Hold the Line

by marco.p.v.vezzoli · February 16, 2025

We made quite a journey so far! Starting from Jupyter and Pandas we explored our datasets and created independent scripts.

It is now the time to learn the basics of a very powerful tool: Linear Regression.

Linearity is a key concept in mathematics: it goes very far from the naive idea of a straight line into more abstract concepts like the independence of two effecs into a dynamic system.

The jupyter notebooks for this series of posts, the datasets, their source and their attribution are available in this Github Repo

Correlations and linear models

A linear model estimate a response from the linear combination of one or more inputs

$y \approx x_1 \alpha_1 + y_2 \alpha_2 + ... + y_n \alpha_n$

$y \approx \vec{x} \cdot \vec{\alpha}$

given an estimation $\hat{y} = \vec{x} \cdot \vec{\alpha}$

we look for the best $\vec{\alpha}$ which minimize all residuals

$\epsilon(\hat{y}) = y - \hat{y}$

import pandas as pd

This dataset collects the yearly water and energy consuption estimation per capita in Milan collected by the italian government

Data is grouped by

water consumption
methan consumption
electricity consumption

let’s first load this resource usage dataset

cons = pd.read_csv("ds523_consumoacquaenergia.csv",sep=";")

We can quickly overview its content

cons.describe(include="all")

              anno              Consumo pro capite tipo  Consumo pro capite
count     36.00000                                   36           36.000000
unique         NaN                                    3                 NaN
top            NaN  Energia elettrica per uso domestico                 NaN
freq           NaN                                   12                 NaN
mean    2005.50000                                  NaN          573.072222
std        3.50102                                  NaN          471.777743
min     2000.00000                                  NaN           80.400000
25%     2002.75000                                  NaN           89.625000
50%     2005.50000                                  NaN          432.900000
75%     2008.25000                                  NaN         1195.650000
max     2011.00000                                  NaN         1228.600000

It requires some cleanup: first let’s check the consumption type

cons["Consumo pro capite tipo"].unique()

array(['Energia elettrica per uso domestico',
       'Gas metano per uso domestico e riscaldamento',
       'Acqua fatturata per uso domestico'], dtype=object)

Now let’s translate these categories

translate = {
    'Energia elettrica per uso domestico':'electricity',
    'Gas metano per uso domestico e riscaldamento':'methan',
    'Acqua fatturata per uso domestico':'water'
}
cons["type"] = cons["Consumo pro capite tipo"].map(translate)

Finally we can reshape the dataset to split the different kind of resources

cons2 = cons.pivot(index="anno",columns="type",values="Consumo pro capite").reset_index()
cons2 = cons2.rename({"anno":"year"}, axis="columns")
cons2

type  year  electricity  methan  water
0     2000       1130.2   509.0   92.1
1     2001       1143.9   500.7   91.3
2     2002       1195.5   504.2   90.4
3     2003       1222.8   480.2   87.3
4     2004       1228.6   442.4   80.4
5     2005       1225.0   434.5   81.3
6     2006       1219.7   431.3   82.2
7     2007       1197.0   381.1   81.6
8     2008       1203.0   384.9   84.5
9     2009       1202.9   389.6   85.8
10    2010       1200.7   406.2   83.2
11    2011       1196.1   377.9   83.1

Now we can make use of our scatter matrix to further investigate this dataset

import seaborn as sns
sns.pairplot(cons2)

Looks like there is some kind of variation of the methan usage in time: we can try to make a linear regression and see how it does look like

sns.regplot(cons2,x="year",y="methan")

<Axes: xlabel='year', ylabel='methan'>

Covariance and correlation

The covariance matrix, defined as

$cov[X_i, X_j] = E[(X_i - E[X_i])(X_j - E[X_j])]$

is the multidimensional extension of the variance, elements of the diagonal being the variance of the corrisponding dimension; its eigenvectors define an ellipsoid representing the most important combinations of the dimensional features; this is used in Principal Compaonent Analysis, a technique which helps to define the most impactful features.

By dividing each element with the product of the standard deviations we have the correlation matrix

$corr[X_i, X_j] = \frac{E[(X_i - E[X_i])(X_j - E[X_j])]}{\sigma_i\sigma_j}$

The elements outside the diagonal are numbers between -1 and 1; 0 represents no correlation (like a spherical cloud) while 1 and -1 represent positive and negative correlation respectively; this gives us a first estimation of the possible linear dependecies within a set of observation features;

import numpy as np
# numpy expects a matrix where each feature is in a row instead of a column
# thus we need to transpose it
np.corrcoef(np.transpose(np.array(cons2)))

array([[ 1.        ,  0.44786015, -0.93548315, -0.65540971],
       [ 0.44786015,  1.        , -0.46029677, -0.77514369],
       [-0.93548315, -0.46029677,  1.        ,  0.75208366],
       [-0.65540971, -0.77514369,  0.75208366,  1.        ]])

we can see that the negative correlation between year and methan is about -0.9 which makes it a good candidate for a linear correlation

from scipy import stats

Regression calculation

in this simple case we have

few observations
only one input value so we may directly use the Ordinary Least Squares regression method to evaluate the best fit

result = stats.linregress(x=cons2.year, y=cons2.methan)
result

LinregressResult(slope=np.float64(-13.141258741258738), intercept=np.float64(26791.62773892773), rvalue=np.float64(-0.9354831530794605), pvalue=np.float64(7.894692952340763e-06), stderr=np.float64(1.5697563928623894), intercept_stderr=np.float64(3148.151109622701))

the returned object contains some interesting values; let’s check the first two:

slope
intercept

allows us to write a simple prediction formula

def predict_methan(year):
    return result.slope * year + result.intercept

with this formula we can build a chart of our linear regression

import matplotlib.pyplot as plt
import seaborn as sns

# create a plot canvas
fig, ax = plt.subplots(1,1)

#first plot the points into our canvas
sns.scatterplot(x=cons2.year, y=cons2.methan, ax=ax)

# then plot a line from the first to the last point on the same canvas
year0 = min(cons2.year)
year1 = max(cons2.year)
ax.plot((year0,year1),(predict_methan(year0),predict_methan(year1)))

note: the polymorphism allows to properly use the prodict_methan function also with pandas Series

Assessing the quality of a regression

residuals = cons2.methan - predict_methan(cons2.year)

looking at residuals distribution may show some pattern; in this case we may assume there is a better way to represent the relation between the features under investigation.

In our example looks like there is no apparent pattern

ax = sns.scatterplot(x=cons2.year, y=residuals)
ax.plot((year0,year1),(0,0))
ax.set_ylabel("residuals")

Text(0, 0.5, 'residuals')

The next step would be to assess the variance of residuals respect to the total variance of the distribution of the output variable Y:

$\frac{var[\epsilon]}{var[Y]} = \frac{E[(\epsilon - E[\epsilon])^2]}{E[(Y - E[Y])^2]}$

let’s use $\hat{Y}$ to represent the predicted values; by knowing that the mean of the residuals is 0 and their definition

$E[\epsilon] = 0$

$\epsilon = Y - \hat{Y}$

we have

$\frac{var[\epsilon]}{var[Y]} = \frac{E[(Y - \hat{Y})^2]}{E[(Y - E[Y])^2]}$

now the quantity

$R^2 = 1 - \frac{E[(Y - \hat{Y})^2]}{E[(Y - E[Y])^2]}$

represent the fraction of the variance of the original dataset explained by the linear relation: this is a real number between 0 and 1 where 0 represents no actual explaination (i.e. the mean has the same prediction power) to 1 representing all the relation is explained

Multiple input parameters

in order to perform this regression with multiple inputs we are going to use the statmodels library (see documentation)

Execute the following cell only the first time

!pip install statsmodels

import statsmodels as sm
from statsmodels.api import formula as smf
import requests
import pandas as pd

We will use a crime dataset from UCLA

headers = "crimerat maleteen south educ police60 police59 labor  males pop nonwhite unemp1  unemp2 median belowmed".split()
crime = pd.read_csv(
    "https://stats.idre.ucla.edu/wp-content/uploads/2016/02/crime.txt",
    sep=r"\s+",
    names=headers,
    dtype=float
)

This is the description of the content of this table

Columns	meaning
CrimeRat	Crime rate: # of offenses reported to police per million population
MaleTeen	The number of males of age 14-24 per 1000 population
South	Indicator variable for Southern states (0 = No, 1 = Yes)
Educ	Mean # of years of schooling for rpersons of age 25 or older
Police60	1960 per capita expenditure on police by state and local government
Police59	1959 per capita expenditure on police by state and local government
Labor	Labor force participation rate per 1000 civilian urban males age 14-24
Males	The number of males per 1000 females
Pop	State population size in hundred thousands
NonWhite	The number of non-whites per 1000 population
Unemp1	Unemployment rate of urban males per 1000 of age 14-24
Unemp2	Unemployment rate of urban males per 1000 of age 35-39
Median	Median value of transferable goods and assets or family income in tens of $
BelowMed	The number of families per 1000 earning below 1/2 the median income

crime.head()

   crimerat  maleteen  south  educ  police60  police59  labor   males    pop  \
0      79.1     151.0    1.0   9.1      58.0      56.0  510.0   950.0   33.0   
1     163.5     143.0    0.0  11.3     103.0      95.0  583.0  1012.0   13.0   
2      57.8     142.0    1.0   8.9      45.0      44.0  533.0   969.0   18.0   
3     196.9     136.0    0.0  12.1     149.0     141.0  577.0   994.0  157.0   
4     123.4     141.0    0.0  12.1     109.0     101.0  591.0   985.0   18.0   
   nonwhite  unemp1  unemp2  median  belowmed  
0     301.0   108.0    41.0   394.0     261.0  
1     102.0    96.0    36.0   557.0     194.0  
2     219.0    94.0    33.0   318.0     250.0  
3      80.0   102.0    39.0   673.0     167.0  
4      30.0    91.0    20.0   578.0     174.0

The south feature is actually categorical and cannot be treated in the same way as others but let’s pretend it is not different for this exercise

crime.describe()

         crimerat    maleteen      south      educ    police60    police59  \
count   47.000000   47.000000  47.000000  47.00000   47.000000   47.000000   
mean    90.508511  138.574468   0.340426  10.56383   85.000000   80.234043   
std     38.676270   12.567634   0.478975   1.11870   29.718974   27.961319   
min     34.200000  119.000000   0.000000   8.70000   45.000000   41.000000   
25%     65.850000  130.000000   0.000000   9.75000   62.500000   58.500000   
50%     83.100000  136.000000   0.000000  10.80000   78.000000   73.000000   
75%    105.750000  146.000000   1.000000  11.45000  104.500000   97.000000   
max    199.300000  177.000000   1.000000  12.20000  166.000000  157.000000   
            labor        males         pop    nonwhite      unemp1     unemp2  \
count   47.000000    47.000000   47.000000   47.000000   47.000000  47.000000   
mean   561.191489   983.021277   36.617021  101.127660   95.468085  33.978723   
std     40.411814    29.467365   38.071188  102.828819   18.028783   8.445450   
min    480.000000   934.000000    3.000000    2.000000   70.000000  20.000000   
25%    530.500000   964.500000   10.000000   24.000000   80.500000  27.500000   
50%    560.000000   977.000000   25.000000   76.000000   92.000000  34.000000   
75%    593.000000   992.000000   41.500000  132.500000  104.000000  38.500000   
max    641.000000  1071.000000  168.000000  423.000000  142.000000  58.000000   
           median    belowmed  
count   47.000000   47.000000  
mean   525.382979  194.000000  
std     96.490944   39.896061  
min    288.000000  126.000000  
25%    459.500000  165.500000  
50%    537.000000  176.000000  
75%    591.500000  227.500000  
max    689.000000  276.000000

Note that there are some very skewed distributions like the non white which has a very large standard deviation respect to the mean; this value also shows a long queue according to the percentiles.

Moreover, due to their definitions some features have very different ranges.

This may have an impact in evaluating the eigenvectors as some dimensions may appear as more relevant then others due to their scale.

For these reasons we may expect that renormalizing all distributions respect to their standard deviation may change our findings.

Evaluating correlations and covariance

import numpy as np
crime_array = np.transpose(np.array(crime))
covariance = np.cov(crime_array)
correlation = np.corrcoef(crime_array)
pd.DataFrame({"correlation":correlation[0,1:],"features":headers[1:]})

    correlation  features
0     -0.089472  maleteen
1     -0.090637     south
2      0.322835      educ
3      0.687604  police60
4      0.666714  police59
5      0.188866     labor
6      0.213914     males
7      0.337474       pop
8      0.032599  nonwhite
9     -0.050478    unemp1
10     0.177321    unemp2
11     0.441320    median
12    -0.179024  belowmed

apparently the crime rate most relevant correlation seems to be the increase in police expenditure which may probably be more a consequence than a causation

from numpy.linalg._linalg import EigResult
# eigenvectors will be returned already sorted from the most to the least relevant
result :EigResult = np.linalg.eig(covariance)

def relevant(headers: [str], result: EigResult, rank: int):
    """retruns the features of the rank-th eigenvalue sorted from the largest descending"""
    # extract the rank-th eigenvector
    vector = result.eigenvectors[:,rank] 
    # square it to get rid of sign
    vector_sq = vector * vector
    # get the order from smallest to largest
    order = vector_sq.argsort()
    # reverse order and return the features from the most relevant
    return [headers[int(i)] for i in reversed(order)]

let’s grab the 5 most relevant set of features

for i in range(5):
    print(relevant(headers, result, i))

['nonwhite', 'median', 'belowmed', 'police60', 'police59', 'labor', 'crimerat', 'maleteen', 'males', 'pop', 'unemp1', 'educ', 'south', 'unemp2']
['nonwhite', 'median', 'crimerat', 'pop', 'police60', 'police59', 'males', 'belowmed', 'labor', 'unemp1', 'unemp2', 'maleteen', 'south', 'educ']
['labor', 'males', 'pop', 'crimerat', 'nonwhite', 'belowmed', 'maleteen', 'unemp2', 'unemp1', 'police59', 'educ', 'police60', 'median', 'south']
['pop', 'crimerat', 'median', 'belowmed', 'nonwhite', 'labor', 'police60', 'police59', 'males', 'unemp1', 'unemp2', 'maleteen', 'educ', 'south']
['labor', 'crimerat', 'pop', 'males', 'unemp1', 'unemp2', 'belowmed', 'police60', 'nonwhite', 'police59', 'median', 'maleteen', 'south', 'educ']

Performing regression from multiple inputs

In the following multilinear correlation we construct a formula representing the features which may impact to the expected output

output ~ feature1 + feature2 + feature3

I chose to use all of the features which appear as most relevant in the first eigenvector and appear before our output

formula = "crimerat ~ "+ (" + ".join(relevant(headers, result, 0)[:6]))
print(formula)
model = smf.ols(formula,crime)
regression = model.fit()
regression.summary()

crimerat ~ nonwhite + median + belowmed + police60 + police59 + labor

<class 'statsmodels.iolib.summary.Summary'>
"""
                            OLS Regression Results                            
==============================================================================
Dep. Variable:               crimerat   R-squared:                       0.638
Model:                            OLS   Adj. R-squared:                  0.584
Method:                 Least Squares   F-statistic:                     11.75
Date:                Sun, 05 Jan 2025   Prob (F-statistic):           1.48e-07
Time:                        21:48:11   Log-Likelihood:                -214.10
No. Observations:                  47   AIC:                             442.2
Df Residuals:                      40   BIC:                             455.2
Df Model:                           6                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept   -304.9695     96.968     -3.145      0.003    -500.950    -108.989
nonwhite       0.0050      0.056      0.088      0.930      -0.109       0.119
median         0.1588      0.112      1.419      0.164      -0.067       0.385
belowmed       0.6875      0.223      3.085      0.004       0.237       1.138
police60       1.3928      1.140      1.222      0.229      -0.910       3.696
police59      -0.3685      1.239     -0.297      0.768      -2.872       2.135
labor          0.1592      0.100      1.594      0.119      -0.043       0.361
==============================================================================
Omnibus:                        2.339   Durbin-Watson:                   2.004
Prob(Omnibus):                  0.311   Jarque-Bera (JB):                1.581
Skew:                          -0.436   Prob(JB):                        0.454
Kurtosis:                       3.220   Cond. No.                     2.16e+04
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.16e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
"""

The result of the fit method which is shown here above displays a wealth of information; most notably

some quality evaluation of the regression e.g. $R^2$
all the evaluated parameters and the intercept

Exploring features

it is also important to not blindly accept the result of a regression without a further analysis of the dataset

In the following code I will check how the output variable depends on the features we examined; as this plot does not really show the interdipendence of all features some images may be difficult to interpret

fig, axs = mpl.subplots(1,6,sharey=True,figsize=(18,3))
features = relevant(headers, result, 0)[:6]
for i in range(6):
    sns.scatterplot(x=crime[features[i]],y=crime.crimerat,ax=axs[i])

Correcting eigenvector bias with correlation matrix

by using the correlation instead of the covariance, the range of all features is normalized now between -1 and 1

As we can see the most interesting eigenvectors change

result2 = np.linalg.eig(correlation)

for i in range(5):
    print(relevant(headers, result2, i))

['median', 'belowmed', 'educ', 'police59', 'police60', 'south', 'maleteen', 'nonwhite', 'crimerat', 'labor', 'males', 'pop', 'unemp1', 'unemp2']
['pop', 'labor', 'unemp2', 'males', 'police60', 'police59', 'nonwhite', 'crimerat', 'south', 'educ', 'median', 'belowmed', 'unemp1', 'maleteen']
['unemp1', 'unemp2', 'labor', 'maleteen', 'crimerat', 'males', 'nonwhite', 'police59', 'police60', 'pop', 'south', 'educ', 'belowmed', 'median']
['males', 'crimerat', 'maleteen', 'labor', 'nonwhite', 'belowmed', 'unemp1', 'pop', 'south', 'unemp2', 'police60', 'police59', 'educ', 'median']
['pop', 'labor', 'belowmed', 'south', 'maleteen', 'police59', 'median', 'police60', 'unemp2', 'educ', 'unemp1', 'nonwhite', 'crimerat', 'males']

rank_no = 0
features_count = 8
formula = "crimerat ~ "+ (" + ".join(relevant(headers, result2, rank_no)[:features_count]))
print(formula)
model = smf.ols(formula,crime)
regression = model.fit()
regression.summary()

crimerat ~ median + belowmed + educ + police59 + police60 + south + maleteen + nonwhite

<class 'statsmodels.iolib.summary.Summary'>
"""
                            OLS Regression Results                            
==============================================================================
Dep. Variable:               crimerat   R-squared:                       0.730
Model:                            OLS   Adj. R-squared:                  0.673
Method:                 Least Squares   F-statistic:                     12.82
Date:                Sat, 04 Jan 2025   Prob (F-statistic):           1.02e-08
Time:                        21:27:44   Log-Likelihood:                -207.24
No. Observations:                  47   AIC:                             432.5
Df Residuals:                      38   BIC:                             449.1
Df Model:                           8                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept   -537.5940    108.276     -4.965      0.000    -756.786    -318.402
median         0.1764      0.101      1.740      0.090      -0.029       0.382
belowmed       0.8438      0.211      3.994      0.000       0.416       1.271
educ          14.4615      5.068      2.853      0.007       4.201      24.722
police59      -0.8715      1.099     -0.793      0.433      -3.096       1.353
police60       1.8952      1.015      1.868      0.069      -0.159       3.949
south         -1.9020     12.426     -0.153      0.879     -27.057      23.253
maleteen       0.9286      0.379      2.451      0.019       0.161       1.696
nonwhite      -0.0025      0.060     -0.041      0.967      -0.124       0.119
==============================================================================
Omnibus:                        0.285   Durbin-Watson:                   1.792
Prob(Omnibus):                  0.867   Jarque-Bera (JB):                0.010
Skew:                          -0.016   Prob(JB):                        0.995
Kurtosis:                       3.064   Cond. No.                     2.02e+04
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.02e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
"""

Interestingly this correlation shows a better $R^2$ than the previous one thus demonstrating the effectiveness of using normalized distributions

rank_no = 0
features_count = 8
fig, axs = mpl.subplots(1,features_count,sharey=True,figsize=(features_count * 3,3))
features = relevant(headers, result2, 0)[:features_count]
for i in range(features_count):
    sns.scatterplot(x=crime[features[i]],y=crime.crimerat,ax=axs[i])

More visualization of the correlations

in the following examples I will show a couple of scatter plots of the most relevant features and use colors for the output variable; while this visualization does not add a great insight, nonetheless can raise interesting questions about the mutual connections of the features

sns.scatterplot(x=crime.belowmed,y=crime["median"],hue=crime.crimerat)

<Axes: xlabel='belowmed', ylabel='median'>

This image shows that some of the highest crime rate seems to show in an area where economic indicators seems more favorable, which just demonstrates how complex and controversial this analysis may be: deciding which features to include may have important consequences.

A 3d version of the same plot adding the education feature

#sns.scatterplot(x=crime.belowmed,y=crime["median"],hue=crime.crimerat)
from mpl_toolkits.mplot3d import Axes3D
sns.set_style("whitegrid", {'axes.grid' : False})

fig = plt.figure()

ax = Axes3D(fig) 
fig.add_axes(ax)
x=crime.belowmed
y=crime["median"]
z=crime.educ

ax.scatter(x, y, z, c=crime.crimerat, marker='o')
ax.set_xlabel('belowmed')
ax.set_ylabel('median')
ax.set_zlabel('educ')
ax

<Axes3D: xlabel='belowmed', ylabel='median', zlabel='educ'>

Non-Linear features

the linearity of linear models is defined by the interaction between different features but this may be used in with non linear cases e.g. trying to fit a polynomial model

# this library is used to read excel files
!pip install openpyxl

import pandas as pd

The following dataset describes financial performances metrics across many countries

financial = pd.read_excel("20220909-global-financial-development-database.xlsx",sheet_name="Data - August 2022")

let’s first set some attributes as categorical: we may use them eventually as a filter

for col in ["iso3", "iso2", "imfn", "country", "region", "income"]:
    financial[col] = financial[col].astype("category")

In this example we will focus on a particular financial metric di01

financial[["country","region","di01"]].describe(include="all")

            country                 region         di01
count         13268                  13268  8594.000000
unique          214                      7          NaN
top     Afghanistan  Europe & Central Asia          NaN
freq             62                   3596          NaN
mean            NaN                    NaN    37.321250
std             NaN                    NaN    34.811684
min             NaN                    NaN     0.010371
25%             NaN                    NaN    13.054380
50%             NaN                    NaN    26.018790
75%             NaN                    NaN    50.293530
max             NaN                    NaN   304.574500

import seaborn as sns

sns.scatterplot(financial,x="year",y="di01",hue="region")

<Axes: xlabel='year', ylabel='di01'>

Let’s first narrow it to a single country and show its dependency from time

fin_italy = financial.loc[financial["country"]=="Italy",:]
sns.scatterplot(fin_italy,x="year",y="di01")

<Axes: xlabel='year', ylabel='di01'>

This shows some kind of growing trend: let’s first try a simple linear regression respect to the years

sns.regression.regplot(fin_italy,x="year",y="di01")

<Axes: xlabel='year', ylabel='di01'>

from scipy.stats import linregress

fin_italy[["year","di01"]].describe()

              year       di01
count    62.000000  57.000000
mean   1990.500000  65.643938
std      18.041619  13.908666
min    1960.000000  46.931830
25%    1975.250000  54.484480
50%    1990.500000  62.710020
75%    2005.750000  75.462590
max    2021.000000  93.921490

We see this dataset does not contain metrics for all years so let’s remove rows without values

fin_italy_valued = fin_italy.loc[~fin_italy.di01.isna(),["year","di01"]]

Here we see the results of the regression

result = linregress(y=fin_italy_valued.di01,x=fin_italy_valued.year)
result

LinregressResult(slope=np.float64(0.4768662371619365), intercept=np.float64(-884.1481148904821), rvalue=np.float64(0.5972453694029883), pvalue=np.float64(9.37869396904616e-07), stderr=np.float64(0.08635123015454191), intercept_stderr=np.float64(171.99538887670158))

The $R^2$ looks poor:

rsquare = result.rvalue ** 2
rsquare

np.float64(0.356702031273312)

let’s plot the residuals to see any clear behavior

residuals = fin_italy_valued.di01 - (fin_italy_valued.year * result.slope + result.intercept)
sns.scatterplot(x=fin_italy_valued.year, y=residuals)

<Axes: xlabel='year', ylabel='None'>

Adding nonlinear features

For simplicity of the fit we will use a column with years calculated as a difference from the first one.

In this case residuals suggests a kind of oscillatory behavior but this is way too complex for this tutorial as it involves the evaluation of periods of the oscillations and phase shifts.

The simpler way to increase the fit can be to use a higher degree polynomial.

import statsmodels as sm
from statsmodels.api import formula as smf
import requests
import pandas as pd

Let’s create the nonlinear feature columns for a polynomial of degree 3. The higher the degree the lower the error: choosing an excessively large degree can lead to overfitting without adding much more insight

fin_italy_valued["dy"] = fin_italy_valued.year - min(fin_italy_valued.year)
fin_italy_valued["dy2"] = fin_italy_valued.dy ** 2
fin_italy_valued["dy3"] = fin_italy_valued.dy ** 3

now we can fit and get the coefficients for these features

model = smf.ols("di01 ~ dy + dy2 + dy3",fin_italy_valued)
regression = model.fit()
regression.summary()

<class 'statsmodels.iolib.summary.Summary'>
"""
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                   di01   R-squared:                       0.598
Model:                            OLS   Adj. R-squared:                  0.575
Method:                 Least Squares   F-statistic:                     26.30
Date:                Sun, 19 Jan 2025   Prob (F-statistic):           1.48e-10
Time:                        19:48:07   Log-Likelihood:                -204.44
No. Observations:                  57   AIC:                             416.9
Df Residuals:                      53   BIC:                             425.1
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     69.6046      4.436     15.690      0.000      60.707      78.503
dy            -1.8193      0.670     -2.715      0.009      -3.163      -0.475
dy2            0.0614      0.027      2.262      0.028       0.007       0.116
dy3           -0.0004      0.000     -1.350      0.183      -0.001       0.000
==============================================================================
Omnibus:                        6.613   Durbin-Watson:                   0.134
Prob(Omnibus):                  0.037   Jarque-Bera (JB):                3.703
Skew:                           0.419   Prob(JB):                        0.157
Kurtosis:                       2.075   Cond. No.                     2.84e+05
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.84e+05. This might indicate that there are
strong multicollinearity or other numerical problems.
"""

The $R^2$ value improved from .39 to .59

Here are our coefficients: the third degree adds very little contribution

regression.params

Intercept    69.604622
dy           -1.819269
dy2           0.061431
dy3          -0.000417
dtype: float64

With them we can now plot the polynomial and verify the new fit

fin_italy_valued["predicted"] = regression.params[0] + \
    regression.params[1] * fin_italy_valued.dy + \
    regression.params[2] * fin_italy_valued.dy2 + \
    regression.params[3] * fin_italy_valued.dy3

fig, ax = plt.subplots(1, 1)
sns.scatterplot(data=fin_italy_valued,x="year",y="di01", ax=ax)
sns.lineplot(data=fin_italy_valued,x="year",y="predicted", ax=ax)

Hold the Line

Correlations and linear models

Covariance and correlation

Regression calculation

Assessing the quality of a regression

Multiple input parameters

Evaluating correlations and covariance

Performing regression from multiple inputs

Exploring features

Correcting eigenvector bias with correlation matrix

More visualization of the correlations

Non-Linear features

Adding nonlinear features

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Hold the Line

Correlations and linear models

Covariance and correlation

Regression calculation

Assessing the quality of a regression

Multiple input parameters

Evaluating correlations and covariance

Performing regression from multiple inputs

Exploring features

Correcting eigenvector bias with correlation matrix

More visualization of the correlations

Non-Linear features

Adding nonlinear features

Share this:

Like this:

Related

You may also like...

Python Tutorial: Variables

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