Nonlinear Least Square

When
modeling real world data for regression analysis, we observe that it is
rarely the case that the equation of the model is a linear equation
giving a linear graph. Most of the time, the equation of the model of
real world data involves mathematical functions of higher degree like an
exponent of 3 or a sin function. In such a scenario, the plot of the
model gives a curve rather than a line. The goal of both linear and
non-linear regression is to adjust the values of the model's parameters
to find the line or curve that comes closest to your data. On finding
these values we will be able to estimate the response variable with good
accuracy.

In
Least Square regression, we establish a regression model in which the
sum of the squares of the vertical distances of different points from
the regression curve is minimized. We generally start with a defined
model and assume some values for the coefficients. We then apply the

**nls()**function of R to get the more accurate values along with the confidence intervals.## Syntax

The basic syntax for creating a nonlinear least square test in R is −

```
nls(formula, data, start)
```

Following is the description of the parameters used −

**formula**is a nonlinear model formula including variables and parameters.**data**is a data frame used to evaluate the variables in the formula.**start**is a named list or named numeric vector of starting estimates.

## Example

We
will consider a nonlinear model with assumption of initial values of
its coefficients. Next we will see what is the confidence intervals of
these assumed values so that we can judge how well these values fir into
the model.

So let's consider the below equation for this purpose −

```
a = b1*x^2+b2
```

Let's assume the initial coefficients to be 1 and 3 and fit these values into nls() function.

xvalues <- c(1.6,2.1,2,2.23,3.71,3.25,3.4,3.86,1.19,2.21) yvalues <- c(5.19,7.43,6.94,8.11,18.75, 14.88,16.06,19.12,3.21,7.58) # Give the chart file a name. png(file = "nls.png") # Plot these values. plot(xvalues,yvalues) # Take the assumed values and fit into the model. model <- nls(yvalues ~ b1*xvalues^2+b2,start = list(b1 = 1,b2 = 3)) # Plot the chart with new data by fitting it to a prediction from 100 data points. new.data <- data.frame(xvalues = seq(min(xvalues),max(xvalues), len = 100)) lines(new.data$xvalues, predict(model,newdata = new.data)) # Save the file. dev.off() # Get the sum of the squared residuals. print(sum(resid(model)^2)) # Get the confidence intervals on the chosen values of the coefficients. print(confint(model))

When we execute the above code, it produces the following result −

```
[1] 1.081935
Waiting for profiling to be done...
2.5% 97.5%
b1 1.137708 1.253135
b2 1.497364 2.496484
```

We can conclude that the value of b1 is more close to 1 while the value of b2 is more close to 2 and not 3.

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