Fix typos

Manual integration. Closes #225

model-basics.Rmd

@@ -186,7 +186,7 @@ ggplot(sim1, aes(x, y)) +
   geom_abline(intercept = best$par[1], slope = best$par[2])
 ```
 
-Don't worry too much about the details of how `optim()` works. It's the intuition that's important here. If you have a function that defines the distance between a model and a dataset, an algorthim that can minimise that distance by modifying the parameters of the model, you can find the best model. The neat thing about this approach is that it will work for any family of models that you can write an equation for. 
+Don't worry too much about the details of how `optim()` works. It's the intuition that's important here. If you have a function that defines the distance between a model and a dataset, an algorithm that can minimise that distance by modifying the parameters of the model, you can find the best model. The neat thing about this approach is that it will work for any family of models that you can write an equation for. 
 
 There's one more approach that we can use for this model, because it's is a special case of a broader family: linear models. A linear model has the general form `y = a_1 + a_2 * x_1 + a_3 * x_2 + ... + a_n * x_(n - 1)`. So this simple model is equivalent to a general linear model where n is 2 and `x_1` is `x`. R has a tool specifically designed for fitting linear models called `lm()`. `lm()` has a special way to specify the model family: formulas. Formulas look like `y ~ x`, which `lm()` will translate to a function like `y = a_1 + a_2 * x`. We can fit the model and look at the output:
 
@@ -239,7 +239,7 @@ These are exactly the same values we got with `optim()`! Behind the scenes `lm()
 
 For simple models, like the one above, you can figure out what pattern the model captures by carefully studying the model family and the fitted coefficients. And if you ever take a statistics course on modelling, you're likely to spend a lot of time doing just that. Here, however, we're going to take a different tack. We're going to focus on understanding a model by looking at its predictions. This has a big advantage: every type of predictive model makes predictions (otherwise what use would it be?) so we can use the same set of techniques to understand any type of predictive model.
 
-It's also useful to see what the model doesn't capture, the so called residuals which are left after subtracting the predictions from the data. Residuals are powerful because they allow us to use models to remove striking patterns so we can study the subtler trends that remain.
+It's also useful to see what the model doesn't capture, the so-called residuals which are left after subtracting the predictions from the data. Residuals are powerful because they allow us to use models to remove striking patterns so we can study the subtler trends that remain.
 
 ### Predictions
 
@@ -727,4 +727,4 @@ This chapter has focussed exclusively on the class of linear models, which assum
   by models like __random forests__ (e.g. `randomForest::randomForest()`) or 
   __gradient boosting machines__ (e.g. `xgboost::xgboost`.)
 
-These models all work similarly from a programming perspective. Once you've mastered linear models, you should find it easy to master the mechanics of these other model classes. Being a skilled modeller is a mixture of having some good general principles and having a big toolbox of techniques. Now that you've learned some general tools and one useful class of models, you can go on and learn more classes from other sources.
+These models all work similarly from a programming perspective. Once you've mastered linear models, you should find it easy to master the mechanics of these other model classes. Being a skilled modeller is a mixture of some good general principles and having a big toolbox of techniques. Now that you've learned some general tools and one useful class of models, you can go on and learn more classes from other sources.