: David needed to go shopping
In the example above, person B never states where the car is. Rather, they share what someone is doing with the car. By doing this, the speaker never shares where the car is.
Conclusions
The interaction of a question and response can be interesting if it is examined more closely from a sociolinguistic perspective. The categories provided here can support the deeper analysis of conversation.
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Conversational Analysis
Conversational analysis is a tool used by sociolinguist to examine dialog between two or more people. The analysis can include such aspects as social 
One unique tool in conversational analysis identifying adjacency pairs. Adjacency pairs are two-part utterances in which the second speaker is replying to something the first speaker said. In this post, we will look at the following examples of adjacency pairs.
- Request-agreement
- Question-Answer
- Assessment-Agreement
- Greeting-Greeting
- Compliment-Acceptance
- Conversational Concluder
- Complaint-Apology
- Blame-Denial
- Threat-Counterthreat
- Warning-Acknowledgement
- Offer-Acceptance
Request-Agreement
Request involves asking someone to do something and agreement indicates that the person will do it. Below is an example
A: Could you open the window?
B: No problem
Question-Answer
One person request information from another. THis is different from request agreement because there is no need to agree. Below is an example
A: Where are you from?
B: I am from Laos
Assessment-Agreement
Assessment seeks an opinion from someone and agreement is a positive position on the subject. The example is below
A: Do you like the food?
B: Yeah, it taste great!
Greeting-Greeting
Two people say hello to one another.
A: Hello
B: Hello
Compliment-Acceptance
One person commends something about the other who shows appreciation for the comment.
A: I really like your shoes
B: Thank you
Conversational Concluder
This is a comment that singles the end of a conversation.
A: Goodbye
B: See you later
Complaint-Apology
One person indicates they are not happy with something and the other person express regret over this.
A: The food is too spicy
B: We’re so sorry
Blame-Denial
One person accuses another who tries to defend himself.
A: You lost the phone?
B: No I didn’t!
Threat-Counterthreat
Two people mutually resist each other.
A: Sit down or I will call your parents!
B: Make me
Warning-Acknowledgement
One person issues a threat or danger and the other indicates they understand
A: Look both ways before crossing the street
B: No problem
Offer-Acceptance
One person gives something and the other person shows appreciation
A: Here’s the money
B: Thank you so much
Conclusion
These kinds of conversational pairs appear whenever people talk. For the average person, this is not important. However, when trying to look at the context of a conversation tot understanding what is affecting the way people are speaking understanding and identifying adjacency pairs can be useful.
The Structure of Academic Writing
“The book is boring.” This is a common complaint many lecturers receive from students about the assigned reading in a class. Although this is discouraging to hear it is usually a cry for help. What the student is really saying is that they cannot understand what they are reading. Yes, the read it but they didn’t get it.
This post will try to explain the structure of academic writing in a general sense.
How it Works
Below is a brief outline of a common structure for an academic textbook.
- Preface
- Purpose of the book
- Big themes of the book (chapters)
- Chapter
- Objectives/headings provide themes of the chapter
- Headings
- Provides theme of a section of a chapter
Here is what I consider to be a major secret of writing. The structure is highly redundant but at different levels of abstraction. The preface, chapter, and headings of a book are all the same in terms of purpose but at different levels of scope. The preface is the biggest picture you can get of the text. It’s similar to the map of a country. The chapter zooms in somewhat and is similar to the map of a city. Lastly, the headings within a chapter are similar to have a neighborhood map of a city.
The point is that academic writing is highly structured and organized. Students often think a text is boring. However, when they see the structure, they may not fall in love with academics but at least they will understand what the author is trying to say. A student must see the structure in order to appreciate the details.
Another way to look at this is as follows.
- The paragraphs of a heading support the heading
- The headings of a chapter support the chapter
- The chapters of a book support the title of the book
A book is like a body, you have cells, you have tissues, and you have organs. Each is an abstraction of a higher level. Cells combine to make tissue, tissues combine to make organs, etc. This structure is how academic writing takes place.
The goal of academic writing is not to be entertaining. That role is normally set aside for fiction writing. Since most students enjoy entertainment they expect academic writing to use the same formula of fun. However, few authors place fun as one of the purposes in their preface. This yet another disconnect between students and textbooks.
Conclusion
Academic writing is repetitive in terms of its structure. Each sub-section supports a higher section in the book. This repetitive structure is probably one aspect of academic writing students find so boring. However, this repetitive nature makes the write highly efficient in terms of understanding giving that the reader is aware of this.
Understanding the Preface of a Textbook
A major problem students have in school is understanding what they read. However, the problem often is not reading in itself. By this I mean the student know what they read but they do not know what it means. In other words, they will read the text but cannot explain what the text was about.
There are several practical things a student can do to overcome this problem without having to make significant changes to their study habits. Some of the strategies that they can use involve looking at the structure of how the writing is developed. Examples of this include the following.
- Reading the preface
- Reading the chapter titles
- Reading the chapter objectives
- Reading the headings in the chapters
- Make some questions
- Now read & answer the questions
In this post, we will look at the benefits of reading the preface to a book.
Reading the Preface
One of the first things a student should do is read the preface of a book. The preface gives you some of the following information
- Information about the author
- The purpose of the book
- The audience of the book
- The major themes of the text
- Assumptions
Knowing the purpose of the text is beneficial to understanding the author’s viewpoint. This is often more important in graduate studies than in undergrad.
Knowing the main themes of the book helps from a psychological perspective as well. These themes serve as mental hooks in your mind in which you can hang the details of the chapters that you will read. It is critical to see the overview and big picture of the text so that you have a framework in which to place the ideas of the chapters you will read.
Many books do not have a preface. Instead what they often do is make chapter one the “introduction” and include all the aspects of the preface in the first chapter. Both strategies are fine. However, it is common for teachers to skip the introduction chapter in order to get straight to the “content.” This is fast but can inhibit understanding of the text.
There are also usually an explanation of assumptions. The assumptions serve to tell the reader what they should already know as well as the biases of the author. This is useful as it communicates the position the author takes from the outset with the readers trying to infer this.
Conclusion
The preface serves the purpose of introducing the reader to the text. One of the goals if the preface is to convince the reader why they should read the book. It provides the big picture of the text, shares about the author, and indicates who the book is for, as well as sharing the author’s viewpoint.