First you should create 3x3 confusion matrix and then calculate statistics, we have two type of calculation (macro and micro) for overall statistics (overall precision, overall recall and ...) look at these links for formula :
Overall Accuracy
$$ACC_{Overall}=\frac{\sum_{i=1}^{|C|}TP_i}{Population}$$
Precision Micro
$$PPV_{Micro}=\frac{\sum_{i=1}^{|C|}TP_i}{\sum_{i=1}^{|C|}TP_i+FP_i}$$
Precision Macro
$$PPV_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}\frac{TP_i}{TP_i+FP_i}$$
Recall Micro
$$TPR_{Micro}=\frac{\sum_{i=1}^{|C|}TP_i}{\sum_{i=1}^{|C|}TP_i+FN_i}$$
Recall Macro
$$TPR_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}\frac{TP_i}{TP_i+FN_i}$$
I suggest my lib for your purpose : PyCM
Example Usage :
>>> from pycm import *
>>> y_actu = [2, 0, 2, 2, 0, 1, 1, 2, 2, 0, 1, 2] # or y_actu = numpy.array([2, 0, 2, 2, 0, 1, 1, 2, 2, 0, 1, 2])
>>> y_pred = [0, 0, 2, 1, 0, 2, 1, 0, 2, 0, 2, 2] # or y_pred = numpy.array([0, 0, 2, 1, 0, 2, 1, 0, 2, 0, 2, 2])
>>> cm = ConfusionMatrix(actual_vector=y_actu, predict_vector=y_pred) # Create CM From Data
>>> cm.classes
[0, 1, 2]
>>> cm.table
{0: {0: 3, 1: 0, 2: 0}, 1: {0: 0, 1: 1, 2: 2}, 2: {0: 2, 1: 1, 2: 3}}
>>> print(cm)
Predict 0 1 2
Actual
0 3 0 0
1 0 1 2
2 2 1 3
Overall Statistics :
95% CI (0.30439,0.86228)
Bennett_S 0.375
Chi-Squared 6.6
Chi-Squared DF 4
Conditional Entropy 0.95915
Cramer_V 0.5244
Cross Entropy 1.59352
Gwet_AC1 0.38931
Joint Entropy 2.45915
KL Divergence 0.09352
Kappa 0.35484
Kappa 95% CI (-0.07708,0.78675)
Kappa No Prevalence 0.16667
Kappa Standard Error 0.22036
Kappa Unbiased 0.34426
Lambda A 0.16667
Lambda B 0.42857
Mutual Information 0.52421
Overall_ACC 0.58333
Overall_RACC 0.35417
Overall_RACCU 0.36458
PPV_Macro 0.56667
PPV_Micro 0.58333
Phi-Squared 0.55
Reference Entropy 1.5
Response Entropy 1.48336
Scott_PI 0.34426
Standard Error 0.14232
Strength_Of_Agreement(Altman) Fair
Strength_Of_Agreement(Cicchetti) Poor
Strength_Of_Agreement(Fleiss) Poor
Strength_Of_Agreement(Landis and Koch) Fair
TPR_Macro 0.61111
TPR_Micro 0.58333
Class Statistics :
Classes 0 1 2
ACC(Accuracy) 0.83333 0.75 0.58333
BM(Informedness or bookmaker informedness) 0.77778 0.22222 0.16667
DOR(Diagnostic odds ratio) None 4.0 2.0
ERR(Error rate) 0.16667 0.25 0.41667
F0.5(F0.5 score) 0.65217 0.45455 0.57692
F1(F1 score - harmonic mean of precision and sensitivity) 0.75 0.4 0.54545
F2(F2 score) 0.88235 0.35714 0.51724
FDR(False discovery rate) 0.4 0.5 0.4
FN(False negative/miss/type 2 error) 0 2 3
FNR(Miss rate or false negative rate) 0.0 0.66667 0.5
FOR(False omission rate) 0.0 0.2 0.42857
FP(False positive/type 1 error/false alarm) 2 1 2
FPR(Fall-out or false positive rate) 0.22222 0.11111 0.33333
G(G-measure geometric mean of precision and sensitivity) 0.7746 0.40825 0.54772
LR+(Positive likelihood ratio) 4.5 3.0 1.5
LR-(Negative likelihood ratio) 0.0 0.75 0.75
MCC(Matthews correlation coefficient) 0.68313 0.2582 0.16903
MK(Markedness) 0.6 0.3 0.17143
N(Condition negative) 9 9 6
NPV(Negative predictive value) 1.0 0.8 0.57143
P(Condition positive) 3 3 6
POP(Population) 12 12 12
PPV(Precision or positive predictive value) 0.6 0.5 0.6
PRE(Prevalence) 0.25 0.25 0.5
RACC(Random accuracy) 0.10417 0.04167 0.20833
RACCU(Random accuracy unbiased) 0.11111 0.0434 0.21007
TN(True negative/correct rejection) 7 8 4
TNR(Specificity or true negative rate) 0.77778 0.88889 0.66667
TON(Test outcome negative) 7 10 7
TOP(Test outcome positive) 5 2 5
TP(True positive/hit) 3 1 3
TPR(Sensitivity, recall, hit rate, or true positive rate) 1.0 0.33333 0.5
>>> cm.matrix()
Predict 0 1 2
Actual
0 3 0 0
1 0 1 2
2 2 1 3
>>> cm.normalized_matrix()
Predict 0 1 2
Actual
0 1.0 0.0 0.0
1 0.0 0.33333 0.66667
2 0.33333 0.16667 0.5
Best Answer
As far as I know there isn't a "de facto" way of calculating precision and recall for multi-class classification.
Your approaches are what I too would try:
There are also other metrics to evaluate the performance of your mode, besides precision and recall: